{"id":2833,"date":"2023-07-18T14:42:11","date_gmt":"2023-07-18T12:42:11","guid":{"rendered":"http:\/\/ff.mhrooz.xyz\/?p=2833"},"modified":"2023-07-18T14:42:15","modified_gmt":"2023-07-18T12:42:15","slug":"dsp_matlab_lian_xi","status":"publish","type":"post","link":"https:\/\/blog.mhrooz.xyz\/index.php\/2023\/07\/18\/dsp_matlab_lian_xi\/","title":{"rendered":"DSP MATLAB\u7ec3\u4e60"},"content":{"rendered":"\n<!doctype html>\n<html>\n<head>\n<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>\n\n<link href='https:\/\/fonts.googleapis.com\/css?family=PT+Serif:400,400italic,700,700italic&#038;subset=latin,cyrillic-ext,cyrillic,latin-ext' rel='stylesheet' type='text\/css' \/><style type='text\/css'>html {overflow-x: initial !important;}:root { --bg-color: #ffffff; --text-color: #333333; --select-text-bg-color: #B5D6FC; --select-text-font-color: auto; --monospace: \"Lucida Console\",Consolas,\"Courier\",monospace; --title-bar-height: 20px; }\n.mac-os-11 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#f3f2ee;\n\t--text-color: #1f0909;\n\t--control-text-color: #444;\n\t--rawblock-edit-panel-bd: #e5e5e5;\n\n\t--select-text-bg-color: rgba(32, 43, 51, 0.63);\n  --select-text-font-color: white;\n}\n\npre {\n\t--select-text-bg-color: #36284e;\n\t--select-text-font-color: #fff;\n}\n\nhtml {\n\tfont-size: 16px;\n\t-webkit-font-smoothing: antialiased;\n}\n\nhtml, body {\n\tbackground-color: #f3f2ee;\n\tfont-family: \"PT Serif\", 'Times New Roman', Times, serif;\n\tcolor: #1f0909;\n\tline-height: 1.5em;\n}\n\n\/*#write {\n\toverflow-x: auto;\n    max-width: initial;\n\tpadding-left: calc(50% - 17em);\n    padding-right: calc(50% - 17em);\n}\n\n@media (max-width: 36em) {\n \t#write {\n \t\tpadding-left: 1em;\n    \tpadding-right: 1em;\n \t}\n}*\/\n\n#write {\n\tmax-width: 40em;\n}\n\n@media only screen and (min-width: 1400px) {\n\t#write {\n\t\t\tmax-width: 914px;\n\t}\n}\n\nol li {\n\tlist-style-type: decimal;\n\tlist-style-position: outside;\n}\nul li {\n\tlist-style-type: disc;\n\tlist-style-position: outside;\n}\n\nol,\nul {\n\tlist-style: none;\n}\n\nblockquote,\nq {\n\tquotes: none;\n}\nblockquote:before,\nblockquote:after,\nq:before,\nq:after {\n\tcontent: '';\n\tcontent: none;\n}\ntable {\n\tborder-collapse: collapse;\n\tborder-spacing: 0;\n}\n\/* styles *\/\n\n\/* ====== *\/\n\n\/* headings *\/\n\nh1,\nh2,\nh3,\nh4,\nh5,\nh6 {\n\tfont-weight: bold;\n}\nh1 {\n\tfont-size: 1.875em;\n\t\/*30 \/ 16*\/\n\tline-height: 1.6em;\n\t\/* 48 \/ 30*\/\n\tmargin-top: 2em;\n}\nh2,\nh3 {\n\tfont-size: 1.3125em;\n\t\/*21 \/ 16*\/\n\tline-height: 1.15;\n\t\/*24 \/ 21*\/\n\tmargin-top: 2.285714em;\n\t\/*48 \/ 21*\/\n\tmargin-bottom: 1.15em;\n\t\/*24 \/ 21*\/\n}\nh3 {\n\tfont-weight: normal;\n}\nh4 {\n\tfont-size: 1.125em;\n\t\/*18 \/ 16*\/\n\tmargin-top: 2.67em;\n\t\/*48 \/ 18*\/\n}\nh5,\nh6 {\n\tfont-size: 1em;\n\t\/*16*\/\n}\nh1 {\n\tborder-bottom: 1px solid;\n\tmargin-bottom: 1.875em;\n\tpadding-bottom: 0.8125em;\n}\n\/* links *\/\n\na {\n\ttext-decoration: none;\n\tcolor: #065588;\n}\na:hover,\na:active {\n\ttext-decoration: underline;\n}\n\/* block spacing *\/\n\np,\nblockquote,\n.md-fences {\n\tmargin-bottom: 1.5em;\n}\nh1,\nh2,\nh3,\nh4,\nh5,\nh6 {\n\tmargin-bottom: 1.5em;\n}\n\/* blockquote *\/\n\nblockquote {\n\tfont-style: italic;\n\tborder-left: 5px solid;\n\tmargin-left: 2em;\n\tpadding-left: 1em;\n}\n\/* lists *\/\n\nul,\nol {\n\tmargin: 0 0 1.5em 1.5em;\n}\n\/* tables *\/\n.md-meta,.md-before, .md-after {\n\tcolor:#999;\n}\n\ntable {\n\tmargin-bottom: 1.5em;\n\t\/*24 \/ 16*\/\n\tfont-size: 1em;\n\t\/* width: 100%; *\/\n}\nthead th,\ntfoot th {\n\tpadding: .25em .25em .25em .4em;\n\ttext-transform: uppercase;\n}\nth {\n\ttext-align: left;\n}\ntd {\n\tvertical-align: top;\n\tpadding: .25em .25em .25em .4em;\n}\n\ncode,\n.md-fences {\n\tbackground-color: #dadada;\n}\n\ncode {\n\tpadding-left: 2px;\n\tpadding-right: 2px;\n}\n\n.md-fences {\n\tmargin-left: 2em;\n\tmargin-bottom: 3em;\n\tpadding-left: 1ch;\n\tpadding-right: 1ch;\n}\n\npre,\ncode,\ntt {\n\tfont-size: .875em;\n\tline-height: 1.714285em;\n}\n\/* some fixes *\/\n\nh1 {\n\tline-height: 1.3em;\n\tfont-weight: normal;\n\tmargin-bottom: 0.5em;\n}\n\np + ul,\np + ol{\n\tmargin-top: .5em;\n}\n\nh3 + ul,\nh4 + ul,\nh5 + ul,\nh6 + ul,\nh3 + ol,\nh4 + ol,\nh5 + ol,\nh6 + ol {\n\tmargin-top: .5em;\n}\n\nli > ul,\nli > ol {\n\tmargin-top: inherit;\n\tmargin-bottom: 0;\n}\n\nli ol>li {\n\tlist-style-type: lower-alpha;\n}\n\nli li ol>li{\n\tlist-style-type: lower-roman;\n}\n\nh2,\nh3 {\n\tmargin-bottom: .75em;\n}\nhr {\n\tborder-top: none;\n\tborder-right: none;\n\tborder-bottom: 1px solid;\n\tborder-left: none;\n}\nh1 {\n\tborder-color: #c5c5c5;\n}\nblockquote {\n\tborder-color: #bababa;\n\tcolor: #656565;\n}\n\nblockquote ul,\nblockquote ol {\n\tmargin-left:0;\n}\n\n.ty-table-edit {\n\tbackground-color: transparent;\n}\nthead {\n\tbackground-color: #dadada;\n}\ntr:nth-child(even) {\n\tbackground: #e8e7e7;\n}\nhr {\n\tborder-color: #c5c5c5;\n}\n.task-list{\n\tpadding-left: 1rem;\n}\n\n.md-task-list-item {\n\tpadding-left: 1.5rem;\n\tlist-style-type: none;\n}\n\n.md-task-list-item > input:before {\n\tcontent: '\\221A';\n\tdisplay: inline-block;\n\twidth: 1.25rem;\n  \theight: 1.6rem;\n\tvertical-align: middle;\n\ttext-align: center;\n\tcolor: #ddd;\n\tbackground-color: #F3F2EE;\n}\n\n.md-task-list-item > input:checked:before,\n.md-task-list-item > input[checked]:before{\n\tcolor: inherit;\n}\n\n#write pre.md-meta-block {\n\tmin-height: 1.875rem;\n\tcolor: #555;\n\tborder: 0px;\n\tbackground: transparent;\n\tmargin-top: -4px;\n\tmargin-left: 1em;\n\tmargin-top: 1em;\n}\n\n.md-image>.md-meta {\n\tcolor: #9B5146;\n}\n\n.md-image>.md-meta{\n\tfont-family: Menlo, 'Ubuntu Mono', Consolas, 'Courier New', 'Microsoft Yahei', 'Hiragino Sans GB', 'WenQuanYi Micro Hei', serif;\n}\n\n\n#write>h3.md-focus:before{\n\tleft: -1.5rem;\n\tcolor:#999;\n\tborder-color:#999;\n}\n#write>h4.md-focus:before{\n\tleft: -1.5rem;\n\ttop: .25rem;\n\tcolor:#999;\n\tborder-color:#999;\n}\n#write>h5.md-focus:before{\n\tleft: -1.5rem;\n\ttop: .0.3125rem;\n\tcolor:#999;\n\tborder-color:#999;\n}\n#write>h6.md-focus:before{\n\tleft: -1.5rem;\n\ttop: 0.3125rem;\n\tcolor:#999;\n\tborder-color:#999;\n}\n\n.md-toc:focus .md-toc-content{\n\tmargin-top: 19px;\n}\n\n.md-toc-content:empty:before{\n\tcolor: #065588;\n}\n.md-toc-item {\n\tcolor: #065588;\n}\n#write div.md-toc-tooltip {\n\tbackground-color: #f3f2ee;\n}\n\n#typora-sidebar {\n\tbackground-color: #f3f2ee;\n\t-webkit-box-shadow: 0 6px 12px rgba(0, 0, 0, 0.375);\n  \tbox-shadow: 0 6px 12px rgba(0, 0, 0, 0.375);\n}\n\n.pin-outline #typora-sidebar {\n\tbackground: inherit;\n\tbox-shadow: none;\n\tborder-right: 1px dashed;\n}\n\n.pin-outline #typora-sidebar:hover .outline-title-wrapper {\n\tborder-left:1px dashed;\n}\n\n.outline-item:hover {\n  background-color: #dadada;\n  border-left: 28px solid #dadada;\n  border-right: 18px solid #dadada;\n}\n\n.typora-node .outline-item:hover {\n  \tborder-right: 28px solid #dadada;\n}\n\n.outline-expander:before {\n  content: \"\\f0da\";\n  font-family: FontAwesome;\n  font-size:14px;\n  top: 1px;\n}\n\n.outline-expander:hover:before,\n.outline-item-open>.outline-item>.outline-expander:before {\n  content: \"\\f0d7\";\n}\n\n.modal-content {\n\tbackground-color: #f3f2ee;\n}\n\n.auto-suggest-container ul li {\n\tlist-style-type: none;\n}\n\n\/** UI for electron *\/\n\n.megamenu-menu,\n#top-titlebar, #top-titlebar *,\n.megamenu-content {\n\tbackground: #f3f2ee;\n\tcolor: #1f0909;\n}\n\n.megamenu-menu-header {\n\tborder-bottom: 1px dashed #202B33;\n}\n\n.megamenu-menu {\n\tbox-shadow: none;\n\tborder-right: 1px dashed;\n}\n\nheader, .context-menu, .megamenu-content, footer {\n\tfont-family: \"PT Serif\", 'Times New Roman', Times, serif;\n    color: #1f0909;\n}\n\n#megamenu-back-btn {\n\tcolor: #1f0909;\n\tborder-color: #1f0909;\n}\n\n.megamenu-menu-header #megamenu-menu-header-title:before {\n\tcolor: #1f0909;\n}\n\n.megamenu-menu-list li a:hover, .megamenu-menu-list li a.active {\n\tcolor: inherit;\n\tbackground-color: #e8e7df;\n}\n\n.long-btn:hover {\n\tbackground-color: #e8e7df;\n}\n\n#recent-file-panel tbody tr:nth-child(2n-1) {\n    background-color: transparent !important;\n}\n\n.megamenu-menu-panel tbody tr:hover td:nth-child(2) {\n    color: inherit;\n}\n\n.megamenu-menu-panel .btn {\n\tbackground-color: #D2D1D1;\n}\n\n.btn-default {\n\tbackground-color: transparent;\n}\n\n.typora-sourceview-on #toggle-sourceview-btn,\n.ty-show-word-count #footer-word-count {\n\tbackground: #c7c5c5;\n}\n\n#typora-quick-open {\n    background-color: inherit;\n}\n\n.md-diagram-panel {\n\tmargin-top: 8px;\n}\n\n.file-list-item-file-name {\n\tfont-weight: initial;\n}\n\n.file-list-item-summary {\n\topacity: 1;\n}\n\n.file-list-item {\n\tcolor: #777;\n}\n\n.file-list-item.active {\n\tbackground-color: inherit;\n\tcolor: black;\n}\n\n.ty-side-sort-btn.active {\n\tbackground-color: inherit;\n}\n\n.file-list-item.active .file-list-item-file-name  {\n\tfont-weight: bold;\n}\n\n.file-list-item{\n    opacity:1 !important;\n}\n\n.file-library-node.active>.file-node-background{\n\tbackground-color: rgba(32, 43, 51, 0.63);\n\tbackground-color: var(--active-file-bg-color);\n}\n\n.file-tree-node.active>.file-node-content{\n\tcolor: white;\n\tcolor: var(--active-file-text-color);\n}\n\n.md-task-list-item>input {\n\tmargin-left: -1.7em;\n\tmargin-top: calc(1rem - 12px);\n\t-webkit-appearance: button;\n}\n\ninput {\n\tborder: 1px solid #aaa;\n}\n\n.megamenu-menu-header #megamenu-menu-header-title,\n.megamenu-menu-header:hover, \n.megamenu-menu-header:focus {\n\tcolor: inherit;\n}\n\n.dropdown-menu .divider {\n\tborder-color: #e5e5e5;\n\topacity: 1;\n}\n\n\/* https:\/\/github.com\/typora\/typora-issues\/issues\/2046 *\/\n.os-windows-7 strong,\n.os-windows-7 strong  {\n\tfont-weight: 760;\n}\n\n.ty-preferences .btn-default {\n\tbackground: transparent;\n}\n\n.ty-preferences .window-header {\n\tborder-bottom: 1px dashed #202B33;\n\tbox-shadow: none;\n}\n\n#sidebar-loading-template, #sidebar-loading-template.file-list-item {\n\tcolor: #777;\n}\n\n.searchpanel-search-option-btn.active {\n\tbackground: #777;\n\tcolor: white;\n}\n\n.export-detail, .light .export-detail, \n.light .export-item.active, \n.light .export-items-list-control {\n\tbackground: #e0e0e0;\n\tborder-radius: 2px;\n\tfont-weight: 700;\n\tcolor: inherit\n}\n\n\nmjx-container[jax=\"SVG\"] {\n  direction: ltr;\n}\n\nmjx-container[jax=\"SVG\"] > svg {\n  overflow: visible;\n  min-height: 1px;\n  min-width: 1px;\n}\n\nmjx-container[jax=\"SVG\"] > svg a {\n  fill: blue;\n  stroke: blue;\n}\n\nmjx-assistive-mml {\n  position: absolute !important;\n  top: 0px;\n  left: 0px;\n  clip: rect(1px, 1px, 1px, 1px);\n  padding: 1px 0px 0px 0px !important;\n  border: 0px !important;\n  display: block !important;\n  width: auto !important;\n  overflow: hidden !important;\n  -webkit-touch-callout: none;\n  -webkit-user-select: none;\n  -khtml-user-select: none;\n  -moz-user-select: none;\n  -ms-user-select: none;\n  user-select: none;\n}\n\nmjx-assistive-mml[display=\"block\"] {\n  width: 100% !important;\n}\n\nmjx-container[jax=\"SVG\"][display=\"true\"] {\n  display: block;\n  text-align: center;\n  margin: 1em 0;\n}\n\nmjx-container[jax=\"SVG\"][display=\"true\"][width=\"full\"] {\n  display: flex;\n}\n\nmjx-container[jax=\"SVG\"][justify=\"left\"] {\n  text-align: left;\n}\n\nmjx-container[jax=\"SVG\"][justify=\"right\"] {\n  text-align: right;\n}\n\ng[data-mml-node=\"merror\"] > g {\n  fill: red;\n  stroke: red;\n}\n\ng[data-mml-node=\"merror\"] > rect[data-background] {\n  fill: yellow;\n  stroke: none;\n}\n\ng[data-mml-node=\"mtable\"] > line[data-line], svg[data-table] > g > line[data-line] {\n  stroke-width: 70px;\n  fill: none;\n}\n\ng[data-mml-node=\"mtable\"] > rect[data-frame], svg[data-table] > g > rect[data-frame] {\n  stroke-width: 70px;\n  fill: none;\n}\n\ng[data-mml-node=\"mtable\"] > .mjx-dashed, svg[data-table] > g > .mjx-dashed {\n  stroke-dasharray: 140;\n}\n\ng[data-mml-node=\"mtable\"] > .mjx-dotted, svg[data-table] > g > .mjx-dotted {\n  stroke-linecap: round;\n  stroke-dasharray: 0,140;\n}\n\ng[data-mml-node=\"mtable\"] > g > svg {\n  overflow: visible;\n}\n\n[jax=\"SVG\"] mjx-tool {\n  display: inline-block;\n  position: relative;\n  width: 0;\n  height: 0;\n}\n\n[jax=\"SVG\"] mjx-tool > mjx-tip {\n  position: absolute;\n  top: 0;\n  left: 0;\n}\n\nmjx-tool > mjx-tip {\n  display: inline-block;\n  padding: .2em;\n  border: 1px solid #888;\n  font-size: 70%;\n  background-color: #F8F8F8;\n  color: black;\n  box-shadow: 2px 2px 5px #AAAAAA;\n}\n\ng[data-mml-node=\"maction\"][data-toggle] {\n  cursor: pointer;\n}\n\nmjx-status {\n  display: block;\n  position: fixed;\n  left: 1em;\n  bottom: 1em;\n  min-width: 25%;\n  padding: .2em .4em;\n  border: 1px solid #888;\n  font-size: 90%;\n  background-color: #F8F8F8;\n  color: black;\n}\n\nforeignObject[data-mjx-xml] {\n  font-family: initial;\n  line-height: normal;\n  overflow: visible;\n}\n\nmjx-container[jax=\"SVG2\"] path[data-c], mjx-container[jax=\"SVG2\"] use[data-c] {\n  stroke-width: 3;\n}\n\ng[data-mml-node=\"xypic\"] path {\n  stroke-width: inherit;\n}\n\n.MathJax g[data-mml-node=\"xypic\"] path {\n  stroke-width: inherit;\n}\nmjx-container[jax=\"SVG\"] path[data-c], mjx-container[jax=\"SVG\"] use[data-c] {\n\t\t\t\t\t\t\tstroke-width: 0;\n\t\t\t\t\t\t} @media print { @page {margin: 0 0 0 0;} body.typora-export {padding-left: 0; padding-right: 0;} #write {padding:0;}}\n<\/style><title>HW1.1 Image Manipulation<\/title>\n<\/head>\n<body class='typora-export'><div class='typora-export-content'>\n<div id='write'  class=''><h2 id='hw11-image-manipulation'><span>HW1.1 Image Manipulation<\/span><\/h2><h3 id='a11-load-image-and-determine-information-of-this-image'><span>A1.1. Load image and determine information of this image.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im<\/span>=<span class=\"cm-variable\">imread<\/span>(<span class=\"cm-string\">'pears.jpg'<\/span>);<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">im<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">datatype<\/span>=<span class=\"cm-variable\">class<\/span>(<span class=\"cm-variable\">im<\/span>); <span class=\"cm-comment\">% uint8, unsigned integer<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_min<\/span>=<span class=\"cm-builtin\">min<\/span>(<span class=\"cm-variable\">im<\/span>(:));<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_max<\/span>=<span class=\"cm-builtin\">max<\/span>(<span class=\"cm-variable\">im<\/span>(:));<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_stddev<\/span>=<span class=\"cm-builtin\">std<\/span>(<span class=\"cm-variable\">double<\/span>(<span class=\"cm-variable\">im<\/span>(:)));<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">imhist<\/span>(<span class=\"cm-variable\">im<\/span>);<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 192px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 192px;\"><\/div><\/div><\/div><\/pre><h3 id='a12-convert-the-data-type-of-image-to-double-and-save-it-to-imd-which-is-matlabs-default-data-type-required-by-many-built-in-functions-data-of-this-type-lies-in-range-\\n\\n0\\n\\n1\\n\\n'><span>A1.2. Convert the data type of image to <\/span><code>double<\/code><span> and save it to <\/span><code>imd<\/code><span>, which is MATLAB&#39;s default data type required by many built-in functions. Data of this type lies in range <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.526ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2000.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-88-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-88-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-88-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-88-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-88-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"5B\" xlink:href=\"#MJX-88-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(278,0)\"><use data-c=\"30\" xlink:href=\"#MJX-88-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(778,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-88-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1222.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-88-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1722.7,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-88-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">[<\/mo><mn>0<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">[0,1]<\/script><span>.<\/span><\/h3><ul><li><p><span>There is no difference when displaying <\/span><code>im<\/code><span> and <\/span><code>imd<\/code><span> using <\/span><code>imshow()<\/code><span>.<\/span><\/p><\/li><li><p><span>Because <\/span><code>imshow()<\/code><span> automatically recognizes the type of image and devises the correct range based on that for display.<\/span><\/p><\/li><\/ul><h3 id='draw-a-\\n2\\n-pixel-wide-black-border-without-changing-the-size-save-the-new-image-to-imframed'><span>Draw a <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.507ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 500 666\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-89-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"32\" xlink:href=\"#MJX-89-TEX-N-32\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>2<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">2<\/script><span> pixel wide black border without changing the size, save the new image to <\/span><code>im_framed<\/code><span>.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">color_black<\/span>=<span class=\"cm-number\">0<\/span>;<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_framed<\/span>=<span class=\"cm-variable\">im<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% rows of matrix \/ width of image<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_framed<\/span>([<span class=\"cm-number\">1<\/span>:<span class=\"cm-number\">2<\/span>,<span class=\"cm-keyword\">end<\/span><span class=\"cm-number\">-1<\/span>:<span class=\"cm-keyword\">end<\/span>],:)=<span class=\"cm-variable\">color_black<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% columns of matrix \/ height of image<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_framed<\/span>(:,[<span class=\"cm-number\">1<\/span>:<span class=\"cm-number\">2<\/span>,<span class=\"cm-keyword\">end<\/span><span class=\"cm-number\">-1<\/span>:<span class=\"cm-keyword\">end<\/span>])=<span class=\"cm-variable\">color_black<\/span>;<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 144px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 144px;\"><\/div><\/div><\/div><\/pre><h3 id='a13-coursework-platform-accepts-a-tolerance-of-\\n001\\n-add-gaussian-noise-with-\\n\u03c3\\n\\n0005\\n-to-the-image-how-many-pixels-do-you-expect-to-lie-outside-the-toleranc-range'><span>A1.3. Coursework platform accepts a tolerance of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.023ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 1778 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-90-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-90-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-90-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"30\" xlink:href=\"#MJX-90-TEX-N-30\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-90-TEX-N-2E\" transform=\"translate(500,0)\"><\/use><use data-c=\"30\" xlink:href=\"#MJX-90-TEX-N-30\" transform=\"translate(778,0)\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-90-TEX-N-31\" transform=\"translate(1278,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0.01<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">0.01<\/script><span>. Add Gaussian noise with <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.463ex\" height=\"1.692ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 4182.6 748\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-91-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-91-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-91-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-91-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-91-TEX-N-35\" d=\"M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-91-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(848.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-91-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1904.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-91-TEX-N-30\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-91-TEX-N-2E\" transform=\"translate(500,0)\"><\/use><use data-c=\"30\" xlink:href=\"#MJX-91-TEX-N-30\" transform=\"translate(778,0)\"><\/use><use data-c=\"30\" xlink:href=\"#MJX-91-TEX-N-30\" transform=\"translate(1278,0)\"><\/use><use data-c=\"35\" xlink:href=\"#MJX-91-TEX-N-35\" transform=\"translate(1778,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03c3<\/mi><mo>=<\/mo><mn>0.005<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\sigma=0.005<\/script><span> to the image. How many pixels do you expect to lie outside the toleranc range?<\/span><\/h3><ul><li><p><span>Given<\/span><\/p><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n15\" cid=\"n15\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"1\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 53.011ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.476ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.173ex; min-width: 53.011ex;\"><defs><path id=\"MJX-1-TEX-I-1D452\" d=\"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z\"><\/path><path id=\"MJX-1-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-1-TEX-I-1D450\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path id=\"MJX-1-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-1-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-1-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 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display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mtext>(1)<\/mtext><\/mtd><mtd><mi>e<\/mi><mi>r<\/mi><mi>f<\/mi><mi>c<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>e<\/mi><mi>r<\/mi><mi>f<\/mi><mo stretchy=\"false\">(<\/mo><mi>n<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>2<\/mn><msqrt><mi>\u03c0<\/mi><\/msqrt><\/mfrac><msubsup><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mi>n<\/mi><mi mathvariant=\"normal\">\u221e<\/mi><\/msubsup><mi>exp<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><msup><mi>t<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>t<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><p><span>We use<\/span><\/p><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" 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xlink:href=\"#MJX-2-TEX-N-28\"><\/use><use data-c=\"32\" xlink:href=\"#MJX-2-TEX-N-32\" transform=\"translate(389,0)\"><\/use><use data-c=\"29\" xlink:href=\"#MJX-2-TEX-N-29\" transform=\"translate(889,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mtext>(2)<\/mtext><\/mtd><mtd><mi>n<\/mi><mo>=<\/mo><mfrac><mrow><mi>x<\/mi><mo>\u2212<\/mo><mi>\u03bc<\/mi><\/mrow><mrow><mi>\u03c3<\/mi><msqrt><mn>2<\/mn><\/msqrt><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mn>0.01<\/mn><mrow><mi>\u03c3<\/mi><msqrt><mn>2<\/mn><\/msqrt><\/mrow><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='c12-copy-each-second-row-and-each-third-column-of-imframed-save-it-to-impart'><span>C1.2. Copy each second row and each third column of <\/span><code>im_framed<\/code><span>, save it to <\/span><code>im_part<\/code><span>.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_part<\/span>=<span class=\"cm-variable\">im_framed<\/span>(<span class=\"cm-number\">1<\/span>:<span class=\"cm-number\">2<\/span>:<span class=\"cm-keyword\">end<\/span>,<span class=\"cm-number\">1<\/span>:<span class=\"cm-number\">3<\/span>:<span class=\"cm-keyword\">end<\/span>);<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 24px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 24px;\"><\/div><\/div><\/div><\/pre><h3 id='a14-draw-a-red-frame-around-the-image-using-plot-function'><span>A1.4. Draw a red frame around the image using <\/span><code>plot()<\/code><span> function.<\/span><\/h3><ul><li><p><code>plot()<\/code><span> does not change the image data, and is not constrained to integer pixel positions.<\/span><\/p><\/li><\/ul><h3 id='c13-use-kronecker-product-to-create-colorized-copies-of-grayscale-image-im'><span>C1.3. Use Kronecker product to create colorized copies of grayscale image <\/span><code>im<\/code><span> <\/span><\/h3><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n28\" cid=\"n28\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"3\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 24.357ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 24.357ex;\"><defs><path id=\"MJX-3-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-3-TEX-I-1D45A\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-3-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 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626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z\"><\/path><path id=\"MJX-3-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.0181,-0.0181) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(2078,0) translate(-2078,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(55.25)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3305 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-3-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(345,0)\"><use data-c=\"1D45A\" xlink:href=\"#MJX-3-TEX-I-1D45A\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1223,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-3-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(554,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-3-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2348.7,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-3-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3404.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D450\" xlink:href=\"#MJX-3-TEX-I-1D450\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(466,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-3-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4386.7,0)\"><use data-c=\"2297\" xlink:href=\"#MJX-3-TEX-N-2297\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5386.9,0)\"><use data-c=\"1D456\" xlink:href=\"#MJX-3-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5731.9,0)\"><use data-c=\"1D45A\" xlink:href=\"#MJX-3-TEX-I-1D45A\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"1278 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:3\"><g data-mml-node=\"mtext\"><use data-c=\"28\" xlink:href=\"#MJX-3-TEX-N-28\"><\/use><use data-c=\"33\" xlink:href=\"#MJX-3-TEX-N-33\" transform=\"translate(389,0)\"><\/use><use data-c=\"29\" xlink:href=\"#MJX-3-TEX-N-29\" transform=\"translate(889,0)\"><\/use><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mtext>(3)<\/mtext><\/mtd><mtd><mi>i<\/mi><mi>m<\/mi><msub><mi>k<\/mi><mi>i<\/mi><\/msub><mo>=<\/mo><msub><mi>c<\/mi><mi>i<\/mi><\/msub><mo>\u2297<\/mo><mi>i<\/mi><mi>m<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='calculate-each-color-channel-separately-using'><span>Calculate each color channel separately using<\/span><\/h3><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n30\" cid=\"n30\" mdtype=\"math_block\" data-math-tag-before=\"3\" data-math-tag-after=\"4\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 71.179ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"8.597ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -3.733ex; min-width: 71.179ex;\"><defs><path id=\"MJX-4-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-4-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 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0V602H403V0H319Z\"><\/path><path id=\"MJX-4-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-4-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-4-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-4-TEX-N-33\" d=\"M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 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id=\"MJX-4-TEX-I-1D43A\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q492 659 471 656T418 643T357 615T294 567T236 496T189 394T158 260Q156 242 156 221Q156 173 170 136T206 79T256 45T308 28T353 24Q407 24 452 47T514 106Q517 114 529 161T541 214Q541 222 528 224T468 227H431Q425 233 425 235T427 254Q431 267 437 273H454Q494 271 594 271Q634 271 659 271T695 272T707 272Q721 272 721 263Q721 261 719 249Q714 230 709 228Q706 227 694 227Q674 227 653 224Q646 221 643 215T629 164Q620 131 614 108Q589 6 586 3Q584 1 581 1Q571 1 553 21T530 52Q530 53 528 52T522 47Q448 -22 322 -22Q201 -22 126 55T50 252Z\"><\/path><path id=\"MJX-4-TEX-N-37\" d=\"M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z\"><\/path><path id=\"MJX-4-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-4-TEX-N-34\" d=\"M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z\"><\/path><path id=\"MJX-4-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 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data-mml-node=\"mtr\"><g data-mml-node=\"mtd\" transform=\"translate(389,0)\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-4-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(2278,0)\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-4-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(3778,0)\"><g data-mml-node=\"mn\"><use data-c=\"30\" xlink:href=\"#MJX-4-TEX-N-30\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-1400)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mn\"><use data-c=\"30\" xlink:href=\"#MJX-4-TEX-N-30\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-4-TEX-N-2E\" transform=\"translate(500,0)\"><\/use><use data-c=\"37\" xlink:href=\"#MJX-4-TEX-N-37\" transform=\"translate(778,0)\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(2278,0)\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-4-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" 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xlink:href=\"#MJX-4-TEX-N-29\" transform=\"translate(889,0)\"><\/use><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mtext>(4)<\/mtext><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><msub><mi>C<\/mi><mi>R<\/mi><\/msub><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>1<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0.3<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>0.3<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mo>,<\/mo><msub><mi>C<\/mi><mi>G<\/mi><\/msub><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0.3<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0.7<\/mn><\/mtd><mtd><mn>0.3<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mo>,<\/mo><msub><mi>C<\/mi><mi>R<\/mi><\/msub><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>1<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0.7<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='merge-these-channels-to-a-color-image'><span>Merge these channels to a color image.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\" style=\"break-inside: unset;\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">im<\/span>);<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_R<\/span>=<span class=\"cm-builtin\">zeros<\/span>(<span class=\"cm-number\">3<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_G<\/span>=<span class=\"cm-builtin\">zeros<\/span>(<span class=\"cm-number\">3<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_B<\/span>=<span class=\"cm-builtin\">zeros<\/span>(<span class=\"cm-number\">3<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">row<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-number\">3<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">col<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-number\">3<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">im_R<\/span>(<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span>(<span class=\"cm-variable\">row<\/span><span class=\"cm-number\">-1<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>):<span class=\"cm-variable\">row<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>),<span class=\"cm-error\">...<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span>(<span class=\"cm-variable\">col<\/span><span class=\"cm-number\">-1<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>):<span class=\"cm-variable\">col<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>))=<span class=\"cm-variable\">cR<\/span>(<span class=\"cm-variable\">row<\/span>,<span class=\"cm-variable\">col<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">im<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">im_G<\/span>(<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span>(<span class=\"cm-variable\">row<\/span><span class=\"cm-number\">-1<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>):<span class=\"cm-variable\">row<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>),<span class=\"cm-error\">...<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span>(<span class=\"cm-variable\">col<\/span><span class=\"cm-number\">-1<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>):<span class=\"cm-variable\">col<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>))=<span class=\"cm-variable\">cG<\/span>(<span class=\"cm-variable\">row<\/span>,<span class=\"cm-variable\">col<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">im<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">im_B<\/span>(<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span>(<span class=\"cm-variable\">row<\/span><span class=\"cm-number\">-1<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>):<span class=\"cm-variable\">row<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>),<span class=\"cm-error\">...<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span>(<span class=\"cm-variable\">col<\/span><span class=\"cm-number\">-1<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>):<span class=\"cm-variable\">col<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>))=<span class=\"cm-variable\">cB<\/span>(<span class=\"cm-variable\">row<\/span>,<span class=\"cm-variable\">col<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">im<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_rgb<\/span>=<span class=\"cm-builtin\">zeros<\/span>([<span class=\"cm-number\">3<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span> <span class=\"cm-number\">3<\/span>]);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_rgb<\/span>(:,:,<span class=\"cm-number\">1<\/span>)=<span class=\"cm-variable\">im_R<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_rgb<\/span>(:,:,<span class=\"cm-number\">2<\/span>)=<span class=\"cm-variable\">im_G<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_rgb<\/span>(:,:,<span class=\"cm-number\">3<\/span>)=<span class=\"cm-variable\">im_B<\/span>;<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 480px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 480px;\"><\/div><\/div><\/div><\/pre><h2 id='hw12-periodic-sequences'><span>HW1.2. Periodic Sequences<\/span><\/h2><h3 id='c21-compute-atomic-image-elements-from-im-specified-by-the-periodicity-matrix-pm-given-the-location-locbase-of-the-topmost-pixel-\\n\\nx\\n\\ny\\n\\n'><span>C2.1. Compute atomic image elements from <\/span><code>im<\/code><span> specified by the periodicity matrix <\/span><code>pm<\/code><span>, given the location <\/span><code>loc_base<\/code><span> of the topmost pixel <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.169ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2284.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-92-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-92-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-92-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-92-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-92-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-92-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-92-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(961,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-92-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1405.7,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-92-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1895.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-92-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>,<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(x,y)<\/script><span>.<\/span><\/h3><h3 id='notice-that-index-\\nx\\n-in-an-image-corresponds-to-the-columnsecond-index-in-a-matrix-and-index-\\ny\\n-in-an-image-corresponds-to-the-rowfirst-index-in-a-matrix'><span>Notice that index <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.294ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 572 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-100-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-100-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x<\/script><span> in an image corresponds to the <\/span><strong><span>column\/second<\/span><\/strong><span> index in a matrix, and index <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.109ex\" height=\"1.464ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 490 647\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-101-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D466\" xlink:href=\"#MJX-101-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">y<\/script><span> in an image corresponds to the <\/span><strong><span>row\/first<\/span><\/strong><span> index in a matrix.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\" style=\"break-inside: unset;\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-builtin\">function<\/span> [<span class=\"cm-variable\">atomic<\/span>,<span class=\"cm-variable\">mask<\/span>]=<span class=\"cm-variable\">get_atomic<\/span>(<span class=\"cm-variable\">im<\/span>,<span class=\"cm-variable\">pm<\/span>,<span class=\"cm-variable\">loc_base<\/span>)<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% Define polygon<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 5 points determine a Parallelogramm<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">poly<\/span>=<span class=\"cm-builtin\">zeros<\/span>(<span class=\"cm-number\">2<\/span>,<span class=\"cm-number\">5<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">poly<\/span>(:,<span class=\"cm-number\">1<\/span>)=<span class=\"cm-variable\">loc_base<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">poly<\/span>(:,<span class=\"cm-number\">2<\/span>)=<span class=\"cm-variable\">poly<\/span>(:,<span class=\"cm-number\">1<\/span>)<span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">pm<\/span>(:,<span class=\"cm-number\">1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">poly<\/span>(:,<span class=\"cm-number\">3<\/span>)=<span class=\"cm-variable\">poly<\/span>(:,<span class=\"cm-number\">1<\/span>)<span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">pm<\/span>(:,<span class=\"cm-number\">2<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">poly<\/span>(:,<span class=\"cm-number\">4<\/span>)=<span class=\"cm-variable\">poly<\/span>(:,<span class=\"cm-number\">1<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">pm<\/span>(:,<span class=\"cm-number\">1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">poly<\/span>(:,<span class=\"cm-number\">5<\/span>)=<span class=\"cm-variable\">poly<\/span>(:,<span class=\"cm-number\">1<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">pm<\/span>(:,<span class=\"cm-number\">2<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% Create mask for atomic element (this procedure is given)<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">poly<\/span>=<span class=\"cm-variable\">poly<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">repmat<\/span>([<span class=\"cm-number\">0<\/span>;<span class=\"cm-number\">0.4<\/span>]<span class=\"cm-operator\">-<\/span><span class=\"cm-builtin\">min<\/span>(<span class=\"cm-variable\">poly<\/span>,[],<span class=\"cm-number\">2<\/span>),<span class=\"cm-number\">1<\/span>,<span class=\"cm-number\">5<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz_mask<\/span>=<span class=\"cm-builtin\">round<\/span>(<span class=\"cm-builtin\">max<\/span>(<span class=\"cm-variable\">poly<\/span>,[],<span class=\"cm-number\">2<\/span>));<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">mask<\/span>=<span class=\"cm-variable\">poly2mask<\/span>(<span class=\"cm-variable\">poly<\/span>(<span class=\"cm-number\">1<\/span>,:),<span class=\"cm-variable\">poly<\/span>(<span class=\"cm-number\">2<\/span>,:),<span class=\"cm-variable\">sz_mask<\/span>(<span class=\"cm-number\">2<\/span>),<span class=\"cm-variable\">sz_mask<\/span>(<span class=\"cm-number\">1<\/span>));<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% Create atomic element<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">atomic<\/span>=<span class=\"cm-variable\">im<\/span>(<span class=\"cm-variable\">loc_base<\/span>(<span class=\"cm-number\">2<\/span>):<span class=\"cm-variable\">loc_base<\/span>(<span class=\"cm-number\">2<\/span>)<span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">sz_mask<\/span>(<span class=\"cm-number\">2<\/span>)<span class=\"cm-number\">-1<\/span>,<span class=\"cm-error\">...<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">loc_base<\/span>(<span class=\"cm-number\">1<\/span>):<span class=\"cm-variable\">loc_base<\/span>(<span class=\"cm-number\">1<\/span>)<span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">sz_mask<\/span>(<span class=\"cm-number\">1<\/span>)<span class=\"cm-number\">-1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">atomic<\/span>=<span class=\"cm-variable\">atomic<\/span><span class=\"cm-operator\">.*<\/span><span class=\"cm-variable\">mask<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 480px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 480px;\"><\/div><\/div><\/div><\/pre><h3 id='c22-implement-an-inverse-operation-generate-a-periodic-image-given-atomic-element'><span>C2.2. Implement an inverse operation: generate a periodic image given atomic element.?<\/span><\/h3><h3 id='c23-calculate-ncc-between-two-equal-sized-periodic-images'><span>C2.3. Calculate NCC between two equal-sized periodic images.<\/span><\/h3><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n39\" cid=\"n39\" mdtype=\"math_block\" data-math-tag-before=\"4\" data-math-tag-after=\"5\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 56.523ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"6.599ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.734ex; min-width: 56.523ex;\"><defs><path id=\"MJX-5-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-5-TEX-I-1D450\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path id=\"MJX-5-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 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data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><mo>,<\/mo><mi>y<\/mi><\/mrow><\/munder><mfrac><mrow><mo stretchy=\"false\">(<\/mo><msub><mi>i<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>,<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><msub><mi>i<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>i<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>,<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><msub><mi>i<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><msub><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>i<\/mi><mn>1<\/mn><\/msub><\/mrow><\/msub><msub><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>i<\/mi><mn>2<\/mn><\/msub><\/mrow><\/msub><\/mrow><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='n\\n-is-number-of-elements'><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-121-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-121-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N<\/script><span> is number of elements.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">i1<\/span>);<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">i1_nomi<\/span>=(<span class=\"cm-variable\">i1<\/span>(:)<span class=\"cm-operator\">-<\/span><span class=\"cm-builtin\">mean<\/span>(<span class=\"cm-variable\">i1<\/span>(:)))<span class=\"cm-operator\">\/<\/span><span class=\"cm-builtin\">std<\/span>(<span class=\"cm-variable\">i1<\/span>(:));<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">i2_nomi<\/span>=(<span class=\"cm-variable\">i2<\/span>(:)<span class=\"cm-operator\">-<\/span><span class=\"cm-builtin\">mean<\/span>(<span class=\"cm-variable\">i2<\/span>(:)))<span class=\"cm-operator\">\/<\/span><span class=\"cm-builtin\">std<\/span>(<span class=\"cm-variable\">i2<\/span>(:));<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">ncc<\/span>=<span class=\"cm-variable\">i1_nomi<\/span><span class=\"cm-operator\">.*<\/span><span class=\"cm-variable\">i2_nomi<\/span><span class=\"cm-operator\">\/<\/span>(<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>));<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">ncc<\/span>=<span class=\"cm-builtin\">sum<\/span>(<span class=\"cm-variable\">ncc<\/span>(:));<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 120px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 120px;\"><\/div><\/div><\/div><\/pre><h3 id='c24-find-the-atomic-element-of-a-given-periodic-image-using-exhausive-search-and-correlation'><span>C2.4. Find the atomic element of a given periodic image using exhausive search and correlation.?<\/span><\/h3><h3 id='a21-why-do-you-observe-several-maxima-in-the-ncc-values-of-the-whole-image'><span>A2.1. Why do you observe several maxima in the NCC values of the whole image?<\/span><\/h3><ul><li><p><span>Because the considered image has partial repetitive parts, which can be matched to the whole.<\/span><\/p><\/li><\/ul><h2 id='hw21-image-sampling'><span>HW2.1. Image Sampling<\/span><\/h2><h3 id='c11-generate-a-regular-and-rectangular-sampling-grid-of-dimensions-\\nn\\n\u00d7\\nn\\n-\\nn\\n\\n400\\n-which-starts-at-\\n\\n1\\n\\n1\\n\\n-and-spans-exactly-the-range-\\n\\n1\\n\\n512\\n\\n-for-\\nx\\n-and-\\ny\\n'><span>C1.1. Generate a regular and rectangular sampling grid of dimensions <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.784ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2998.4 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-111-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-111-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-111-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1110.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-111-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2110.4,0)\"><use data-c=\"1D441\" xlink:href=\"#MJX-111-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><mo>\u00d7<\/mo><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N\\times N<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.42ex\" height=\"1.731ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 3721.6 765\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-97-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 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jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.029ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2222.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-102-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-102-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-102-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 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xlink:href=\"#MJX-102-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1833.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-102-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(1,1)<\/script><span> and spans exactly the range <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.789ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3000.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-99-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-99-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-99-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-99-TEX-N-35\" d=\"M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z\"><\/path><path id=\"MJX-99-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-99-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"5B\" xlink:href=\"#MJX-99-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(278,0)\"><use data-c=\"31\" xlink:href=\"#MJX-99-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(778,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-99-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1222.7,0)\"><use data-c=\"35\" xlink:href=\"#MJX-99-TEX-N-35\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-99-TEX-N-31\" transform=\"translate(500,0)\"><\/use><use data-c=\"32\" xlink:href=\"#MJX-99-TEX-N-32\" transform=\"translate(1000,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2722.7,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-99-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">[<\/mo><mn>1<\/mn><mo>,<\/mo><mn>512<\/mn><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">[1,512]<\/script><span> for <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.294ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 572 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-100-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-100-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.109ex\" height=\"1.464ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 490 647\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-101-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D466\" xlink:href=\"#MJX-101-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">y<\/script><span>.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-number\">512<\/span>;<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\">[<span class=\"cm-variable\">gridX<\/span>,<span class=\"cm-variable\">gridY<\/span>]=<span class=\"cm-builtin\">meshgrid<\/span>(<span class=\"cm-builtin\">linspace<\/span>(<span class=\"cm-number\">1<\/span>,<span class=\"cm-variable\">sz<\/span>,<span class=\"cm-variable\">N<\/span>));<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 48px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 48px;\"><\/div><\/div><\/div><\/pre><h3 id='c12-given-gray-scale-image-im-use-interp2-to-find-image-intensity-at-non-integer-coordinates-gridxgridy-try-different-interpolation-options'><span>C1.2. Given gray scale image <\/span><code>im<\/code><span>, use <\/span><code>interp2()<\/code><span> to find image intensity at non-integer coordinates <\/span><code>(gridX,gridY)<\/code><span>. Try different interpolation options.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-builtin\">function<\/span> [<span class=\"cm-variable\">im_linear<\/span>,<span class=\"cm-variable\">im_cubic<\/span>]=<span class=\"cm-variable\">interp<\/span>(<span class=\"cm-variable\">im<\/span>,<span class=\"cm-variable\">gridX<\/span>,<span class=\"cm-variable\">gridY<\/span>)<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">im<\/span>,<span class=\"cm-number\">1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% given image points<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\">[<span class=\"cm-variable\">imX<\/span>,<span class=\"cm-variable\">imY<\/span>]=<span class=\"cm-builtin\">meshgrid<\/span>(<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">sz<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% interpolation<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% imX,imY: sample points<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% im: function values at sample points<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% gridX, gridY: query points<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_linear<\/span>=<span class=\"cm-variable\">interp2<\/span>(<span class=\"cm-variable\">imX<\/span>,<span class=\"cm-variable\">imY<\/span>,<span class=\"cm-variable\">im<\/span>,<span class=\"cm-variable\">gridX<\/span>,<span class=\"cm-variable\">gridY<\/span>,<span class=\"cm-string\">'linear'<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_cubic<\/span>=<span class=\"cm-variable\">interp2<\/span>(<span class=\"cm-variable\">imX<\/span>,<span class=\"cm-variable\">imY<\/span>,<span class=\"cm-variable\">im<\/span>,<span class=\"cm-variable\">gridX<\/span>,<span class=\"cm-variable\">gridY<\/span>,<span class=\"cm-string\">'cubic'<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 264px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 264px;\"><\/div><\/div><\/div><\/pre><h3 id='a11-describe-how-above-images-differ-and-compare-the-runtime'><span>A1.1. Describe how above images differ and compare the runtime.<\/span><\/h3><ul><li><p><span>Image interpolated using <\/span><code>cubic<\/code><span> can better preserve edges and other fine details, but takes more time.<\/span><\/p><\/li><li><p><code>linear<\/code><span> interpolation loses some details but runs faster.<\/span>\n<span>F=TriScatteredInterp(gridX(:),gridY(:),im_scatter(:),&#39;linear&#39;);<\/span><\/p><\/li><\/ul><h3 id='c13-given-image-intensity-at-non-integer-coordinates-gridxgridy-reconstruct-image-with-size-sz-from-imscatter-using-triscatteredinterp'><span>C1.3. Given image intensity at non-integer coordinates <\/span><code>(gridX,gridY)<\/code><span>, reconstruct image with size <\/span><code>sz<\/code><span> from <\/span><code>im_scatter<\/code><span> using <\/span><code>TriScatteredInterp()<\/code><span>.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-builtin\">function<\/span> <span class=\"cm-variable\">im_rec<\/span>=<span class=\"cm-variable\">reconstruct<\/span>(<span class=\"cm-variable\">im_scatter<\/span>,<span class=\"cm-variable\">sz<\/span>,<span class=\"cm-variable\">gridX<\/span>,<span class=\"cm-variable\">gridY<\/span>)<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% Reconstruct function<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% gridX(:),gridY(:): sample points<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% im_scatter(:): function values at sample points<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">F<\/span>=<span class=\"cm-variable\">TriScatteredInterp<\/span>(<span class=\"cm-variable\">gridX<\/span>(:),<span class=\"cm-variable\">gridY<\/span>(:),<span class=\"cm-variable\">im_scatter<\/span>(:),<span class=\"cm-string\">'linear'<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% query image points<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\">[<span class=\"cm-variable\">imX<\/span>,<span class=\"cm-variable\">imY<\/span>]=<span class=\"cm-builtin\">meshgrid<\/span>(<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">sz<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_rec<\/span>=<span class=\"cm-variable\">F<\/span>(<span class=\"cm-variable\">imX<\/span>,<span class=\"cm-variable\">imY<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 216px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 216px;\"><\/div><\/div><\/div><\/pre><h3 id='c14-generate-a-horizontally-aligned-hexagonal-grid-with-the-first-sampling-point-located-at-\\n\\n1\\n\\n1\\n\\n-the-horizontal-and-vertical-sampling-distance-is-\\nd\\nx\\n-and-\\nd\\ny\\n-the-sampling-locations-must-span-horizontally-\\nx\\n\u2208\\n\\n1\\n\\n511\\n\\n-and-vertically-\\ny\\n\u2208\\n\\n1\\n\\n512\\n\\n-given'><span>C1.4. Generate a horizontally aligned hexagonal grid with the first sampling point located at <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.029ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2222.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-102-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-102-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-102-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-102-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-102-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(389,0)\"><use data-c=\"31\" xlink:href=\"#MJX-102-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(889,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-102-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1333.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-102-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1833.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-102-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(1,1)<\/script><span>. The horizontal and vertical sampling distance is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.279ex\" height=\"1.927ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 1007.5 851.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-107-TEX-I-1D451\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-107-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D451\" xlink:href=\"#MJX-107-TEX-I-1D451\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(553,-150) scale(0.707)\"><use data-c=\"1D465\" xlink:href=\"#MJX-107-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>d<\/mi><mi>x<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">d_x<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.148ex\" height=\"2.237ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 949.5 989\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.667ex;\"><defs><path id=\"MJX-108-TEX-I-1D451\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-108-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D451\" xlink:href=\"#MJX-108-TEX-I-1D451\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(553,-150) scale(0.707)\"><use data-c=\"1D466\" xlink:href=\"#MJX-108-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>d<\/mi><mi>y<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">d_y<\/script><span>. The sampling locations must span horizontally <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.849ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4795.2 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-105-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-105-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-105-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-105-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-105-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-105-TEX-N-35\" d=\"M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z\"><\/path><path id=\"MJX-105-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-105-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(849.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-105-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1794.6,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-105-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2072.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-105-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2572.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-105-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3017.2,0)\"><use data-c=\"35\" xlink:href=\"#MJX-105-TEX-N-35\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-105-TEX-N-31\" transform=\"translate(500,0)\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-105-TEX-N-31\" transform=\"translate(1000,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4517.2,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-105-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><mo>\u2208<\/mo><mo stretchy=\"false\">[<\/mo><mn>1<\/mn><mo>,<\/mo><mn>511<\/mn><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x\\in[1,511]<\/script><span> and vertically <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.663ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4713.2 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-106-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-106-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-106-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-106-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-106-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-106-TEX-N-35\" d=\"M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z\"><\/path><path id=\"MJX-106-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-106-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D466\" xlink:href=\"#MJX-106-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(767.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-106-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mo\" 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stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">y\\in[1,512]<\/script><span>, given<\/span><\/h3><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n61\" cid=\"n61\" mdtype=\"math_block\" data-math-tag-before=\"5\" data-math-tag-after=\"6\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 40.849ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"4.774ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.821ex; min-width: 40.849ex;\"><defs><path id=\"MJX-6-TEX-I-1D451\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 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data-mml-node=\"mn\"><use data-c=\"33\" xlink:href=\"#MJX-6-TEX-N-33\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(0,136.7)\"><use data-c=\"221A\" xlink:href=\"#MJX-6-TEX-N-221A\"><\/use><\/g><rect width=\"500\" height=\"60\" x=\"853\" y=\"876.7\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(10306.4,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-6-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(10806.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-6-TEX-N-30\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-6-TEX-N-2E\" transform=\"translate(500,0)\"><\/use><use data-c=\"35\" xlink:href=\"#MJX-6-TEX-N-35\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12306.8,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-6-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(12807.1,0)\"><use data-c=\"1D451\" xlink:href=\"#MJX-6-TEX-I-1D451\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13327.1,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-6-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"1278 -1305 1 2110\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:6\" transform=\"translate(0,713)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,-750)\"><g data-mml-node=\"mtext\"><use data-c=\"28\" xlink:href=\"#MJX-6-TEX-N-28\"><\/use><use data-c=\"36\" xlink:href=\"#MJX-6-TEX-N-36\" transform=\"translate(389,0)\"><\/use><use data-c=\"29\" xlink:href=\"#MJX-6-TEX-N-29\" transform=\"translate(889,0)\"><\/use><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mtext>(6)<\/mtext><\/mtd><mtd><msub><mi>d<\/mi><mi>x<\/mi><\/msub><mo>=<\/mo><mfrac><mrow><mn>511<\/mn><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><mrow><mn>372<\/mn><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mo>,<\/mo><msub><mi>d<\/mi><mi>y<\/mi><\/msub><mo>=<\/mo><msqrt><mn>3<\/mn><\/msqrt><mo>\u22c5<\/mo><mn>0.5<\/mn><mo>\u22c5<\/mo><mi>d<\/mi><mi>x<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><p><span>First generate points of grid using <\/span><code>meshgrid()<\/code><span> with given <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.279ex\" height=\"1.927ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 1007.5 851.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-107-TEX-I-1D451\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-107-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D451\" xlink:href=\"#MJX-107-TEX-I-1D451\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(553,-150) scale(0.707)\"><use data-c=\"1D465\" xlink:href=\"#MJX-107-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>d<\/mi><mi>x<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">d_x<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.148ex\" height=\"2.237ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 949.5 989\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.667ex;\"><defs><path id=\"MJX-108-TEX-I-1D451\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-108-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D451\" xlink:href=\"#MJX-108-TEX-I-1D451\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(553,-150) scale(0.707)\"><use data-c=\"1D466\" xlink:href=\"#MJX-108-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>d<\/mi><mi>y<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">d_y<\/script><\/p><\/li><li><p><span>Then shift each second row by <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.171ex\" height=\"1.927ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2285.5 851.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-109-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-109-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-109-TEX-N-35\" d=\"M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z\"><\/path><path id=\"MJX-109-TEX-I-1D451\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><path id=\"MJX-109-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"30\" xlink:href=\"#MJX-109-TEX-N-30\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-109-TEX-N-2E\" transform=\"translate(500,0)\"><\/use><use data-c=\"35\" xlink:href=\"#MJX-109-TEX-N-35\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1278,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D451\" xlink:href=\"#MJX-109-TEX-I-1D451\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(553,-150) scale(0.707)\"><use data-c=\"1D465\" xlink:href=\"#MJX-109-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0.5<\/mn><msub><mi>d<\/mi><mi>x<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">0.5d_x<\/script><span> to obtain the hexagonal grid<\/span><\/p><\/li><\/ul><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-number\">512<\/span>;<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">dx<\/span>=(<span class=\"cm-variable\">sz<\/span><span class=\"cm-number\">-1-1<\/span>)<span class=\"cm-operator\">\/<\/span>(<span class=\"cm-number\">372-1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">gridX<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">dx<\/span>:<span class=\"cm-variable\">sz<\/span><span class=\"cm-number\">-1<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">dy<\/span>=<span class=\"cm-builtin\">sqrt<\/span>(<span class=\"cm-number\">3<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-number\">0.5<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">dx<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">gridY<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">dy<\/span>:<span class=\"cm-variable\">sz<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 1. meshgrid<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\">[<span class=\"cm-variable\">hexaX<\/span>,<span class=\"cm-variable\">hexaY<\/span>]=<span class=\"cm-builtin\">meshgrid<\/span>(<span class=\"cm-variable\">gridX<\/span>,<span class=\"cm-variable\">gridY<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 2. shift rows<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">hexaX<\/span>(<span class=\"cm-number\">2<\/span>:<span class=\"cm-number\">2<\/span>:<span class=\"cm-keyword\">end<\/span>,:)=<span class=\"cm-variable\">hexaX<\/span>(<span class=\"cm-number\">2<\/span>:<span class=\"cm-number\">2<\/span>:<span class=\"cm-keyword\">end<\/span>,:)<span class=\"cm-number\">+0.5<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">dx<\/span>;<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 216px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 216px;\"><\/div><\/div><\/div><\/pre><h3 id='c15-interpolate-image-im-using-interp2-at-hexagonal-sampling-positions-above-then-reconstruct-the-image-using-triscatteredinterp'><span>C1.5. Interpolate image <\/span><code>im<\/code><span> using <\/span><code>interp2()<\/code><span> at hexagonal sampling positions above, then reconstruct the image using <\/span><code>TriScatteredInterp()<\/code><span>.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-builtin\">function<\/span> <span class=\"cm-variable\">im_rec<\/span>=<span class=\"cm-variable\">inter_recon<\/span>(<span class=\"cm-variable\">im<\/span>,<span class=\"cm-variable\">hexaX<\/span>,<span class=\"cm-variable\">hexaY<\/span>)<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">im<\/span>,<span class=\"cm-number\">1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 1. Interpolation<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\">[<span class=\"cm-variable\">imX<\/span>,<span class=\"cm-variable\">imY<\/span>]=<span class=\"cm-builtin\">meshgrid<\/span>(<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">sz<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_interp<\/span>=<span class=\"cm-variable\">interp2<\/span>(<span class=\"cm-variable\">imX<\/span>,<span class=\"cm-variable\">imY<\/span>,<span class=\"cm-variable\">im<\/span>,<span class=\"cm-variable\">hexaX<\/span>,<span class=\"cm-variable\">hexaY<\/span>,<span class=\"cm-string\">'linear'<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 2. Reconstruction<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">F<\/span>=<span class=\"cm-variable\">TriScatteredInterp<\/span>(<span class=\"cm-variable\">hexaX<\/span>(:),<span class=\"cm-variable\">hexaY<\/span>(:),<span class=\"cm-variable\">im_interp<\/span>(:),<span class=\"cm-string\">'linear'<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_rec<\/span>=<span class=\"cm-variable\">F<\/span>(<span class=\"cm-variable\">imX<\/span>,<span class=\"cm-variable\">imY<\/span>);<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 192px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 192px;\"><\/div><\/div><\/div><\/pre><h3 id='c16-to-know-the-construction-quality-compute-psnr-remove-a-border-of-\\n5\\n-pixels-to-neglect-border-effect'><span>C1.6. To know the construction quality, compute PSNR, remove a border of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 500 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-153-TEX-N-35\" d=\"M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"35\" xlink:href=\"#MJX-153-TEX-N-35\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>5<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">5<\/script><span> pixels to neglect border effect.<\/span><\/h3><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n71\" cid=\"n71\" mdtype=\"math_block\" data-math-tag-before=\"6\" data-math-tag-after=\"7\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 43.044ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"11.735ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -5.302ex; min-width: 43.044ex;\"><defs><path id=\"MJX-7-TEX-SS-1D5AF\" d=\"M88 0V694H230Q347 693 370 692T410 686Q487 667 535 611T583 485Q583 409 527 348T379 276Q369 274 279 274H192V0H88ZM486 485Q486 523 471 551T432 593T391 612T357 621Q350 622 268 623H189V347H268Q350 348 357 349Q370 351 383 354T416 368T450 391T475 429T486 485Z\"><\/path><path id=\"MJX-7-TEX-SS-1D5B2\" d=\"M55 514Q55 589 115 652T283 716Q315 716 345 711T396 699T432 685T457 672T467 667Q467 666 459 618T449 568Q383 634 282 634Q214 634 182 600T150 525Q150 507 155 492T172 465T194 446T222 432T247 423T272 416T289 412Q353 396 378 384Q432 358 466 307T500 194Q500 110 438 44T272 -22Q215 -22 159 -5T73 28T44 50Q45 51 49 75T57 122T62 146L65 143Q68 140 74 136T88 125T107 111T131 98T160 85T194 74T232 66T274 63H286Q327 63 366 96T406 182Q406 245 352 280Q329 296 265 310T173 339Q124 363 90 409T55 514Z\"><\/path><path 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stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-builtin\">function<\/span> <span class=\"cm-variable\">psnr_diff<\/span>=<span class=\"cm-variable\">psnr_diff<\/span>(<span class=\"cm-variable\">i1<\/span>,<span class=\"cm-variable\">i2<\/span>)<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">i_diff<\/span>=<span class=\"cm-variable\">i1<\/span><span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">i2<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">border<\/span>=<span class=\"cm-number\">5<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">mse<\/span>=<span class=\"cm-variable\">i_diff<\/span>(<span class=\"cm-variable\">border<\/span><span class=\"cm-number\">+1<\/span>;<span class=\"cm-keyword\">end<\/span><span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">border<\/span>,<span class=\"cm-variable\">border<\/span><span class=\"cm-number\">+1<\/span>:<span class=\"cm-keyword\">end<\/span><span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">border<\/span>)<span class=\"cm-operator\">.^<\/span><span class=\"cm-number\">2<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">psnr_diff<\/span>=<span class=\"cm-number\">10<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-builtin\">log10<\/span>(<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">\/<\/span><span class=\"cm-variable\">mse<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 144px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 144px;\"><\/div><\/div><\/div><\/pre><h3 id='a12-which-image-has-betterhigher-psnr'><span>A1.2. Which image has better\/higher PSNR?<\/span><\/h3><ul><li><p><span>Hexagonal sampling has better PSNR as it exhibits a better spectral efficiency.<\/span><\/p><\/li><\/ul><h2 id='hw22-zone-plate'><span>HW2.2. Zone Plate<\/span><\/h2><h3 id='c21-generate-an-\\nn\\n\u00d7\\nn\\n-image-matrix-whose-entries-represent-the-euclidean-distance-in-pixel-unit-to-the-top-left-pixel-then-calculate-zone-plate-image-using'><span>C2.1. Generate an <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.784ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2998.4 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-111-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-111-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-111-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1110.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-111-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2110.4,0)\"><use data-c=\"1D441\" xlink:href=\"#MJX-111-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><mo>\u00d7<\/mo><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N\\times N<\/script><span> image matrix whose entries represent the Euclidean distance in pixel unit to the top-left pixel. Then calculate zone plate image using <\/span><\/h3><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n79\" cid=\"n79\" mdtype=\"math_block\" data-math-tag-before=\"7\" data-math-tag-after=\"8\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 41.928ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"4.588ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.729ex; min-width: 41.928ex;\"><defs><path id=\"MJX-8-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 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stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>,<\/mo><mi>y<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='choose-\\nk\\n-such-that-the-maximum-frequencyphase-change-along-last-two-diagonal-pixels-is-\\n\u03c0\\n'><span>Choose <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-112-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-112-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span> such that the maximum frequency\/phase change along last two diagonal pixels is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.29ex\" height=\"1ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -431 570 442\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-113-TEX-I-1D70B\" d=\"M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" 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transform=\"translate(2610.4,0)\"><use data-c=\"36\" xlink:href=\"#MJX-9-TEX-N-36\"><\/use><\/g><\/g><rect width=\"3310.4\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"1278 -2374.6 1 4249.3\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:9\"><g data-mml-node=\"mtext\"><use data-c=\"28\" xlink:href=\"#MJX-9-TEX-N-28\"><\/use><use data-c=\"39\" xlink:href=\"#MJX-9-TEX-N-39\" transform=\"translate(389,0)\"><\/use><use data-c=\"29\" xlink:href=\"#MJX-9-TEX-N-29\" transform=\"translate(889,0)\"><\/use><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mtext>(9)<\/mtext><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><mi>\u03c0<\/mi><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><msqrt><mn>2<\/mn><\/msqrt><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mi>\u03c0<\/mi><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><msqrt><mn>2<\/mn><\/msqrt><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><mi>\u03c0<\/mi><mi>k<\/mi><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>N<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo stretchy=\"false\">)<\/mo><mover><mo>=<\/mo><mo>!<\/mo><\/mover><mi>\u03c0<\/mi><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><mi>k<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mn>4<\/mn><mi>N<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-builtin\">function<\/span> [<span class=\"cm-variable\">D<\/span>,<span class=\"cm-variable\">k<\/span>,<span class=\"cm-variable\">Izp<\/span>]=<span class=\"cm-variable\">zone_plate<\/span>(<span class=\"cm-variable\">N<\/span>)<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\">[<span class=\"cm-variable\">x<\/span>, <span class=\"cm-variable\">y<\/span>] = <span class=\"cm-builtin\">meshgrid<\/span>(<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">N<\/span>, <span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">N<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">D<\/span> = <span class=\"cm-builtin\">sqrt<\/span>((<span class=\"cm-variable\">x<\/span><span class=\"cm-number\">-1<\/span>)<span class=\"cm-operator\">.^<\/span><span class=\"cm-number\">2<\/span> <span class=\"cm-operator\">+<\/span> (<span class=\"cm-variable\">y<\/span><span class=\"cm-number\">-1<\/span>)<span class=\"cm-operator\">.^<\/span><span class=\"cm-number\">2<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">k<\/span> = <span class=\"cm-number\">1<\/span><span class=\"cm-operator\">\/<\/span>(<span class=\"cm-number\">4<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">N<\/span><span class=\"cm-number\">-6<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">Izp<\/span> = <span class=\"cm-number\">1<\/span><span class=\"cm-operator\">\/<\/span><span class=\"cm-number\">2<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-builtin\">cos<\/span>(<span class=\"cm-variable\">D<\/span><span class=\"cm-operator\">.^<\/span><span class=\"cm-number\">2<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">pi<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">k<\/span>)<span class=\"cm-operator\">\/<\/span><span class=\"cm-number\">2<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 144px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 144px;\"><\/div><\/div><\/div><\/pre><h3 id='a21-subsample-izp-with-\\n2\\n\\n1\\n-and-\\n4\\n\\n1\\n-what-effects-do-you-observe'><span>A2.1. Subsample <\/span><code>Izp<\/code><span> with <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.148ex\" height=\"1.507ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 1833.6 666\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-155-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-155-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-155-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"32\" xlink:href=\"#MJX-155-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(777.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-155-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1333.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-155-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>2<\/mn><mo>:<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">2:1<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.148ex\" height=\"1.532ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 1833.6 677\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-115-TEX-N-34\" d=\"M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z\"><\/path><path id=\"MJX-115-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-115-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"34\" xlink:href=\"#MJX-115-TEX-N-34\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(777.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-115-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1333.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-115-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>4<\/mn><mo>:<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">4:1<\/script><span>, what effects do you observe?<\/span><\/h3><ul><li><p><span>Higher frequency components are projected to lower frequency after subsampling.<\/span><\/p><\/li><li><p><span>Images after subsampling have low frquency components on three corners, which are similar to the top-left corner.<\/span><\/p><\/li><\/ul><h2 id='hw23-image-filters'><span>HW2.3. Image Filters<\/span><\/h2><h3 id='a31-analyze-different-filters-plot-the-filter-using-surf-and-analyze-its-frequency-response-using-freqz2'><span>A3.1. Analyze different filters, plot the filter using <\/span><code>surf()<\/code><span> and analyze its frequency response using <\/span><code>freqz2()<\/code><span>.<\/span><\/h3><ul><li><p><span>Gaussian: low pass<\/span><\/p><\/li><li><p><span>Smaller Gaussian filters in space have a wider frequency spectrum,<\/span><\/p><\/li><li><p><span>Laplacian: high pass<\/span><\/p><\/li><li><p><span>Prewitt\/Laplacian of Gaussian: high pass, and reduces sensitivity to noise.<\/span><\/p><\/li><\/ul><h3 id='c31-find-out-how-freqz2-generates-its-plot-and-implement-it-manually-using-fft2'><span>C3.1. Find out how <\/span><code>freqz2()<\/code><span> generates its plot and implement it manually using <\/span><code>fft2()<\/code><span>.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">filter<\/span>=<span class=\"cm-variable\">fspecial<\/span>(<span class=\"cm-string\">'prewitt'<\/span>);<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">N<\/span>=<span class=\"cm-number\">64<\/span>; <span class=\"cm-comment\">% default for freqz2()<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">freq_resp<\/span>=<span class=\"cm-variable\">fft2<\/span>(<span class=\"cm-variable\">filter<\/span>,<span class=\"cm-variable\">N<\/span>,<span class=\"cm-variable\">N<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">freq_plot<\/span>=<span class=\"cm-builtin\">abs<\/span>(<span class=\"cm-variable\">fftshift<\/span>(<span class=\"cm-variable\">freq_resp<\/span>));<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 96px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 96px;\"><\/div><\/div><\/div><\/pre><h3 id='c32-filter-the-image-with-a-gaussian-filter-\\n\\n16\\n\u00d7\\n16\\n\\n\u03c3\\n\\n3\\n\\n-and-replicate-border-processing'><span>C3.2. Filter the image with a Gaussian filter <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"15.497ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6849.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-116-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-116-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-116-TEX-N-36\" d=\"M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z\"><\/path><path id=\"MJX-116-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><path id=\"MJX-116-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-116-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-116-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-116-TEX-N-33\" d=\"M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z\"><\/path><path id=\"MJX-116-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-116-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(389,0)\"><use data-c=\"31\" xlink:href=\"#MJX-116-TEX-N-31\"><\/use><use data-c=\"36\" xlink:href=\"#MJX-116-TEX-N-36\" transform=\"translate(500,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1611.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-116-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2611.4,0)\"><use data-c=\"31\" xlink:href=\"#MJX-116-TEX-N-31\"><\/use><use data-c=\"36\" xlink:href=\"#MJX-116-TEX-N-36\" transform=\"translate(500,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3611.4,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-116-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4056.1,0)\"><use data-c=\"1D70E\" xlink:href=\"#MJX-116-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4904.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-116-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(5960.7,0)\"><use data-c=\"33\" xlink:href=\"#MJX-116-TEX-N-33\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6460.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-116-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mn>16<\/mn><mo>\u00d7<\/mo><mn>16<\/mn><mo>,<\/mo><mi>\u03c3<\/mi><mo>=<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(16\\times 16,\\sigma=3)<\/script><span> and <\/span><code>replicate<\/code><span> border processing.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-builtin\">function<\/span> <span class=\"cm-variable\">im_filtered<\/span>=<span class=\"cm-variable\">filter_im<\/span>(<span class=\"cm-variable\">im<\/span>)<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">filter<\/span>=<span class=\"cm-variable\">fspecial<\/span>(<span class=\"cm-string\">'gaussian'<\/span>,<span class=\"cm-number\">16<\/span>,<span class=\"cm-number\">3<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_filtered<\/span>=<span class=\"cm-variable\">imfilter<\/span>(<span class=\"cm-variable\">im<\/span>,<span class=\"cm-variable\">filter<\/span>,<span class=\"cm-string\">'replicate'<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 96px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 96px;\"><\/div><\/div><\/div><\/pre><h3 id='a32-'><span>A3.2. &#8230;.<\/span><\/h3><h2 id='hw32-separable-block-transforms'><span>HW3.2. Separable Block Transforms<\/span><\/h2><h3 id='c21-implement-function-blocksplitter-with-special-support-for-border-blocks-image-is-like-\\n\u2218\\n\u25b3\\n\\n\u2218\\n'><span>C2.1. Implement function <\/span><code>block_splitter<\/code><span> with special support for <\/span><strong><span>border blocks<\/span><\/strong><span>. Image is like <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.482ex\" height=\"3.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -983 2423 1567\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.321ex;\"><defs><path id=\"MJX-117-TEX-N-2218\" d=\"M55 251Q55 328 112 386T249 444T386 388T444 249Q444 171 388 113T250 55Q170 55 113 112T55 251ZM245 403Q188 403 142 361T96 250Q96 183 141 140T250 96Q284 96 313 109T354 135T375 160Q403 197 403 250Q403 313 360 358T245 403Z\"><\/path><path id=\"MJX-117-TEX-N-25B3\" d=\"M75 0L72 2Q69 3 67 5T62 11T59 20Q59 24 62 30Q65 37 245 370T428 707Q428 708 430 710T436 714T444 716Q451 716 455 712Q459 710 644 368L828 27V20Q828 7 814 0H75ZM610 347L444 653Q443 653 278 347T113 40H775Q775 42 610 347Z\"><\/path><path id=\"MJX-117-TEX-S4-2013\" d=\"M0 248V285H499V248H0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"menclose\"><g transform=\"translate(267, 0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mstyle\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"2218\" xlink:href=\"#MJX-117-TEX-N-2218\"><\/use><\/g><g data-mml-node=\"munder\" transform=\"translate(500,0)\"><g data-mml-node=\"mi\"><use data-c=\"25B3\" xlink:href=\"#MJX-117-TEX-N-25B3\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(0,-465)\"><svg width=\"889\" height=\"237\" x=\"0\" y=\"148\" viewBox=\"222.2 148 889 237\"><use data-c=\"2013\" xlink:href=\"#MJX-117-TEX-S4-2013\" transform=\"scale(2.667,1)\"><\/use><\/svg><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1389,0)\"><use data-c=\"2218\" xlink:href=\"#MJX-117-TEX-N-2218\"><\/use><\/g><\/g><\/g><\/g><\/g><rect x=\"33.5\" y=\"-550.5\" width=\"2356\" height=\"1500\" fill=\"none\" stroke-width=\"67\"><\/rect><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><menclose notation=\"box\"><mrow data-mjx-texclass=\"ORD\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mo>\u2218<\/mo><munder><mi mathvariant=\"normal\">\u25b3<\/mi><mo accent=\"true\">\u2015<\/mo><\/munder><mo>\u2218<\/mo><\/mrow><\/mstyle><\/mrow><\/menclose><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">{\\boxed{\\circ \\underline\\triangle\\circ}}<\/script><span>.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\" style=\"break-inside: unset;\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-builtin\">function<\/span> <span class=\"cm-variable\">im_out<\/span>=<span class=\"cm-variable\">block_splitter<\/span>(<span class=\"cm-variable\">im<\/span>,<span class=\"cm-variable\">block_sz<\/span>,<span class=\"cm-variable\">block_func<\/span>,<span class=\"cm-variable\">block_arg<\/span>)<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 1. Pad the input image based on block_sz=16<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">im<\/span>); <span class=\"cm-comment\">% [486,732]<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">pad<\/span>=<span class=\"cm-variable\">block_sz<\/span><span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">mod<\/span>(<span class=\"cm-variable\">sz<\/span>,<span class=\"cm-variable\">block_sz<\/span>); <span class=\"cm-comment\">% [16,16]-[6,12]=[10,4]<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">pad<\/span>(<span class=\"cm-variable\">pad<\/span><span class=\"cm-operator\">==<\/span><span class=\"cm-variable\">block_sz<\/span>)=<span class=\"cm-number\">0<\/span> <span class=\"cm-comment\">% if mod=0, no padding needed<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im<\/span>=<span class=\"cm-variable\">padarray<\/span>(<span class=\"cm-variable\">im<\/span>,<span class=\"cm-variable\">pad<\/span>,<span class=\"cm-string\">'replicate'<\/span>,<span class=\"cm-string\">'post'<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-variable\">sz<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">pad<\/span>; <span class=\"cm-comment\">% [486,732]+[10,4]=[496,736]<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 2. Loop over blocks and apply block_func<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">x<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">+1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">y<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">+1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">block_in<\/span>=<span class=\"cm-variable\">im<\/span>(<span class=\"cm-variable\">y<\/span>:<span class=\"cm-variable\">y<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span>,<span class=\"cm-variable\">x<\/span>:<span class=\"cm-variable\">x<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">block_out<\/span>=<span class=\"cm-variable\">block_func<\/span>(<span class=\"cm-variable\">block_in<\/span>,<span class=\"cm-variable\">block_arg<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">im_out<\/span>(<span class=\"cm-variable\">y<\/span>:<span class=\"cm-variable\">y<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span>,<span class=\"cm-variable\">x<\/span>:<span class=\"cm-variable\">x<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span>)=<span class=\"cm-variable\">block_out<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 3. Cut away padding<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_out<\/span>(<span class=\"cm-keyword\">end<\/span><span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">pad<\/span>(<span class=\"cm-number\">1<\/span>)<span class=\"cm-number\">+1<\/span>:<span class=\"cm-keyword\">end<\/span>,:)=[]; <span class=\"cm-comment\">% assign an empty matrix<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_out<\/span>(:,<span class=\"cm-keyword\">end<\/span><span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">pad<\/span>(<span class=\"cm-number\">2<\/span>)<span class=\"cm-number\">+1<\/span>:<span class=\"cm-keyword\">end<\/span>)=[];<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 504px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 504px;\"><\/div><\/div><\/div><\/pre><h3 id='c22-write-a-simple-blockfunc-that-flips-the-image-block-left-to-right-and-decreases-the-brightness-by-substracting-03-from-each-pixel'><span>C2.2. Write a simple <\/span><code>block_func<\/code><span> that flips the image block left to right and decreases the brightness by substracting 0.3 from each pixel.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">block_out<\/span>=<span class=\"cm-builtin\">fliplr<\/span>(<span class=\"cm-variable\">block_in<\/span>)<span class=\"cm-number\">-0.3<\/span>;<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">block_out<\/span>(<span class=\"cm-variable\">block_out<\/span><span class=\"cm-operator\">&lt;<\/span><span class=\"cm-number\">0<\/span>)=<span class=\"cm-number\">0<\/span>;<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 48px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 48px;\"><\/div><\/div><\/div><\/pre><h3 id='a21-above-operation-ensures-that-pixel-value-is-non-negative-so-its-nonlinear'><span>A2.1. Above operation ensures that pixel value is non-negative, so it&#39;s <\/span><strong><span>nonlinear<\/span><\/strong><span>.<\/span><\/h3><h3 id='c23-apply-given-haar-transform-matrix-to-the-image'><span>C2.3. Apply given Haar transform matrix to the image.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">fcnHandleBlockSplitter<\/span>=<span class=\"cm-operator\">@<\/span><span class=\"cm-variable\">block_splitter<\/span>;;<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">block_func<\/span>=<span class=\"cm-operator\">@<\/span>(<span class=\"cm-variable\">block_in<\/span>,<span class=\"cm-variable\">A<\/span>) <span class=\"cm-variable\">A<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">block_in<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">A<\/span><span class=\"cm-operator\">'<\/span>; <span class=\"cm-comment\">% create a function<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">haar_im<\/span>=<span class=\"cm-variable\">fcnHandleBlockSplitter<\/span>(<span class=\"cm-variable\">im<\/span>,<span class=\"cm-variable\">block_sz<\/span>,<span class=\"cm-error\">...<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">block_func<\/span>,<span class=\"cm-variable\">haar_4<\/span>);<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 96px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 96px;\"><\/div><\/div><\/div><\/pre><h3 id='c24-reorder-the-transformed-image-to-make-it-meaningful-for-human'><span>C2.4. Reorder the transformed image to make it meaningful for human.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\" style=\"break-inside: unset;\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">im<\/span>);<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_reorder<\/span>=<span class=\"cm-builtin\">zeros<\/span>(<span class=\"cm-variable\">sz<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">rows<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% or rows=1:block_sz:sz(1)-block_sz+1;<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">cols<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% or cols=1:block_sz:sz(2)-block_sz+1;<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">lr<\/span>=<span class=\"cm-variable\">length<\/span>(<span class=\"cm-variable\">rows<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">lc<\/span>=<span class=\"cm-variable\">length<\/span>(<span class=\"cm-variable\">cols<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">y<\/span>=<span class=\"cm-number\">0<\/span>:<span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">x<\/span>=<span class=\"cm-number\">0<\/span>:<span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">block<\/span>=<span class=\"cm-variable\">im<\/span>(<span class=\"cm-variable\">rows<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">y<\/span>,<span class=\"cm-variable\">cols<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">x<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">im_reorder<\/span>(<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">y<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">lr<\/span>:(<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">y<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">lr<\/span>,<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">x<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">lc<\/span>:(<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">x<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">lc<\/span>)=<span class=\"cm-variable\">block<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 360px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 360px;\"><\/div><\/div><\/div><\/pre><h3 id='c25-implement-calculatestatis-to-calculate-the-mean-and-variance-for-each-coefficient-over-all-blocks'><span>C2.5. Implement <\/span><code>calculate_statis<\/code><span> to calculate the mean and variance for each coefficient over all blocks.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\" style=\"break-inside: unset;\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-builtin\">function<\/span> [<span class=\"cm-variable\">im_mean<\/span>, <span class=\"cm-variable\">im_var<\/span>]=<span class=\"cm-variable\">calculate_statis<\/span>(<span class=\"cm-variable\">haar_im<\/span>, <span class=\"cm-variable\">block_sz<\/span>)<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">%%%%%%%%%%%%%%%%%%<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% After padding: %<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% &nbsp; sz=sz+pad &nbsp;  %<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">%%%%%%%%%%%%%%%%%%<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_out<\/span>=<span class=\"cm-builtin\">zeros<\/span>(<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">im<\/span>));<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">all_blocks<\/span>=<span class=\"cm-builtin\">zeros<\/span>(<span class=\"cm-variable\">block_sz<\/span>,<span class=\"cm-variable\">block_sz<\/span>,<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>)<span class=\"cm-operator\">\/<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-operator\">^<\/span><span class=\"cm-number\">2<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% Loop over blocks<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">ibl<\/span>=<span class=\"cm-number\">0<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">x<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">+1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">y<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">+1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">ibl<\/span>=<span class=\"cm-variable\">ibl<\/span><span class=\"cm-number\">+1<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">block<\/span>=<span class=\"cm-variable\">im<\/span>(<span class=\"cm-variable\">y<\/span>:<span class=\"cm-variable\">y<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span>,<span class=\"cm-variable\">x<\/span>:<span class=\"cm-variable\">x<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">all_blocks<\/span>(:,:,<span class=\"cm-variable\">ibl<\/span>)=<span class=\"cm-variable\">block<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% Calculate statistics<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_mean<\/span>=<span class=\"cm-builtin\">mean<\/span>(<span class=\"cm-variable\">all_blocks<\/span>,<span class=\"cm-number\">3<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_var<\/span>=<span class=\"cm-variable\">var<\/span>(<span class=\"cm-variable\">all_blocks<\/span>,[],<span class=\"cm-number\">3<\/span>); <span class=\"cm-comment\">% []:no weights assigned<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 576px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 576px;\"><\/div><\/div><\/div><\/pre><h3 id='a22-what-can-be-observed-by-analyzing-imvar'><span>A2.2. What can be observed by analyzing <\/span><code>im_var<\/code><span>?<\/span><\/h3><ul><li><p><span>For natural image, variances of block elements are similar to each other, also similar to the global statistics. This may change once a transformation is applied.<\/span><\/p><\/li><li><p><span>After Haar transformation, variances decay considerably from (0,0) coefficient into higher coefficients such as (block_sz,block_sz), which means it packs most energy\/variance into low-order coefficients.<\/span><\/p><\/li><\/ul><h3 id='c26-implement-an-image-blur-using-a-symmetric-impulse-response-of-\\n\\n025\\n\\n05\\n\\n025\\n\\n-to-the-image-horizonally-and-vertically'><span>C2.6. Implement an image blur using a symmetric impulse response of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.207ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6279.3 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-118-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-118-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-118-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-118-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-118-TEX-N-35\" d=\"M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z\"><\/path><path id=\"MJX-118-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-118-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"5B\" xlink:href=\"#MJX-118-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(278,0)\"><use data-c=\"30\" xlink:href=\"#MJX-118-TEX-N-30\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-118-TEX-N-2E\" transform=\"translate(500,0)\"><\/use><use data-c=\"32\" xlink:href=\"#MJX-118-TEX-N-32\" transform=\"translate(778,0)\"><\/use><use data-c=\"35\" xlink:href=\"#MJX-118-TEX-N-35\" transform=\"translate(1278,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2056,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-118-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2500.7,0)\"><use data-c=\"30\" xlink:href=\"#MJX-118-TEX-N-30\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-118-TEX-N-2E\" transform=\"translate(500,0)\"><\/use><use data-c=\"35\" xlink:href=\"#MJX-118-TEX-N-35\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3778.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-118-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(4223.3,0)\"><use data-c=\"30\" xlink:href=\"#MJX-118-TEX-N-30\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-118-TEX-N-2E\" transform=\"translate(500,0)\"><\/use><use data-c=\"32\" xlink:href=\"#MJX-118-TEX-N-32\" transform=\"translate(778,0)\"><\/use><use data-c=\"35\" xlink:href=\"#MJX-118-TEX-N-35\" transform=\"translate(1278,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6001.3,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-118-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">[<\/mo><mn>0.25<\/mn><mo>,<\/mo><mn>0.5<\/mn><mo>,<\/mo><mn>0.25<\/mn><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">[0.25,0.5,0.25]<\/script><span> to the image horizonally and vertically.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">blur_A<\/span>=<span class=\"cm-number\">0.5<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">diag<\/span>(<span class=\"cm-builtin\">ones<\/span>(<span class=\"cm-number\">1<\/span>,<span class=\"cm-variable\">block_sz<\/span>),<span class=\"cm-number\">0<\/span>)<span class=\"cm-operator\">+<\/span><span class=\"cm-error\">...<\/span><\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-number\">0.25<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">diag<\/span>(<span class=\"cm-builtin\">ones<\/span>(<span class=\"cm-number\">1<\/span>,<span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span>),<span class=\"cm-number\">-1<\/span>)<span class=\"cm-operator\">+<\/span><span class=\"cm-error\">...<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-number\">0.25<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">diag<\/span>(<span class=\"cm-builtin\">ones<\/span>(<span class=\"cm-number\">1<\/span>,<span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span>),<span class=\"cm-number\">1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">block_func<\/span>=<span class=\"cm-operator\">@<\/span>(<span class=\"cm-variable\">block_in<\/span>,<span class=\"cm-variable\">A<\/span>) <span class=\"cm-variable\">A<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">block_in<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">A<\/span><span class=\"cm-operator\">'<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">blur_im<\/span>=<span class=\"cm-variable\">fcnHandleBlockSplitter<\/span>(<span class=\"cm-variable\">im<\/span>,<span class=\"cm-variable\">block_sz<\/span>,<span class=\"cm-variable\">block_func<\/span>,<span class=\"cm-variable\">blur_A<\/span>);<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 120px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 120px;\"><\/div><\/div><\/div><\/pre><h3 id='a23-observe-the-blurred-image-and-discuss-the-limitation-of-\\na\\nx\\na\\nt\\n-along-the-block-borders'><span>A2.3. Observe the blurred image and discuss the limitation of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.635ex\" height=\"1.904ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -841.7 2932.8 841.7\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-120-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-120-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-120-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-120-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(750,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-120-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(1602,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-120-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(783,363) scale(0.707)\"><use data-c=\"1D447\" xlink:href=\"#MJX-120-TEX-I-1D447\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mi>X<\/mi><msup><mi>A<\/mi><mi>T<\/mi><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">AXA^T<\/script><span> along the block borders.<\/span><\/h3><ul><li><p><span>Image is blurred, i.e., low-pass filtered so that fine details cannot be observed anymore.<\/span><\/p><\/li><li><p><span>The block processor <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.635ex\" height=\"1.904ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -841.7 2932.8 841.7\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-120-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-120-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-120-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-120-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(750,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-120-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(1602,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-120-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(783,363) scale(0.707)\"><use data-c=\"1D447\" xlink:href=\"#MJX-120-TEX-I-1D447\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mi>X<\/mi><msup><mi>A<\/mi><mi>T<\/mi><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">AXA^T<\/script><span> does not have access to neighbor pixels outside the block, so border effect such as dark frame is observed around blocks, thus not &quot;smooth&quot;.<\/span><\/p><\/li><li><p><span>In order to mitigate the border effect, one could use overlapping blocks at the cost of performance.<\/span><\/p><\/li><\/ul><h2 id='hw41-discrete-cosine-transform'><span>HW4.1. Discrete Cosine Transform<\/span><\/h2><h3 id='c11-transform-images-to-dct-domain-using-block-processing'><span>C1.1. Transform images to DCT domain using block processing.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">block_sz<\/span>=<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">dct_matrix<\/span>,<span class=\"cm-number\">1<\/span>);<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">block_func<\/span>=<span class=\"cm-operator\">@<\/span>(<span class=\"cm-variable\">block_in<\/span>,<span class=\"cm-variable\">A<\/span>) <span class=\"cm-variable\">A<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">block_in<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">A<\/span><span class=\"cm-operator\">'<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">dct_im<\/span>=<span class=\"cm-variable\">fcnHandleBlkSplitter<\/span>(<span class=\"cm-variable\">im<\/span>,<span class=\"cm-variable\">block_sz<\/span>,<span class=\"cm-variable\">block_func<\/span>,<span class=\"cm-variable\">dct_matrix<\/span>);<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 72px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 72px;\"><\/div><\/div><\/div><\/pre><h3 id='a11-what-can-be-observed-by-analyzing-immean-and-imvar-of-all-blocks'><span>A1.1. What can be observed by analyzing <\/span><code>im_mean<\/code><span> and <\/span><code>im_var<\/code><span> of all blocks?<\/span><\/h3><ul><li><p><span>The mean of the DC-coefficient is proportional to the mean of the untransformed image in the respective area.<\/span><\/p><\/li><li><p><span>The variance of the DC coefficient is the largest, and it decreases with increasing spatial frequency.<\/span><\/p><\/li><li><p><span>Variances in horizontal directions exhibit slightly higher amplitude compared to the vertical direction.<\/span><\/p><\/li><li><p><span>AC-coefficients exhibit zero mean.<\/span><\/p><\/li><\/ul><h3 id='c12-perform-compression-of-a-dct-transformed-image-by-keeping-only-the-first-\\nn\\n-coefficients-of-each-block-others-are-set-to-zero-return-in-image-in-spatial-domain-resulting-from-compression'><span>C1.2. Perform compression of a DCT-transformed image by keeping only the first <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-121-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-121-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N<\/script><span> coefficients of each block, others are set to zero. Return in image in spatial domain resulting from compression.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% Remove all but the first N coefficients<\/span><\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">d<\/span>=<span class=\"cm-builtin\">zeros<\/span>(<span class=\"cm-variable\">block_sz<\/span>,<span class=\"cm-number\">1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">d<\/span>(<span class=\"cm-number\">1<\/span>:<span class=\"cm-builtin\">sqrt<\/span>(<span class=\"cm-variable\">N<\/span>))=<span class=\"cm-number\">1<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">comp_matrix<\/span>=<span class=\"cm-variable\">diag<\/span>(<span class=\"cm-variable\">d<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">block_func<\/span>=<span class=\"cm-operator\">@<\/span>(<span class=\"cm-variable\">block_in<\/span>,<span class=\"cm-variable\">A<\/span>) <span class=\"cm-variable\">A<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">block_in<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">A<\/span><span class=\"cm-operator\">'<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">dct_im_comp<\/span>=<span class=\"cm-variable\">fcnHandleBlkSplitter<\/span>(<span class=\"cm-variable\">dct_im<\/span>,<span class=\"cm-variable\">block_sz<\/span>,<span class=\"cm-error\">...<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">block_func<\/span>,<span class=\"cm-variable\">comp_matrix<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% Back-transform the compressed image<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">dct_im_comp_rec<\/span>=<span class=\"cm-variable\">fcnHandleBlkSplitter<\/span>(<span class=\"cm-variable\">dct_im_comp<\/span>,<span class=\"cm-variable\">block_sz<\/span>,<span class=\"cm-error\">...<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">block_func<\/span>,<span class=\"cm-variable\">dct_image<\/span><span class=\"cm-operator\">'<\/span>);<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 288px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 288px;\"><\/div><\/div><\/div><\/pre><h3 id='a12-compare-the-compressed-image-and-original-image-what-effects-do-you-observe-when-using-different-compression-level-\\n\\nn\\n\\n9\\n\\n16\\n\\n64\\n\\n'><span>A1.2. Compare the compressed image and original image, what effects do you observe when using different compression level <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.454ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 6388.9 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-122-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-122-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-122-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-122-TEX-N-39\" d=\"M352 287Q304 211 232 211Q154 211 104 270T44 396Q42 412 42 436V444Q42 537 111 606Q171 666 243 666Q245 666 249 666T257 665H261Q273 665 286 663T323 651T370 619T413 560Q456 472 456 334Q456 194 396 97Q361 41 312 10T208 -22Q147 -22 108 7T68 93T121 149Q143 149 158 135T173 96Q173 78 164 65T148 49T135 44L131 43Q131 41 138 37T164 27T206 22H212Q272 22 313 86Q352 142 352 280V287ZM244 248Q292 248 321 297T351 430Q351 508 343 542Q341 552 337 562T323 588T293 615T246 625Q208 625 181 598Q160 576 154 546T147 441Q147 358 152 329T172 282Q197 248 244 248Z\"><\/path><path id=\"MJX-122-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-122-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-122-TEX-N-36\" d=\"M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z\"><\/path><path id=\"MJX-122-TEX-N-34\" d=\"M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z\"><\/path><path id=\"MJX-122-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-122-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D441\" xlink:href=\"#MJX-122-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1554.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-122-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2610.6,0)\"><use data-c=\"39\" xlink:href=\"#MJX-122-TEX-N-39\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3110.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-122-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3555.2,0)\"><use data-c=\"31\" xlink:href=\"#MJX-122-TEX-N-31\"><\/use><use data-c=\"36\" xlink:href=\"#MJX-122-TEX-N-36\" transform=\"translate(500,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4555.2,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-122-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(4999.9,0)\"><use data-c=\"36\" xlink:href=\"#MJX-122-TEX-N-36\"><\/use><use data-c=\"34\" xlink:href=\"#MJX-122-TEX-N-34\" transform=\"translate(500,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5999.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-122-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>=<\/mo><mn>9<\/mn><mo>,<\/mo><mn>16<\/mn><mo>,<\/mo><mn>64<\/mn><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">(N=9,16,64)<\/script><span>?<\/span><\/h3><ul><li><p><span>If only <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 500 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-123-TEX-N-39\" d=\"M352 287Q304 211 232 211Q154 211 104 270T44 396Q42 412 42 436V444Q42 537 111 606Q171 666 243 666Q245 666 249 666T257 665H261Q273 665 286 663T323 651T370 619T413 560Q456 472 456 334Q456 194 396 97Q361 41 312 10T208 -22Q147 -22 108 7T68 93T121 149Q143 149 158 135T173 96Q173 78 164 65T148 49T135 44L131 43Q131 41 138 37T164 27T206 22H212Q272 22 313 86Q352 142 352 280V287ZM244 248Q292 248 321 297T351 430Q351 508 343 542Q341 552 337 562T323 588T293 615T246 625Q208 625 181 598Q160 576 154 546T147 441Q147 358 152 329T172 282Q197 248 244 248Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"39\" xlink:href=\"#MJX-123-TEX-N-39\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>9<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">9<\/script><span> coefficients are kept, one can observe significant block effects in image regions with much details, such as hair and eyes.<\/span><\/p><\/li><li><p><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.262ex\" height=\"1.581ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 1000 699\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-124-TEX-N-36\" d=\"M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z\"><\/path><path id=\"MJX-124-TEX-N-34\" d=\"M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"36\" xlink:href=\"#MJX-124-TEX-N-36\"><\/use><use data-c=\"34\" xlink:href=\"#MJX-124-TEX-N-34\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>64<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">64<\/script><span> coefficients are kept, one observes only minor block effects.<\/span><\/p><\/li><li><p><span>Different image regions require different numbers of coefficients to look well for human.<\/span><\/p><\/li><\/ul><h3 id='c13-because-of-its-separable-property-dct-transformation-can-be-efficiently-extended-to-\\n3\\n-dimensions-implement-apply3ddct-to-a-block-along-each-dimension-subsequently-to-get-the-3d-transformed-dct-image'><span>C1.3. Because of its separable property, DCT transformation can be efficiently extended to <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.554ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -665 500 687\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-125-TEX-N-33\" d=\"M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"33\" xlink:href=\"#MJX-125-TEX-N-33\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>3<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">3<\/script><span> dimensions. Implement <\/span><code>apply_3ddct<\/code><span> to a block along each dimension subsequently to get the 3D transformed DCT image.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\" style=\"break-inside: unset;\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">block_dct<\/span>=<span class=\"cm-builtin\">zeros<\/span>(<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">block_in<\/span>));<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 1. vertical columns<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">i<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">block_in<\/span>,<span class=\"cm-number\">3<\/span>) <\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">j<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">block_in<\/span>,<span class=\"cm-number\">2<\/span>)<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">vert<\/span>(:,<span class=\"cm-variable\">j<\/span>,<span class=\"cm-variable\">i<\/span>)=<span class=\"cm-variable\">dct_mat<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">block_in<\/span>(:,<span class=\"cm-variable\">j<\/span>,<span class=\"cm-variable\">i<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 2. horizontal rows<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">i<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">block_in<\/span>,<span class=\"cm-number\">3<\/span>) <\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">j<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">block_in<\/span>,<span class=\"cm-number\">1<\/span>)<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">hor<\/span>(<span class=\"cm-variable\">j<\/span>,:,<span class=\"cm-variable\">i<\/span>)=<span class=\"cm-variable\">dct_mat<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">vert<\/span>(<span class=\"cm-variable\">j<\/span>,:,<span class=\"cm-variable\">i<\/span>)<span class=\"cm-operator\">'<\/span>; <span class=\"cm-comment\">% transpose!<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 3. out-of-plane rotations<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">i<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">block_in<\/span>,<span class=\"cm-number\">1<\/span>) <\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">j<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">block_in<\/span>,<span class=\"cm-number\">2<\/span>)<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">vec<\/span>=<span class=\"cm-variable\">squeeze<\/span>(<span class=\"cm-variable\">hor<\/span>(<span class=\"cm-variable\">i<\/span>,<span class=\"cm-variable\">j<\/span>,<span class=\"cm-keyword\">end<\/span>:<span class=\"cm-number\">-1<\/span>:<span class=\"cm-number\">1<\/span>)); <span class=\"cm-comment\">% out-of-plane<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">block_dct<\/span>(<span class=\"cm-variable\">i<\/span>,<span class=\"cm-variable\">j<\/span>,<span class=\"cm-keyword\">end<\/span>:<span class=\"cm-number\">-1<\/span>:<span class=\"cm-number\">1<\/span>)=<span class=\"cm-variable\">dct_mat<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">vec<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 552px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 552px;\"><\/div><\/div><\/div><\/pre><h3 id='c14-apply-above-3d-dct-to-any-3d-sequence-video-ignore-the-padding-return-the-dc-and-high-frequency-components-of-the-resulting-sequence'><span>C1.4. Apply above 3D DCT to any 3D sequence <\/span><code>video<\/code><span>. Ignore the padding. Return the DC and high-frequency components of the resulting sequence.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\" style=\"break-inside: unset;\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">fcnHandle3DDCT<\/span>=<span class=\"cm-operator\">@<\/span><span class=\"cm-variable\">apply_3ddct<\/span>;<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">video<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">i<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">+1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">j<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">+1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">k<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">3<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">+1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">tmp<\/span>=<span class=\"cm-variable\">video<\/span>(<span class=\"cm-variable\">i<\/span>:<span class=\"cm-variable\">i<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span>,<span class=\"cm-variable\">j<\/span>:<span class=\"cm-variable\">j<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span>,<span class=\"cm-variable\">k<\/span>:<span class=\"cm-variable\">k<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">block_dct<\/span>=<span class=\"cm-variable\">fcnHandle3DDCT<\/span>(<span class=\"cm-variable\">tmp<\/span>,<span class=\"cm-variable\">dct_mat<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">video_out<\/span>(<span class=\"cm-variable\">i<\/span>:<span class=\"cm-variable\">i<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span>,<span class=\"cm-variable\">j<\/span>:<span class=\"cm-variable\">j<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span>,<span class=\"cm-variable\">k<\/span>:<span class=\"cm-variable\">k<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">block_sz<\/span><span class=\"cm-number\">-1<\/span>)=<span class=\"cm-variable\">block_dct<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">video_dc<\/span>=<span class=\"cm-variable\">video_out<\/span>(<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-keyword\">end<\/span>,<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-keyword\">end<\/span>,<span class=\"cm-number\">2<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-keyword\">end<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">video_hf<\/span>=<span class=\"cm-variable\">video_out<\/span>(<span class=\"cm-number\">2<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-keyword\">end<\/span>,<span class=\"cm-number\">2<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-keyword\">end<\/span>,<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">block_sz<\/span>:<span class=\"cm-keyword\">end<\/span>);<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 360px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 360px;\"><\/div><\/div><\/div><\/pre><h3 id='a13-which-dct-component-of-the-3d-dct-transformed-sequence-contains-the-most-energy-what-is-the-best-strategy-for-video-compression'><span>A1.3. Which DCT component of the 3D DCT transformed sequence contains the most energy? What is the best strategy for video compression?<\/span><\/h3><ul><li><p><span>The most energy is stored in DC component. <\/span><\/p><\/li><li><p><span>The energy decreases with increasing coefficient positions.<\/span><\/p><\/li><li><p><span>For sequences with <\/span><strong><span>dynamic<\/span><\/strong><span> content but with ares of same color, it is best to preserve more DCT coefficients corresponding to <\/span><strong><span>time<\/span><\/strong><span> dimension.<\/span><\/p><\/li><li><p><span>For sequences with <\/span><strong><span>static<\/span><\/strong><span> content but with high level of spatial detail, it is best to preserve the coefficients corresponding to the <\/span><strong><span>spatial<\/span><\/strong><span> domain.<\/span><\/p><\/li><\/ul><h2 id='hw42-karhunen-loeve-transform'><span>HW4.2. Karhunen-Loeve-Transform<\/span><\/h2><h3 id='c21-implement-a-block-processor-which-reshapes-a-\\n32\\n\u00d7\\n32\\n-block-into-a-vector-and-adds-\\n01\\n-to-the-vector-elements-52954427232784-and-reshapes-the-result-back-to-a-block-perform-above-block-processor-to-an-image'><span>C2.1. Implement a block processor which reshapes a <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.291ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 3222.4 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-154-TEX-N-33\" d=\"M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z\"><\/path><path id=\"MJX-154-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-154-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"33\" xlink:href=\"#MJX-154-TEX-N-33\"><\/use><use data-c=\"32\" xlink:href=\"#MJX-154-TEX-N-32\" transform=\"translate(500,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1222.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-154-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2222.4,0)\"><use data-c=\"33\" xlink:href=\"#MJX-154-TEX-N-33\"><\/use><use data-c=\"32\" xlink:href=\"#MJX-154-TEX-N-32\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>32<\/mn><mo>\u00d7<\/mo><mn>32<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">32\\times 32<\/script><span> block into a vector, and adds <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.891ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 1278 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-127-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-127-TEX-N-2E\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-127-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"30\" xlink:href=\"#MJX-127-TEX-N-30\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-127-TEX-N-2E\" transform=\"translate(500,0)\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-127-TEX-N-31\" transform=\"translate(778,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0.1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">0.1<\/script><span> to the vector elements <\/span><code>[529:544,272:32:784]<\/code><span> and reshapes the result back to a block. Perform above block processor to an image.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">add<\/span>=<span class=\"cm-builtin\">zeros<\/span>(<span class=\"cm-variable\">N<\/span><span class=\"cm-operator\">^<\/span><span class=\"cm-number\">2<\/span>,<span class=\"cm-number\">1<\/span>);<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">add<\/span>([<span class=\"cm-number\">529<\/span>:<span class=\"cm-number\">544<\/span>,<span class=\"cm-number\">272<\/span>:<span class=\"cm-number\">32<\/span>:<span class=\"cm-number\">784<\/span>])=<span class=\"cm-number\">0.1<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">block_func<\/span>=<span class=\"cm-operator\">@<\/span>(<span class=\"cm-variable\">block<\/span>,<span class=\"cm-variable\">add<\/span>) <span class=\"cm-builtin\">reshape<\/span>(<span class=\"cm-variable\">block<\/span>(:)<span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">add<\/span>,<span class=\"cm-variable\">N<\/span>,<span class=\"cm-variable\">N<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_reshape<\/span>=<span class=\"cm-variable\">block_splitter<\/span>(<span class=\"cm-variable\">im<\/span>,<span class=\"cm-variable\">N<\/span>,<span class=\"cm-variable\">block_func<\/span>,<span class=\"cm-variable\">add<\/span>);<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 96px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 96px;\"><\/div><\/div><\/div><\/pre><h3 id='c22-calculate-the-klt-basis-functions-for-\\n8\\n\u00d7\\n8\\n-blocks-using-following-steps'><span>C2.2. Calculate the KLT basis functions for <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.028ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 2222.4 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-130-TEX-N-38\" d=\"M70 417T70 494T124 618T248 666Q319 666 374 624T429 515Q429 485 418 459T392 417T361 389T335 371T324 363L338 354Q352 344 366 334T382 323Q457 264 457 174Q457 95 399 37T249 -22Q159 -22 101 29T43 155Q43 263 172 335L154 348Q133 361 127 368Q70 417 70 494ZM286 386L292 390Q298 394 301 396T311 403T323 413T334 425T345 438T355 454T364 471T369 491T371 513Q371 556 342 586T275 624Q268 625 242 625Q201 625 165 599T128 534Q128 511 141 492T167 463T217 431Q224 426 228 424L286 386ZM250 21Q308 21 350 55T392 137Q392 154 387 169T375 194T353 216T330 234T301 253T274 270Q260 279 244 289T218 306L210 311Q204 311 181 294T133 239T107 157Q107 98 150 60T250 21Z\"><\/path><path id=\"MJX-130-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"38\" xlink:href=\"#MJX-130-TEX-N-38\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(722.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-130-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1722.4,0)\"><use data-c=\"38\" xlink:href=\"#MJX-130-TEX-N-38\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>8<\/mn><mo>\u00d7<\/mo><mn>8<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">8\\times 8<\/script><span> blocks using following steps:<\/span><\/h3><ul><li><p><span>Reconstructure image into blocks of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.028ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 2222.4 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-130-TEX-N-38\" d=\"M70 417T70 494T124 618T248 666Q319 666 374 624T429 515Q429 485 418 459T392 417T361 389T335 371T324 363L338 354Q352 344 366 334T382 323Q457 264 457 174Q457 95 399 37T249 -22Q159 -22 101 29T43 155Q43 263 172 335L154 348Q133 361 127 368Q70 417 70 494ZM286 386L292 390Q298 394 301 396T311 403T323 413T334 425T345 438T355 454T364 471T369 491T371 513Q371 556 342 586T275 624Q268 625 242 625Q201 625 165 599T128 534Q128 511 141 492T167 463T217 431Q224 426 228 424L286 386ZM250 21Q308 21 350 55T392 137Q392 154 387 169T375 194T353 216T330 234T301 253T274 270Q260 279 244 289T218 306L210 311Q204 311 181 294T133 239T107 157Q107 98 150 60T250 21Z\"><\/path><path id=\"MJX-130-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"38\" xlink:href=\"#MJX-130-TEX-N-38\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(722.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-130-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1722.4,0)\"><use data-c=\"38\" xlink:href=\"#MJX-130-TEX-N-38\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>8<\/mn><mo>\u00d7<\/mo><mn>8<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">8\\times 8<\/script><span> and save results to <\/span><code>blocks<\/code><\/p><\/li><li><p><span>Calculate the autocorrelation matrix for those blocks and save results to <\/span><code>C_im<\/code><\/p><\/li><li><p><span>Perform EVD of <\/span><code>C_im<\/code><\/p><\/li><li><p><span>Sort the resulting eigenvalues with descending order and correspondingly the eigenvectors<\/span><\/p><\/li><li><p><span>The resulting eigenvectors constitutes the KLT basis images. Reshape each eigenvector to an <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.028ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 2222.4 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-130-TEX-N-38\" d=\"M70 417T70 494T124 618T248 666Q319 666 374 624T429 515Q429 485 418 459T392 417T361 389T335 371T324 363L338 354Q352 344 366 334T382 323Q457 264 457 174Q457 95 399 37T249 -22Q159 -22 101 29T43 155Q43 263 172 335L154 348Q133 361 127 368Q70 417 70 494ZM286 386L292 390Q298 394 301 396T311 403T323 413T334 425T345 438T355 454T364 471T369 491T371 513Q371 556 342 586T275 624Q268 625 242 625Q201 625 165 599T128 534Q128 511 141 492T167 463T217 431Q224 426 228 424L286 386ZM250 21Q308 21 350 55T392 137Q392 154 387 169T375 194T353 216T330 234T301 253T274 270Q260 279 244 289T218 306L210 311Q204 311 181 294T133 239T107 157Q107 98 150 60T250 21Z\"><\/path><path id=\"MJX-130-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"38\" xlink:href=\"#MJX-130-TEX-N-38\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(722.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-130-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1722.4,0)\"><use data-c=\"38\" xlink:href=\"#MJX-130-TEX-N-38\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>8<\/mn><mo>\u00d7<\/mo><mn>8<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">8\\times 8<\/script><span> block and pack them into a <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.291ex\" height=\"1.581ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 3222.4 699\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-131-TEX-N-36\" d=\"M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z\"><\/path><path id=\"MJX-131-TEX-N-34\" d=\"M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z\"><\/path><path id=\"MJX-131-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"36\" xlink:href=\"#MJX-131-TEX-N-36\"><\/use><use data-c=\"34\" xlink:href=\"#MJX-131-TEX-N-34\" transform=\"translate(500,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1222.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-131-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2222.4,0)\"><use data-c=\"36\" xlink:href=\"#MJX-131-TEX-N-36\"><\/use><use data-c=\"34\" xlink:href=\"#MJX-131-TEX-N-34\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>64<\/mn><mo>\u00d7<\/mo><mn>64<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">64\\times 64<\/script><span> image in <\/span><strong><span>column major order<\/span><\/strong><\/p><\/li><\/ul><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\" style=\"break-inside: unset;\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">N<\/span>=<span class=\"cm-number\">8<\/span>;<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">im<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 1. Restructure im into blocks<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">ibl<\/span>=<span class=\"cm-number\">0<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">x<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">N<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">N<\/span><span class=\"cm-number\">+1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">y<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">N<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">N<\/span><span class=\"cm-number\">+1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">ibl<\/span>=<span class=\"cm-variable\">ibl<\/span><span class=\"cm-number\">+1<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">block<\/span>=<span class=\"cm-variable\">im<\/span>(<span class=\"cm-variable\">y<\/span>:<span class=\"cm-variable\">y<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">N<\/span><span class=\"cm-number\">-1<\/span>,<span class=\"cm-variable\">x<\/span>:<span class=\"cm-variable\">x<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">N<\/span><span class=\"cm-number\">-1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">blocks<\/span>(<span class=\"cm-variable\">ibl<\/span>,:)=<span class=\"cm-variable\">block<\/span>(:);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-comment\">% size(blocks,1)=...<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-comment\">% &nbsp; length(1:N:sz(2)-N+1)*length(1:N:sz(1)-N+1)<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-comment\">% size(blocks,2)=64<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 2. Compute autocorrelation over blocks, i.e., rows of blocks<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% C_XX=E[XX^H]<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">C_im<\/span>=<span class=\"cm-variable\">blocks<\/span><span class=\"cm-operator\">'*<\/span><span class=\"cm-variable\">blocks<\/span><span class=\"cm-operator\">\/<\/span><span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">blocks<\/span>,<span class=\"cm-number\">1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 3. EVD of C_im<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\">[<span class=\"cm-variable\">evec<\/span>,<span class=\"cm-builtin\">eval<\/span>]=<span class=\"cm-builtin\">eig<\/span>(<span class=\"cm-variable\">C_im<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\">[<span class=\"cm-builtin\">eval<\/span>,<span class=\"cm-variable\">eval_idx<\/span>]=<span class=\"cm-variable\">sort<\/span>(<span class=\"cm-variable\">diag<\/span>(<span class=\"cm-builtin\">eval<\/span>),<span class=\"cm-string\">'descend'<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% eval is sorted vector<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% eval_idx is the permutation<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">klt_base_v<\/span>=<span class=\"cm-variable\">evec<\/span>(:,<span class=\"cm-variable\">eval_idx<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 4. Reshape in column major order<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">idx<\/span>=<span class=\"cm-number\">0<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">x<\/span>=<span class=\"cm-number\">0<\/span>:<span class=\"cm-variable\">N<\/span><span class=\"cm-number\">-1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">y<\/span>=<span class=\"cm-number\">0<\/span>:<span class=\"cm-variable\">N<\/span><span class=\"cm-number\">-1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">idx<\/span>=<span class=\"cm-variable\">idx<\/span><span class=\"cm-number\">+1<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">klt_base<\/span>(<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">y<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">N<\/span>,(<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">y<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">N<\/span>,<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">x<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">N<\/span>:(<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">x<\/span>)<span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">N<\/span>)=<span class=\"cm-error\">...<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-builtin\">reshape<\/span>(<span class=\"cm-variable\">klt_base_v<\/span>(:,<span class=\"cm-variable\">idx<\/span>),<span class=\"cm-variable\">N<\/span>,<span class=\"cm-variable\">N<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 864px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 864px;\"><\/div><\/div><\/div><\/pre><h3 id='a21-compare-klt-basis-images-when-using-different-images-for-training'><span>A2.1. Compare KLT basis images when using different images for training.<\/span><\/h3><ul><li><p><span>KLT is the most efficient transform in terms of decorrelation and energy packing.<\/span><\/p><\/li><li><p><span>But it depends on image statistics. As shown above, autocorrelation of a specific image is calculated to derive KLT basis images.<\/span><\/p><\/li><li><p><span>For different images with different statistics, the resulting KLT basis images would be different.<\/span><\/p><\/li><\/ul><h3 id='c23-project-an-image-to-the-klt-basis-above-compare-the-back-transformed-image-with-the-original-one-and-calculte-the-psnr'><span>C2.3. Project an image to the KLT basis above. Compare the back-transformed image with the original one and calculte the PSNR.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% Projection is obtained by inner product<\/span><\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">block_func<\/span>=<span class=\"cm-operator\">@<\/span>(<span class=\"cm-variable\">block_in<\/span>,<span class=\"cm-variable\">base<\/span>) <span class=\"cm-builtin\">reshape<\/span>(<span class=\"cm-variable\">base<\/span><span class=\"cm-operator\">'*<\/span><span class=\"cm-variable\">block_in<\/span>(:),<span class=\"cm-variable\">N<\/span>,<span class=\"cm-variable\">N<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% Project the image<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_proj<\/span>=<span class=\"cm-variable\">block_splitter<\/span>(<span class=\"cm-variable\">im<\/span>,<span class=\"cm-variable\">N<\/span>,<span class=\"cm-variable\">block_func<\/span>,<span class=\"cm-variable\">klt_base_v<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% Back project, using base*base'=I<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_back_proj<\/span>=<span class=\"cm-variable\">block_splitter<\/span>(<span class=\"cm-variable\">im_proj<\/span>,<span class=\"cm-variable\">N<\/span>,<span class=\"cm-variable\">block_func<\/span>,<span class=\"cm-variable\">klt_base_v<\/span><span class=\"cm-operator\">'<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% Compute PSNR<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_diff<\/span>=(<span class=\"cm-variable\">im_back_proj<\/span><span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">im<\/span>)<span class=\"cm-operator\">.^<\/span><span class=\"cm-number\">2<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">klt_psnr<\/span>=<span class=\"cm-number\">10<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-builtin\">log10<\/span>(<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">\/<\/span><span class=\"cm-builtin\">mean<\/span>(<span class=\"cm-variable\">im_diff<\/span>(:)));<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 216px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 216px;\"><\/div><\/div><\/div><\/pre><h3 id='c24-calculate-the-autocorrelation-matrix-of-all-blocks-from-klt-transformed-image-same-as-c22'><span>C2.4. Calculate the autocorrelation matrix of all blocks from KLT transformed image. (same as C2.2.)<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\" style=\"break-inside: unset;\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">im_proj<\/span>);<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 1. Restructure im into blocks<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">ibl<\/span>=<span class=\"cm-number\">0<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">x<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">N<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">N<\/span><span class=\"cm-number\">+1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">y<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">N<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">N<\/span><span class=\"cm-number\">+1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">ibl<\/span>=<span class=\"cm-variable\">ibl<\/span><span class=\"cm-number\">+1<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">block<\/span>=<span class=\"cm-variable\">im<\/span>(<span class=\"cm-variable\">y<\/span>:<span class=\"cm-variable\">y<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">N<\/span><span class=\"cm-number\">-1<\/span>,<span class=\"cm-variable\">x<\/span>:<span class=\"cm-variable\">x<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">N<\/span><span class=\"cm-number\">-1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">blocks<\/span>(<span class=\"cm-variable\">ibl<\/span>,:)=<span class=\"cm-variable\">block<\/span>(:);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-comment\">% size(blocks,1)=...<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-comment\">% &nbsp; length(1:N:sz(2)-N+1)*length(1:N:sz(1)-N+1)<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-comment\">% size(blocks,2)=64<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% 2. Compute autocorrelation over blocks, i.e., rows of blocks<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-comment\">% C_XX=E[XX^H]<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">autocorr<\/span>=<span class=\"cm-variable\">blocks<\/span><span class=\"cm-operator\">'*<\/span><span class=\"cm-variable\">blocks<\/span><span class=\"cm-operator\">\/<\/span><span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">blocks<\/span>,<span class=\"cm-number\">1<\/span>);<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 408px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 408px;\"><\/div><\/div><\/div><\/pre><h3 id='a22-how-should-the-autocorrelation-matrix-of-klt-blocks-look-like'><span>A2.2. How should the autocorrelation matrix of KLT blocks look like?<\/span><\/h3><ul><li><p><span>When KLT learned from Lenna image is applied to Lenna image, it allows for perfect decorrelation (except for numerical errors).<\/span><\/p><\/li><li><p><span>Thus, the autocorrelation matrix exhibits only elements on its diagonal and zero elsewhere.<\/span><\/p><\/li><li><p><span>If this KLT is applied to other images, the diagonal of the autocorrelation matrix still dominates, but there are significant non-diagonal elements.<\/span><\/p><\/li><li><p><span>KLT is not suitable as a generic compression method.<\/span><\/p><\/li><\/ul><h3 id='a23-calculate-the-statistics-mean-and-variance-of-above-klt-transformed-image'><span>A2.3. Calculate the statistics (mean and variance) of above KLT transformed image.<\/span><\/h3><ul><li><p><span>KLT of Lenna on its own base exhibits a monotomic decrease of the variance coefficients.<\/span><\/p><\/li><li><p><span>For other images, the energy compaction is not optimal, there is a noisy behavior in the variance plot.<\/span><\/p><\/li><\/ul><h3 id='a24-recall-the-variance-plot-for-haar-transform-plot-the-variance-of-dct-and-klt-what-do-you-observe'><span>A2.4. Recall the variance plot for Haar transform, plot the variance of DCT and KLT, what do you observe?<\/span><\/h3><ul><li><p><span>The KLT of Lenna on its own base has a monotonic decrease and higher energy compaction efficiency than DCT and Haar.<\/span><\/p><\/li><\/ul><h2 id='hw51-autoregressive-models'><span>HW5.1. Autoregressive Models<\/span><\/h2><h3 id='c11-calculate-the-autocovariance-of-a-natural-image-i-using-neighboring-\\n11\\n\u00d7\\n11\\n-for-each-pixel-as-follows'><span>C1.1. Calculate the autocovariance of a natural image <\/span><code>I<\/code><span>, using neighboring <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.291ex\" height=\"1.507ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 3222.4 666\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-132-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-132-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-132-TEX-N-31\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-132-TEX-N-31\" transform=\"translate(500,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1222.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-132-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2222.4,0)\"><use data-c=\"31\" xlink:href=\"#MJX-132-TEX-N-31\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-132-TEX-N-31\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>11<\/mn><mo>\u00d7<\/mo><mn>11<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script 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xlink:href=\"#MJX-10-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(21791.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-10-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(22236.3,0)\"><use data-c=\"1D459\" xlink:href=\"#MJX-10-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(22812.1,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-10-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(23867.9,0)\"><use data-c=\"35\" xlink:href=\"#MJX-10-TEX-N-35\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"1778 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:10\"><g data-mml-node=\"mtext\"><use data-c=\"28\" xlink:href=\"#MJX-10-TEX-N-28\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-10-TEX-N-31\" transform=\"translate(389,0)\"><\/use><use data-c=\"30\" xlink:href=\"#MJX-10-TEX-N-30\" transform=\"translate(889,0)\"><\/use><use data-c=\"29\" xlink:href=\"#MJX-10-TEX-N-29\" transform=\"translate(1389,0)\"><\/use><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mtext>(10)<\/mtext><\/mtd><mtd><msub><mi>C<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>I<\/mi><mi>I<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>k<\/mi><mo>,<\/mo><mi>l<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>,<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mi>I<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mi>k<\/mi><mo>,<\/mo><mi>y<\/mi><mo>+<\/mo><mi>l<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mtext>&nbsp;with&nbsp;<\/mtext><mo>\u2212<\/mo><mn>5<\/mn><mo>\u2264<\/mo><mi>k<\/mi><mo>,<\/mo><mi>l<\/mi><mo>\u2264<\/mo><mn>5<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><p><span>Don&#39;t forget to substract the mean value of the image.<\/span><\/p><\/li><li><p><span>Take into account a <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 500 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-153-TEX-N-35\" d=\"M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"35\" xlink:href=\"#MJX-153-TEX-N-35\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>5<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">5<\/script><span> pixel border.<\/span><\/p><\/li><li><p><span>Extract the variance <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.395ex\" height=\"2.463ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 1058.5 1088.7\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.576ex;\"><defs><path id=\"MJX-134-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-134-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-134-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-134-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-134-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(604,-247) scale(0.707)\"><use data-c=\"1D465\" xlink:href=\"#MJX-134-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mi>\u03c3<\/mi><mi>x<\/mi><mn>2<\/mn><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\sigma_x^2<\/script><span> and the covariance for <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.507ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 500 666\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-151-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-151-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">1<\/script><span> pixel horizontal and vertical displacement.<\/span><\/p><\/li><\/ul><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\" style=\"break-inside: unset;\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">im<\/span>);<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">border<\/span>=<span class=\"cm-number\">5<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">y<\/span>=<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">border<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">1<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">border<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">x<\/span>=<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">border<\/span>:<span class=\"cm-variable\">sz<\/span>(<span class=\"cm-number\">2<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">border<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">imm<\/span>=<span class=\"cm-variable\">im<\/span><span class=\"cm-operator\">-<\/span><span class=\"cm-builtin\">mean<\/span>(<span class=\"cm-variable\">im<\/span>(:));<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">t_y<\/span>=<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">border<\/span>:<span class=\"cm-variable\">border<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">t_x<\/span>=<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">border<\/span>:<span class=\"cm-variable\">border<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">data_mat<\/span>=<span class=\"cm-variable\">imm<\/span>(<span class=\"cm-variable\">y<\/span>,<span class=\"cm-variable\">x<\/span>)<span class=\"cm-operator\">.*<\/span><span class=\"cm-variable\">imm<\/span>(<span class=\"cm-variable\">t_y<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">y<\/span>,<span class=\"cm-variable\">t_x<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">x<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"cm-variable\">acov<\/span>(<span class=\"cm-variable\">t_y<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">border<\/span><span class=\"cm-number\">+1<\/span>,<span class=\"cm-variable\">t_x<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">border<\/span><span class=\"cm-number\">+1<\/span>)=<span class=\"cm-builtin\">mean<\/span>(<span class=\"cm-variable\">data_mat<\/span>(:));<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">var<\/span>=<span class=\"cm-variable\">acov<\/span>(<span class=\"cm-variable\">border<\/span><span class=\"cm-number\">+1<\/span>,<span class=\"cm-variable\">border<\/span><span class=\"cm-number\">+1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">cov_h<\/span>=<span class=\"cm-variable\">acov<\/span>(<span class=\"cm-variable\">border<\/span><span class=\"cm-number\">+1<\/span>,<span class=\"cm-variable\">border<\/span><span class=\"cm-number\">+1+1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">cov_v<\/span>=<span class=\"cm-variable\">acov<\/span>(<span class=\"cm-variable\">border<\/span><span class=\"cm-number\">+1+1<\/span>,<span class=\"cm-variable\">border<\/span><span class=\"cm-number\">+1<\/span>);<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 384px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 384px;\"><\/div><\/div><\/div><\/pre><h3 id='a11-discuss-above-values-and-explain-the-structure-of-the-autocovariance-matrix'><span>A1.1. Discuss above values and explain the structure of the autocovariance matrix.<\/span><\/h3><ul><li><p><span>The center of autocovariance matrix is the average energy of pixels.<\/span><\/p><\/li><li><p><span>Since we measure all pixels, the autocovariance matrix should be point symmetrix around its center point.<\/span><\/p><\/li><li><p><span>(With increasing distance between pixels, the autocovariance drops almost linearly. A slightly higher covariance is observed in vertical direction)<\/span><\/p><\/li><\/ul><h3 id='c12-a-separable-\\na\\nr\\n\\n1\\n\\n-process-can-be-used-to-generate-a-2d-signal-and-has-autocovariance'><span>C1.2. A separable <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.305ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2787 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-147-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-147-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 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xlink:href=\"#MJX-12-TEX-N-29\" transform=\"translate(1389,0)\"><\/use><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mtext>(12)<\/mtext><\/mtd><mtd><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>,<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>\u03c1<\/mi><mi>h<\/mi><\/msub><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>,<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>\u03c1<\/mi><mi>v<\/mi><\/msub><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>,<\/mo><mi>y<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><msub><mi>\u03c1<\/mi><mi>h<\/mi><\/msub><msub><mi>\u03c1<\/mi><mi>v<\/mi><\/msub><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>,<\/mo><mi>y<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>n<\/mi><mi>z<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>,<\/mo><mi>y<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='where-\\nn\\nz\\n-is-a-zero-mean-gaussian-noise-with-standard-deviation-\\n\u03c3\\nz\\n'><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.289ex\" height=\"1.357ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 1011.8 599.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-137-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-137-TEX-I-1D467\" d=\"M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-137-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(633,-150) scale(0.707)\"><use data-c=\"1D467\" xlink:href=\"#MJX-137-TEX-I-1D467\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>n<\/mi><mi>z<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n_z<\/script><span> is a zero-mean Gaussian noise with standard deviation <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.224ex\" height=\"1.332ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -431 982.8 588.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-138-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-138-TEX-I-1D467\" d=\"M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-138-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(604,-150) scale(0.707)\"><use data-c=\"1D467\" xlink:href=\"#MJX-138-TEX-I-1D467\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>\u03c3<\/mi><mi>z<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\sigma_z<\/script><span>.<\/span><\/h3><h3 id='generate-a-\\n512\\n\u00d7\\n512\\n-image-imout-using-above-process-the-process-will-need-some-pixels-to-start-up-so-add-a-temporary-border-of-\\n100\\n-pixels-filled-with-zeros-to-the-top-and-to-the-left-notice-that-indexes-of-matrix-and-indexes-of-image-are-reversed-so-\\n\u03c1\\nh\\n-can-be-obtained-by-\\nc\\nx\\nx\\n\\n10\\n\\nc\\nx\\nx\\n\\n00\\n\\nc\\ni\\ni\\n\\n67\\n\\nc\\ni\\ni\\n\\n66\\n\\ncovhvar'><span>Generate a <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.553ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 4222.4 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-152-TEX-N-35\" d=\"M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z\"><\/path><path id=\"MJX-152-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-152-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-152-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"35\" xlink:href=\"#MJX-152-TEX-N-35\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-152-TEX-N-31\" transform=\"translate(500,0)\"><\/use><use data-c=\"32\" xlink:href=\"#MJX-152-TEX-N-32\" transform=\"translate(1000,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1722.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-152-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2722.4,0)\"><use data-c=\"35\" xlink:href=\"#MJX-152-TEX-N-35\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-152-TEX-N-31\" transform=\"translate(500,0)\"><\/use><use data-c=\"32\" xlink:href=\"#MJX-152-TEX-N-32\" transform=\"translate(1000,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>512<\/mn><mo>\u00d7<\/mo><mn>512<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">512\\times 512<\/script><span> image <\/span><code>im_out<\/code><span> using above process. The process will need some pixels to start up, so add a temporary border of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.394ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 1500 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-140-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-140-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-140-TEX-N-31\"><\/use><use data-c=\"30\" xlink:href=\"#MJX-140-TEX-N-30\" transform=\"translate(500,0)\"><\/use><use data-c=\"30\" xlink:href=\"#MJX-140-TEX-N-30\" transform=\"translate(1000,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>100<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">100<\/script><span> pixels filled with zeros to the top and to the left. Notice that indexes of matrix and indexes of image are reversed, so <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.279ex\" height=\"1.489ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 1007.3 658\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.489ex;\"><defs><path id=\"MJX-141-TEX-I-1D70C\" d=\"M58 -216Q25 -216 23 -186Q23 -176 73 26T127 234Q143 289 182 341Q252 427 341 441Q343 441 349 441T359 442Q432 442 471 394T510 276Q510 219 486 165T425 74T345 13T266 -10H255H248Q197 -10 165 35L160 41L133 -71Q108 -168 104 -181T92 -202Q76 -216 58 -216ZM424 322Q424 359 407 382T357 405Q322 405 287 376T231 300Q217 269 193 170L176 102Q193 26 260 26Q298 26 334 62Q367 92 389 158T418 266T424 322Z\"><\/path><path id=\"MJX-141-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D70C\" xlink:href=\"#MJX-141-TEX-I-1D70C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(550,-150) scale(0.707)\"><use data-c=\"210E\" xlink:href=\"#MJX-141-TEX-I-210E\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>\u03c1<\/mi><mi>h<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\rho_h<\/script><span> can be obtained by <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"29.954ex\" height=\"2.347ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 13239.5 1037.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path id=\"MJX-142-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-142-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-142-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-142-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-142-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-142-TEX-N-2F\" d=\"M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z\"><\/path><path id=\"MJX-142-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-142-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-142-TEX-N-36\" d=\"M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 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xlink:href=\"#MJX-142-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(504,0)\"><use data-c=\"1D43C\" xlink:href=\"#MJX-142-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1008,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-142-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1286,0)\"><use data-c=\"36\" xlink:href=\"#MJX-142-TEX-N-36\"><\/use><use data-c=\"36\" xlink:href=\"#MJX-142-TEX-N-36\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(12461.5,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-142-TEX-N-3D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>C<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><mi>x<\/mi><mo>,<\/mo><mn>10<\/mn><\/mrow><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msub><mi>C<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>x<\/mi><mi>x<\/mi><mo>,<\/mo><mn>00<\/mn><\/mrow><\/msub><mo>=<\/mo><msub><mi>C<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>I<\/mi><mi>I<\/mi><mo>,<\/mo><mn>67<\/mn><\/mrow><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msub><mi>C<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>I<\/mi><mi>I<\/mi><mo>,<\/mo><mn>66<\/mn><\/mrow><\/msub><mo>=<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">C_{xx,10}\/C_{xx,00}=C_{II,67}\/C_{II,66}=<\/script><code>cov_h<\/code><span>\/<\/span><code>var<\/code><span>.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\" style=\"break-inside: unset;\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">sz<\/span>=<span class=\"cm-number\">512<\/span>;<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">border<\/span>=<span class=\"cm-number\">100<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">rh<\/span>=<span class=\"cm-variable\">cov_h<\/span><span class=\"cm-operator\">\/<\/span><span class=\"cm-variable\">var<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">rv<\/span>=<span class=\"cm-variable\">cov_v<\/span><span class=\"cm-operator\">\/<\/span><span class=\"cm-variable\">var<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">std_z<\/span>=<span class=\"cm-builtin\">sqrt<\/span>(<span class=\"cm-variable\">var<\/span><span class=\"cm-operator\">*<\/span>(<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">rh<\/span><span class=\"cm-operator\">^<\/span><span class=\"cm-number\">2<\/span>)<span class=\"cm-operator\">*<\/span>(<span class=\"cm-number\">1<\/span><span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">rv<\/span><span class=\"cm-operator\">^<\/span><span class=\"cm-number\">2<\/span>));<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im<\/span>=<span class=\"cm-builtin\">zeros<\/span>(<span class=\"cm-variable\">border<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">sz<\/span>,<span class=\"cm-variable\">border<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">sz<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">i<\/span>=<span class=\"cm-number\">2<\/span>:<span class=\"cm-variable\">border<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">sz<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">j<\/span>=<span class=\"cm-number\">2<\/span>:<span class=\"cm-variable\">border<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">sz<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">im<\/span>(<span class=\"cm-variable\">i<\/span>,<span class=\"cm-variable\">j<\/span>)=<span class=\"cm-variable\">rh<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">im<\/span>(<span class=\"cm-variable\">i<\/span>,<span class=\"cm-variable\">j<\/span><span class=\"cm-number\">-1<\/span>)<span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">rv<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">im<\/span>(<span class=\"cm-variable\">i<\/span><span class=\"cm-number\">-1<\/span>,<span class=\"cm-variable\">j<\/span>)<span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">rv<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">rh<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">im<\/span>(<span class=\"cm-variable\">i<\/span><span class=\"cm-number\">-1<\/span>,<span class=\"cm-variable\">j<\/span><span class=\"cm-number\">-1<\/span>)<span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">std_z<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">randn<\/span>(<span class=\"cm-number\">1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">im_out<\/span>=<span class=\"cm-variable\">im<\/span>(<span class=\"cm-variable\">border<\/span><span class=\"cm-number\">+1<\/span>:<span class=\"cm-variable\">border<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">sz<\/span>,<span class=\"cm-variable\">border<\/span><span class=\"cm-number\">+1<\/span>:<span class=\"cm-variable\">border<\/span><span class=\"cm-operator\">+<\/span><span class=\"cm-variable\">sz<\/span>);<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 360px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 360px;\"><\/div><\/div><\/div><\/pre><h3 id='compute-the-autocovariance-of-above-generated-image-compare-it-to-the-autocovariance-obtained-by-the-equation-in-c12'><span>Compute the autocovariance of above generated image, compare it to the autocovariance obtained by the equation in C1.2.<\/span><\/h3><ul><li><p><span>The separable <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.305ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2787 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-147-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-147-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-147-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-147-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-147-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-147-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(750,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-147-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1509,0)\"><use data-c=\"28\" xlink:href=\"#MJX-147-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1898,0)\"><use data-c=\"31\" xlink:href=\"#MJX-147-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2398,0)\"><use data-c=\"29\" xlink:href=\"#MJX-147-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mi>R<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">AR(1)<\/script><span> process produces an autocovariance which is piecewise planar in four quadrants.<\/span><\/p><\/li><li><p><span>This results in a modelling error which increases from the center towards the border along <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.327ex\" height=\"1.756ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2354.6 776\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-144-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-144-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-144-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-144-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(798.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-144-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1854.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-144-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><mo>=<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k=0<\/script><span> or <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.823ex\" height=\"1.756ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 2131.6 776\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-145-TEX-I-1D459\" d=\"M117 59Q117 26 142 26Q179 26 205 131Q211 151 215 152Q217 153 225 153H229Q238 153 241 153T246 151T248 144Q247 138 245 128T234 90T214 43T183 6T137 -11Q101 -11 70 11T38 85Q38 97 39 102L104 360Q167 615 167 623Q167 626 166 628T162 632T157 634T149 635T141 636T132 637T122 637Q112 637 109 637T101 638T95 641T94 647Q94 649 96 661Q101 680 107 682T179 688Q194 689 213 690T243 693T254 694Q266 694 266 686Q266 675 193 386T118 83Q118 81 118 75T117 65V59Z\"><\/path><path id=\"MJX-145-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-145-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D459\" xlink:href=\"#MJX-145-TEX-I-1D459\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(575.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-145-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1631.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-145-TEX-N-30\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>l<\/mi><mo>=<\/mo><mn>0<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">l=0<\/script><span>.<\/span><\/p><\/li><li><p><span>There is a constant offset between the measured autocovairance and expected autocovariance.<\/span><\/p><\/li><li><p><span>This error can be minimized using a larger startup border.<\/span><\/p><\/li><\/ul><h3 id='a13-repeat-c11-with-another-image-explain-why-this-autocovariance-matrix-looks-considerably-different-how-would-an-image-generated-by-an-\\na\\nr\\n\\n1\\n\\n-process-with-parameters-measured-from-this-image-look-like'><span>A1.3. Repeat C1.1. with another image. Explain why this autocovariance matrix looks considerably different. How would an image generated by an <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.305ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2787 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-147-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-147-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-147-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-147-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-147-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-147-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(750,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-147-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1509,0)\"><use data-c=\"28\" xlink:href=\"#MJX-147-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1898,0)\"><use data-c=\"31\" xlink:href=\"#MJX-147-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2398,0)\"><use data-c=\"29\" xlink:href=\"#MJX-147-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mi>R<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">AR(1)<\/script><span> process with parameters measured from this image look like?<\/span><\/h3><ul><li><p><span>The resulting autocovariance matrix exhibits a significantly higher correlation in vertical direction, due to the dominant vartical structures in the image.<\/span><\/p><\/li><li><p><span>An <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.305ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2787 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-147-TEX-I-1D434\" d=\"M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z\"><\/path><path id=\"MJX-147-TEX-I-1D445\" d=\"M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z\"><\/path><path id=\"MJX-147-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-147-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-147-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D434\" xlink:href=\"#MJX-147-TEX-I-1D434\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(750,0)\"><use data-c=\"1D445\" xlink:href=\"#MJX-147-TEX-I-1D445\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1509,0)\"><use data-c=\"28\" xlink:href=\"#MJX-147-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1898,0)\"><use data-c=\"31\" xlink:href=\"#MJX-147-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2398,0)\"><use data-c=\"29\" xlink:href=\"#MJX-147-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>A<\/mi><mi>R<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">AR(1)<\/script><span> image created with corresponding parameters will exhibit vertically dominating structures.<\/span><\/p><\/li><\/ul><h2 id='hw52-pyramid-representations'><span>HW5.2. Pyramid Representations<\/span><\/h2><h3 id='c21-a-\\n3\\n\u00d7\\n3\\n-approximation-of-gaussian-lowpass-filter-is-given'><span>C2.1. A <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.028ex\" height=\"1.554ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -665 2222.4 687\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-149-TEX-N-33\" d=\"M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z\"><\/path><path id=\"MJX-149-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"33\" xlink:href=\"#MJX-149-TEX-N-33\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(722.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-149-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1722.4,0)\"><use data-c=\"33\" xlink:href=\"#MJX-149-TEX-N-33\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>3<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">3\\times 3<\/script><span> approximation of Gaussian lowpass filter is given<\/span><\/h3><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n262\" cid=\"n262\" mdtype=\"math_block\" data-math-tag-before=\"12\" data-math-tag-after=\"13\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 29.44ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"8.597ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -3.733ex; min-width: 29.44ex;\"><defs><path id=\"MJX-13-TEX-I-1D439\" d=\"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z\"><\/path><path id=\"MJX-13-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-13-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 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rowspacing=\"4pt\"><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>1<\/mn><\/mtd><mtd><mn>4<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='compute-an-appropriate-\\n3\\n\u00d7\\n3\\n-reconstruction-filter-g-for-12-upsampling-using-linear-interpolation'><span>Compute an appropriate <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.028ex\" height=\"1.554ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -665 2222.4 687\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-149-TEX-N-33\" d=\"M127 463Q100 463 85 480T69 524Q69 579 117 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29Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"33\" xlink:href=\"#MJX-149-TEX-N-33\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(722.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-149-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1722.4,0)\"><use data-c=\"33\" xlink:href=\"#MJX-149-TEX-N-33\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>3<\/mn><mo>\u00d7<\/mo><mn>3<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">3\\times 3<\/script><span> reconstruction filter <\/span><code>G<\/code><span> for 1:2 upsampling using linear interpolation.<\/span><\/h3><ul><li><p><span>First, zeros are filled between pixels<\/span><\/p><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" 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data-c=\"28\" xlink:href=\"#MJX-14-TEX-N-28\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-14-TEX-N-31\" transform=\"translate(389,0)\"><\/use><use data-c=\"34\" xlink:href=\"#MJX-14-TEX-N-34\" transform=\"translate(889,0)\"><\/use><use data-c=\"29\" xlink:href=\"#MJX-14-TEX-N-29\" transform=\"translate(1389,0)\"><\/use><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mtext>(14)<\/mtext><\/mtd><mtd><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mi>a<\/mi><\/mtd><mtd><mi>b<\/mi><\/mtd><\/mtr><mtr><mtd><mi>c<\/mi><\/mtd><mtd><mi>d<\/mi><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mover><mo stretchy=\"false\">\u27f6<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">upsampling<\/mi><\/mrow><\/mover><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mi>a<\/mi><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mi>b<\/mi><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mi>c<\/mi><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mi>d<\/mi><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><p><span>The reconstructed matrix should be<\/span><\/p><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block 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id=\"mjx-eqn:16\"><g data-mml-node=\"mtext\"><use data-c=\"28\" xlink:href=\"#MJX-16-TEX-N-28\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-16-TEX-N-31\" transform=\"translate(389,0)\"><\/use><use data-c=\"36\" xlink:href=\"#MJX-16-TEX-N-36\" transform=\"translate(889,0)\"><\/use><use data-c=\"29\" xlink:href=\"#MJX-16-TEX-N-29\" transform=\"translate(1389,0)\"><\/use><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mtext>(16)<\/mtext><\/mtd><mtd><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mi>a<\/mi><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><menclose notation=\"box\"><mrow data-mjx-texclass=\"ORD\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mn>0<\/mn><\/mrow><\/mstyle><\/mrow><\/menclose><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mi>c<\/mi><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mo>\u2217<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>2<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>2<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mi>a<\/mi><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><menclose notation=\"box\"><mrow data-mjx-texclass=\"ORD\"><mstyle displaystyle=\"true\" scriptlevel=\"0\"><mrow data-mjx-texclass=\"ORD\"><mfrac><mrow><mi>a<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><mn>2<\/mn><\/mfrac><\/mrow><\/mstyle><\/mrow><\/menclose><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mi>c<\/mi><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><p><span>Continuing with this technique, we get the filter<\/span><\/p><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n278\" cid=\"n278\" mdtype=\"math_block\" data-math-tag-before=\"16\" data-math-tag-after=\"17\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" 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250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-17-TEX-N-37\" d=\"M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z\"><\/path><path id=\"MJX-17-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" 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xlink:href=\"#MJX-17-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4167,0)\"><use data-c=\"23A4\" xlink:href=\"#MJX-17-TEX-S4-23A4\" transform=\"translate(0,996)\"><\/use><use data-c=\"23A6\" xlink:href=\"#MJX-17-TEX-S4-23A6\" transform=\"translate(0,-1006)\"><\/use><svg width=\"667\" height=\"402\" y=\"49\" x=\"0\" viewBox=\"0 100.5 667 402\"><use data-c=\"23A5\" xlink:href=\"#MJX-17-TEX-S4-23A5\" transform=\"scale(1,1.002)\"><\/use><\/svg><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"1778 -2150 1 3800\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:17\"><g data-mml-node=\"mtext\"><use data-c=\"28\" xlink:href=\"#MJX-17-TEX-N-28\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-17-TEX-N-31\" transform=\"translate(389,0)\"><\/use><use data-c=\"37\" xlink:href=\"#MJX-17-TEX-N-37\" transform=\"translate(889,0)\"><\/use><use data-c=\"29\" xlink:href=\"#MJX-17-TEX-N-29\" transform=\"translate(1389,0)\"><\/use><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mtext>(17)<\/mtext><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><mi>G<\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mn>4<\/mn><\/mfrac><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>1<\/mn><\/mtd><mtd><mn>2<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>2<\/mn><\/mtd><mtd><mn>4<\/mn><\/mtd><mtd><mn>2<\/mn><\/mtd><\/mtr><mtr><mtd><mn>1<\/mn><\/mtd><mtd><mn>2<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='c22-generate-a-laplacian-pyramid-of-given-image-im-using-above-filter-with-\\n5\\n-levels-level-\\n1\\n-holds-the-finest-detail-at-full-resolution-\\n512\\n\u00d7\\n512\\n-while-level-\\n5\\n-corresponds-to-the-lowest-level-of-detail-\\n32\\n\u00d7\\n32\\n-the-scaling-factor-is-\\n2\\n\\n1\\n-horizontally-and-vertically'><span>C2.2. Generate a Laplacian pyramid of given image <\/span><code>im<\/code><span> using above filter with <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 500 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-153-TEX-N-35\" d=\"M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"35\" xlink:href=\"#MJX-153-TEX-N-35\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>5<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">5<\/script><span> levels. Level <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.507ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 500 666\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-151-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-151-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">1<\/script><span> holds the finest detail at full resolution <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.553ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 4222.4 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-152-TEX-N-35\" d=\"M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z\"><\/path><path id=\"MJX-152-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-152-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-152-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"35\" xlink:href=\"#MJX-152-TEX-N-35\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-152-TEX-N-31\" transform=\"translate(500,0)\"><\/use><use data-c=\"32\" xlink:href=\"#MJX-152-TEX-N-32\" transform=\"translate(1000,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1722.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-152-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2722.4,0)\"><use data-c=\"35\" xlink:href=\"#MJX-152-TEX-N-35\"><\/use><use data-c=\"31\" xlink:href=\"#MJX-152-TEX-N-31\" transform=\"translate(500,0)\"><\/use><use data-c=\"32\" xlink:href=\"#MJX-152-TEX-N-32\" transform=\"translate(1000,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>512<\/mn><mo>\u00d7<\/mo><mn>512<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">512\\times 512<\/script><span>, while level <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 500 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-153-TEX-N-35\" d=\"M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"35\" xlink:href=\"#MJX-153-TEX-N-35\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>5<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">5<\/script><span> corresponds to the lowest level of detail <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.291ex\" height=\"1.557ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 3222.4 688\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-154-TEX-N-33\" d=\"M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z\"><\/path><path id=\"MJX-154-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-154-TEX-N-D7\" d=\"M630 29Q630 9 609 9Q604 9 587 25T493 118L389 222L284 117Q178 13 175 11Q171 9 168 9Q160 9 154 15T147 29Q147 36 161 51T255 146L359 250L255 354Q174 435 161 449T147 471Q147 480 153 485T168 490Q173 490 175 489Q178 487 284 383L389 278L493 382Q570 459 587 475T609 491Q630 491 630 471Q630 464 620 453T522 355L418 250L522 145Q606 61 618 48T630 29Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"33\" xlink:href=\"#MJX-154-TEX-N-33\"><\/use><use data-c=\"32\" xlink:href=\"#MJX-154-TEX-N-32\" transform=\"translate(500,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1222.2,0)\"><use data-c=\"D7\" xlink:href=\"#MJX-154-TEX-N-D7\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2222.4,0)\"><use data-c=\"33\" xlink:href=\"#MJX-154-TEX-N-33\"><\/use><use data-c=\"32\" xlink:href=\"#MJX-154-TEX-N-32\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>32<\/mn><mo>\u00d7<\/mo><mn>32<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">32\\times 32<\/script><span>. The scaling factor is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.148ex\" height=\"1.507ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -666 1833.6 666\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-155-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-155-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-155-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"32\" xlink:href=\"#MJX-155-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(777.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-155-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1333.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-155-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>2<\/mn><mo>:<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">2:1<\/script><span> horizontally and vertically.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\" style=\"break-inside: unset;\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">NL<\/span>=<span class=\"cm-number\">5<\/span>;<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">lap_pyr<\/span>=<span class=\"cm-variable\">cell<\/span>(<span class=\"cm-number\">1<\/span>,<span class=\"cm-number\">5<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">tmp_im<\/span>=<span class=\"cm-variable\">im<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span cm-text=\"\" cm-zwsp=\"\">\n<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">level<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-variable\">NL<\/span><span class=\"cm-number\">-1<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-comment\">% filter &amp; downsample<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">imf<\/span>=<span class=\"cm-variable\">imfilter<\/span>(<span class=\"cm-variable\">tmp_im<\/span>,<span class=\"cm-variable\">F<\/span>,<span class=\"cm-string\">'replicate'<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">imf<\/span>=<span class=\"cm-variable\">imf<\/span>(<span class=\"cm-number\">1<\/span>:<span class=\"cm-number\">2<\/span>:<span class=\"cm-keyword\">end<\/span>,<span class=\"cm-number\">1<\/span>:<span class=\"cm-number\">2<\/span>:<span class=\"cm-keyword\">end<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-comment\">% upsample and filter<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">imr<\/span>=<span class=\"cm-builtin\">zeros<\/span>(<span class=\"cm-number\">2<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-builtin\">size<\/span>(<span class=\"cm-variable\">imf<\/span>));<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">imr<\/span>(<span class=\"cm-number\">1<\/span>:<span class=\"cm-number\">2<\/span>:<span class=\"cm-keyword\">end<\/span>,<span class=\"cm-number\">1<\/span>:<span class=\"cm-number\">2<\/span>:<span class=\"cm-keyword\">end<\/span>)=<span class=\"cm-variable\">imf<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">imr<\/span>=<span class=\"cm-variable\">imfilter<\/span>(<span class=\"cm-variable\">imr<\/span>,<span class=\"cm-variable\">G<\/span>,<span class=\"cm-string\">'replicate'<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-comment\">% difference image<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">lap_pyr<\/span>{<span class=\"cm-variable\">level<\/span>}=<span class=\"cm-variable\">tmp_im<\/span><span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">imr<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-comment\">% for next loop<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">tmp_im<\/span>=<span class=\"cm-variable\">imf<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">lap_pyr<\/span>{<span class=\"cm-variable\">level<\/span><span class=\"cm-number\">+1<\/span>}=<span class=\"cm-variable\">tmp_im<\/span>;<\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 432px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 432px;\"><\/div><\/div><\/div><\/pre><h3 id='c23-obtain-the-2d-filter-kernel-for-analysis-and-synthesis-steps-given-1d-lowpass-filter-for-9-7-cohen-daubechies-feauveau-wavelet'><span>C2.3. Obtain the 2D filter kernel for analysis and synthesis steps given 1D lowpass filter for 9-7 Cohen-Daubechies-Feauveau wavelet.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-builtin\">function<\/span> [<span class=\"cm-variable\">F<\/span>,<span class=\"cm-variable\">G<\/span>]=<span class=\"cm-variable\">compute_k<\/span>(<span class=\"cm-variable\">f1d<\/span>,<span class=\"cm-variable\">g1d<\/span>)<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">F<\/span>=<span class=\"cm-variable\">f1d<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">f1d<\/span><span class=\"cm-operator\">'<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">G<\/span>=<span class=\"cm-variable\">g1d<\/span><span class=\"cm-operator\">*<\/span><span class=\"cm-variable\">g1d<\/span><span class=\"cm-operator\">'<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 96px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 96px;\"><\/div><\/div><\/div><\/pre><h3 id='c24-calculate-the-variances-of-each-level-of-the-pyramid-to-neglect-border-effects-remove-a-border-of-\\n4\\n-pixels'><span>C2.4. Calculate the variances of each level of the pyramid. To neglect border effects, remove a border of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.532ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 500 677\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-157-TEX-N-34\" d=\"M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"34\" xlink:href=\"#MJX-157-TEX-N-34\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>4<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">4<\/script><span> pixels.<\/span><\/h3><pre class=\"md-fences md-end-block ty-contain-cm modeLoaded\" spellcheck=\"false\" lang=\"matlab\"><div class=\"CodeMirror cm-s-inner cm-s-null-scroll CodeMirror-wrap\" lang=\"matlab\"><div style=\"overflow: hidden; position: relative; width: 3px; height: 0px; top: 10px; left: 4px;\"><textarea autocorrect=\"off\" autocapitalize=\"off\" spellcheck=\"false\" tabindex=\"0\" style=\"position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;\"><\/textarea><\/div><div class=\"CodeMirror-scrollbar-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-gutter-filler\" cm-not-content=\"true\"><\/div><div class=\"CodeMirror-scroll\" tabindex=\"-1\"><div class=\"CodeMirror-sizer\" style=\"margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;\"><div style=\"position: relative; top: 0px;\"><div class=\"CodeMirror-lines\" role=\"presentation\"><div role=\"presentation\" style=\"position: relative; outline: none;\"><div class=\"CodeMirror-measure\"><pre><span>xxxxxxxxxx<\/span><\/pre><\/div><div class=\"CodeMirror-measure\"><\/div><div style=\"position: relative; z-index: 1;\"><\/div><div class=\"CodeMirror-code\" role=\"presentation\" style=\"\"><div class=\"CodeMirror-activeline\" style=\"position: relative;\"><div class=\"CodeMirror-activeline-background CodeMirror-linebackground\"><\/div><div class=\"CodeMirror-gutter-background CodeMirror-activeline-gutter\" style=\"left: 0px; width: 0px;\"><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-builtin\">function<\/span> <span class=\"cm-variable\">var_pyr<\/span>=<span class=\"cm-variable\">pyr_stat<\/span>(<span class=\"cm-variable\">pyr<\/span>)<\/span><\/pre><\/div><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">border<\/span>=<span class=\"cm-number\">4<\/span>;<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-variable\">var_pyr<\/span>=<span class=\"cm-builtin\">zeros<\/span>(<span class=\"cm-number\">5<\/span>,<span class=\"cm-number\">1<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">for<\/span> <span class=\"cm-variable\">i<\/span>=<span class=\"cm-number\">1<\/span>:<span class=\"cm-number\">5<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">tmp<\/span>=<span class=\"cm-variable\">pyr<\/span>{<span class=\"cm-variable\">i<\/span>};<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">tmp<\/span>=<span class=\"cm-variable\">tmp<\/span>(<span class=\"cm-variable\">border<\/span><span class=\"cm-number\">+1<\/span>;<span class=\"cm-keyword\">end<\/span><span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">border<\/span>,<span class=\"cm-variable\">border<\/span><span class=\"cm-number\">+1<\/span>:<span class=\"cm-keyword\">end<\/span><span class=\"cm-operator\">-<\/span><span class=\"cm-variable\">border<\/span>);<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"> &nbsp; &nbsp;<span class=\"cm-variable\">var_pyr<\/span>(<span class=\"cm-variable\">i<\/span>)=<span class=\"cm-variable\">var<\/span>(<span class=\"cm-variable\">tmp<\/span>(:));<\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><pre class=\" CodeMirror-line \" role=\"presentation\"><span role=\"presentation\" style=\"padding-right: 0.1px;\"><span class=\"cm-keyword\">end<\/span><\/span><\/pre><\/div><\/div><\/div><\/div><\/div><div style=\"position: absolute; height: 0px; width: 1px; border-bottom-width: 0px; border-bottom-style: solid; border-bottom-color: transparent; top: 216px;\"><\/div><div class=\"CodeMirror-gutters\" style=\"display: none; height: 216px;\"><\/div><\/div><\/div><\/pre><h3 id='a21-explain-the-difference-you-observe-for-varpyr-when-using-9-7-cohen-daubechies-feauveau-wavelet-and-gaussian-filters'><span>A2.1. Explain the difference you observe for <\/span><code>var_pyr<\/code><span> when using 9-7 Cohen-Daubechies-Feauveau wavelet and Gaussian filters.<\/span><\/h3><ul><li><p><span>In both cases, the variance\/energy of levels 1-4 are lower than the last level.<\/span><\/p><\/li><li><p><span>With CDF filters, a little more energy is packed into level 5.<\/span><\/p><\/li><\/ul><h3 id='c25-resonstruct-image-from-lappyr-calculate-the-psnr-neglect-a-border-of-\\n4\\n-pixels'><span>C2.5. Resonstruct image from <\/span><code>lap_pyr<\/code><span>. Calculate the PSNR, neglect a border of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.131ex\" height=\"1.532ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 500 677\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-157-TEX-N-34\" d=\"M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"34\" xlink:href=\"#MJX-157-TEX-N-34\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>4<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">4<\/script><span> pixels.<\/span><\/h3><\/div><\/div>\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>HW1.1 Image Manipulation HW1.1 Image Manipulation A1.1.<a class=\"more-link\" href=\"https:\/\/blog.mhrooz.xyz\/index.php\/2023\/07\/18\/dsp_matlab_lian_xi\/\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">&#8220;DSP MATLAB\u7ec3\u4e60&#8221;<\/span><\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[49],"tags":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2833"}],"collection":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/comments?post=2833"}],"version-history":[{"count":7,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2833\/revisions"}],"predecessor-version":[{"id":2840,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2833\/revisions\/2840"}],"wp:attachment":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/media?parent=2833"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/categories?post=2833"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/tags?post=2833"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}