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href="https://ca.wikipedia.org/wiki/Desconvoluci%C3%B3_(matem%C3%A0tica)" title="Desconvolució (matemàtica) – Catalan" lang="ca" hreflang="ca" data-title="Desconvolució (matemàtica)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Dekonvoluce" title="Dekonvoluce – Czech" lang="cs" hreflang="cs" data-title="Dekonvoluce" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dekonvolution" title="Dekonvolution – German" lang="de" hreflang="de" data-title="Dekonvolution" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Dekonvolutsioon" title="Dekonvolutsioon – Estonian" lang="et" hreflang="et" data-title="Dekonvolutsioon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Deconvoluci%C3%B3n" title="Deconvolución – Spanish" lang="es" hreflang="es" data-title="Deconvolución" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%88%D8%A7%D9%87%D9%85%E2%80%8C%DA%AF%D8%B4%D8%AA" title="واهم‌گشت – Persian" lang="fa" hreflang="fa" data-title="واهم‌گشت" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9convolution" title="Déconvolution – French" lang="fr" hreflang="fr" data-title="Déconvolution" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Deconvoluzione" title="Deconvoluzione – Italian" lang="it" hreflang="it" data-title="Deconvoluzione" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%94_%D7%A7%D7%95%D7%A0%D7%91%D7%95%D7%9C%D7%95%D7%A6%D7%99%D7%94" title="דה קונבולוציה – Hebrew" lang="he" hreflang="he" data-title="דה קונבולוציה" data-language-autonym="עברית" data-language-local-name="Hebrew" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Reconstruction of a filtered signal</div><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Upsampling" title="Upsampling">Upsampling</a>.</div><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Deconvolution_of_an_astronomical_image.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Deconvolution_of_an_astronomical_image.png/220px-Deconvolution_of_an_astronomical_image.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Deconvolution_of_an_astronomical_image.png/330px-Deconvolution_of_an_astronomical_image.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Deconvolution_of_an_astronomical_image.png/440px-Deconvolution_of_an_astronomical_image.png 2x" data-file-width="2000" data-file-height="1000" /></a><figcaption>Before and after deconvolution of an image of the lunar crater <a href="/wiki/Copernicus_(lunar_crater)" title="Copernicus (lunar crater)">Copernicus</a> using the <a href="/wiki/Richardson%E2%80%93Lucy_deconvolution" title="Richardson–Lucy deconvolution">Richardson-Lucy</a> algorithm.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>deconvolution</b> is the <a href="/wiki/Inverse_function" title="Inverse function">inverse</a> of <a href="/wiki/Convolution" title="Convolution">convolution</a>. Both operations are used in <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a> and <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a>. For example, it may be possible to recover the original signal after a filter (convolution) by using a deconvolution method with a certain degree of accuracy.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Due to the measurement error of the recorded signal or image, it can be demonstrated that the worse the <a href="/wiki/Signal-to-noise_ratio" title="Signal-to-noise ratio">signal-to-noise ratio</a> (SNR), the worse the reversing of a filter will be; hence, inverting a filter is not always a good solution as the error amplifies. Deconvolution offers a solution to this problem. </p><p>The foundations for deconvolution and <a href="/wiki/Time-series_analysis" class="mw-redirect" title="Time-series analysis">time-series analysis</a> were largely laid by <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a> of the <a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">Massachusetts Institute of Technology</a> in his book <i>Extrapolation, Interpolation, and Smoothing of Stationary Time Series</i> (1949).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The book was based on work Wiener had done during <a href="/wiki/World_War_II" title="World War II">World War II</a> but that had been classified at the time. Some of the early attempts to apply these theories were in the fields of <a href="/wiki/Weather_forecasting" title="Weather forecasting">weather forecasting</a> and <a href="/wiki/Economics" title="Economics">economics</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deconvolution&action=edit&section=1" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In general, the objective of deconvolution is to find the solution <i>f</i> of a convolution equation of the form: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g^{-1}=h\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>h</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*g^{-1}=h\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6aeb7716fb8b468e85ca3ce578b11450e411b6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.749ex; height:3.009ex;" alt="{\displaystyle f*g^{-1}=h\,}"></span></dd></dl> <p>Usually, <i>h</i> is some recorded signal, and <i>f</i> is some signal that we wish to recover, but has been convolved with a filter or distortion function <i>g</i>, before we recorded it. Usually, <i>h</i> is a distorted version of <i>f</i> and the shape of <i>f</i> can't be easily recognized by the eye or simpler time-domain operations. The function <i>g</i> represents the <a href="/wiki/Impulse_response" title="Impulse response">impulse response</a> of an instrument or a driving force that was applied to a physical system. If we know <i>g</i>, or at least know the form of <i>g</i>, then we can perform deterministic deconvolution. However, if we do not know <i>g</i> in advance, then we need to estimate it. This can be done using methods of <a href="/wiki/Statistics" title="Statistics">statistical</a> <a href="/wiki/Estimation_theory" title="Estimation theory">estimation</a> or building the physical principles of the underlying system, such as the electrical circuit equations or diffusion equations. </p><p>There are several deconvolution techniques, depending on the choice of the measurement error and deconvolution parameters: </p> <div class="mw-heading mw-heading3"><h3 id="Raw_deconvolution">Raw deconvolution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deconvolution&action=edit&section=2" title="Edit section: Raw deconvolution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When the measurement error is very low (ideal case), deconvolution collapses into a filter reversing. This kind of deconvolution can be performed in the Laplace domain. By computing the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> of the recorded signal <i>h</i> and the system response function <i>g</i>, you get <i>H</i> and <i>G</i>, with <i>G</i> as the <a href="/wiki/Transfer_function" title="Transfer function">transfer function</a>. Using the <a href="/wiki/Convolution_theorem" title="Convolution theorem">Convolution theorem</a>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=H/G\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>G</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=H/G\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b565c0ef3094e9c7e016ac3fb3b261ce36708890" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.279ex; height:2.843ex;" alt="{\displaystyle F=H/G\,}"></span></dd></dl> <p>where <i>F</i> is the estimated Fourier transform of <i>f</i>. Finally, the <a href="/wiki/Fourier_inversion_theorem" title="Fourier inversion theorem">inverse Fourier transform</a> of the function <i>F</i> is taken to find the estimated deconvolved signal <i>f</i>. Note that <i>G</i> is at the denominator and could amplify elements of the error model if present. </p> <div class="mw-heading mw-heading3"><h3 id="Deconvolution_with_noise">Deconvolution with noise</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deconvolution&action=edit&section=3" title="Edit section: Deconvolution with noise"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In physical measurements, the situation is usually closer to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g^{-1})+\varepsilon =h\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ε</mi> <mo>=</mo> <mi>h</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g^{-1})+\varepsilon =h\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d93dac99c8dac0922658ae77677cadc07476792" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.482ex; height:3.176ex;" alt="{\displaystyle (f*g^{-1})+\varepsilon =h\,}"></span></dd></dl> <p>In this case <i>ε</i> is <a href="/wiki/Noise_(physics)" class="mw-redirect" title="Noise (physics)">noise</a> that has entered our recorded signal. If a noisy signal or image is assumed to be noiseless, the statistical estimate of <i>g</i> will be incorrect. In turn, the estimate of <i>ƒ</i> will also be incorrect. The lower the <a href="/wiki/Signal-to-noise_ratio" title="Signal-to-noise ratio">signal-to-noise ratio</a>, the worse the estimate of the deconvolved signal will be. That is the reason why <a href="/wiki/Inverse_filter" title="Inverse filter">inverse filtering</a> the signal (as in the "raw deconvolution" above) is usually not a good solution. However, if at least some knowledge exists of the type of noise in the data (for example, <a href="/wiki/White_noise" title="White noise">white noise</a>), the estimate of <i>ƒ</i> can be improved through techniques such as <a href="/wiki/Wiener_deconvolution" title="Wiener deconvolution">Wiener deconvolution</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deconvolution&action=edit&section=4" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Seismology">Seismology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deconvolution&action=edit&section=5" title="Edit section: Seismology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The concept of deconvolution had an early application in <a href="/wiki/Reflection_seismology" title="Reflection seismology">reflection seismology</a>. In 1950, <a href="/w/index.php?title=Enders_Robinson&action=edit&redlink=1" class="new" title="Enders Robinson (page does not exist)">Enders Robinson</a> was a graduate student at <a href="/wiki/MIT" class="mw-redirect" title="MIT">MIT</a>. He worked with others at MIT, such as <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a>, <a href="/wiki/Norman_Levinson" title="Norman Levinson">Norman Levinson</a>, and economist <a href="/wiki/Paul_Samuelson" title="Paul Samuelson">Paul Samuelson</a>, to develop the "convolutional model" of a reflection <a href="/wiki/Seismogram" title="Seismogram">seismogram</a>. This model assumes that the recorded seismogram <i>s</i>(<i>t</i>) is the convolution of an Earth-reflectivity function <i>e</i>(<i>t</i>) and a <a href="/wiki/Seismic" class="mw-redirect" title="Seismic">seismic</a> <a href="/wiki/Wavelet" title="Wavelet">wavelet</a> <i>w</i>(<i>t</i>) from a <a href="/wiki/Point_source" title="Point source">point source</a>, where <i>t</i> represents recording time. Thus, our convolution equation is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(t)=(e*w)(t).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mo>∗</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(t)=(e*w)(t).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29d50abbedf33129e290d3357635a7f1798f3fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.272ex; height:2.843ex;" alt="{\displaystyle s(t)=(e*w)(t).\,}"></span></dd></dl> <p>The seismologist is interested in <i>e</i>, which contains information about the Earth's structure. By the <a href="/wiki/Convolution_theorem" title="Convolution theorem">convolution theorem</a>, this equation may be <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transformed</a> to </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\omega )=E(\omega )W(\omega )\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mi>W</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(\omega )=E(\omega )W(\omega )\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a4c5094d015d789adcd02785a35e40207d5b81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.961ex; height:2.843ex;" alt="{\displaystyle S(\omega )=E(\omega )W(\omega )\,}"></span></dd></dl> <p>in the <a href="/wiki/Frequency_domain" title="Frequency domain">frequency domain</a>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> is the frequency variable. By assuming that the reflectivity is white, we can assume that the <a href="/wiki/Spectral_density" title="Spectral density">power spectrum</a> of the reflectivity is constant, and that the power spectrum of the seismogram is the spectrum of the wavelet multiplied by that constant. Thus, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |S(\omega )|\approx k|W(\omega )|.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≈</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>W</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |S(\omega )|\approx k|W(\omega )|.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8938e1a9a124a4830645b8d60148e945bd3c5136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.376ex; height:2.843ex;" alt="{\displaystyle |S(\omega )|\approx k|W(\omega )|.\,}"></span></dd></dl> <p>If we assume that the wavelet is <a href="/wiki/Minimum_phase" title="Minimum phase">minimum phase</a>, we can recover it by calculating the minimum phase equivalent of the power spectrum we just found. The reflectivity may be recovered by designing and applying a <a href="/wiki/Wiener_filter" title="Wiener filter">Wiener filter</a> that shapes the estimated wavelet to a <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a> (i.e., a spike). The result may be seen as a series of scaled, shifted delta functions (although this is not mathematically rigorous): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e(t)=\sum _{i=1}^{N}r_{i}\delta (t-\tau _{i}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e(t)=\sum _{i=1}^{N}r_{i}\delta (t-\tau _{i}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95fff9e980ba54d7b072dea4367485862431718d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.422ex; height:7.343ex;" alt="{\displaystyle e(t)=\sum _{i=1}^{N}r_{i}\delta (t-\tau _{i}),}"></span></dd></dl> <p>where <i>N</i> is the number of reflection events, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}"></span> are the <a href="/wiki/Reflection_coefficient" title="Reflection coefficient">reflection coefficients</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t-\tau _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>−</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t-\tau _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6dda274d95dcaeec4685d31f7d89ddaf27344c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.496ex; height:2.343ex;" alt="{\displaystyle t-\tau _{i}}"></span> are the reflection times of each event, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span> is the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>. </p><p>In practice, since we are dealing with noisy, finite <a href="/wiki/Bandwidth_(computing)" title="Bandwidth (computing)">bandwidth</a>, finite length, <a href="/wiki/Sampling_(signal_processing)" title="Sampling (signal processing)">discretely sampled</a> datasets, the above procedure only yields an approximation of the filter required to deconvolve the data. However, by formulating the problem as the solution of a <a href="/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz matrix</a> and using <a href="/wiki/Levinson_recursion" title="Levinson recursion">Levinson recursion</a>, we can relatively quickly estimate a filter with the smallest <a href="/wiki/Mean_squared_error" title="Mean squared error">mean squared error</a> possible. We can also do deconvolution directly in the frequency domain and get similar results. The technique is closely related to <a href="/wiki/Linear_prediction" title="Linear prediction">linear prediction</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Optics_and_other_imaging">Optics and other imaging</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deconvolution&action=edit&section=6" title="Edit section: Optics and other imaging"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Depth_Coded_Phalloidin_Stained_Actin_Filaments_Cancer_Cell.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Depth_Coded_Phalloidin_Stained_Actin_Filaments_Cancer_Cell.png/245px-Depth_Coded_Phalloidin_Stained_Actin_Filaments_Cancer_Cell.png" decoding="async" width="245" height="191" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Depth_Coded_Phalloidin_Stained_Actin_Filaments_Cancer_Cell.png/368px-Depth_Coded_Phalloidin_Stained_Actin_Filaments_Cancer_Cell.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Depth_Coded_Phalloidin_Stained_Actin_Filaments_Cancer_Cell.png/490px-Depth_Coded_Phalloidin_Stained_Actin_Filaments_Cancer_Cell.png 2x" data-file-width="3176" data-file-height="2472" /></a><figcaption>Example of a deconvolved microscope image.</figcaption></figure> <p>In optics and imaging, the term "deconvolution" is specifically used to refer to the process of reversing the <a href="/wiki/Aberration_in_optical_systems#Distortion_of_the_image" class="mw-redirect" title="Aberration in optical systems">optical distortion</a> that takes place in an optical <a href="/wiki/Microscope" title="Microscope">microscope</a>, <a href="/wiki/Electron_microscope" title="Electron microscope">electron microscope</a>, <a href="/wiki/Telescope" title="Telescope">telescope</a>, or other imaging instrument, thus creating clearer images. It is usually done in the digital domain by a <a href="/wiki/Software" title="Software">software</a> <a href="/wiki/Algorithm" title="Algorithm">algorithm</a>, as part of a suite of <a href="/wiki/Microscope_image_processing" title="Microscope image processing">microscope image processing</a> techniques. Deconvolution is also practical to sharpen images that suffer from fast motion or jiggles during capturing. Early <a href="/wiki/Hubble_Space_Telescope" title="Hubble Space Telescope">Hubble Space Telescope</a> images were distorted by a <a href="/wiki/Hubble_Space_Telescope#Flawed_mirror" title="Hubble Space Telescope">flawed mirror</a> and were sharpened by deconvolution. </p><p>The usual method is to assume that the optical path through the instrument is optically perfect, convolved with a <a href="/wiki/Point_spread_function" title="Point spread function">point spread function</a> (PSF), that is, a <a href="/wiki/Mathematical_function" class="mw-redirect" title="Mathematical function">mathematical function</a> that describes the distortion in terms of the pathway a theoretical <a href="/wiki/Point_source" title="Point source">point source</a> of light (or other waves) takes through the instrument.<sup id="cite_ref-Pawley_2006_3-0" class="reference"><a href="#cite_note-Pawley_2006-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Usually, such a point source contributes a small area of fuzziness to the final image. If this function can be determined, it is then a matter of computing its <a href="/wiki/Inverse_function" title="Inverse function">inverse</a> or complementary function, and convolving the acquired image with that. The result is the original, undistorted image. </p><p>In practice, finding the true PSF is impossible, and usually an approximation of it is used, theoretically calculated<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> or based on some experimental estimation by using known probes. Real optics may also have different PSFs at different focal and spatial locations, and the PSF may be non-linear. The accuracy of the approximation of the PSF will dictate the final result. Different algorithms can be employed to give better results, at the price of being more computationally intensive. Since the original convolution discards data, some algorithms use additional data acquired at nearby focal points to make up some of the lost information. <a href="/wiki/Regularization_(mathematics)" title="Regularization (mathematics)">Regularization</a> in iterative algorithms (as in <a href="/wiki/Expectation-maximization_algorithm" class="mw-redirect" title="Expectation-maximization algorithm">expectation-maximization algorithms</a>) can be applied to avoid unrealistic solutions. </p><p>When the PSF is unknown, it may be possible to deduce it by systematically trying different possible PSFs and assessing whether the image has improved. This procedure is called <i><a href="/wiki/Blind_deconvolution" title="Blind deconvolution">blind deconvolution</a></i>.<sup id="cite_ref-Pawley_2006_3-1" class="reference"><a href="#cite_note-Pawley_2006-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Blind deconvolution is a well-established <a href="/wiki/Iterative_reconstruction" title="Iterative reconstruction">image restoration</a> technique in <a href="/wiki/Astronomy" title="Astronomy">astronomy</a>, where the point nature of the objects photographed exposes the PSF thus making it more feasible. It is also used in <a href="/wiki/Fluorescence_microscopy" class="mw-redirect" title="Fluorescence microscopy">fluorescence microscopy</a> for image restoration, and in fluorescence <a href="/wiki/Spectral_imaging" title="Spectral imaging">spectral imaging</a> for spectral separation of multiple unknown <a href="/wiki/Fluorophore" title="Fluorophore">fluorophores</a>. The most common <a href="/wiki/Iteration" title="Iteration">iterative</a> algorithm for the purpose is the <a href="/wiki/Richardson%E2%80%93Lucy_deconvolution" title="Richardson–Lucy deconvolution">Richardson–Lucy deconvolution</a> algorithm; the <a href="/wiki/Wiener_deconvolution" title="Wiener deconvolution">Wiener deconvolution</a> (and approximations) are the most common non-iterative algorithms. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:High_Resolution_THz_image.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/High_Resolution_THz_image.png/316px-High_Resolution_THz_image.png" decoding="async" width="316" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/High_Resolution_THz_image.png/474px-High_Resolution_THz_image.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/High_Resolution_THz_image.png/632px-High_Resolution_THz_image.png 2x" data-file-width="2512" data-file-height="1192" /></a><figcaption>High Resolution THz image is achieved by deconvolution of the THz image and the mathematically modeled THz PSF. <b>(a)</b> THz image of an integrated circuit (IC) before enhancement; <b>(b)</b> Mathematically modeled THz PSF; <b>(c)</b> High resolution THz image which is achieved as a result of deconvolution of the THz image shown in (a) and the PSF which is shown in (b); <b>(d)</b> High resolution X-ray image confirms the accuracy of the measured values.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>For some specific imaging systems such as laser pulsed terahertz systems, PSF can be modeled mathematically.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> As a result, as shown in the figure, deconvolution of the modeled PSF and the terahertz image can give a higher resolution representation of the terahertz image. </p> <div class="mw-heading mw-heading3"><h3 id="Radio_astronomy">Radio astronomy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deconvolution&action=edit&section=7" title="Edit section: Radio astronomy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When performing image synthesis in radio <a href="/wiki/Interferometry" title="Interferometry">interferometry</a>, a specific kind of <a href="/wiki/Radio_astronomy" title="Radio astronomy">radio astronomy</a>, one step consists of deconvolving the produced image with the "dirty beam", which is a different name for the <a href="/wiki/Point_spread_function" title="Point spread function">point spread function</a>. A commonly used method is the <a href="/wiki/CLEAN_(algorithm)" title="CLEAN (algorithm)">CLEAN algorithm</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Biology,_physiology_and_medical_devices"><span id="Biology.2C_physiology_and_medical_devices"></span>Biology, physiology and medical devices</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deconvolution&action=edit&section=8" title="Edit section: Biology, physiology and medical devices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Typical use of deconvolution is in tracer kinetics. For example, when measuring a hormone concentration in the blood, its secretion rate can be estimated by deconvolution. Another example is the estimation of the blood glucose concentration from the measured interstitial glucose, which is a distorted version in time and amplitude of the real blood glucose. <sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Absorption_spectra">Absorption spectra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deconvolution&action=edit&section=9" title="Edit section: Absorption spectra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Deconvolution has been applied extensively to <a href="/wiki/Absorption_spectroscopy" title="Absorption spectroscopy">absorption spectra</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The <a href="https://de.wikipedia.org/wiki/Van-Cittert-Dekonvolution" class="extiw" title="de:Van-Cittert-Dekonvolution">Van Cittert algorithm</a> (article in German) may be used.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Fourier_transform_aspects">Fourier transform aspects</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deconvolution&action=edit&section=10" title="Edit section: Fourier transform aspects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Deconvolution maps to division in the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier co-domain</a>. This allows deconvolution to be easily applied with experimental data that are subject to a <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>. An example is <a href="/wiki/NMR_spectroscopy" class="mw-redirect" title="NMR spectroscopy">NMR spectroscopy</a> where the data are recorded in the time domain, but analyzed in the frequency domain. Division of the time-domain data by an exponential function has the effect of reducing the width of Lorentzian lines in the frequency domain. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deconvolution&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Convolution" title="Convolution">Convolution</a></li> <li><a href="/wiki/Bit_plane" title="Bit plane">Bit plane</a></li> <li><a href="/wiki/Digital_filter" title="Digital filter">Digital filter</a></li> <li><a href="/wiki/Filter_(signal_processing)" title="Filter (signal processing)">Filter (signal processing)</a></li> <li><a href="/wiki/Filter_design" title="Filter design">Filter design</a></li> <li><a href="/wiki/Minimum_phase" title="Minimum phase">Minimum phase</a></li> <li><a href="/wiki/Independent_component_analysis" title="Independent component analysis">Independent component analysis</a></li> <li><a href="/wiki/Wiener_deconvolution" title="Wiener deconvolution">Wiener deconvolution</a></li> <li><a href="/wiki/Richardson%E2%80%93Lucy_deconvolution" title="Richardson–Lucy deconvolution">Richardson–Lucy deconvolution</a></li> <li><a href="/wiki/Digital_room_correction" title="Digital room correction">Digital room correction</a></li> <li><a href="/wiki/Free_deconvolution" class="mw-redirect" title="Free deconvolution">Free deconvolution</a></li> <li><a href="/wiki/Point_spread_function" title="Point spread function">Point spread function</a></li> <li><a href="/wiki/Deblurring" title="Deblurring">Deblurring</a></li> <li><a href="/wiki/Unsharp_masking" title="Unsharp masking">Unsharp masking</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Deconvolution&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFO'Haver" class="citation web cs1">O'Haver, T. <a rel="nofollow" class="external text" href="http://www.wam.umd.edu/~toh/spectrum/Deconvolution.html">"Intro to Signal Processing - Deconvolution"</a>. University of Maryland at College Park<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-08-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Intro+to+Signal+Processing+-+Deconvolution&rft.pub=University+of+Maryland+at+College+Park&rft.aulast=O%27Haver&rft.aufirst=T.&rft_id=http%3A%2F%2Fwww.wam.umd.edu%2F~toh%2Fspectrum%2FDeconvolution.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeconvolution" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWiener1964" class="citation book cs1">Wiener, N. (1964). <i>Extrapolation, Interpolation, and Smoothing of Stationary Time Series</i>. Cambridge, Mass: MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-73005-7" title="Special:BookSources/0-262-73005-7"><bdi>0-262-73005-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Extrapolation%2C+Interpolation%2C+and+Smoothing+of+Stationary+Time+Series&rft.place=Cambridge%2C+Mass&rft.pub=MIT+Press&rft.date=1964&rft.isbn=0-262-73005-7&rft.aulast=Wiener&rft.aufirst=N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeconvolution" class="Z3988"></span></span> </li> <li id="cite_note-Pawley_2006-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Pawley_2006_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Pawley_2006_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCheng2006" class="citation book cs1">Cheng, P. C. (2006). "The Contrast Formation in Optical Microscopy". In Pawley, J. B. (ed.). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/handbookbiologic00pawl"><i>Handbook of Biological Confocal Microscopy</i></a></span> (3rd ed.). 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