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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Spatial frequency</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 12 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-12" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">12 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%B1%D8%AF%D8%AF_%D9%81%D8%B1%D8%A7%D8%BA%D9%8A" title="تردد فراغي – Arabic" lang="ar" hreflang="ar" data-title="تردد فراغي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Prostorna_frekvencija" title="Prostorna frekvencija – Bosnian" lang="bs" hreflang="bs" data-title="Prostorna frekvencija" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Freq%C3%BC%C3%A8ncia_espacial" title="Freqüència espacial – Catalan" lang="ca" hreflang="ca" data-title="Freqüència espacial" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ortsfrequenz" title="Ortsfrequenz – German" lang="de" hreflang="de" data-title="Ortsfrequenz" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A8%D8%B3%D8%A7%D9%85%D8%AF_%D9%81%D8%B6%D8%A7%DB%8C%DB%8C" title="بسامد فضایی – Persian" lang="fa" hreflang="fa" data-title="بسامد فضایی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Fr%C3%A9quence_spatiale" title="Fréquence spatiale – French" lang="fr" hreflang="fr" data-title="Fréquence spatiale" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80%D0%BD%D0%B0_%D1%87%D0%B5%D1%81%D1%82%D0%BE%D1%82%D0%B0" title="Просторна честота – Macedonian" lang="mk" hreflang="mk" data-title="Просторна честота" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%A9%BA%E9%96%93%E5%91%A8%E6%B3%A2%E6%95%B0" title="空間周波数 – Japanese" lang="ja" hreflang="ja" data-title="空間周波数" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Frekuenca_hap%C3%ABsinore" title="Frekuenca hapësinore – Albanian" lang="sq" hreflang="sq" data-title="Frekuenca hapësinore" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Prostorna_u%C4%8Destanost" title="Prostorna učestanost – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Prostorna učestanost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%96%E0%B8%B5%E0%B9%88%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%9E%E0%B8%B7%E0%B9%89%E0%B8%99%E0%B8%97%E0%B8%B5%E0%B9%88" title="ความถี่เชิงพื้นที่ – Thai" lang="th" hreflang="th" data-title="ความถี่เชิงพื้นที่" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%A9%BA%E9%96%93%E9%A0%BB%E7%8E%87" title="空間頻率 – Chinese" lang="zh" hreflang="zh" data-title="空間頻率" data-language-autonym="中文" data-language-local-name="Chinese" 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metre</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Characteristic of any structure that is periodic across a position in space</div> <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:186px;max-width:186px"><div class="trow"><div class="tsingle" style="width:184px;max-width:184px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:160_by_160_thumbnail_of_%27Green_Sea_Shell%27.png" class="mw-file-description"><img alt="Green Sea Shell image" src="//upload.wikimedia.org/wikipedia/commons/a/a6/160_by_160_thumbnail_of_%27Green_Sea_Shell%27.png" decoding="async" width="182" height="182" class="mw-file-element" data-file-width="160" data-file-height="160" /></a></span></div><div class="thumbcaption">Green Sea Shell image</div></div></div><div class="trow"><div class="tsingle" style="width:184px;max-width:184px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:160_by_160_thumbnail_of_%27Green_Sea_Shell%27_-_0._in_fourier_domain_all_components_(RGB).png" class="mw-file-description"><img alt="Spatial frequency representation of the Green Sea Shell image" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/160_by_160_thumbnail_of_%27Green_Sea_Shell%27_-_0._in_fourier_domain_all_components_%28RGB%29.png/250px-160_by_160_thumbnail_of_%27Green_Sea_Shell%27_-_0._in_fourier_domain_all_components_%28RGB%29.png" decoding="async" width="182" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/160_by_160_thumbnail_of_%27Green_Sea_Shell%27_-_0._in_fourier_domain_all_components_%28RGB%29.png/330px-160_by_160_thumbnail_of_%27Green_Sea_Shell%27_-_0._in_fourier_domain_all_components_%28RGB%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/160_by_160_thumbnail_of_%27Green_Sea_Shell%27_-_0._in_fourier_domain_all_components_%28RGB%29.png/364px-160_by_160_thumbnail_of_%27Green_Sea_Shell%27_-_0._in_fourier_domain_all_components_%28RGB%29.png 2x" data-file-width="435" data-file-height="342" /></a></span></div><div class="thumbcaption">Spatial frequency representation of the Green Sea Shell image</div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Image and its spatial frequencies: Magnitude of frequency domain is logarithmically scaled, and zero frequency is in the center. Notable is the clustering of the content on the lower frequencies, a typical property of natural images.</div></div></div></div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Physics" title="Physics">physics</a>, and <a href="/wiki/Engineering" title="Engineering">engineering</a>, <b>spatial frequency</b> is a characteristic of any structure that is <a href="/wiki/Periodic_function" title="Periodic function">periodic</a> across position in <a href="/wiki/Space" title="Space">space</a>. The spatial frequency is a measure of how often <a href="/wiki/Sine_wave" title="Sine wave">sinusoidal components</a> (as determined by the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>) of the structure repeat per unit of distance. </p><p>The <a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a> of spatial frequency is the <a href="/wiki/Reciprocal_metre" class="mw-redirect" title="Reciprocal metre">reciprocal metre</a> (m<sup>−1</sup>),<sup id="cite_ref-ISO80000-3_2019_1-0" class="reference"><a href="#cite_note-ISO80000-3_2019-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> although <a href="/wiki/Cycle_(rotational_unit)" class="mw-redirect" title="Cycle (rotational unit)">cycles</a> per <a href="/wiki/Meter" class="mw-redirect" title="Meter">meter</a> (c/m) is also common. In <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image-processing</a> applications, spatial frequency is often expressed in units of cycles per <a href="/wiki/Millimeter" class="mw-redirect" title="Millimeter">millimeter</a> (c/mm) or also <a href="/wiki/Image_line_pairs" class="mw-redirect" title="Image line pairs">line pairs</a> per millimeter (LP/mm). </p><p>In <a href="/wiki/Wave_propagation" class="mw-redirect" title="Wave propagation">wave propagation</a>, the spatial frequency is also known as <i><a href="/wiki/Wavenumber" title="Wavenumber">wavenumber</a></i>. Ordinary wavenumber is defined as the reciprocal of <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" /></span> and is commonly denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b461aaf61091abd5d2c808931c48b8ff9647db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.03ex; height:2.509ex;" alt="{\displaystyle \xi }" /></span><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> or sometimes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" /></span>:<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi ={\frac {1}{\lambda }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>λ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi ={\frac {1}{\lambda }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36e0679bb3c874911eb295d3c22443a9c8fe6c42" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.967ex; height:5.343ex;" alt="{\displaystyle \xi ={\frac {1}{\lambda }}.}" /></span> Angular wavenumber <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span>, expressed in <a href="/wiki/Radian" title="Radian">radian</a> per metre (rad/m), is related to ordinary wavenumber and wavelength by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2\pi \xi ={\frac {2\pi }{\lambda }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>ξ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2\pi \xi ={\frac {2\pi }{\lambda }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8824ab0d1e3ae4d7b624a3a49118914f3835f1fb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.91ex; height:5.343ex;" alt="{\displaystyle k=2\pi \xi ={\frac {2\pi }{\lambda }}.}" /></span> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Visual_perception">Visual perception</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_frequency&action=edit&section=1" title="Edit section: Visual perception"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the study of <a href="/wiki/Visual_perception" title="Visual perception">visual perception</a>, <a href="/wiki/Sine_wave" title="Sine wave">sinusoidal</a> <a href="/wiki/Grating" title="Grating">gratings</a> are frequently used to probe the capabilities of the <a href="/wiki/Visual_system" title="Visual system">visual system</a>, such as <a href="/wiki/Contrast_sensitivity" class="mw-redirect" title="Contrast sensitivity">contrast sensitivity</a>. In these <a href="/wiki/Stimulus_(physiology)" title="Stimulus (physiology)">stimuli</a>, spatial frequency is expressed as the number of cycles per <a href="/wiki/Degree_(angle)" title="Degree (angle)">degree</a> of <a href="/wiki/Visual_angle" title="Visual angle">visual angle</a>. Sine-wave gratings also differ from one another in amplitude (the magnitude of difference in intensity between light and dark stripes), <a href="/wiki/Orientation_(geometry)" title="Orientation (geometry)">orientation</a>, and <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Spatial-frequency_theory">Spatial-frequency theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_frequency&action=edit&section=2" title="Edit section: Spatial-frequency theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The spatial-frequency theory refers to the theory that the <a href="/wiki/Visual_cortex" title="Visual cortex">visual cortex</a> operates on a code of spatial frequency, not on the code of straight edges and lines hypothesised by Hubel and Wiesel on the basis of early experiments on <a href="/wiki/Visual_cortex#Primary_visual_cortex_(V1)" title="Visual cortex">V1 neurons</a> in the cat.<sup id="cite_ref-pmc2556291_4-0" class="reference"><a href="#cite_note-pmc2556291-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In support of this theory is the experimental observation that the visual cortex neurons respond even more robustly to sine-wave gratings that are placed at specific angles in their <a href="/wiki/Receptive_field#Visual_system" title="Receptive field">receptive fields</a> than they do to edges or bars. Most neurons in the primary visual cortex respond best when a sine-wave grating of a particular frequency is presented at a particular angle in a particular location in the visual field.<sup id="cite_ref-pmc2412904_6-0" class="reference"><a href="#cite_note-pmc2412904-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> (However, as noted by Teller (1984),<sup id="cite_ref-pmid6395480_7-0" class="reference"><a href="#cite_note-pmid6395480-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> it is probably not wise to treat the highest firing rate of a particular neuron as having a special significance with respect to its role in the perception of a particular stimulus, given that the neural code is known to be linked to relative firing rates. For example, in color coding by the three cones in the human retina, there is no special significance to the cone that is firing most strongly – what matters is the relative rate of firing of all three simultaneously. Teller (1984) similarly noted that a strong firing rate in response to a particular stimulus should not be interpreted as indicating that the neuron is somehow specialized for that stimulus, since there is an unlimited equivalence class of stimuli capable of producing similar firing rates.) </p><p>The spatial-frequency theory of vision is based on two physical principles: </p> <ul><li>Any visual stimulus can be represented by plotting the intensity of the light along lines running through it.</li> <li>Any curve can be broken down into constituent sine waves by <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>.</li></ul> <p>The theory (for which empirical support has yet to be developed) states that in each functional module of the visual cortex, Fourier analysis (or its piecewise form <sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>) is performed on the receptive field and the neurons in each module are thought to respond selectively to various orientations and frequencies of sine wave gratings.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> When all of the visual cortex neurons that are influenced by a specific scene respond together, the perception of the scene is created by the summation of the various sine-wave gratings. (This procedure, however, does not address the problem of the organization of the products of the summation into figures, grounds, and so on. It effectively recovers the original (pre-Fourier analysis) distribution of photon intensity and wavelengths across the retinal projection, but does not add information to this original distribution. So the functional value of such a hypothesized procedure is unclear. Some other objections to the "Fourier theory" are discussed by Westheimer (2001) <sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>). One is generally not aware of the individual spatial frequency components since all of the elements are essentially blended together into one smooth representation. However, computer-based filtering procedures can be used to deconstruct an image into its individual spatial frequency components.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Research on spatial frequency detection by visual neurons complements and extends previous research using straight edges rather than refuting it.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Further research shows that different spatial frequencies convey different information about the appearance of a stimulus. High spatial frequencies represent abrupt spatial changes in the image, such as edges, and generally correspond to featural information and fine detail. M. Bar (2004) has proposed that low spatial frequencies represent global information about the shape, such as general orientation and proportions.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Rapid and specialised perception of faces is known to rely more on low spatial frequency information.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> In the general population of adults, the threshold for spatial frequency discrimination is about 7%. It is often poorer in dyslexic individuals.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Spatial_frequency_in_MRI">Spatial frequency in MRI</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_frequency&action=edit&section=3" title="Edit section: Spatial frequency in MRI"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/K-space_in_magnetic_resonance_imaging" title="K-space in magnetic resonance imaging"><i>k</i>-space in magnetic resonance imaging</a></div> <p>When spatial frequency is used as a variable in a mathematical function, the function is said to be in <a href="/wiki/K-space_(magnetic_resonance_imaging)" class="mw-redirect" title="K-space (magnetic resonance imaging)"><i>k-space</i></a>. Two dimensional k-space has been introduced into <a href="/wiki/MRI" class="mw-redirect" title="MRI">MRI</a> as a raw data storage space. The value of each data point in k-space is measured in the unit of 1/meter, i.e. the unit of spatial frequency. </p><p>It is very common that the raw data in k-space shows features of periodic functions. The periodicity is not spatial frequency, but is temporal frequency. An MRI raw data matrix is composed of a series of phase-variable spin-echo signals. Each of the spin-echo signal is a <a href="/wiki/Sinc_function" title="Sinc function">sinc function</a> of time, which can be described by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spin-Echo}}={\frac {M_{\mathrm {0} }\sin \omega _{\mathrm {r} }t}{\omega _{\mathrm {r} }t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spin-Echo</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mi>t</mi> </mrow> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Spin-Echo}}={\frac {M_{\mathrm {0} }\sin \omega _{\mathrm {r} }t}{\omega _{\mathrm {r} }t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/139d438679f02b69fc401556d7ddf3e306ab53cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:24.404ex; height:5.676ex;" alt="{\displaystyle {\text{Spin-Echo}}={\frac {M_{\mathrm {0} }\sin \omega _{\mathrm {r} }t}{\omega _{\mathrm {r} }t}}}" /></span> Where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\mathrm {r} }=\omega _{\mathrm {0} }+{\bar {\gamma }}rG}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>r</mi> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {r} }=\omega _{\mathrm {0} }+{\bar {\gamma }}rG}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c94e571fe9f0ff7f06752940ff7422ea58bc3aa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.899ex; height:2.676ex;" alt="{\displaystyle \omega _{\mathrm {r} }=\omega _{\mathrm {0} }+{\bar {\gamma }}rG}" /></span> Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\gamma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\gamma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3c5e5512da2917274fb6d82579ec44edcb7e18d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.509ex;" alt="{\displaystyle {\bar {\gamma }}}" /></span> is the gyromagnetic ratio constant, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\mathrm {0} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {0} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec6031e2185b5bf1c2424ae634637bd9062f5b76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{\mathrm {0} }}" /></span> is the basic resonance frequency of the spin. Due to the presence of the gradient <i>G</i>, the spatial information <b>r</b> is encoded onto the frequency <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" /></span>. The periodicity seen in the MRI raw data is just this frequency <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\mathrm {r} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {r} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e702f835f387b8b280c57c927744e22d7b8d4b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.323ex; height:2.009ex;" alt="{\displaystyle \omega _{\mathrm {r} }}" /></span>, which is basically the temporal frequency in nature. </p><p>In a rotating frame, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\mathrm {0} }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {0} }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e0084c0ea7e8c2a11d4b670e1c36d1352327500" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.761ex; height:2.509ex;" alt="{\displaystyle \omega _{\mathrm {0} }=0}" /></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\mathrm {r} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {r} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e702f835f387b8b280c57c927744e22d7b8d4b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.323ex; height:2.009ex;" alt="{\displaystyle \omega _{\mathrm {r} }}" /></span> is simplified to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\gamma }}rG}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>r</mi> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\gamma }}rG}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d379fd44bae72c0a70f20ed88a2dd295f1ee9d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.138ex; height:2.676ex;" alt="{\displaystyle {\bar {\gamma }}rG}" /></span>. Just by letting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\bar {\gamma }}Gt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mi>G</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\bar {\gamma }}Gt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd105666937eee1dc54b16861f2b16e89090799c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.238ex; height:2.676ex;" alt="{\displaystyle k={\bar {\gamma }}Gt}" /></span>, the spin-echo signal is expressed in an alternative form <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spin-Echo}}={\frac {M_{\mathrm {0} }\sin rk}{rk}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spin-Echo</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>r</mi> <mi>k</mi> </mrow> <mrow> <mi>r</mi> <mi>k</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Spin-Echo}}={\frac {M_{\mathrm {0} }\sin rk}{rk}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a57fddb5a94ddfe0a10e9aa9c72bc0aad34f68af" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.502ex; height:5.509ex;" alt="{\displaystyle {\text{Spin-Echo}}={\frac {M_{\mathrm {0} }\sin rk}{rk}}}" /></span> </p><p>Now, the spin-echo signal is in the k-space. It becomes a periodic function of <i>k</i> with <i>r</i> as the <i>k-space frequency</i> but not as the "spatial frequency", since "spatial frequency" is reserved for the name of the periodicity seen in the real space r. </p><p>The k-space domain and the space domain form a Fourier pair. Two pieces of information are found in each domain, the spatial information and the spatial frequency information. The spatial information, which is of great interest to all medical doctors, is seen as periodic functions in the k-space domain and is seen as the image in the space domain. The spatial frequency information, which might be of interest to some MRI engineers, is not easily seen in the space domain but is readily seen as the data points in the k-space domain. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_frequency&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a></li> <li><a href="/wiki/Superlens" title="Superlens">Superlens</a></li> <li><a href="/wiki/Visual_perception" title="Visual perception">Visual perception</a></li> <li><a href="/wiki/Fringe_visibility" class="mw-redirect" title="Fringe visibility">Fringe visibility</a></li> <li><a href="/wiki/Reciprocal_space" class="mw-redirect" title="Reciprocal space">Reciprocal space</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_frequency&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-ISO80000-3_2019-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-ISO80000-3_2019_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.iso.org/standard/64974.html">"ISO 80000-3:2019 Quantities and units — Part 3: Space and time"</a> (2 ed.). <a href="/wiki/International_Organization_for_Standardization" title="International Organization for Standardization">International Organization for Standardization</a>. 2019<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-10-23</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=ISO+80000-3%3A2019+Quantities+and+units+%E2%80%94+Part+3%3A+Space+and+time&rft.edition=2&rft.pub=International+Organization+for+Standardization&rft.date=2019&rft_id=https%3A%2F%2Fwww.iso.org%2Fstandard%2F64974.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpatial+frequency" class="Z3988"></span> <a rel="nofollow" class="external autonumber" href="https://www.iso.org/obp/ui/#iso:std:iso:80000:-3:ed-2:v1:en">[1]</a> (11 pages)</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">SPIE Optipedia article: <a rel="nofollow" class="external text" href="http://spie.org/x34301.xml">"Spatial Frequency"</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">The symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" /></span> is also used to represent temporal <a href="/wiki/Frequency" title="Frequency">frequency</a>, as in, e.g., <a href="/wiki/Planck_constant" title="Planck constant">Planck's formula</a>.</span> </li> <li id="cite_note-pmc2556291-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-pmc2556291_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMartinezAlonso2003" class="citation journal cs1">Martinez LM, Alonso JM (2003). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2556291">"Complex receptive fields in primary visual cortex"</a>. <i>Neuroscientist</i>. <b>9</b> (5): <span class="nowrap">317–</span>31. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1177%2F1073858403252732">10.1177/1073858403252732</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2556291">2556291</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/14580117">14580117</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Neuroscientist&rft.atitle=Complex+receptive+fields+in+primary+visual+cortex&rft.volume=9&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E317-%3C%2Fspan%3E31&rft.date=2003&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2556291%23id-name%3DPMC&rft_id=info%3Apmid%2F14580117&rft_id=info%3Adoi%2F10.1177%2F1073858403252732&rft.aulast=Martinez&rft.aufirst=LM&rft.au=Alonso%2C+JM&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC2556291&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpatial+frequency" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDe_ValoisDe_Valois1988" class="citation book cs1">De Valois, R. L.; De Valois, K. K. (1988). <i>Spatial vision</i>. 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D. (1995). Vision and mind: Modeling mental functions. Lawrence Erlbaum Associates, Inc. <a rel="nofollow" class="external free" href="https://doi.org/10.4324/9780203773932">https://doi.org/10.4324/9780203773932</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBarghout2014" class="citation book cs1">Barghout, Lauren (2014). <a rel="nofollow" class="external text" href="https://www.morebooks.de/store/gb/book/vision/isbn/978-3-639-70962-9"><i>Vision: How Global Perceptual Context Changes Local Contrast Processing (Ph.D. Dissertation 2003). Updated for Computer Vision Techniques</i></a>. Scholars' Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-639-70962-9" title="Special:BookSources/978-3-639-70962-9"><bdi>978-3-639-70962-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vision%3A+How+Global+Perceptual+Context+Changes+Local+Contrast+Processing+%28Ph.D.+Dissertation+2003%29.+Updated+for+Computer+Vision+Techniques&rft.pub=Scholars%27+Press&rft.date=2014&rft.isbn=978-3-639-70962-9&rft.aulast=Barghout&rft.aufirst=Lauren&rft_id=https%3A%2F%2Fwww.morebooks.de%2Fstore%2Fgb%2Fbook%2Fvision%2Fisbn%2F978-3-639-70962-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpatial+frequency" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://journals.sagepub.com/doi/abs/10.1068/p3193?url_ver=Z39.88-2003&rfr_id=ori:rid:crossref.org&rfr_dat=cr_pub%3dpubmed">Westheimer, G. "The Fourier Theory of Vision"</a></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Blake, R. and Sekuler, R., <i>Perception</i>, 3rd ed. Chapter 3. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-072-88760-0" title="Special:BookSources/978-0-072-88760-0">978-0-072-88760-0</a></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Pinel, J. P. J., <i>Biopsychology</i>, 6th ed. 293–294. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-205-42651-4" title="Special:BookSources/0-205-42651-4">0-205-42651-4</a></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBar_M2004" class="citation journal cs1">Bar M (Aug 2004). "Visual objects in context". <i>Nat. Rev. Neurosci</i>. <b>5</b> (8): <span class="nowrap">617–</span>29. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnrn1476">10.1038/nrn1476</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/15263892">15263892</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:205499985">205499985</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nat.+Rev.+Neurosci.&rft.atitle=Visual+objects+in+context&rft.volume=5&rft.issue=8&rft.pages=%3Cspan+class%3D%22nowrap%22%3E617-%3C%2Fspan%3E29&rft.date=2004-08&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A205499985%23id-name%3DS2CID&rft_id=info%3Apmid%2F15263892&rft_id=info%3Adoi%2F10.1038%2Fnrn1476&rft.au=Bar+M&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpatial+frequency" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAwasthiFriedmanWilliams2011" class="citation journal cs1">Awasthi B, Friedman J, Williams MA (2011). "Faster, stronger, lateralized: Low spatial frequency information supports face processing". <i>Neuropsychologia</i>. <b>49</b> (13): <span class="nowrap">3583–</span>3590. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.neuropsychologia.2011.08.027">10.1016/j.neuropsychologia.2011.08.027</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21939676">21939676</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10037045">10037045</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Neuropsychologia&rft.atitle=Faster%2C+stronger%2C+lateralized%3A+Low+spatial+frequency+information+supports+face+processing&rft.volume=49&rft.issue=13&rft.pages=%3Cspan+class%3D%22nowrap%22%3E3583-%3C%2Fspan%3E3590&rft.date=2011&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10037045%23id-name%3DS2CID&rft_id=info%3Apmid%2F21939676&rft_id=info%3Adoi%2F10.1016%2Fj.neuropsychologia.2011.08.027&rft.aulast=Awasthi&rft.aufirst=B&rft.au=Friedman%2C+J&rft.au=Williams%2C+MA&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpatial+frequency" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBen-YehudahAhissar2004" class="citation journal cs1">Ben-Yehudah G, Ahissar M (May 2004). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.visres.2003.12.001">"Sequential spatial frequency discrimination is consistently impaired among adult dyslexics"</a>. <i>Vision Res</i>. <b>44</b> (10): <span class="nowrap">1047–</span>63. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.visres.2003.12.001">10.1016/j.visres.2003.12.001</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/15031099">15031099</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:12605281">12605281</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Vision+Res.&rft.atitle=Sequential+spatial+frequency+discrimination+is+consistently+impaired+among+adult+dyslexics&rft.volume=44&rft.issue=10&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1047-%3C%2Fspan%3E63&rft.date=2004-05&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A12605281%23id-name%3DS2CID&rft_id=info%3Apmid%2F15031099&rft_id=info%3Adoi%2F10.1016%2Fj.visres.2003.12.001&rft.aulast=Ben-Yehudah&rft.aufirst=G&rft.au=Ahissar%2C+M&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.visres.2003.12.001&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpatial+frequency" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spatial_frequency&action=edit&section=6" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.haberdar.org/Spatial-Frequency-of-an-Image-Tutorial.htm">"Tutorial: Spatial Frequency of an Image"</a>. Hakan Haberdar, University of Houston<span class="reference-accessdate">. Retrieved <span class="nowrap">22 March</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Tutorial%3A+Spatial+Frequency+of+an+Image&rft.pub=Hakan+Haberdar%2C+University+of+Houston&rft_id=http%3A%2F%2Fwww.haberdar.org%2FSpatial-Frequency-of-an-Image-Tutorial.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpatial+frequency" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKalloniatisLuu2007" class="citation web cs1">Kalloniatis, Michael; Luu, Charles (2007). <a rel="nofollow" class="external text" href="http://webvision.med.utah.edu/KallSpatial.html#contrast">"Webvision: Part IX Psychophysics of Vision. 2 Visual Acuity, Contrast Sensitivity"</a>. University of Utah<span class="reference-accessdate">. Retrieved <span class="nowrap">19 July</span> 2009</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Webvision%3A+Part+IX+Psychophysics+of+Vision.+2+Visual+Acuity%2C+Contrast+Sensitivity&rft.pub=University+of+Utah&rft.date=2007&rft.aulast=Kalloniatis&rft.aufirst=Michael&rft.au=Luu%2C+Charles&rft_id=http%3A%2F%2Fwebvision.med.utah.edu%2FKallSpatial.html%23contrast&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpatial+frequency" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Spatial_frequency&oldid=1282285872">https://en.wikipedia.org/w/index.php?title=Spatial_frequency&oldid=1282285872</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Mathematical_physics" title="Category:Mathematical physics">Mathematical physics</a></li><li><a href="/wiki/Category:Space" title="Category:Space">Space</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 25 March 2025, at 14:20<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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