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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Monogenes Signal</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Das <b>monogene Signal</b> ist eine Verallgemeinerung des <a href="Analytisches_Signal" title="Analytisches Signal">analytischen Signals</a> auf mehr als eine <a href="Dimension_(Mathematik)" title="Dimension (Mathematik)">Dimension</a> auf Basis der <a href="Marcel_Riesz" title="Marcel Riesz">Riesz</a>-Transformation. Das monogene Signal findet Anwendung in der <a href="Bildverarbeitung" title="Bildverarbeitung">Bildverarbeitung</a>. Mit ihm können Bilder in lokale <a href="Amplitude" title="Amplitude">Amplitude</a> und lokale <a href="Phase_(Schwingung)" class="mw-redirect" title="Phase (Schwingung)">Phase</a> zerlegt werden.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definitionen">Definitionen</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Monogenes_Signal">Monogenes Signal</h3></div>
<p>Es sei d eine natürliche Zahl und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{2}(\mathbb {R} ^{d})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>∈<!-- ∈ --></mo>
<msup>
<mi>L</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\in L^{2}(\mathbb {R} ^{d})}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/949f5203c6d5d0950764e98a8c095d8d55f8fd02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.336ex; height:3.176ex;" alt="{\displaystyle f\in L^{2}(\mathbb {R} ^{d})}" loading="lazy"></span> eine Funktion. Dann ist das monogene Signal <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{M}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>M</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{M}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b53fcb87b0f4d69be22cc6a9d2fb227dd2da91d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.509ex;" alt="{\displaystyle f_{M}}" loading="lazy"></span> definiert durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{M}:=(f,R_{1}f,\ldots ,R_{d}f),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>M</mi>
</mrow>
</msub>
<mo>:=</mo>
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo>,</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mi>f</mi>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f_{M}:=(f,R_{1}f,\ldots ,R_{d}f),}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27f805206bac231dedfd811171a1577cf1ead3f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.022ex; height:2.843ex;" alt="{\displaystyle f_{M}:=(f,R_{1}f,\ldots ,R_{d}f),}" loading="lazy"></span></dd></dl>
<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{j},j=1,\ldots ,d}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>,</mo>
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>d</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R_{j},j=1,\ldots ,d}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85511df090ada6a89381d1292b54b6661b436bd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.321ex; height:2.843ex;" alt="{\displaystyle R_{j},j=1,\ldots ,d}" loading="lazy"></span> die j-te Komponente der Riesz-Transformation bezeichnet.
</p>
<div class="mw-heading mw-heading3"><h3 id="Riesz-Transformation">Riesz-Transformation</h3></div>
<p>Es sei d eine natürliche Zahl. Die j-te Komponente, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,\ldots ,d,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mi>d</mi>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle j=1,\ldots ,d,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e191552709f9410223f186e10e0a200d0783d9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:12.287ex; height:2.509ex;" alt="{\displaystyle j=1,\ldots ,d,}" loading="lazy"></span> der Riesz-Transformation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R_{j}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bc93d61e1436bb2c2fb771a13d0892784754998.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.674ex; height:2.843ex;" alt="{\displaystyle R_{j}}" loading="lazy"></span> ist definiert durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{j}f(x):=c_{d}\lim _{\epsilon \to 0}\int _{0<\epsilon \leq \left|y\right|}{\frac {y_{j}}{\left|y\right|^{d+1}}}f(x-y)\,\mathrm {d} y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<munder>
<mo movablelimits="true" form="prefix">lim</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>ϵ<!-- ϵ --></mi>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mrow>
</munder>
<msub>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
<mo><</mo>
<mi>ϵ<!-- ϵ --></mi>
<mo>≤<!-- ≤ --></mo>
<mrow>
<mo>|</mo>
<mi>y</mi>
<mo>|</mo>
</mrow>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msub>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<msup>
<mrow>
<mo>|</mo>
<mi>y</mi>
<mo>|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
</mrow>
<mi>y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R_{j}f(x):=c_{d}\lim _{\epsilon \to 0}\int _{0<\epsilon \leq \left|y\right|}{\frac {y_{j}}{\left|y\right|^{d+1}}}f(x-y)\,\mathrm {d} y}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72ea67446cfc1fa714cd9313d1c6450994fad6eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.356ex; height:6.509ex;" alt="{\displaystyle R_{j}f(x):=c_{d}\lim _{\epsilon \to 0}\int _{0<\epsilon \leq \left|y\right|}{\frac {y_{j}}{\left|y\right|^{d+1}}}f(x-y)\,\mathrm {d} y}" loading="lazy"></span></dd></dl>
<p>mit
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{d}={\frac {\Gamma \left({\frac {d+1}{2}}\right)}{\pi ^{(d+1)/2}}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">Γ<!-- Γ --></mi>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<msup>
<mi>π<!-- π --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">(</mo>
<mi>d</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle c_{d}={\frac {\Gamma \left({\frac {d+1}{2}}\right)}{\pi ^{(d+1)/2}}},}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a7f536b30e39f5bf033b6ae27f99d60e5dec31c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.092ex; height:7.843ex;" alt="{\displaystyle c_{d}={\frac {\Gamma \left({\frac {d+1}{2}}\right)}{\pi ^{(d+1)/2}}},}" loading="lazy"></span></dd></dl>
<p>wobei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Γ<!-- Γ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"></span> die <a href="Gammafunktion" title="Gammafunktion">Gammafunktion</a> bezeichnet.
</p><p>Die <i>Riesz-Transformation</i> ist definiert als d-dimensionaler Vektor der j-Komponenten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R_{j}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bc93d61e1436bb2c2fb771a13d0892784754998.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.674ex; height:2.843ex;" alt="{\displaystyle R_{j}}" loading="lazy"></span>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Rf(x):=(R_{1}f(x),\ldots ,R_{d}f(x)).}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Rf(x):=(R_{1}f(x),\ldots ,R_{d}f(x)).}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f224644c8e4b7a44dafe13cce93aa1d20ab31fb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.071ex; height:2.843ex;" alt="{\displaystyle Rf(x):=(R_{1}f(x),\ldots ,R_{d}f(x)).}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Zusammenhang_mit_dem_analytischen_Signal_und_der_Hilberttransformation">Zusammenhang mit dem analytischen Signal und der Hilberttransformation</h2></div>
<p>Für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle d=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/958b426c5ace91d4fb7f5a3becd7b21dba288d50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d=1}" loading="lazy"></span> ist die Riesz-Transformation die <a href="Hilbert-Transformation" title="Hilbert-Transformation">Hilbert-Transformation</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19ef4c7b923a5125ac91aa491838a95ee15b804f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}" loading="lazy"></span> und das monogene Signal entspricht in diesem Fall dem analytischen Signal, wenn man den Vektor des monogenen Signals als komplexe Zahl auffasst, d. h.
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(f(x),R_{1}f(x)\right)=f(x)+iR_{1}f(x)=f(x)+i{\mathcal {H}}f(x)}">
<semantics>
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<mo>)</mo>
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<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
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<mo>+</mo>
<mi>i</mi>
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<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>i</mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
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<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \left(f(x),R_{1}f(x)\right)=f(x)+iR_{1}f(x)=f(x)+i{\mathcal {H}}f(x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2eec6abd0a2111f8a10cecfc47ceb84e69847e63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.431ex; height:2.843ex;" alt="{\displaystyle \left(f(x),R_{1}f(x)\right)=f(x)+iR_{1}f(x)=f(x)+i{\mathcal {H}}f(x)}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Zerlegung_in_Phase_und_Amplitude">Zerlegung in Phase und Amplitude</h2></div>
<p>Das monogene Signal erlaubt eine Zerlegung eines mehrdimensionalen Signals in lokale <a href="Amplitude" title="Amplitude">Amplitude</a> und lokale <a href="Phase_(Schwingung)" class="mw-redirect" title="Phase (Schwingung)">Phase</a>. Die lokale Amplitude <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> ist in diesem Falle definiert durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x)={\sqrt {f^{2}(x)+(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<msup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mo>…<!-- … --></mo>
<mo>+</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A(x)={\sqrt {f^{2}(x)+(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}},}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ce816c9d9a860f55b721952f6807fa665e3fe9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:47.946ex; height:4.843ex;" alt="{\displaystyle A(x)={\sqrt {f^{2}(x)+(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}},}" loading="lazy"></span></dd></dl>
<p>der lokale <a href="Phasenwinkel" title="Phasenwinkel">Phasenwinkel</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (x)=\left|\operatorname {atan2} ({\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}},f)\right|,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>|</mo>
<mrow>
<mi>atan2</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mo stretchy="false">(</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mo>…<!-- … --></mo>
<mo>+</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mrow>
<mo>,</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>|</mo>
</mrow>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha (x)=\left|\operatorname {atan2} ({\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}},f)\right|,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99ed7aa9cb7aadc15e16a13907f15b002824e7f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.823ex; height:5.176ex;" alt="{\displaystyle \alpha (x)=\left|\operatorname {atan2} ({\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}},f)\right|,}" loading="lazy"></span></dd></dl>
<p>die lokale <a href="Ausbreitungsrichtung" class="mw-redirect" title="Ausbreitungsrichtung">Phasenrichtung</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x)={\frac {(R_{1}f(x),\ldots ,R_{d}f(x))}{\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mrow>
<msqrt>
<mo stretchy="false">(</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mo>…<!-- … --></mo>
<mo>+</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle u(x)={\frac {(R_{1}f(x),\ldots ,R_{d}f(x))}{\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4cabf0914bf4f084cc908e05922c4d3a720ed46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.368ex; height:7.009ex;" alt="{\displaystyle u(x)={\frac {(R_{1}f(x),\ldots ,R_{d}f(x))}{\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}}}}" loading="lazy"></span></dd></dl>
<p>und die lokale Phase <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"></span> durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (x)=\alpha (x)u(x)=\left|\operatorname {atan2} ({\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}},f)\right|{\frac {(R_{1}f(x),\ldots ,R_{d}f(x))}{\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>α<!-- α --></mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mi>u</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>|</mo>
<mrow>
<mi>atan2</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mo stretchy="false">(</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mo>…<!-- … --></mo>
<mo>+</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
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<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mrow>
<mo>,</mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>|</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mrow>
<msqrt>
<mo stretchy="false">(</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mo>…<!-- … --></mo>
<mo>+</mo>
<mo stretchy="false">(</mo>
<msub>
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>d</mi>
</mrow>
</msub>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</msqrt>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi (x)=\alpha (x)u(x)=\left|\operatorname {atan2} ({\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}},f)\right|{\frac {(R_{1}f(x),\ldots ,R_{d}f(x))}{\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}}}.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b94c56521175921babcebd86dbd025babfae48f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:94.715ex; height:7.009ex;" alt="{\displaystyle \phi (x)=\alpha (x)u(x)=\left|\operatorname {atan2} ({\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}},f)\right|{\frac {(R_{1}f(x),\ldots ,R_{d}f(x))}{\sqrt {(R_{1}f(x))^{2}+\ldots +(R_{d}f(x))^{2}}}}.}" loading="lazy"></span></dd></dl>
<table class="centered">
<caption>Beispiel
</caption>
<tbody><tr style="vertical-align:top">
<td>
</td>
<td>
</td>
<td>
</td>
<td>
</td></tr></tbody></table>
<div class="mw-heading mw-heading2"><h2 id="Anwendung_in_der_Bildanalyse">Anwendung in der Bildanalyse</h2></div>
<p>Fasst man die Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> als zwei- oder dreidimensionales Bild auf, hat das monogene Signal folgende mögliche Anwendungen:
</p>
<ul><li>Die lokale Phase kann als eine Art optischer Fluss eines Bildes aufgefasst werden. Dabei gibt die lokale Phasenrichtung eine Flussrichtung an, der lokale Phasenwinkel eine Flussstärke.</li>
<li>Unter Verwendung einer <a href="Multiskalenanalyse" title="Multiskalenanalyse">Multiskalenanalyse</a> kann das monogene Signal dazu verwendet werden, Strukturen aus Bildern unabhängig von Helligkeit und Beleuchtungsstärke zu extrahieren.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>M. Felsberg, G. Sommer: <cite style="font-style:italic">The monogenic signal</cite>. In: <cite style="font-style:italic">IEEE Transactions on Signal Processing</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>49</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>12</span>, 2001, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>3136–3144</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Monogenes+Signal&rft.atitle=The+monogenic+signal&rft.au=M.+Felsberg%2C+G.+Sommer&rft.date=2001&rft.genre=journal&rft.issue=12&rft.jtitle=IEEE+Transactions+on+Signal+Processing&rft.pages=3136-3144&rft.volume=49" style="display:none"> </span></li>
<li>S. Held, M. Storath, P. Massopust, B. Forster: <cite style="font-style:italic">Steerable Wavelet Frames Based on the Riesz Transform</cite>. In: <cite style="font-style:italic">IEEE Transactions on Image Processing</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>19</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>3</span>, 2010, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>653–667</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Monogenes+Signal&rft.atitle=Steerable+Wavelet+Frames+Based+on+the+Riesz+Transform&rft.au=S.+Held%2C+M.+Storath%2C+P.+Massopust%2C+...&rft.date=2010&rft.genre=journal&rft.issue=3&rft.jtitle=IEEE+Transactions+on+Image+Processing&rft.pages=653-667&rft.volume=19" style="display:none"> </span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Software">Software</h2></div>
<p>Die folgenden Softwarepakete implementieren das monogene Signal auf Multiskalenbasis
</p>
<ul><li><a rel="nofollow" class="external text" href="https://www-m6.ma.tum.de/Mamebia/MonogenicWaveletToolbox%7C">Monogenic Wavelet Toolbox for ImageJ,</a> Technische Universität München</li>
<li><a rel="nofollow" class="external text" href="https://bigwww.epfl.ch/demo/monogenic/">MonogenicJ: A ImageJ plugin for wavelet-based monogenic analysis of images</a> EPF Lausanne</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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