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detection_corners.html
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<!DOCTYPE html>
<html lang="en">
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<meta charset="UTF-8" />
<title>Detection » Corner detectors module | DIPlib | a library for quantitative image analysis</title>
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<a href="https://diplib.org"><img src="DIPlib_logo.svg" alt="" />DIPlib</a><span class="m-breadcrumb">┃</span><a href="index.html" class="m-thin">a library for quantitative image analysis</a><span class="m-breadcrumb">┃</span><a href="https://github.com/DIPlib/diplib/releases/tag/3.5.1" class="m-thin">version 3.5.1</a> </span>
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<h1>
<span class="m-breadcrumb"><a href="detection.html">Detection</a> »</span>
Corner detectors <span class="m-thin">module</span> <div class="m-doc-include m-code m-inverted m-text-right"><span class="cp">#include</span> <a class="cpf" href="file--diplib--detection-h.html">"diplib/detection.h"</a></div>
</h1>
<p>Corner detection algorithms</p>
<div class="m-block m-default">
<h3>Contents</h3>
<ul>
<li>
Reference
<ul>
<li><a href="#function-members">Functions</a></li>
</ul>
</li>
</ul>
</div>
<section id="function-members">
<h2>Functions</h2>
<dl class="m-doc">
<dt>
<span class="m-doc-wrap-bumper">void <a href="#dip-HarrisCornerDetector-Image-CL-Image-L-dfloat--FloatArray-CL-StringArray-CL" class="m-doc">dip::<wbr />HarrisCornerDetector</a>(</span><span class="m-doc-wrap"><a href="dip-Image.html" class="m-doc">dip::Image</a> const& in,
<a href="dip-Image.html" class="m-doc">dip::Image</a>& out,
<a href="pixeltypes.html#dip-dfloat" class="m-doc">dip::dfloat</a> kappa = 0.04,
<a href="supporttypes.html#dip-FloatArray" class="m-doc">dip::FloatArray</a> const& sigmas = {2.0},
<a href="supporttypes.html#dip-StringArray" class="m-doc">dip::StringArray</a> const& boundaryCondition = {})</span>
</dt>
<dd>Harris corner detector</dd>
<dt>
<span class="m-doc-wrap-bumper">void <a href="#dip-ShiTomasiCornerDetector-Image-CL-Image-L-FloatArray-CL-StringArray-CL" class="m-doc">dip::<wbr />ShiTomasiCornerDetector</a>(</span><span class="m-doc-wrap"><a href="dip-Image.html" class="m-doc">dip::Image</a> const& in,
<a href="dip-Image.html" class="m-doc">dip::Image</a>& out,
<a href="supporttypes.html#dip-FloatArray" class="m-doc">dip::FloatArray</a> const& sigmas = {2.0},
<a href="supporttypes.html#dip-StringArray" class="m-doc">dip::StringArray</a> const& boundaryCondition = {})</span>
</dt>
<dd>Shi-Tomasi corner detector</dd>
<dt>
<span class="m-doc-wrap-bumper">void <a href="#dip-NobleCornerDetector-Image-CL-Image-L-FloatArray-CL-StringArray-CL" class="m-doc">dip::<wbr />NobleCornerDetector</a>(</span><span class="m-doc-wrap"><a href="dip-Image.html" class="m-doc">dip::Image</a> const& in,
<a href="dip-Image.html" class="m-doc">dip::Image</a>& out,
<a href="supporttypes.html#dip-FloatArray" class="m-doc">dip::FloatArray</a> const& sigmas = {2.0},
<a href="supporttypes.html#dip-StringArray" class="m-doc">dip::StringArray</a> const& boundaryCondition = {})</span>
</dt>
<dd>Noble’s corner detector</dd>
<dt>
<span class="m-doc-wrap-bumper">void <a href="#dip-WangBradyCornerDetector-Image-CL-Image-L-dfloat--FloatArray-CL-StringArray-CL" class="m-doc">dip::<wbr />WangBradyCornerDetector</a>(</span><span class="m-doc-wrap"><a href="dip-Image.html" class="m-doc">dip::Image</a> const& in,
<a href="dip-Image.html" class="m-doc">dip::Image</a>& out,
<a href="pixeltypes.html#dip-dfloat" class="m-doc">dip::dfloat</a> threshold = 0.1,
<a href="supporttypes.html#dip-FloatArray" class="m-doc">dip::FloatArray</a> const& sigmas = {2.0},
<a href="supporttypes.html#dip-StringArray" class="m-doc">dip::StringArray</a> const& boundaryCondition = {})</span>
</dt>
<dd>Wang-Brady corner detector</dd>
</dl>
</section>
<section>
<h2>Function documentation</h2>
<section class="m-doc-details" id="dip-HarrisCornerDetector-Image-CL-Image-L-dfloat--FloatArray-CL-StringArray-CL"><div>
<h3>
<span class="m-doc-wrap-bumper">void
</span><span class="m-doc-wrap"><span class="m-doc-wrap-bumper"><a href="#dip-HarrisCornerDetector-Image-CL-Image-L-dfloat--FloatArray-CL-StringArray-CL" class="m-doc-self">dip::<wbr />HarrisCornerDetector</a>(</span><span class="m-doc-wrap"><a href="dip-Image.html" class="m-doc">dip::Image</a> const& in,
<a href="dip-Image.html" class="m-doc">dip::Image</a>& out,
<a href="pixeltypes.html#dip-dfloat" class="m-doc">dip::dfloat</a> kappa = 0.04,
<a href="supporttypes.html#dip-FloatArray" class="m-doc">dip::FloatArray</a> const& sigmas = {2.0},
<a href="supporttypes.html#dip-StringArray" class="m-doc">dip::StringArray</a> const& boundaryCondition = {})</span></span>
</h3>
<p>Harris corner detector</p>
<p>The Harris corner detector is defined as</p>
<div class="m-math"><svg style="width: 9.962em; height: 1.325em; vertical-align: -0.006em; " viewBox="144.06 -13.19 99.62 13.25">
<title>
\[ \text{Det}(M) - \kappa \text{Tr}(M)^2 \; , \]
</title>
<defs>
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</svg></div>
<p>where <span class="m-math"><svg style="width: 1.061em; height: 0.778em; vertical-align: -0.000em; " viewBox=".38 -7.78 10.61 7.78">
<title>
\(M\)
</title>
<defs>
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<use x='.6' y='0' xlink:href='#eq96-g1-34'/>
</g>
</svg></span> is the structure tensor, and <span class="m-math"><svg style="width: 0.532em; height: 0.538em; vertical-align: -0.013em; " viewBox=".32 -5.25 5.32 5.38">
<title>
\(\kappa\)
</title>
<defs>
<path id='eq97-g1-94' d='M5.32-4.91C5.32-5.07 5.26-5.23 5.03-5.23C4.11-5.23 3.39-3.11 1.92-2.7L2.58-5.25L1.6-5.09L.48-.96C.38-.61 .15-.14 .08 0H1.17L1.77-2.17C1.79-2.17 1.81-2.17 1.98-2.25L3.04-.37C3.26 .02 3.62 .13 3.95 .13C4.26 .13 4.55 .01 4.6 .01L4.67-.2C4.25-.2 4.14-.23 3.98-.54L2.82-2.69C3.45-3.1 4.19-4.29 4.55-4.29C4.88-4.29 5.32-4.51 5.32-4.91Z'/>
</defs>
<g id='eq97-page1'>
<use x='.32' y='0' xlink:href='#eq97-g1-94'/>
</g>
</svg></span> is a constant typically set to 0.04, in this function
controlled by parameter <code>kappa</code>. Harris and Stephens noted in their paper that corners are locations in the image
where both eigenvalues of <span class="m-math"><svg style="width: 1.061em; height: 0.778em; vertical-align: -0.000em; " viewBox=".38 -7.78 10.61 7.78">
<title>
\(M\)
</title>
<defs>
<path id='eq98-g1-34' d='M8.74 0V-.19C7.93-.26 7.83-.32 7.83-.7C7.83-.91 7.85-1.05 7.96-1.43L9.42-6.71C9.64-7.42 9.7-7.48 10.4-7.59V-7.78H8.41L4.41-1.91L3.75-7.78H1.6V-7.59C2.19-7.55 2.48-7.42 2.48-7.15C2.48-7.05 2.43-6.87 2.36-6.66L2.25-6.3L2.22-6.17L.95-1.74C.58-.5 .4-.26-.21-.19V0H2.14V-.19C1.54-.24 1.3-.38 1.3-.71C1.3-.85 1.35-1.13 1.41-1.35L2.81-6.5L3.54 0H3.74L8.23-6.68L6.69-1.08C6.5-.45 6.26-.27 5.49-.19V0H8.74Z'/>
</defs>
<g id='eq98-page1'>
<use x='.6' y='0' xlink:href='#eq98-g1-34'/>
</g>
</svg></span> are large. But they considered eigenvalue computation too expensive, and therefore
proposed this cheaper alternative. <a href="detection_corners.html#dip-ShiTomasiCornerDetector-Image-CL-Image-L-FloatArray-CL-StringArray-CL"><code>dip::ShiTomasiCornerDetector</code></a> returns the smallest eigenvalue of <span class="m-math"><svg style="width: 1.061em; height: 0.778em; vertical-align: -0.000em; " viewBox=".38 -7.78 10.61 7.78">
<title>
\(M\)
</title>
<defs>
<path id='eq99-g1-34' d='M8.74 0V-.19C7.93-.26 7.83-.32 7.83-.7C7.83-.91 7.85-1.05 7.96-1.43L9.42-6.71C9.64-7.42 9.7-7.48 10.4-7.59V-7.78H8.41L4.41-1.91L3.75-7.78H1.6V-7.59C2.19-7.55 2.48-7.42 2.48-7.15C2.48-7.05 2.43-6.87 2.36-6.66L2.25-6.3L2.22-6.17L.95-1.74C.58-.5 .4-.26-.21-.19V0H2.14V-.19C1.54-.24 1.3-.38 1.3-.71C1.3-.85 1.35-1.13 1.41-1.35L2.81-6.5L3.54 0H3.74L8.23-6.68L6.69-1.08C6.5-.45 6.26-.27 5.49-.19V0H8.74Z'/>
</defs>
<g id='eq99-page1'>
<use x='.6' y='0' xlink:href='#eq99-g1-34'/>
</g>
</svg></span>.</p>
<p>The structure tensor <span class="m-math"><svg style="width: 1.061em; height: 0.778em; vertical-align: -0.000em; " viewBox=".38 -7.78 10.61 7.78">
<title>
\(M\)
</title>
<defs>
<path id='eq100-g1-34' d='M8.74 0V-.19C7.93-.26 7.83-.32 7.83-.7C7.83-.91 7.85-1.05 7.96-1.43L9.42-6.71C9.64-7.42 9.7-7.48 10.4-7.59V-7.78H8.41L4.41-1.91L3.75-7.78H1.6V-7.59C2.19-7.55 2.48-7.42 2.48-7.15C2.48-7.05 2.43-6.87 2.36-6.66L2.25-6.3L2.22-6.17L.95-1.74C.58-.5 .4-.26-.21-.19V0H2.14V-.19C1.54-.24 1.3-.38 1.3-.71C1.3-.85 1.35-1.13 1.41-1.35L2.81-6.5L3.54 0H3.74L8.23-6.68L6.69-1.08C6.5-.45 6.26-.27 5.49-.19V0H8.74Z'/>
</defs>
<g id='eq100-page1'>
<use x='.6' y='0' xlink:href='#eq100-g1-34'/>
</g>
</svg></span> is computed using <a href="analysis.html#dip-StructureTensor-Image-CL-Image-CL-Image-L-FloatArray-CL-FloatArray-CL-String-CL-StringArray-CL-dfloat-"><code>dip::StructureTensor</code></a>, with <code>gradientSigmas</code> equal to 1.0 and
<code>tensorSigmas</code> set through this function’s <code>sigmas</code> parameter.</p>
<p>This function generalizes the corner measure above to any number of dimensions. <code>in</code> must be scalar and real-valued.</p>
<p>This function is equivalent to:</p>
<div class="m-code"><pre><span></span><span class="n">dip</span><span class="o">::</span><span class="n">Image</span><span class="w"> </span><span class="n">M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">StructureTensor</span><span class="p">(</span><span class="w"> </span><span class="n">in</span><span class="p">,</span><span class="w"> </span><span class="p">{},</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mf">1.0</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="n">sigmas</span><span class="p">,</span><span class="w"> </span><span class="n">S</span><span class="o">::</span><span class="n">BEST</span><span class="p">,</span><span class="w"> </span><span class="n">boundaryCondition</span><span class="w"> </span><span class="p">);</span>
<span class="n">Image</span><span class="w"> </span><span class="n">out</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dip</span><span class="o">::</span><span class="n">Determinant</span><span class="p">(</span><span class="w"> </span><span class="n">M</span><span class="w"> </span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">k</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">dip</span><span class="o">::</span><span class="n">Square</span><span class="p">(</span><span class="w"> </span><span class="n">dip</span><span class="o">::</span><span class="n">Trace</span><span class="p">(</span><span class="w"> </span><span class="n">M</span><span class="w"> </span><span class="p">));</span>
<span class="n">dip</span><span class="o">::</span><span class="n">ClipLow</span><span class="p">(</span><span class="w"> </span><span class="n">out</span><span class="p">,</span><span class="w"> </span><span class="n">out</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">);</span>
</pre></div>
<aside class="m-note m-default">
<h4>Literature</h4>
<ul>
<li>C. Harris and M. Stephens, “A combined corner and edge detector”, Proceedings of the 4<sup>th</sup> Alvey Vision Conference, pp. 147–151, 1988.</li>
</ul>
</aside>
</div></section>
<section class="m-doc-details" id="dip-ShiTomasiCornerDetector-Image-CL-Image-L-FloatArray-CL-StringArray-CL"><div>
<h3>
<span class="m-doc-wrap-bumper">void
</span><span class="m-doc-wrap"><span class="m-doc-wrap-bumper"><a href="#dip-ShiTomasiCornerDetector-Image-CL-Image-L-FloatArray-CL-StringArray-CL" class="m-doc-self">dip::<wbr />ShiTomasiCornerDetector</a>(</span><span class="m-doc-wrap"><a href="dip-Image.html" class="m-doc">dip::Image</a> const& in,
<a href="dip-Image.html" class="m-doc">dip::Image</a>& out,
<a href="supporttypes.html#dip-FloatArray" class="m-doc">dip::FloatArray</a> const& sigmas = {2.0},
<a href="supporttypes.html#dip-StringArray" class="m-doc">dip::StringArray</a> const& boundaryCondition = {})</span></span>
</h3>
<p>Shi-Tomasi corner detector</p>
<p>The Shi-Tomasi corner detector is defined as</p>
<div class="m-math"><svg style="width: 6.184em; height: 1.095em; vertical-align: -0.006em; " viewBox="162.95 -10.89 61.84 10.95">
<title>
\[ \text{min}(\lambda_1, \lambda_2) \; , \]
</title>
<defs>
<path id='eq101-g8-49' d='M3.44 0V-.13C2.75-.14 2.61-.23 2.61-.65V-5.89L2.54-5.9L.97-5.11V-4.99C1.07-5.03 1.17-5.07 1.21-5.08C1.36-5.14 1.51-5.18 1.6-5.18C1.78-5.18 1.86-5.05 1.86-4.77V-.81C1.86-.52 1.79-.32 1.65-.24C1.52-.17 1.4-.14 1.03-.13V0H3.44Z'/>
<path id='eq101-g8-50' d='M4.15-1.2L4.04-1.24C3.71-.74 3.6-.66 3.21-.66H1.12L2.59-2.2C3.36-3.01 3.7-3.68 3.7-4.36C3.7-5.23 3-5.9 2.09-5.9C1.61-5.9 1.15-5.71 .83-5.36C.55-5.07 .42-4.79 .27-4.17L.45-4.12C.8-4.98 1.12-5.26 1.72-5.26C2.45-5.26 2.95-4.76 2.95-4.03C2.95-3.35 2.55-2.53 1.82-1.76L.26-.1V0H3.67L4.15-1.2Z'/>
<path id='eq101-g4-95' d='M6.28-.87L6.26-.89C6.1-.89 5.9-.49 5.4-.49C3.14-.49 7.24-7.8 3.74-7.8H3.61C3.39-7.8 2.89-7.62 2.74-7.48C2.73-7.47 2.82-7.36 2.83-7.36C2.89-7.41 3.16-7.52 3.39-7.52C4.55-7.52 4.24-5.61 4.24-5.3C4.25-5.26 4.24-5.24 4.23-5.19C3.99-4.09 .48-.96 .19 0H1.38C1.38-.12 1.39-.24 1.42-.37C1.76-1.89 4.1-4.2 4.1-4.22H4.11C4.04-3.6 2.97 .13 4.91 .13C5.43 .13 6.25-.73 6.28-.87Z'/>
<path id='eq101-g4-150' d='M2.32-.07C2.32-1.06 1.63-1.21 1.36-1.21C1.06-1.21 .67-1.04 .67-.52C.67-.05 1.1 .07 1.41 .07C1.49 .07 1.55 .06 1.58 .05C1.63 .04 1.67 .02 1.69 .02C1.77 .02 1.86 .08 1.86 .19C1.86 .42 1.67 .95 .88 1.45L.99 1.68C1.35 1.56 2.32 .77 2.32-.07Z'/>
<path id='eq101-g1-185' d='M3.51-8.54C1.79-7.42 .57-5.49 .57-3.06C.57-.85 1.83 1.39 3.48 2.41L3.62 2.22C2.05 .98 1.6-.46 1.6-3.1C1.6-5.74 2.08-7.11 3.62-8.35L3.51-8.54Z'/>
<path id='eq101-g1-186' d='M.45-8.54L.35-8.35C1.88-7.11 2.37-5.74 2.37-3.1C2.37-.46 1.92 .98 .35 2.22L.49 2.41C2.13 1.39 3.39-.85 3.39-3.06C3.39-5.49 2.18-7.42 .45-8.54Z'/>
<path id='eq101-g9-105' d='M3.02 0V-.18C2.24-.24 2.14-.36 2.14-1.22V-5.46L2.09-5.5L.24-4.84V-4.66L.33-4.67C.48-4.7 .63-4.71 .74-4.71C1.03-4.71 1.14-4.52 1.14-3.99V-1.22C1.14-.36 1.02-.23 .19-.18V0H3.02ZM2.25-7.56C2.25-7.89 1.98-8.17 1.64-8.17C1.3-8.17 1.03-7.89 1.03-7.56C1.03-7.21 1.3-6.95 1.64-6.95C1.98-6.95 2.25-7.21 2.25-7.56Z'/>
<path id='eq101-g9-109' d='M9.27 0V-.18L8.95-.2C8.6-.23 8.44-.44 8.44-.91V-3.37C8.44-4.78 7.97-5.5 7.05-5.5C6.36-5.5 5.75-5.19 5.1-4.5C4.89-5.18 4.48-5.5 3.84-5.5C3.31-5.5 2.98-5.33 1.98-4.58V-5.48L1.9-5.5C1.29-5.27 .88-5.14 .23-4.96V-4.76C.38-4.79 .48-4.81 .61-4.81C.92-4.81 1.03-4.61 1.03-4.04V-1.02C1.03-.37 .86-.19 .19-.18V0H2.85V-.18C2.21-.2 2.03-.33 2.03-.8V-4.17C2.03-4.17 2.13-4.32 2.21-4.4C2.51-4.67 3.02-4.88 3.44-4.88C3.97-4.88 4.23-4.46 4.23-3.62V-1.03C4.23-.36 4.1-.23 3.42-.18V0H6.1V-.18C5.42-.19 5.24-.39 5.24-1.14V-4.15C5.6-4.66 5.99-4.88 6.54-4.88C7.22-4.88 7.44-4.55 7.44-3.56V-1.04C7.44-.36 7.34-.26 6.65-.18V0H9.27Z'/>
<path id='eq101-g9-110' d='M5.8 0V-.18C5.21-.24 5.07-.38 5.07-.97V-3.71C5.07-4.83 4.54-5.5 3.66-5.5C3.11-5.5 2.74-5.3 1.92-4.53V-5.48L1.84-5.5C1.26-5.28 .85-5.15 .19-4.96V-4.76C.26-4.79 .38-4.81 .51-4.81C.85-4.81 .96-4.63 .96-4.04V-1.08C.96-.39 .82-.23 .22-.18V0H2.75V-.18C2.14-.23 1.96-.37 1.96-.8V-4.16C2.53-4.7 2.8-4.84 3.19-4.84C3.78-4.84 4.06-4.47 4.06-3.68V-1.18C4.06-.43 3.91-.23 3.31-.18V0H5.8Z'/>
</defs>
<g id='eq101-page1'>
<use x='162.95' y='-2.35' xlink:href='#eq101-g9-109'/>
<use x='172.26' y='-2.35' xlink:href='#eq101-g9-105'/>
<use x='175.58' y='-2.35' xlink:href='#eq101-g9-110'/>
<use x='182.15' y='-2.35' xlink:href='#eq101-g1-185'/>
<use x='186.46' y='-2.35' xlink:href='#eq101-g4-95'/>
<use x='193.15' y='-.56' xlink:href='#eq101-g8-49'/>
<use x='198.03' y='-2.35' xlink:href='#eq101-g4-150'/>
<use x='202.99' y='-2.35' xlink:href='#eq101-g4-95'/>
<use x='209.69' y='-.56' xlink:href='#eq101-g8-50'/>
<use x='214.57' y='-2.35' xlink:href='#eq101-g1-186'/>
<use x='222.47' y='-2.35' xlink:href='#eq101-g4-150'/>
</g>
</svg></div>
<p>where the <span class="m-math"><svg style="width: 0.724em; height: 0.793em; vertical-align: -0.013em; " viewBox="-.16 -7.8 7.24 7.93">
<title>
\(\lambda\)
</title>
<defs>
<path id='eq102-g1-95' d='M6.28-.87L6.26-.89C6.1-.89 5.9-.49 5.4-.49C3.14-.49 7.24-7.8 3.74-7.8H3.61C3.39-7.8 2.89-7.62 2.74-7.48C2.73-7.47 2.82-7.36 2.83-7.36C2.89-7.41 3.16-7.52 3.39-7.52C4.55-7.52 4.24-5.61 4.24-5.3C4.25-5.26 4.24-5.24 4.23-5.19C3.99-4.09 .48-.96 .19 0H1.38C1.38-.12 1.39-.24 1.42-.37C1.76-1.89 4.1-4.2 4.1-4.22H4.11C4.04-3.6 2.97 .13 4.91 .13C5.43 .13 6.25-.73 6.28-.87Z'/>
</defs>
<g id='eq102-page1'>
<use x='-.16' y='0' xlink:href='#eq102-g1-95'/>
</g>
</svg></span> are the eigenvalues of <span class="m-math"><svg style="width: 1.061em; height: 0.778em; vertical-align: -0.000em; " viewBox=".38 -7.78 10.61 7.78">
<title>
\(M\)
</title>
<defs>
<path id='eq103-g1-34' d='M8.74 0V-.19C7.93-.26 7.83-.32 7.83-.7C7.83-.91 7.85-1.05 7.96-1.43L9.42-6.71C9.64-7.42 9.7-7.48 10.4-7.59V-7.78H8.41L4.41-1.91L3.75-7.78H1.6V-7.59C2.19-7.55 2.48-7.42 2.48-7.15C2.48-7.05 2.43-6.87 2.36-6.66L2.25-6.3L2.22-6.17L.95-1.74C.58-.5 .4-.26-.21-.19V0H2.14V-.19C1.54-.24 1.3-.38 1.3-.71C1.3-.85 1.35-1.13 1.41-1.35L2.81-6.5L3.54 0H3.74L8.23-6.68L6.69-1.08C6.5-.45 6.26-.27 5.49-.19V0H8.74Z'/>
</defs>
<g id='eq103-page1'>
<use x='.6' y='0' xlink:href='#eq103-g1-34'/>
</g>
</svg></span>, the structure tensor. Corners are locations in the image
where both eigenvalues of <span class="m-math"><svg style="width: 1.061em; height: 0.778em; vertical-align: -0.000em; " viewBox=".38 -7.78 10.61 7.78">
<title>
\(M\)
</title>
<defs>
<path id='eq104-g1-34' d='M8.74 0V-.19C7.93-.26 7.83-.32 7.83-.7C7.83-.91 7.85-1.05 7.96-1.43L9.42-6.71C9.64-7.42 9.7-7.48 10.4-7.59V-7.78H8.41L4.41-1.91L3.75-7.78H1.6V-7.59C2.19-7.55 2.48-7.42 2.48-7.15C2.48-7.05 2.43-6.87 2.36-6.66L2.25-6.3L2.22-6.17L.95-1.74C.58-.5 .4-.26-.21-.19V0H2.14V-.19C1.54-.24 1.3-.38 1.3-.71C1.3-.85 1.35-1.13 1.41-1.35L2.81-6.5L3.54 0H3.74L8.23-6.68L6.69-1.08C6.5-.45 6.26-.27 5.49-.19V0H8.74Z'/>
</defs>
<g id='eq104-page1'>
<use x='.6' y='0' xlink:href='#eq104-g1-34'/>
</g>
</svg></span> are large.</p>
<p>The structure tensor <span class="m-math"><svg style="width: 1.061em; height: 0.778em; vertical-align: -0.000em; " viewBox=".38 -7.78 10.61 7.78">
<title>
\(M\)
</title>
<defs>
<path id='eq105-g1-34' d='M8.74 0V-.19C7.93-.26 7.83-.32 7.83-.7C7.83-.91 7.85-1.05 7.96-1.43L9.42-6.71C9.64-7.42 9.7-7.48 10.4-7.59V-7.78H8.41L4.41-1.91L3.75-7.78H1.6V-7.59C2.19-7.55 2.48-7.42 2.48-7.15C2.48-7.05 2.43-6.87 2.36-6.66L2.25-6.3L2.22-6.17L.95-1.74C.58-.5 .4-.26-.21-.19V0H2.14V-.19C1.54-.24 1.3-.38 1.3-.71C1.3-.85 1.35-1.13 1.41-1.35L2.81-6.5L3.54 0H3.74L8.23-6.68L6.69-1.08C6.5-.45 6.26-.27 5.49-.19V0H8.74Z'/>
</defs>
<g id='eq105-page1'>
<use x='.6' y='0' xlink:href='#eq105-g1-34'/>
</g>
</svg></span> is computed using <a href="analysis.html#dip-StructureTensor-Image-CL-Image-CL-Image-L-FloatArray-CL-FloatArray-CL-String-CL-StringArray-CL-dfloat-"><code>dip::StructureTensor</code></a>, with <code>gradientSigmas</code> equal to 1.0 and
<code>tensorSigmas</code> set through this function’s <code>sigmas</code> parameter.</p>
<p>This function generalizes the corner measure above to any number of dimensions. <code>in</code> must be scalar and real-valued.</p>
<p>This function is equivalent to:</p>
<div class="m-code"><pre><span></span><span class="n">dip</span><span class="o">::</span><span class="n">Image</span><span class="w"> </span><span class="n">M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">StructureTensor</span><span class="p">(</span><span class="w"> </span><span class="n">in</span><span class="p">,</span><span class="w"> </span><span class="p">{},</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mf">1.0</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="n">sigmas</span><span class="p">,</span><span class="w"> </span><span class="n">S</span><span class="o">::</span><span class="n">BEST</span><span class="p">,</span><span class="w"> </span><span class="n">boundaryCondition</span><span class="w"> </span><span class="p">);</span>
<span class="n">out</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dip</span><span class="o">::</span><span class="n">SmallestEigenvalue</span><span class="p">(</span><span class="w"> </span><span class="n">M</span><span class="w"> </span><span class="p">);</span>
</pre></div>
<aside class="m-note m-default">
<h4>Literature</h4>
<ul>
<li>J. Shi and C. Tomasi, “Good features to track”, 9<sup>th</sup> IEEE Conference on Computer Vision and Pattern Recognition, pp. 593–600, 1994.</li>
</ul>
</aside>
</div></section>
<section class="m-doc-details" id="dip-NobleCornerDetector-Image-CL-Image-L-FloatArray-CL-StringArray-CL"><div>
<h3>
<span class="m-doc-wrap-bumper">void
</span><span class="m-doc-wrap"><span class="m-doc-wrap-bumper"><a href="#dip-NobleCornerDetector-Image-CL-Image-L-FloatArray-CL-StringArray-CL" class="m-doc-self">dip::<wbr />NobleCornerDetector</a>(</span><span class="m-doc-wrap"><a href="dip-Image.html" class="m-doc">dip::Image</a> const& in,
<a href="dip-Image.html" class="m-doc">dip::Image</a>& out,
<a href="supporttypes.html#dip-FloatArray" class="m-doc">dip::FloatArray</a> const& sigmas = {2.0},
<a href="supporttypes.html#dip-StringArray" class="m-doc">dip::StringArray</a> const& boundaryCondition = {})</span></span>
</h3>
<p>Noble’s corner detector</p>
<p>Noble defined a corner detector as</p>
<div class="m-math"><svg style="width: 8.130em; height: 1.095em; vertical-align: -0.006em; " viewBox="153.22 -10.89 81.3 10.95">
<title>
\[ \text{Det}(M) / \text{Tr}(M) \; , \]
</title>
<defs>
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</defs>
<g id='eq106-page1'>
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<use x='167.16' y='-2.35' xlink:href='#eq106-g6-116'/>
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<use x='232.2' y='-2.35' xlink:href='#eq106-g4-150'/>
</g>
</svg></div>
<p>This is similar to the Harris corner detector (see <a href="detection_corners.html#dip-HarrisCornerDetector-Image-CL-Image-L-dfloat--FloatArray-CL-StringArray-CL"><code>dip::HarrisCornerDetector</code></a>), except it has no parameter to tune. The ratio
of the determinant to the trace is equivalent to the harmonic mean of the eigenvalues.</p>
<p>This function generalizes the corner measure above to any number of dimensions. <code>in</code> must be scalar and real-valued.</p>
<p>This function is equivalent to:</p>
<div class="m-code"><pre><span></span><span class="n">dip</span><span class="o">::</span><span class="n">Image</span><span class="w"> </span><span class="n">M</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">StructureTensor</span><span class="p">(</span><span class="w"> </span><span class="n">in</span><span class="p">,</span><span class="w"> </span><span class="p">{},</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="mf">1.0</span><span class="w"> </span><span class="p">},</span><span class="w"> </span><span class="n">sigmas</span><span class="p">,</span><span class="w"> </span><span class="n">S</span><span class="o">::</span><span class="n">BEST</span><span class="p">,</span><span class="w"> </span><span class="n">boundaryCondition</span><span class="w"> </span><span class="p">);</span>
<span class="n">Image</span><span class="w"> </span><span class="n">out</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dip</span><span class="o">::</span><span class="n">SafeDivide</span><span class="p">(</span><span class="w"> </span><span class="n">dip</span><span class="o">::</span><span class="n">Determinant</span><span class="p">(</span><span class="w"> </span><span class="n">M</span><span class="w"> </span><span class="p">),</span><span class="w"> </span><span class="n">dip</span><span class="o">::</span><span class="n">Trace</span><span class="p">(</span><span class="w"> </span><span class="n">M</span><span class="w"> </span><span class="p">));</span>
</pre></div>
<aside class="m-note m-default">
<h4>Literature</h4>
<ul>
<li>J.A. Noble, “Finding corners”, Proceedings of the Alvey Vision Conference, pp. 37.1-37.8, 1987.</li>
</ul>
</aside>
</div></section>
<section class="m-doc-details" id="dip-WangBradyCornerDetector-Image-CL-Image-L-dfloat--FloatArray-CL-StringArray-CL"><div>
<h3>
<span class="m-doc-wrap-bumper">void
</span><span class="m-doc-wrap"><span class="m-doc-wrap-bumper"><a href="#dip-WangBradyCornerDetector-Image-CL-Image-L-dfloat--FloatArray-CL-StringArray-CL" class="m-doc-self">dip::<wbr />WangBradyCornerDetector</a>(</span><span class="m-doc-wrap"><a href="dip-Image.html" class="m-doc">dip::Image</a> const& in,
<a href="dip-Image.html" class="m-doc">dip::Image</a>& out,
<a href="pixeltypes.html#dip-dfloat" class="m-doc">dip::dfloat</a> threshold = 0.1,
<a href="supporttypes.html#dip-FloatArray" class="m-doc">dip::FloatArray</a> const& sigmas = {2.0},
<a href="supporttypes.html#dip-StringArray" class="m-doc">dip::StringArray</a> const& boundaryCondition = {})</span></span>
</h3>
<p>Wang-Brady corner detector</p>
<p>Wang and Brady (1995) define a corner operator as</p>
<div class="m-math"><svg style="width: 16.893em; height: 5.832em; vertical-align: 0.068em; " viewBox="103.45 -59 168.93 58.32">
<title>
\[ \begin{cases}
\Gamma = \left( \frac{\delta^2 F}{\delta \bf{t}^2} \right)^2 - s|\nabla F|^2 = \text{maximum}
\\ \frac{\delta^2 F}{\delta \bf{n}^2} = 0
\\ |\nabla F|^2 > T_1 , \Gamma > T_2
\end{cases} \]
</title>
<defs>
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<use id='eq107-g19-41' xlink:href='#eq107-g16-41' transform='scale(1.36)'/>
<use id='eq107-g19-66' xlink:href='#eq107-g16-66' transform='scale(1.36)'/>
<use id='eq107-g19-150' xlink:href='#eq107-g16-150' transform='scale(1.36)'/>
<use id='eq107-g19-161' xlink:href='#eq107-g16-161' transform='scale(1.36)'/>
<path id='eq107-g13-0' d='M6.84-2.73V-3.39H.74V-2.73H6.84Z'/>
<path id='eq107-g13-106' d='M1.52 2.3V-8.42H.86V2.3H1.52Z'/>
<path id='eq107-g13-114' d='M7.49-8.05H.37L3.88 0H4.12L7.49-8.05ZM6.5-7.02L4.31-1.73L2.11-7.02H6.5Z'/>
<use id='eq107-g26-49' xlink:href='#eq107-g23-49' transform='scale(1.33)'/>
<use id='eq107-g26-50' xlink:href='#eq107-g23-50' transform='scale(1.33)'/>
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<path id='eq107-g10-60' d='M2.78 10.18L2.75 14.53H3.82V10.81C3.82 9.75 3.67 7.93 1.75 7.13C3.67 6.3 3.82 4.49 3.82 3.44V-.24H2.75L2.74 4.32C2.74 5.1 1.99 6.56 .79 7C.73 7.03 .69 7.06 .69 7.13C.69 7.22 .71 7.25 .81 7.29C2.41 7.87 2.78 9.56 2.78 10.18Z'/>
<path id='eq107-g10-62' d='M3.82 .24V-3.81H2.75V.24H3.82Z'/>
</defs>
<g id='eq107-page1'>
<use x='103.45' y='-51.43' xlink:href='#eq107-g10-56'/>
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<use x='139.66' y='-37.66' xlink:href='#eq107-g16-88'/>
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<use x='146.91' y='-40.2' xlink:href='#eq107-g2-50'/>
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<use x='177.08' y='-42.6' xlink:href='#eq107-g19-66'/>
<use x='182.51' y='-42.6' xlink:href='#eq107-g13-106'/>
<use x='185.5' y='-42.6' xlink:href='#eq107-g13-114'/>
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<use x='130.65' y='-22.54' xlink:href='#eq107-g7-61'/>
<use x='141.05' y='-22.54' xlink:href='#eq107-g29-48'/>
<use x='111.01' y='-5.2' xlink:href='#eq107-g13-106'/>
<use x='113.99' y='-5.2' xlink:href='#eq107-g13-114'/>
<use x='122.4' y='-5.2' xlink:href='#eq107-g19-27'/>
<use x='130.96' y='-5.2' xlink:href='#eq107-g13-106'/>
<use x='133.83' y='-9.54' xlink:href='#eq107-g26-50'/>
<use x='142.28' y='-5.2' xlink:href='#eq107-g19-161'/>
<use x='153.05' y='-5.2' xlink:href='#eq107-g19-41'/>
<use x='159.18' y='-3.41' xlink:href='#eq107-g26-49'/>
<use x='164.07' y='-5.2' xlink:href='#eq107-g19-150'/>
<use x='169.18' y='-5.2' xlink:href='#eq107-g7-0'/>
<use x='179.77' y='-5.2' xlink:href='#eq107-g19-161'/>
<use x='190.54' y='-5.2' xlink:href='#eq107-g19-41'/>
<use x='196.67' y='-3.41' xlink:href='#eq107-g26-50'/>
</g>
</svg></div>
<p>Here, <span class="m-math"><svg style="width: 0.687em; height: 0.788em; vertical-align: -0.000em; " viewBox="0 -7.88 6.87 7.88">
<title>
\(\Gamma\)
</title>
<defs>
<path id='eq108-g1-0' d='M6.87-5.88L6.81-7.88H.14V-7.66C1.06-7.59 1.18-7.43 1.18-6.56V-1.48C1.18-.43 1.04-.29 .13-.23V0H3.48V-.23C2.55-.27 2.39-.42 2.39-1.35V-6.96C2.39-7.34 2.49-7.43 2.82-7.43H4.91C6.11-7.43 6.46-7.06 6.57-5.88H6.87Z'/>
</defs>
<g id='eq108-page1'>
<use x='0' y='0' xlink:href='#eq108-g1-0'/>
</g>
</svg></span> is composed of the square of the second derivative of the image <span class="m-math"><svg style="width: 0.768em; height: 0.778em; vertical-align: -0.000em; " viewBox=".35 -7.78 7.68 7.78">
<title>
\(F\)
</title>
<defs>
<path id='eq109-g1-27' d='M7.3-5.96L7.68-7.78H1.75V-7.59C2.54-7.52 2.67-7.44 2.67-7.11C2.67-6.92 2.62-6.63 2.54-6.34L1.07-1.07C.86-.36 .77-.29 .1-.19V0H3.1V-.19C2.41-.23 2.19-.36 2.19-.71C2.19-.82 2.22-.96 2.3-1.25L3.04-3.91C3.48-3.88 3.74-3.87 4-3.87C4.47-3.87 4.54-3.86 4.66-3.81C4.82-3.73 4.92-3.56 4.92-3.32C4.92-3.13 4.9-2.99 4.82-2.62L5.03-2.56L5.93-5.35L5.72-5.41C5.18-4.34 5.15-4.32 3.14-4.3L3.93-7.07C4-7.31 4.17-7.38 4.66-7.38C6.67-7.38 7.09-7.23 7.09-6.48C7.09-6.4 7.09-6.35 7.07-6.21C7.06-6.12 7.06-6.07 7.05-5.98L7.3-5.96Z'/>
</defs>
<g id='eq109-page1'>
<use x='.35' y='0' xlink:href='#eq109-g1-27'/>
</g>
</svg></span> in the contour direction
(<span class="m-math"><svg style="width: 0.395em; height: 0.765em; vertical-align: -0.015em; " viewBox="0 -7.5 3.95 7.65">
<title>
\(\bf{t}\)
</title>
<defs>
<path id='eq110-g2-116' d='M3.95-1.19L3.66-1.32C3.42-.87 3.23-.7 2.98-.7C2.64-.7 2.51-.91 2.51-1.38V-4.97H3.63V-5.49H2.51V-7.5H2.22C1.49-6.48 1.01-5.94 .24-5.29V-4.97H.86V-1.11C.86-.33 1.37 .14 2.19 .14C2.99 .14 3.47-.21 3.95-1.19Z'/>
</defs>
<g id='eq110-page1'>
<use x='0' y='0' xlink:href='#eq110-g2-116'/>
</g>
</svg></span> is the unit vector perpendicular to the gradient), and the square norm of the gradient. The first
term is a measure for curvature, the second term is a measure for edgeness. <span class="m-math"><svg style="width: 0.436em; height: 0.542em; vertical-align: -0.016em; " viewBox=".31 -5.26 4.36 5.42">
<title>
\(s\)
</title>
<defs>
<path id='eq111-g1-66' d='M3.62-1.42C3.62-1.88 3.37-2.39 2.81-3.08C2.36-3.64 2.17-4.01 2.17-4.34C2.17-4.74 2.42-4.98 2.83-4.98C3.45-4.98 3.81-4.54 3.93-3.61H4.12L4.36-5.26H4.19C4.1-5.1 4.01-5.05 3.84-5.05C3.74-5.05 3.63-5.07 3.41-5.13C3.11-5.23 2.94-5.25 2.74-5.25C1.86-5.25 1.3-4.74 1.3-3.94C1.3-3.56 1.55-3.05 2.05-2.41C2.51-1.8 2.72-1.38 2.72-1.04C2.72-.49 2.36-.12 1.83-.12C1.17-.12 .81-.61 .62-1.74H.43L.19 .15H.38C.49-.05 .54-.1 .67-.1S1.01-.05 1.25 .01C1.52 .1 1.7 .13 1.91 .13C2.91 .13 3.62-.51 3.62-1.42Z'/>
</defs>
<g id='eq111-page1'>
<use x='.31' y='0' xlink:href='#eq111-g1-66'/>
</g>
</svg></span> is a threshold (in this
function defined through <code>threshold</code>) that determines how much larger the curvature must be compared to the
edgeness. Typical values are in the range 0.0 to 0.5, the default is 0.1.</p>
<p>The second equation indicates that the second derivative in the gradient direction must be zero (the zero crossing
of the second derivative indicates the exact sub-pixel location of the edge). The third equation indicates two
thresholds that must be satisfied. This function computes only <span class="m-math"><svg style="width: 0.687em; height: 0.788em; vertical-align: -0.000em; " viewBox="0 -7.88 6.87 7.88">
<title>
\(\Gamma\)
</title>
<defs>
<path id='eq112-g1-0' d='M6.87-5.88L6.81-7.88H.14V-7.66C1.06-7.59 1.18-7.43 1.18-6.56V-1.48C1.18-.43 1.04-.29 .13-.23V0H3.48V-.23C2.55-.27 2.39-.42 2.39-1.35V-6.96C2.39-7.34 2.49-7.43 2.82-7.43H4.91C6.11-7.43 6.46-7.06 6.57-5.88H6.87Z'/>
</defs>
<g id='eq112-page1'>
<use x='0' y='0' xlink:href='#eq112-g1-0'/>
</g>
</svg></span>, the thresholding must be applied
separately.</p>
<p>This function generalizes the corner measure above to any number of dimensions. <code>in</code> must be scalar and real-valued.</p>
<p>Gradients are computed using Gaussian derivatives, with the <code>sigmas</code> parameter. This function is equivalent to:</p>
<div class="m-code"><pre><span></span><span class="n">Image</span><span class="w"> </span><span class="n">out</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">dip</span><span class="o">::</span><span class="n">Square</span><span class="p">(</span><span class="w"> </span><span class="n">dip</span><span class="o">::</span><span class="n">LaplaceMinusDgg</span><span class="p">(</span><span class="w"> </span><span class="n">in</span><span class="p">,</span><span class="w"> </span><span class="n">sigmas</span><span class="w"> </span><span class="p">))</span>
<span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">threshold</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">dip</span><span class="o">::</span><span class="n">SquareNorm</span><span class="p">(</span><span class="w"> </span><span class="n">dip</span><span class="o">::</span><span class="n">Gradient</span><span class="p">(</span><span class="w"> </span><span class="n">in</span><span class="p">,</span><span class="w"> </span><span class="n">sigmas</span><span class="w"> </span><span class="p">));</span>
<span class="n">dip</span><span class="o">::</span><span class="n">ClipLow</span><span class="p">(</span><span class="w"> </span><span class="n">out</span><span class="p">,</span><span class="w"> </span><span class="n">out</span><span class="p">,</span><span class="w"> </span><span class="mi">0</span><span class="w"> </span><span class="p">);</span>
</pre></div>
<aside class="m-note m-default">
<h4>Literature</h4>
<ul>
<li>H. Wang and M. Brady, “Real-time corner detection algorithm for motion estimation”, Image and Vision Computing 13(9):695–703, 1995.</li>
</ul>
</aside>
</div></section>
</section>
</div>
</div>
</div>
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<p>DIPlib, a library for quantitative image analysis. Documentation compiled with <a href="https://crisluengo.github.io/doxpp/">dox++</a> and styled with <a href="https://mcss.mosra.cz/">m.css</a>.</p>
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