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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Van-Cittert-Dekonvolution</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>Die <b>Van-Cittert-Dekonvolution</b> (benannt nach <a href="Pieter_Hendrik_van_Cittert" title="Pieter Hendrik van Cittert">Pieter Hendrik van Cittert</a>) ist ein Verfahren, um die <a href="Faltung_(Mathematik)" title="Faltung (Mathematik)">Faltung</a> eines Bildes <i>g</i> mit einer <a href="Faltungsmatrix" title="Faltungsmatrix">Filtermaske</a> (PSF) <i>h</i> rückgängig zu machen (<a href="Dekonvolution" title="Dekonvolution">Dekonvolution</a>/inverse Filterung). Sie kann zur Verbesserung der Bildqualität benutzt werden, wenn das Bild zum Beispiel durch ein unscharfes Objektiv o. ä. „verwaschen“ wurde. Das Bild <i>g</i> stellt das ideale Bild dar, das man als Ergebnis des Verfahrens erhalten möchte. Das verwaschene Bild <i>f</i>, das den Ausgangspunkt des Verfahrens darstellt, wird beschrieben 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={\mathcal {H}}\ g=g*h}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
<mtext> </mtext>
<mi>g</mi>
<mo>=</mo>
<mi>g</mi>
<mo>∗<!-- ∗ --></mo>
<mi>h</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f={\mathcal {H}}\ g=g*h}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a5cb3b46ca61d915a15b68230714a10a8111f7d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.786ex; height:2.509ex;" alt="{\displaystyle f={\mathcal {H}}\ g=g*h}" loading="lazy"></span></dd></dl>
<p>Hier entspricht <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> dem Filteroperator, der durch Faltung mit <i>h</i> dargestellt wird. Ziel ist es, folgenden Ausdruck zu berechnen:
</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 g={\mathcal {H}}^{-1}\ f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>g</mi>
<mo>=</mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mtext> </mtext>
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g={\mathcal {H}}^{-1}\ f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d90cc5a2598eeaceaa4a4e5389aadbce52272c60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.37ex; height:3.009ex;" alt="{\displaystyle g={\mathcal {H}}^{-1}\ f}" loading="lazy"></span></dd></dl>
<p>Die Van-Cittert-Dekonvolution approximiert diesen durch eine iterative Formel:
</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 g_{0}=f\,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mi>f</mi>
<mspace width="thinmathspace"></mspace>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{0}=f\,}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80e5af311677d15ebadf0885dd9b1e836bb76c0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.927ex; height:2.509ex;" alt="{\displaystyle g_{0}=f\,}" loading="lazy"></span></dd>
<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 g_{k+1}=f+({\mathcal {I}}-{\mathcal {H}})g_{k}=f+(I-h)*g_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<mi>f</mi>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">I</mi>
</mrow>
</mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">H</mi>
</mrow>
</mrow>
<mo stretchy="false">)</mo>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
<mo>=</mo>
<mi>f</mi>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>I</mi>
<mo>−<!-- − --></mo>
<mi>h</mi>
<mo stretchy="false">)</mo>
<mo>∗<!-- ∗ --></mo>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{k+1}=f+({\mathcal {I}}-{\mathcal {H}})g_{k}=f+(I-h)*g_{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/812a3113fffd2fd19ec84acf047732153443c2d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.59ex; height:2.843ex;" alt="{\displaystyle g_{k+1}=f+({\mathcal {I}}-{\mathcal {H}})g_{k}=f+(I-h)*g_{k}}" loading="lazy"></span></dd></dl>
<p>Dabei ist <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 {I}}}">
<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">I</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e9730a0ada0426927ff64141eb9f505eca132d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.069ex; width:1.561ex; height:2.176ex;" alt="{\displaystyle {\mathcal {I}}}" loading="lazy"></span> ein Operator, dessen Punktantwort <i>I</i> einem Delta-Puls entspricht (überall 0, nur in der Mitte 1). Die Operation <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 {I}}g_{k}}">
<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">I</mi>
</mrow>
</mrow>
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}g_{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7637fd738f72d48eeb9bb47d3a9d24f58ece8b99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.069ex; width:3.759ex; height:2.509ex;" alt="{\displaystyle {\mathcal {I}}g_{k}}" loading="lazy"></span> ergibt also gerade <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 g_{k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>g</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle g_{k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de363eb168fab5e16a5acc74d8b0288e07a23aca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.198ex; height:2.009ex;" alt="{\displaystyle g_{k}}" loading="lazy"></span>. Die Stärke der Rückfaltung hängt von der Anzahl der Iterationsschritte <i>k</i> ab. Je mehr Iterationsschritte durchgeführt werden, desto stärker ist die Rückfaltung (Schärfung). Dafür wird das <a href="Bildrauschen" title="Bildrauschen">Bildrauschen</a> bei zu großer Anzahl an Iterationen verstärkt und somit das Bild wieder undeutlich.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>
<p>Die folgenden Bilder zeigen die Anwendung der Van-Cittert-Iteration auf ein weichgezeichnetes Bild (3×3-Gauß-Filter):
</p>
<div class="mw-heading mw-heading2"><h2 id="Herleitung">Herleitung</h2></div>
<p>Im <a href="Fouriertransformation" class="mw-redirect" title="Fouriertransformation">Fourierraum</a> wird die Faltung zu einer punktweisen Multiplikation, sodass gilt:
</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 {\hat {g}}={\hat {f}}\cdot {\hat {h}}^{-1}={\frac {\hat {f}}{\hat {h}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>g</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>f</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>h</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>f</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>h</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {g}}={\hat {f}}\cdot {\hat {h}}^{-1}={\frac {\hat {f}}{\hat {h}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac9041cbcfadc5b781848b4951e272da2299c4ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.014ex; height:6.676ex;" alt="{\displaystyle {\hat {g}}={\hat {f}}\cdot {\hat {h}}^{-1}={\frac {\hat {f}}{\hat {h}}}}" loading="lazy"></span></dd></dl>
<p>Dies lässt sich leicht berechnen, wenn die Übertragungsfunktion <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 {\hat {h}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>h</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {h}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61505780f3740aa55551090a2b23c668c934a82b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.843ex;" alt="{\displaystyle {\hat {h}}}" loading="lazy"></span> keine Nullstellen enthält, da sonst eine Division durch 0 nötig wäre. Um dieses Problem zu umgehen, führt man <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 {\hat {h}}'=1-{\hat {h}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>h</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>′</mo>
</msup>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>h</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {h}}'=1-{\hat {h}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6af849bdf639d3d3aa2df28c546ec5558006253f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.464ex; height:3.343ex;" alt="{\displaystyle {\hat {h}}'=1-{\hat {h}}}" loading="lazy"></span> ein. Damit gilt dann:
</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 {\hat {g}}={\frac {\hat {f}}{\hat {h}}}={\frac {\hat {f}}{1-{\hat {h}}'}}\approx (1+{\hat {h}}'+{\hat {h}}'^{2}+{\hat {h}}'^{3}+\ldots )\cdot {\hat {f}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>g</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>f</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>h</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>f</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>h</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
<mo>′</mo>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>≈<!-- ≈ --></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msup>
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<annotation encoding="application/x-tex">{\displaystyle {\hat {g}}={\frac {\hat {f}}{\hat {h}}}={\frac {\hat {f}}{1-{\hat {h}}'}}\approx (1+{\hat {h}}'+{\hat {h}}'^{2}+{\hat {h}}'^{3}+\ldots )\cdot {\hat {f}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50d8acb0b5f747f501355e3e2cac1776fdd40cb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.075ex; height:7.176ex;" alt="{\displaystyle {\hat {g}}={\frac {\hat {f}}{\hat {h}}}={\frac {\hat {f}}{1-{\hat {h}}'}}\approx (1+{\hat {h}}'+{\hat {h}}'^{2}+{\hat {h}}'^{3}+\ldots )\cdot {\hat {f}}}" loading="lazy"></span></dd></dl>
<p>Im letzten Schritt wurde eine <a href="Taylor-Entwicklung" class="mw-redirect" title="Taylor-Entwicklung">Taylor-Entwicklung</a> durchgeführt. Dabei wird der Term <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 (1-{\hat {h}}')^{-1}}">
<semantics>
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<mo stretchy="false">(</mo>
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<mo stretchy="false">^<!-- ^ --></mo>
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<msup>
<mo stretchy="false">)</mo>
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<mo>−<!-- − --></mo>
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<annotation encoding="application/x-tex">{\displaystyle (1-{\hat {h}}')^{-1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a48dda1af53e889aa639313df6d959aba64d93b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.169ex; height:3.676ex;" alt="{\displaystyle (1-{\hat {h}}')^{-1}}" loading="lazy"></span> um die invariante Abbildung <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 {\hat {h}}=1}">
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<annotation encoding="application/x-tex">{\displaystyle {\hat {h}}=1}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4c27f31e3449fe054798de39c629af4d31eeae09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.6ex; height:2.843ex;" alt="{\displaystyle {\hat {h}}=1}" loading="lazy"></span>, bzw. <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 {\hat {h}}'=0}">
<semantics>
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<annotation encoding="application/x-tex">{\displaystyle {\hat {h}}'=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7cfcb0ebaa4e49fb22a0fd9becebb9860c7e5766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.285ex; height:3.176ex;" alt="{\displaystyle {\hat {h}}'=0}" loading="lazy"></span> entwickelt. Im <a href="Ortsraum" title="Ortsraum">Ortsraum</a> ergibt dieser Ausdruck:
</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 g={\mathcal {H}}^{-1}f\approx ({\mathcal {I}}+{\mathcal {H}}'+{\mathcal {H}}'^{2}+{\mathcal {H}}'^{3}+\ldots )f}">
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<annotation encoding="application/x-tex">{\displaystyle g={\mathcal {H}}^{-1}f\approx ({\mathcal {I}}+{\mathcal {H}}'+{\mathcal {H}}'^{2}+{\mathcal {H}}'^{3}+\ldots )f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c7e18b3bfdecb7fcd60a403eee08c058de782229.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.142ex; height:3.176ex;" alt="{\displaystyle g={\mathcal {H}}^{-1}f\approx ({\mathcal {I}}+{\mathcal {H}}'+{\mathcal {H}}'^{2}+{\mathcal {H}}'^{3}+\ldots )f}" loading="lazy"></span> mit <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}}'={\mathcal {I}}-{\mathcal {H}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}'={\mathcal {I}}-{\mathcal {H}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00f07494a3e44a4c65cf3d7df4d3cdbcca3571fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.043ex; height:2.676ex;" alt="{\displaystyle {\mathcal {H}}'={\mathcal {I}}-{\mathcal {H}}}" loading="lazy"></span>.</dd></dl>
<p>Unter Ausnutzung des <a href="Horner-Schema" title="Horner-Schema">Horner-Schemas</a> für dieses Polynom erhält man obige Iterationsvorschrift:
</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 g_{0}=f}">
<semantics>
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<msub>
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<mn>0</mn>
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<mo>=</mo>
<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle g_{0}=f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/daf19c31dfae70076329660f90097eb338a438af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.54ex; height:2.509ex;" alt="{\displaystyle g_{0}=f}" loading="lazy"></span></dd>
<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 g_{k+1}=f+({\mathcal {I}}-{\mathcal {H}})g_{k}=f+(I-h)*g_{k}}">
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<annotation encoding="application/x-tex">{\displaystyle g_{k+1}=f+({\mathcal {I}}-{\mathcal {H}})g_{k}=f+(I-h)*g_{k}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/812a3113fffd2fd19ec84acf047732153443c2d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.59ex; height:2.843ex;" alt="{\displaystyle g_{k+1}=f+({\mathcal {I}}-{\mathcal {H}})g_{k}=f+(I-h)*g_{k}}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>P. H. van Cittert: <cite style="font-style:italic">Zum Einfluß der Spaltbreite auf die Intensitätsverteilung in Spektrallinien. II</cite>. In: <cite style="font-style:italic">Zeitschrift für Physik</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>69</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>5</span>, 1. Mai 1931, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>298–308</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/BF01391351">10.1007/BF01391351</a></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:Van-Cittert-Dekonvolution&rft.atitle=Zum+Einflu%C3%9F+der+Spaltbreite+auf+die+Intensit%C3%A4tsverteilung+in+Spektrallinien.+II&rft.au=P.+H.+van+Cittert&rft.date=1931-05-01&rft.doi=10.1007%2FBF01391351&rft.genre=journal&rft.issue=5&rft.jtitle=Zeitschrift+f%C3%BCr+Physik&rft.pages=298-308&rft.volume=69" style="display:none"> </span></li>
<li><a href="Bernd_J%C3%A4hne" title="Bernd Jähne">Bernd Jähne</a>: <i>Digitale Bildverarbeitung.</i> 6. Auflage, Springer, 2005, ISBN 3-540-24999-0.</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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