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| <html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Convolution, correlation, and Fourier transform · LocalFilters.jl Package</title><meta name="title" content="Convolution, correlation, and Fourier transform · LocalFilters.jl Package"/><meta property="og:title" content="Convolution, correlation, and Fourier transform · LocalFilters.jl Package"/><meta property="twitter:title" content="Convolution, correlation, and Fourier transform · LocalFilters.jl Package"/><meta name="description" content="Documentation for LocalFilters.jl Package."/><meta property="og:description" content="Documentation for LocalFilters.jl Package."/><meta property="twitter:description" content="Documentation for LocalFilters.jl Package."/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link 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data-theme-name="catppuccin-mocha"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">LocalFilters.jl Package</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Introduction</a></li><li><a class="tocitem" href="../generic/">Generic local filters</a></li><li><a class="tocitem" href="../neighborhoods/">Neighborhoods, structuring elements, and kernels</a></li><li><a class="tocitem" href="../linear/">Linear filters</a></li><li><a class="tocitem" href="../nonlinear/">Non-linear filters</a></li><li><a class="tocitem" href="../morphology/">Non-linear morphological filters</a></li><li><a class="tocitem" href="../separable/">Efficient separable filters for associative operations</a></li><li><a class="tocitem" href="../reference/">Reference</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Convolution, correlation, and Fourier transform</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Convolution, correlation, and Fourier transform</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/emmt/LocalFilters.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/emmt/LocalFilters.jl/blob/master/docs/src/math.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Convolution,-correlation,-and-Fourier-transform"><a class="docs-heading-anchor" href="#Convolution,-correlation,-and-Fourier-transform">Convolution, correlation, and Fourier transform</a><a id="Convolution,-correlation,-and-Fourier-transform-1"></a><a class="docs-heading-anchor-permalink" href="#Convolution,-correlation,-and-Fourier-transform" title="Permalink"></a></h1><h2 id="Fourier-transform"><a class="docs-heading-anchor" href="#Fourier-transform">Fourier transform</a><a id="Fourier-transform-1"></a><a class="docs-heading-anchor-permalink" href="#Fourier-transform" title="Permalink"></a></h2><p>The continuous Fourier transform of <span>$a(x)$</span> is defined by:</p><p class="math-container">\[\hat{a}(u) = \int a(x)\,\mathrm{e}^{-\mathrm{i}\,2\,\pi\,u\,x}\,\mathrm{d}x.\]</p><p>The inverse Fourier transform of <span>$\hat{a}(u)$</span> then writes:</p><p class="math-container">\[a(x) = \int \hat{a}(u)\,\mathrm{e}^{+\mathrm{i}\,2\,\pi\,u\,x}\,\mathrm{d}u.\]</p><h2 id="Convolution-product"><a class="docs-heading-anchor" href="#Convolution-product">Convolution product</a><a id="Convolution-product-1"></a><a class="docs-heading-anchor-permalink" href="#Convolution-product" title="Permalink"></a></h2><p>The convolution product of <span>$a(x)$</span> by <span>$b(x)$</span> is defined by:</p><p class="math-container">\[c(x) = \mathrm{Conv}(a,b)(x) | ||
| = \int a(y)\,b(x - y)\,\mathrm{d}y | ||
| = \int b(z)\,a(x - z)\,\mathrm{d}z,\]</p><p>with <span>$z = x - y$</span>. This also shows that the convolution product is symmetrical:</p><p class="math-container">\[\mathrm{Conv}(b,a) = \mathrm{Conv}(a,b).\]</p><p>Taking <span>$z = x - y$</span>, the Fourier transform of the convolution product can be expanded as follows:</p><p class="math-container">\[\begin{align*} | ||
| \hat{c}(u) &= \int c(x)\,\mathrm{e}^{-\mathrm{i}\,2\,\pi\,u\,x}\,\mathrm{d}x\\ | ||
| &= \iint a(y)\,b(x - y)\,\mathrm{e}^{-\mathrm{i}\,2\,\pi\,u\,x}\,\mathrm{d}x\,\mathrm{d}y\\ | ||
| &= \iint a(y)\,b(z)\,\mathrm{e}^{-\mathrm{i}\,2\,\pi\,u\,(y + z)}\,\mathrm{d}y\,\mathrm{d}z\\ | ||
| &= \int a(y)\,\mathrm{e}^{-\mathrm{i}\,2\,\pi\,u\,y}\,\mathrm{d}y | ||
| \int b(z)\,\mathrm{e}^{-\mathrm{i}\,2\,\pi\,u\,z}\,\mathrm{d}z\\ | ||
| &= \hat{a}(u)\,\hat{b}(u). | ||
| \end{align*}\]</p><h2 id="Correlation-product"><a class="docs-heading-anchor" href="#Correlation-product">Correlation product</a><a id="Correlation-product-1"></a><a class="docs-heading-anchor-permalink" href="#Correlation-product" title="Permalink"></a></h2><p>The correlation product of <span>$a(x)$</span> by <span>$b(x)$</span> is defined by:</p><p class="math-container">\[r(x) = \mathrm{Corr}(a,b)(x) | ||
| = \int a(x + y)\,{b}^\star(y)\,\mathrm{d}y | ||
| = \int {b}^\star(z - x)\,a(z)\,\mathrm{d}z,\]</p><p>where <span>${b}^\star(y)$</span> denotes the complex conjugate of <span>$b(y)$</span> and with <span>$z = x + y$</span>. From this follows that:</p><p class="math-container">\[\mathrm{Corr}(b,a)(x) = {\mathrm{Corr}(a,b)}^\star(-x).\]</p><p>Taking <span>$z = x + y$</span>, the Fourier transform of the correlation product can be expanded as follows:</p><p class="math-container">\[\begin{align*} | ||
| \hat{r}(u) &= \int r(x)\,\mathrm{e}^{-\mathrm{i}\,2\,\pi\,u\,x}\,\mathrm{d}x\\ | ||
| &= \iint a(x + y)\,{b}^\star(y)\,\mathrm{e}^{-\mathrm{i}\,2\,\pi\,u\,x}\,\mathrm{d}x\,\mathrm{d}y\\ | ||
| &= \iint a(z)\,b^\star(y)\,\mathrm{e}^{-\mathrm{i}\,2\,\pi\,u\,(z - y)}\,\mathrm{d}y\,\mathrm{d}z\\ | ||
| &= \int a(z)\,\mathrm{e}^{-\mathrm{i}\,2\,\pi\,u\,z}\,\mathrm{d}z | ||
| \left(\int b(y)\,\mathrm{e}^{-\mathrm{i}\,2\,\pi\,u\,y}\,\mathrm{d}y\right)^\star\\ | ||
| &= \hat{a}(u)\,{\hat{b}}^\star(u). | ||
| \end{align*}\]</p><h2 id="Discrete-convolution-and-correlation"><a class="docs-heading-anchor" href="#Discrete-convolution-and-correlation">Discrete convolution and correlation</a><a id="Discrete-convolution-and-correlation-1"></a><a class="docs-heading-anchor-permalink" href="#Discrete-convolution-and-correlation" title="Permalink"></a></h2><p>Following the continuous definition, the discrete convolution of <span>$a$</span> by <span>$b$</span> is given by:</p><p class="math-container">\[c[i] = \sum_j a[j]\,b[i - j] = \sum_k b[k]\,a[i - k],\]</p><p>with <span>$k = i - j$</span> and where the sums are taken for all possible valid indices.</p><p>Similarly, following the continuous definition, the discrete correlation of <span>$a$</span> by <span>$b$</span> is given by:</p><p class="math-container">\[r[i] = \sum_k a[i + k]\,{b}^\star[k] = \sum_k {b}^\star[j - i]\,a[j],\]</p><p>with <span>$j = i + k$</span> and where the sums are taken for all possible valid indices.</p></article><nav class="docs-footer"><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Friday 15 November 2024 16:53">Friday 15 November 2024</span>. Using Julia version 1.6.7.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html> |
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