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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Daubechies+wavelet%22">"Daubechies wavelet"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Daubechies+wavelet%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Daubechies+wavelet%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Daubechies+wavelet%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Daubechies+wavelet%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Daubechies+wavelet%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">August 2009</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Daubechies20LowPassHighPass2DFilterM.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Daubechies20LowPassHighPass2DFilterM.png/406px-Daubechies20LowPassHighPass2DFilterM.png" decoding="async" width="406" height="296" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Daubechies20LowPassHighPass2DFilterM.png/609px-Daubechies20LowPassHighPass2DFilterM.png 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a3/Daubechies20LowPassHighPass2DFilterM.png 2x" data-file-width="800" data-file-height="583" /></a><figcaption>Daubechies 20 2-d wavelet (Wavelet Fn X Scaling Fn)</figcaption></figure> <p>The <b>Daubechies wavelets</b>, based on the work of <a href="/wiki/Ingrid_Daubechies" title="Ingrid Daubechies">Ingrid Daubechies</a>, are a family of <a href="/wiki/Orthogonal_wavelet" title="Orthogonal wavelet">orthogonal wavelets</a> defining a <a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">discrete wavelet transform</a> and characterized by a maximal number of vanishing <a href="/wiki/Moment_(mathematics)" title="Moment (mathematics)">moments</a> for some given <a href="/wiki/Support_(mathematics)" title="Support (mathematics)">support</a>. With each wavelet type of this class, there is a scaling function (called the <i>father wavelet</i>) which generates an orthogonal <a href="/wiki/Multiresolution_analysis" title="Multiresolution analysis">multiresolution analysis</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Daubechies_wavelet&action=edit&section=1" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In general the Daubechies wavelets are chosen to have the highest number <i>A</i> of vanishing moments, (this does not imply the best smoothness) for given support width (number of coefficients) 2<i>A</i>.<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> There are two naming schemes in use, D<i>N</i> using the length or number of taps, and db<i>A</i> referring to the number of vanishing moments. So D4 and db2 are the same wavelet transform. </p><p>Among the 2<sup><i>A</i>−1</sup> possible solutions of the algebraic equations for the moment and orthogonality conditions, the one is chosen whose scaling filter has extremal phase. The wavelet transform is also easy to put into practice using the <a href="/wiki/Fast_wavelet_transform" title="Fast wavelet transform">fast wavelet transform</a>. Daubechies wavelets are widely used in solving a broad range of problems, e.g. self-similarity properties of a signal or <a href="/wiki/Fractal" title="Fractal">fractal</a> problems, signal discontinuities, etc. </p><p>The Daubechies wavelets are not defined in terms of the resulting scaling and wavelet functions; in fact, they are not possible to write down in <a href="/wiki/Closed_form_expression" class="mw-redirect" title="Closed form expression">closed form</a>. The graphs below are generated using the <a href="/wiki/Cascade_algorithm" title="Cascade algorithm">cascade algorithm</a>, a numeric technique consisting of inverse-transforming [1 0 0 0 0 ... ] an appropriate number of times. </p> <table class="wikitable"> <tbody><tr> <td>Scaling and wavelet functions </td> <td><span typeof="mw:File"><a href="/wiki/File:Daubechies4-functions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Daubechies4-functions.svg/360px-Daubechies4-functions.svg.png" decoding="async" width="360" height="270" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Daubechies4-functions.svg/540px-Daubechies4-functions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Daubechies4-functions.svg/720px-Daubechies4-functions.svg.png 2x" data-file-width="1000" data-file-height="750" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Daubechies12-functions.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/0b/Daubechies12-functions.png" decoding="async" width="360" height="252" class="mw-file-element" data-file-width="360" data-file-height="252" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Daubechies20-functions.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/2/24/Daubechies20-functions.png" decoding="async" width="360" height="252" class="mw-file-element" data-file-width="360" data-file-height="252" /></a></span> </td></tr> <tr> <td>Amplitudes of the frequency spectra of the above functions </td> <td><span typeof="mw:File"><a href="/wiki/File:Daubechies4-spectrum.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Daubechies4-spectrum.svg/360px-Daubechies4-spectrum.svg.png" decoding="async" width="360" height="270" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Daubechies4-spectrum.svg/540px-Daubechies4-spectrum.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Daubechies4-spectrum.svg/720px-Daubechies4-spectrum.svg.png 2x" data-file-width="1000" data-file-height="750" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Daubechies12-spectrum.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/08/Daubechies12-spectrum.png" decoding="async" width="360" height="252" class="mw-file-element" data-file-width="360" data-file-height="252" /></a></span> </td> <td><span typeof="mw:File"><a href="/wiki/File:Daubechies20-spectrum.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/07/Daubechies20-spectrum.png" decoding="async" width="360" height="252" class="mw-file-element" data-file-width="360" data-file-height="252" /></a></span> </td></tr></tbody></table> <p>Note that the spectra shown here are not the frequency response of the high and low pass filters, but rather the amplitudes of the continuous Fourier transforms of the scaling (blue) and wavelet (red) functions. </p><p>Daubechies orthogonal wavelets D2–D20 resp. db1–db10 are commonly used. Each wavelet has a number of <i>zero moments</i> or <i>vanishing moments</i> equal to half the number of coefficients. For example, D2 has one vanishing moment, D4 has two, etc. A vanishing moment limits the wavelets ability to represent <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> behaviour or information in a signal. For example, D2, with one vanishing moment, easily encodes polynomials of one coefficient, or constant signal components. D4 encodes polynomials with two coefficients, i.e. constant and linear signal components; and D6 encodes 3-polynomials, i.e. constant, linear and <a href="/wiki/Quadratic_polynomial" class="mw-redirect" title="Quadratic polynomial">quadratic</a> signal components. This ability to encode signals is nonetheless subject to the phenomenon of <i>scale leakage</i>, and the lack of shift-invariance, which arise from the discrete shifting operation (below) during application of the transform. Sub-sequences which represent linear, <a href="/wiki/Quadratic_polynomial" class="mw-redirect" title="Quadratic polynomial">quadratic</a> (for example) signal components are treated differently by the transform depending on whether the points align with even- or odd-numbered locations in the sequence. The lack of the important property of <a href="/wiki/Translational_invariance" class="mw-redirect" title="Translational invariance">shift-invariance</a>, has led to the development of several different versions of a <a href="/wiki/Shift_invariant_wavelet_transform" class="mw-redirect" title="Shift invariant wavelet transform">shift-invariant (discrete) wavelet transform</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Daubechies_wavelet&action=edit&section=2" title="Edit section: Construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Confusing plainlinks metadata ambox ambox-style ambox-confusing" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>may be <a href="/wiki/Wikipedia:Vagueness" title="Wikipedia:Vagueness">confusing or unclear</a> to readers</b>. In particular, there is undefined math symbols (e.g. a, p, P).<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarify the section</a>. There might be a discussion about this on <a href="/wiki/Talk:Daubechies_wavelet" title="Talk:Daubechies wavelet">the talk page</a>.</span> <span class="date-container"><i>(<span class="date">September 2019</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Both the scaling sequence (low-pass filter) and the wavelet sequence (band-pass filter) (see <a href="/wiki/Orthogonal_wavelet" title="Orthogonal wavelet">orthogonal wavelet</a> for details of this construction) will here be normalized to have sum equal 2 and sum of squares equal 2. In some applications, they are normalised to have sum <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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span>, so that both sequences and all shifts of them by an even number of coefficients are orthonormal to each other. </p><p>Using the general representation for a scaling sequence of an orthogonal discrete wavelet transform with approximation order <i>A</i>, </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 a(Z)=2^{1-A}(1+Z)^{A}p(Z),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>A</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>Z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mi>p</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(Z)=2^{1-A}(1+Z)^{A}p(Z),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f69e9781f6ee5fcdf38f90272217bced8b76ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.809ex; height:3.176ex;" alt="{\displaystyle a(Z)=2^{1-A}(1+Z)^{A}p(Z),}"></span></dd></dl> <p>with <i>N</i> = 2<i>A</i>, <i>p</i> having real coefficients, <i>p</i>(1) = 1 and deg(<i>p</i>) = <i>A</i> − 1, one can write the orthogonality condition as </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 a(Z)a\left(Z^{-1}\right)+a(-Z)a\left(-Z^{-1}\right)=4,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mi>a</mi> <mrow> <mo>(</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>4</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(Z)a\left(Z^{-1}\right)+a(-Z)a\left(-Z^{-1}\right)=4,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3207f29d245d205ff6c434387fc643c424900923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.379ex; height:3.343ex;" alt="{\displaystyle a(Z)a\left(Z^{-1}\right)+a(-Z)a\left(-Z^{-1}\right)=4,}"></span></dd></dl> <p>or equally as </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 (2-X)^{A}P(X)+X^{A}P(2-X)=2^{A}\qquad (*),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>X</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mi>P</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mspace width="2em" /> <mo stretchy="false">(</mo> <mo>∗</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2-X)^{A}P(X)+X^{A}P(2-X)=2^{A}\qquad (*),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daef1982f783ab6e7aa025ee2809305b3d6633c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.621ex; height:3.176ex;" alt="{\displaystyle (2-X)^{A}P(X)+X^{A}P(2-X)=2^{A}\qquad (*),}"></span></dd></dl> <p>with the Laurent-polynomial </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 X:={\frac {1}{2}}\left(2-Z-Z^{-1}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>−</mo> <mi>Z</mi> <mo>−</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X:={\frac {1}{2}}\left(2-Z-Z^{-1}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7689099539ea7f0e209f7292539e9bd93d480e67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.806ex; height:5.176ex;" alt="{\displaystyle X:={\frac {1}{2}}\left(2-Z-Z^{-1}\right)}"></span></dd></dl> <p>generating all symmetric sequences 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 X(-Z)=2-X(Z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>−</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(-Z)=2-X(Z).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/388697e456f97c835b925ae1bcc6512da3fc30f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.496ex; height:2.843ex;" alt="{\displaystyle X(-Z)=2-X(Z).}"></span> Further, <i>P</i>(<i>X</i>) stands for the symmetric Laurent-polynomial </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 P(X(Z))=p(Z)p\left(Z^{-1}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mi>p</mi> <mrow> <mo>(</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X(Z))=p(Z)p\left(Z^{-1}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81cd8f1ffc69ceebf8df9bb606f8dbd9a1d29e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.543ex; height:3.343ex;" alt="{\displaystyle P(X(Z))=p(Z)p\left(Z^{-1}\right).}"></span></dd></dl> <p>Since </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 X(e^{iw})=1-\cos(w)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>w</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(e^{iw})=1-\cos(w)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a46fdf2f71dcc745bf323117105a0b43b903db8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.535ex; height:3.176ex;" alt="{\displaystyle X(e^{iw})=1-\cos(w)}"></span></dd> <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 p(e^{iw})p(e^{-iw})=|p(e^{iw})|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>w</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mi>p</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>w</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>w</mi> </mrow> </msup> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(e^{iw})p(e^{-iw})=|p(e^{iw})|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08cddbf5cf6aac19c71e3e64f8673e39d563ae24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:24.93ex; height:3.343ex;" alt="{\displaystyle p(e^{iw})p(e^{-iw})=|p(e^{iw})|^{2}}"></span></dd></dl> <p><i>P</i> takes nonnegative values on the segment [0,2]. </p><p>Equation (*) has one minimal solution for each <i>A</i>, which can be obtained by division in the ring of truncated <a href="/wiki/Power_series" title="Power series">power series</a> in <i>X</i>, </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 P_{A}(X)=\sum _{k=0}^{A-1}{\binom {A+k-1}{A-1}}2^{-k}X^{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>A</mi> <mo>+</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mi>A</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{A}(X)=\sum _{k=0}^{A-1}{\binom {A+k-1}{A-1}}2^{-k}X^{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b2ef986b8178d705eb3fc9161c7cb5b1828eb23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:34.068ex; height:7.509ex;" alt="{\displaystyle P_{A}(X)=\sum _{k=0}^{A-1}{\binom {A+k-1}{A-1}}2^{-k}X^{k}.}"></span></dd></dl> <p>Obviously, this has positive values on (0,2). </p><p>The homogeneous equation for (*) is antisymmetric about <i>X</i> = 1 and has thus the general solution </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 X^{A}(X-1)R\left((X-1)^{2}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{A}(X-1)R\left((X-1)^{2}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09be942338b00e32965a9bd3edfac7d554b7197a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.415ex; height:3.343ex;" alt="{\displaystyle X^{A}(X-1)R\left((X-1)^{2}\right),}"></span></dd></dl> <p>with <i>R</i> some polynomial with real coefficients. That the sum </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 P(X)=P_{A}(X)+X^{A}(X-1)R\left((X-1)^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>R</mi> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X)=P_{A}(X)+X^{A}(X-1)R\left((X-1)^{2}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5b152cc558207b4400dfcf8fb3db5593c4db2f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.601ex; height:3.343ex;" alt="{\displaystyle P(X)=P_{A}(X)+X^{A}(X-1)R\left((X-1)^{2}\right)}"></span></dd></dl> <p>shall be nonnegative on the interval [0,2] translates into a set of linear restrictions on the coefficients of <i>R</i>. The values of <i>P</i> on the interval [0,2] are bounded by some quantity <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 4^{A-r},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>−</mo> <mi>r</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4^{A-r},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e5b64a615dcd7c1e1ac87e36f0416ec6cd769e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.294ex; height:3.009ex;" alt="{\displaystyle 4^{A-r},}"></span> maximizing <i>r</i> results in a linear program with infinitely many inequality conditions. </p><p>To solve </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 P(X(Z))=p(Z)p\left(Z^{-1}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mi>p</mi> <mrow> <mo>(</mo> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X(Z))=p(Z)p\left(Z^{-1}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e93eee947e34548fa07fc5c10d4efe9e7fac740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.509ex; height:3.343ex;" alt="{\displaystyle P(X(Z))=p(Z)p\left(Z^{-1}\right)}"></span></dd></dl> <p>for <i>p</i> one uses a technique called spectral factorization resp. Fejér-Riesz-algorithm. The polynomial <i>P</i>(<i>X</i>) splits into linear factors </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 P(X)=(X-\mu _{1})\cdots (X-\mu _{N}),\qquad N=A+1+2\deg(R).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <mi>N</mi> <mo>=</mo> <mi>A</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(X)=(X-\mu _{1})\cdots (X-\mu _{N}),\qquad N=A+1+2\deg(R).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb9040d7d0c1ccbf3a43f5cebf77e6fcaa4b57c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.624ex; height:2.843ex;" alt="{\displaystyle P(X)=(X-\mu _{1})\cdots (X-\mu _{N}),\qquad N=A+1+2\deg(R).}"></span></dd></dl> <p>Each linear factor represents a Laurent-polynomial </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 X(Z)-\mu =-{\frac {1}{2}}Z+1-\mu -{\frac {1}{2}}Z^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>μ</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>Z</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>μ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(Z)-\mu =-{\frac {1}{2}}Z+1-\mu -{\frac {1}{2}}Z^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d043d8badc0383756811261dfba74958ffd8519" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.422ex; height:5.176ex;" alt="{\displaystyle X(Z)-\mu =-{\frac {1}{2}}Z+1-\mu -{\frac {1}{2}}Z^{-1}}"></span></dd></dl> <p>that can be factored into two linear factors. One can assign either one of the two linear factors to <i>p</i>(<i>Z</i>), thus one obtains 2<sup><i>N</i></sup> possible solutions. For extremal phase one chooses the one that has all complex roots of <i>p</i>(<i>Z</i>) inside or on the unit circle and is thus real. </p><p>For Daubechies wavelet transform, a pair of linear filters is used. Each filter of the pair should be a <a href="/wiki/Quadrature_mirror_filter" title="Quadrature mirror filter">quadrature mirror filter</a>. Solving the coefficient of the linear filter <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 c_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01acb7953ba52c2aa44264b5d0f8fd223aa178a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}"></span> using the quadrature mirror filter property results in the following solution for the coefficient values for filter of order 4. </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 c_{0}={\frac {1+{\sqrt {3}}}{4{\sqrt {2}}}},\quad c_{1}={\frac {3+{\sqrt {3}}}{4{\sqrt {2}}}},\quad c_{2}={\frac {3-{\sqrt {3}}}{4{\sqrt {2}}}},\quad c_{3}={\frac {1-{\sqrt {3}}}{4{\sqrt {2}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{0}={\frac {1+{\sqrt {3}}}{4{\sqrt {2}}}},\quad c_{1}={\frac {3+{\sqrt {3}}}{4{\sqrt {2}}}},\quad c_{2}={\frac {3-{\sqrt {3}}}{4{\sqrt {2}}}},\quad c_{3}={\frac {1-{\sqrt {3}}}{4{\sqrt {2}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88f906f947d021d6dc73d1f28f60fb9b420882bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:63.104ex; height:6.843ex;" alt="{\displaystyle c_{0}={\frac {1+{\sqrt {3}}}{4{\sqrt {2}}}},\quad c_{1}={\frac {3+{\sqrt {3}}}{4{\sqrt {2}}}},\quad c_{2}={\frac {3-{\sqrt {3}}}{4{\sqrt {2}}}},\quad c_{3}={\frac {1-{\sqrt {3}}}{4{\sqrt {2}}}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="The_scaling_sequences_of_lowest_approximation_order">The scaling sequences of lowest approximation order</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Daubechies_wavelet&action=edit&section=3" title="Edit section: The scaling sequences of lowest approximation order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Below are the coefficients for the scaling functions for D2-20. The wavelet coefficients are derived by reversing the order of the <a href="/wiki/Wavelet#Scaling_function" title="Wavelet">scaling function</a> coefficients and then reversing the sign of every second one, (i.e., D4 wavelet <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 \approx }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>≈</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \approx }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58f4c2b73283ce8a5ad28fb3746f2a8c998789" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.808ex; height:1.509ex;" alt="{\displaystyle \approx }"></span> {−0.1830127, −0.3169873, 1.1830127, −0.6830127}). Mathematically, this looks like <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 b_{k}=(-1)^{k}a_{N-1-k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{k}=(-1)^{k}a_{N-1-k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e37e2a7e85911efa5351711e54e1e74ff8dfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.21ex; height:3.176ex;" alt="{\displaystyle b_{k}=(-1)^{k}a_{N-1-k}}"></span> where <i>k</i> is the coefficient index, <i>b</i> is a coefficient of the wavelet sequence and <i>a</i> a coefficient of the scaling sequence. <i>N</i> is the wavelet index, i.e., 2 for D2. </p> <div style="font-size:83%"> <table class="wikitable"> <caption><b>Orthogonal Daubechies coefficients (normalized to have sum 2)</b> </caption> <tbody><tr> <th>D2 (<a href="/wiki/Haar_wavelet" title="Haar wavelet">Haar</a>) </th> <th>D4 </th> <th>D6 </th> <th>D8 </th> <th>D10 </th> <th>D12 </th> <th>D14 </th> <th>D16 </th> <th>D18 </th> <th>D20 </th></tr> <tr> <td>1 </td> <td>0.6830127 </td> <td>0.47046721 </td> <td>0.32580343 </td> <td>0.22641898 </td> <td>0.15774243 </td> <td>0.11009943 </td> <td>0.07695562 </td> <td>0.05385035 </td> <td>0.03771716 </td></tr> <tr> <td>1 </td> <td>1.1830127 </td> <td>1.14111692 </td> <td>1.01094572 </td> <td>0.85394354 </td> <td>0.69950381 </td> <td>0.56079128 </td> <td>0.44246725 </td> <td>0.34483430 </td> <td>0.26612218 </td></tr> <tr> <td> </td> <td>0.3169873 </td> <td>0.650365 </td> <td>0.89220014 </td> <td>1.02432694 </td> <td>1.06226376 </td> <td>1.03114849 </td> <td>0.95548615 </td> <td>0.85534906 </td> <td>0.74557507 </td></tr> <tr> <td> </td> <td>−0.1830127 </td> <td>−0.19093442 </td> <td>−0.03957503 </td> <td>0.19576696 </td> <td>0.44583132 </td> <td>0.66437248 </td> <td>0.82781653 </td> <td>0.92954571 </td> <td>0.97362811 </td></tr> <tr> <td> </td> <td> </td> <td>−0.12083221 </td> <td>−0.26450717 </td> <td>−0.34265671 </td> <td>−0.31998660 </td> <td>−0.20351382 </td> <td>−0.02238574 </td> <td>0.18836955 </td> <td>0.39763774 </td></tr> <tr> <td> </td> <td> </td> <td>0.0498175 </td> <td>0.0436163 </td> <td>−0.04560113 </td> <td>−0.18351806 </td> <td>−0.31683501 </td> <td>−0.40165863 </td> <td>−0.41475176 </td> <td>−0.35333620 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>0.0465036 </td> <td>0.10970265 </td> <td>0.13788809 </td> <td>0.1008467 </td> <td>6.68194092 × 10<sup>−4</sup> </td> <td>−0.13695355 </td> <td>−0.27710988 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td>−0.01498699 </td> <td>−0.00882680 </td> <td>0.03892321 </td> <td>0.11400345 </td> <td>0.18207636 </td> <td>0.21006834 </td> <td>0.18012745 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td>−0.01779187 </td> <td>−0.04466375 </td> <td>−0.05378245 </td> <td>−0.02456390 </td> <td>0.043452675 </td> <td>0.13160299 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td>4.71742793 × 10<sup>−3</sup> </td> <td>7.83251152 × 10<sup>−4</sup> </td> <td>−0.02343994 </td> <td>−0.06235021 </td> <td>−0.09564726 </td> <td>−0.10096657 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>6.75606236 × 10<sup>−3</sup> </td> <td>0.01774979 </td> <td>0.01977216 </td> <td>3.54892813 × 10<sup>−4</sup> </td> <td>−0.04165925 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>−1.52353381 × 10<sup>−3</sup> </td> <td>6.07514995 × 10<sup>−4</sup> </td> <td>0.01236884 </td> <td>0.03162417 </td> <td>0.04696981 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>−2.54790472 × 10<sup>−3</sup> </td> <td>−6.88771926 × 10<sup>−3</sup> </td> <td>−6.67962023 × 10<sup>−3</sup> </td> <td>5.10043697 × 10<sup>−3</sup> </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>5.00226853 × 10<sup>−4</sup> </td> <td>−5.54004549 × 10<sup>−4</sup> </td> <td>−6.05496058 × 10<sup>−3</sup> </td> <td>−0.01517900 </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>9.55229711 × 10<sup>−4</sup> </td> <td>2.61296728 × 10<sup>−3</sup> </td> <td>1.97332536 × 10<sup>−3</sup> </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>−1.66137261 × 10<sup>−4</sup> </td> <td>3.25814671 × 10<sup>−4</sup> </td> <td>2.81768659 × 10<sup>−3</sup> </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>−3.56329759 × 10<sup>−4</sup> </td> <td>−9.69947840 × 10<sup>−4</sup> </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>5.5645514 × 10<sup>−5</sup> </td> <td>−1.64709006 × 10<sup>−4</sup> </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>1.32354367 × 10<sup>−4</sup> </td></tr> <tr> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td> </td> <td>−1.875841 × 10<sup>−5</sup> </td></tr></tbody></table> </div> <p>Parts of the construction are also used to derive the biorthogonal <a href="/wiki/Cohen%E2%80%93Daubechies%E2%80%93Feauveau_wavelet" title="Cohen–Daubechies–Feauveau wavelet">Cohen–Daubechies–Feauveau wavelets</a> (CDFs). </p> <div class="mw-heading mw-heading2"><h2 id="Implementation">Implementation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Daubechies_wavelet&action=edit&section=4" title="Edit section: Implementation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While software such as <a href="/wiki/Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> supports Daubechies wavelets directly<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> a basic implementation is possible in <a href="/wiki/MATLAB" title="MATLAB">MATLAB</a> (in this case, Daubechies 4). This implementation uses periodization to handle the problem of finite length signals. Other, more sophisticated methods are available, but often it is not necessary to use these as it only affects the very ends of the transformed signal. The periodization is accomplished in the forward transform directly in MATLAB vector notation, and the inverse transform by using the <code>circshift()</code> function: </p> <div class="mw-heading mw-heading3"><h3 id="Transform,_D4"><span id="Transform.2C_D4"></span>Transform, D4</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Daubechies_wavelet&action=edit&section=5" title="Edit section: Transform, D4"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is assumed that <i>S</i>, a column vector with an even number of elements, has been pre-defined as the signal to be analyzed. Note that the D4 coefficients are [1 + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span>, 3 + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span>, 3 − <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span>, 1 − <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span>]/4. </p> <div class="mw-highlight mw-highlight-lang-matlab mw-content-ltr" dir="ltr"><pre><span></span><span class="n">N</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">length</span><span class="p">(</span><span class="n">S</span><span class="p">);</span> <span class="n">sqrt3</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="nb">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">);</span> <span class="n">s_odd</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="o">-</span><span class="mi">1</span><span class="p">);</span> <span class="n">s_even</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="p">);</span> <span class="n">s</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">sqrt3</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">s_odd</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="n">sqrt3</span><span class="p">)</span><span class="o">*</span><span class="n">s_even</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="mi">3</span><span class="o">-</span><span class="n">sqrt3</span><span class="p">)</span><span class="o">*</span><span class="p">[</span><span class="n">s_odd</span><span class="p">(</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">);</span><span class="n">s_odd</span><span class="p">(</span><span class="mi">1</span><span class="p">)]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">sqrt3</span><span class="p">)</span><span class="o">*</span><span class="p">[</span><span class="n">s_even</span><span class="p">(</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">);</span><span class="n">s_even</span><span class="p">(</span><span class="mi">1</span><span class="p">)];</span> <span class="n">d</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">sqrt3</span><span class="p">)</span><span class="o">*</span><span class="p">[</span><span class="n">s_odd</span><span class="p">(</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">);</span><span class="n">s_odd</span><span class="p">(</span><span class="mi">1</span><span class="p">:</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">)]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="n">sqrt3</span><span class="o">-</span><span class="mi">3</span><span class="p">)</span><span class="o">*</span><span class="p">[</span><span class="n">s_even</span><span class="p">(</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="p">);</span><span class="n">s_even</span><span class="p">(</span><span class="mi">1</span><span class="p">:</span><span class="n">N</span><span class="o">/</span><span class="mi">2</span><span class="o">-</span><span class="mi">1</span><span class="p">)]</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="mi">3</span><span class="o">+</span><span class="n">sqrt3</span><span class="p">)</span><span class="o">*</span><span class="n">s_odd</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">-</span><span class="n">sqrt3</span><span class="p">)</span><span class="o">*</span><span class="n">s_even</span> <span class="n">s</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="nb">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">));</span> <span class="n">d</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="p">(</span><span class="mi">4</span><span class="o">*</span><span class="nb">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">));</span> </pre></div> <div class="mw-heading mw-heading3"><h3 id="Inverse_transform,_D4"><span id="Inverse_transform.2C_D4"></span>Inverse transform, D4</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Daubechies_wavelet&action=edit&section=6" title="Edit section: Inverse transform, D4"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-highlight mw-highlight-lang-matlab mw-content-ltr" dir="ltr"><pre><span></span><span class="n">d1</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">((</span><span class="nb">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nb">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">));</span> <span class="n">s2</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="p">((</span><span class="nb">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="nb">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">));</span> <span class="n">s1</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">s2</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nb">circshift</span><span class="p">(</span><span class="n">d1</span><span class="p">,</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span> <span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">d1</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="nb">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">4</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">s1</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="p">(</span><span class="nb">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="o">/</span><span class="w"> </span><span class="mi">4</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="nb">circshift</span><span class="p">(</span><span class="n">s1</span><span class="p">,</span><span class="w"> </span><span class="mi">1</span><span class="p">);</span> <span class="n">S</span><span class="p">(</span><span class="mi">1</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">s1</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="nb">sqrt</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">S</span><span class="p">(</span><span class="mi">2</span><span class="p">:</span><span class="mi">2</span><span class="p">:</span><span class="n">N</span><span class="p">);</span> </pre></div> <div class="mw-heading mw-heading2"><h2 id="Binomial-QMF">Binomial-QMF</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Daubechies_wavelet&action=edit&section=7" title="Edit section: Binomial-QMF"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It was shown by <a href="/wiki/Ali_Akansu" title="Ali Akansu">Ali Akansu</a> in 1990 that the <a href="/wiki/Binomial_QMF" title="Binomial QMF">binomial quadrature mirror filter bank</a> (binomial QMF) is identical to the Daubechies wavelet filter, and its performance was ranked among known subspace solutions from a discrete-time signal processing perspective.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><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> It was an extension of the prior work on <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a> and <a href="/wiki/Hermite_polynomials" title="Hermite polynomials">Hermite polynomials</a> that led to the development of the Modified Hermite Transformation (MHT) in 1987.<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><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> The magnitude square functions of <a href="/wiki/Binomial-QMF" class="mw-redirect" title="Binomial-QMF">Binomial-QMF</a> filters are the unique maximally flat functions in a two-band perfect reconstruction QMF (PR-QMF) design formulation that is related to the wavelet regularity in the continuous domain.<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><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> </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=Daubechies_wavelet&action=edit&section=8" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The application of Daubechies wavelet transform as a watermarking scheme has been proved effective. This approach operates in a proficient multi-resolution frequency domain, enabling the incorporation of an encrypted digital logo in the format of QR codes.<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></li></ul> <ul><li>Daubechies wavelet approximation can be used to analyze Griffith crack behavior in nonlocal magneto-elastic horizontally shear (SH) wave propagation within a finite-thickness, infinitely long homogeneous isotropic strip. <sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>Daubechies wavelet cepstral coefficients can be useful in the context of Parkinson's disease detection. Daubechies wavelets, known for their efficient multi-resolution analysis, are utilized to extract cepstral features from vocal signal data. These wavelet-based coefficients can act as discriminative features for accurately identifying patterns indicative of Parkinson's disease, offering a novel approach to diagnostic methodologies.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>When it comes to analysis and detection of Community Acquired Pneumonia (CAP), Complex Daubechies wavelets can be used to identify intricate details of the CAP affected areas in infected lungs to produce accurate results. <sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>The elastohydrodynamic lubrication problem involves the study of lubrication regimes in which the deformation of the contacting surfaces significantly influences the lubricating film. Daubechies wavelets can address the challenges associated with accurately modeling and simulating such intricate lubrication phenomena. Daubechies wavelets allows for a more detailed and refined exploration of the interactions between the lubricant and the contacting surfaces.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>Daubechies Wavelet can extract intricate details and features from the vibroacoustic signals, offering a comprehensive diagnostic approach for evaluating the condition and performance of diesel engines in combine harvesters. The Daubechies Wavelet spectrum serves as a powerful analytical tool, allowing the researchers to identify patterns, anomalies, and characteristic signatures within the signals associated with different engine conditions. This detailed spectral analysis aids in enhancing the accuracy of diagnostic assessments, enabling a more nuanced understanding of the vibrational and acoustic characteristics indicative of engine health or potential issues.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>In practical terms, the Daubechies wavelets facilitate a finely tuned examination of the temporal and spatial characteristics of dynamic waves within elastic materials. This approach enables a more nuanced understanding of how elastic solids respond to varying dynamic conditions over time. The integration of Daubechies wavelets into the finite wavelet domain method likely contributes to a more versatile and robust analytical framework for studying transient dynamic waves in elastic solids.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li>The brachistochrone problem can be formulated and expressed as a variational problem, emphasizing the importance of finding the optimal curve that minimizes the time of descent. By introducing Daubechies wavelets into the mathematical framework, scaling functions associated with these wavelets can construct an approximation of the optimal curve. Daubechies wavelets, with their ability to capture both high and low-frequency components of a function, prove instrumental in achieving a detailed representation of the brachistochrone curve.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></li></ul> <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=Daubechies_wavelet&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Fast_wavelet_transform" title="Fast wavelet transform">Fast wavelet transform</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=Daubechies_wavelet&action=edit&section=10" 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 mw-references-columns"><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">I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992, p. 194.</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"> <a rel="nofollow" class="external text" href="http://reference.wolfram.com/mathematica/ref/DaubechiesWavelet.html">Daubechies Wavelet in Mathematica. Note that in there <i>n</i> is <i>n</i>/2 from the text.</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">A.N. Akansu, <a rel="nofollow" class="external text" href="http://web.njit.edu/~ali/NJITSYMP1990/AkansuNJIT1STWAVELETSSYMPAPRIL301990.pdf">An Efficient QMF-Wavelet Structure</a> (Binomial-QMF Daubechies Wavelets), Proc. 1st NJIT Symposium on Wavelets, April 1990.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">A.N. Akansu, R.A. Haddad and H. Caglar, <a rel="nofollow" class="external text" href="http://web.njit.edu/~akansu/PAPERS/Akansu-BinomialQMF-SPIESept1990.pdf">Perfect Reconstruction Binomial QMF-Wavelet Transform</a>, Proc. SPIE Visual Communications and Image Processing, pp. 609–618, vol. 1360, Lausanne, Sept. 1990.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">A.N. Akansu, Statistical Adaptive Transform Coding of Speech Signals. Ph.D. Thesis. Polytechnic University, 1987.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">R.A. Haddad and A.N. Akansu, "A New Orthogonal Transform for Signal Coding," IEEE Transactions on Acoustics, Speech and Signal Processing, vol.36, no.9, pp. 1404-1411, September 1988.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">H. Caglar and A.N. Akansu, <a rel="nofollow" class="external text" href="http://web.njit.edu/~akansu/PAPERS/CaglarAkansuBernstein.pdf">A Generalized Parametric PR-QMF Design Technique Based on Bernstein Polynomial Approximation</a>, IEEE Trans. Signal Process., pp. 2314–2321, July 1993.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">O. Herrmann, <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/1083275">On the Approximation Problem in Nonrecursive Digital Filter Design</a>, IEEE Trans. Circuit Theory, vol CT-18, no. 3, pp. 411–413, May 1971.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><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="CITEREFUmerMajidSyeda2019" class="citation journal cs1">Umer, Aziz Waqas; Majid, Khan; Syeda, Iram Batool (18 December 2019). <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/s11042-019-08570-5">"A new watermarking scheme based on Daubechies wavelet and chaotic map for quick response code images"</a>. <i>Multimedia Tools and Applications</i>. <b>79</b> (9–10): 6891–6914. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11042-019-08570-5">10.1007/s11042-019-08570-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Multimedia+Tools+and+Applications&rft.atitle=A+new+watermarking+scheme+based+on+Daubechies+wavelet+and+chaotic+map+for+quick+response+code+images&rft.volume=79&rft.issue=9%E2%80%9310&rft.pages=6891-6914&rft.date=2019-12-18&rft_id=info%3Adoi%2F10.1007%2Fs11042-019-08570-5&rft.aulast=Umer&rft.aufirst=Aziz+Waqas&rft.au=Majid%2C+Khan&rft.au=Syeda%2C+Iram+Batool&rft_id=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs11042-019-08570-5&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADaubechies+wavelet" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJyotirmoyNantuSoumen2023" class="citation journal cs1">Jyotirmoy, Mouley; 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font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Daubechies_wavelets" class="extiw" title="commons:Category:Daubechies wavelets">Daubechies wavelets</a></span>.</div></div> </div> <ul><li>Ingrid Daubechies: <i>Ten Lectures on Wavelets</i>, SIAM 1992.</li> <li><a rel="nofollow" class="external text" href="http://web.njit.edu/~akansu/s1.htm">Proc. 1st NJIT Symposium on Wavelets, Subbands and Transforms, April 1990.</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAkansuHaddad1992" class="citation cs2">Akansu, Ali N.; Haddad, Richard A. (1992), <i>Multiresolution Signal Decomposition: Transforms, Subbands, and Wavelets</i>, Boston, MA: Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-047141-6" title="Special:BookSources/978-0-12-047141-6"><bdi>978-0-12-047141-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multiresolution+Signal+Decomposition%3A+Transforms%2C+Subbands%2C+and+Wavelets&rft.place=Boston%2C+MA&rft.pub=Academic+Press&rft.date=1992&rft.isbn=978-0-12-047141-6&rft.aulast=Akansu&rft.aufirst=Ali+N.&rft.au=Haddad%2C+Richard+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADaubechies+wavelet" class="Z3988"></span></li> <li>A.N. Akansu, <a rel="nofollow" class="external text" href="https://web.njit.edu/~akansu/PAPERS/Akansu-FilterBanksWaveletsSP-SPIEOct1993.pdf">Filter Banks and Wavelets in Signal Processing: A Critical Review</a>, Proc. SPIE Video Communications and PACS for Medical Applications (Invited Paper), pp. 330-341, vol. 1977, Berlin, Oct. 1993.</li> <li><a rel="nofollow" class="external text" href="http://mate.dm.uba.ar/~hafg/">Carlos Cabrelli, Ursula Molter</a>: "Generalized Self-similarity", <i>Journal of Mathematical Analysis and Applications</i>, 230: 251–260, 1999.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20050421032443/http://etd.lib.fsu.edu/theses/available/etd-11242003-185039/">Hardware implementation of wavelets</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Daubechies_wavelets">"Daubechies wavelets"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Daubechies+wavelets&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DDaubechies_wavelets&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADaubechies+wavelet" class="Z3988"></span>.</li> <li>I. Kaplan, <a rel="nofollow" class="external text" href="http://www.bearcave.com/misl/misl_tech/wavelets/daubechies/index.html">The Daubechies D4 Wavelet Transform</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJensenla_Cour-Harbo2001" class="citation book cs1">Jensen; la Cour-Harbo (2001). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081202170023/http://www.control.auc.dk/~alc/ripples.html"><i>Ripples in Mathematics</i></a>. Berlin: Springer. pp. 157–160. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-41662-5" title="Special:BookSources/3-540-41662-5"><bdi>3-540-41662-5</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://www.control.auc.dk/~alc/ripples.html">the original</a> on 2008-12-02<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-12-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ripples+in+Mathematics&rft.place=Berlin&rft.pages=157-160&rft.pub=Springer&rft.date=2001&rft.isbn=3-540-41662-5&rft.au=Jensen&rft.au=la+Cour-Harbo&rft_id=http%3A%2F%2Fwww.control.auc.dk%2F~alc%2Fripples.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADaubechies+wavelet" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://sites.google.com/site/jackieneoshen/">Jianhong (Jackie) Shen</a> and <a href="/wiki/Gilbert_Strang" title="Gilbert Strang">Gilbert Strang</a>, <i>Applied and Computational Harmonic Analysis</i>, <b>5</b>(3), <a rel="nofollow" class="external text" href="https://doi.org/10.1006/acha.1997.0234"><i>Asymptotics of Daubechies Filters, Scaling Functions, and Wavelets</i></a>.</li></ul> <div class="navbox-styles"><style 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abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Compression_methods" title="Template:Compression methods"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Compression_methods" title="Template talk:Compression methods"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Compression_methods" title="Special:EditPage/Template:Compression methods"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Data_compression_methods" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_compression" title="Data compression">Data compression</a> methods</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lossless_compression" title="Lossless compression">Lossless</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Entropy_coding" title="Entropy coding">Entropy type</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adaptive_coding" title="Adaptive coding">Adaptive coding</a></li> <li><a href="/wiki/Arithmetic_coding" title="Arithmetic coding">Arithmetic</a></li> <li><a href="/wiki/Asymmetric_numeral_systems" title="Asymmetric numeral systems">Asymmetric numeral systems</a></li> <li><a href="/wiki/Golomb_coding" title="Golomb coding">Golomb</a></li> <li><a href="/wiki/Huffman_coding" title="Huffman coding">Huffman</a> <ul><li><a href="/wiki/Adaptive_Huffman_coding" title="Adaptive Huffman coding">Adaptive</a></li> <li><a href="/wiki/Canonical_Huffman_code" title="Canonical Huffman code">Canonical</a></li> <li><a href="/wiki/Modified_Huffman_coding" title="Modified Huffman coding">Modified</a></li></ul></li> <li><a href="/wiki/Range_coding" title="Range coding">Range</a></li> <li><a href="/wiki/Shannon_coding" title="Shannon coding">Shannon</a></li> <li><a href="/wiki/Shannon%E2%80%93Fano_coding" title="Shannon–Fano coding">Shannon–Fano</a></li> <li><a href="/wiki/Shannon%E2%80%93Fano%E2%80%93Elias_coding" title="Shannon–Fano–Elias coding">Shannon–Fano–Elias</a></li> <li><a href="/wiki/Tunstall_coding" title="Tunstall coding">Tunstall</a></li> <li><a href="/wiki/Unary_coding" title="Unary coding">Unary</a></li> <li><a href="/wiki/Universal_code_(data_compression)" title="Universal code (data compression)">Universal</a> <ul><li><a href="/wiki/Exponential-Golomb_coding" title="Exponential-Golomb coding">Exp-Golomb</a></li> <li><a href="/wiki/Fibonacci_coding" title="Fibonacci coding">Fibonacci</a></li> <li><a href="/wiki/Elias_gamma_coding" title="Elias gamma coding">Gamma</a></li> <li><a href="/wiki/Levenshtein_coding" title="Levenshtein coding">Levenshtein</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Dictionary_coder" title="Dictionary coder">Dictionary type</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Byte_pair_encoding" title="Byte pair encoding">Byte pair encoding</a></li> <li><a href="/wiki/LZ77_and_LZ78" title="LZ77 and LZ78">Lempel–Ziv</a> <ul><li><a href="/wiki/842_(compression_algorithm)" title="842 (compression algorithm)">842</a></li> <li><a href="/wiki/LZ4_(compression_algorithm)" title="LZ4 (compression algorithm)">LZ4</a></li> <li><a href="/wiki/LZJB" class="mw-redirect" title="LZJB">LZJB</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Oberhumer" title="Lempel–Ziv–Oberhumer">LZO</a></li> <li><a href="/wiki/LZRW" title="LZRW">LZRW</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Storer%E2%80%93Szymanski" title="Lempel–Ziv–Storer–Szymanski">LZSS</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Welch" title="Lempel–Ziv–Welch">LZW</a></li> <li><a href="/wiki/LZWL" title="LZWL">LZWL</a></li> <li><a href="/wiki/Snappy_(compression)" title="Snappy (compression)">Snappy</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Other types</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Burrows%E2%80%93Wheeler_transform" title="Burrows–Wheeler transform">BWT</a></li> <li><a href="/wiki/Context_tree_weighting" title="Context tree weighting">CTW</a></li> <li><a href="/wiki/Context_mixing" title="Context mixing">CM</a></li> <li><a href="/wiki/Delta_encoding" title="Delta encoding">Delta</a> <ul><li><a href="/wiki/Incremental_encoding" title="Incremental encoding">Incremental</a></li></ul></li> <li><a href="/wiki/Dynamic_Markov_compression" title="Dynamic Markov compression">DMC</a></li> <li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a></li> <li><a href="/wiki/Grammar-based_code" title="Grammar-based code">Grammar</a> <ul><li><a href="/wiki/Re-Pair" title="Re-Pair">Re-Pair</a></li> <li><a href="/wiki/Sequitur_algorithm" title="Sequitur algorithm">Sequitur</a></li></ul></li> <li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">LDCT</a></li> <li><a href="/wiki/Move-to-front_transform" title="Move-to-front transform">MTF</a></li> <li><a href="/wiki/PAQ" title="PAQ">PAQ</a></li> <li><a href="/wiki/Prediction_by_partial_matching" title="Prediction by partial matching">PPM</a></li> <li><a href="/wiki/Run-length_encoding" title="Run-length encoding">RLE</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Hybrid</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li>LZ77 + Huffman <ul><li><a href="/wiki/Deflate" title="Deflate">Deflate</a></li> <li><a href="/wiki/LZX" title="LZX">LZX</a></li> <li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Stac" title="Lempel–Ziv–Stac">LZS</a></li></ul></li> <li>LZ77 + ANS <ul><li><a href="/wiki/LZFSE" title="LZFSE">LZFSE</a></li></ul></li> <li>LZ77 + Huffman + ANS <ul><li><a href="/wiki/Zstd" title="Zstd">Zstandard</a></li></ul></li> <li>LZ77 + Huffman + context <ul><li><a href="/wiki/Brotli" title="Brotli">Brotli</a></li></ul></li> <li>LZSS + Huffman <ul><li><a href="/wiki/LHA_(file_format)" title="LHA (file format)">LHA/LZH</a></li></ul></li> <li>LZ77 + Range <ul><li><a href="/wiki/Lempel%E2%80%93Ziv%E2%80%93Markov_chain_algorithm" title="Lempel–Ziv–Markov chain algorithm">LZMA</a></li> <li>LZHAM</li></ul></li> <li>RLE + BWT + MTF + Huffman <ul><li><a href="/wiki/Bzip2" title="Bzip2">bzip2</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Lossy_compression" title="Lossy compression">Lossy</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Transform_coding" title="Transform coding">Transform type</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">Discrete cosine transform</a> <ul><li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">DCT</a></li> <li><a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">MDCT</a></li></ul></li> <li><a href="/wiki/Discrete_sine_transform" title="Discrete sine transform">DST</a></li> <li><a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a></li> <li><a href="/wiki/Wavelet_transform" title="Wavelet transform">Wavelet</a> <ul><li><a class="mw-selflink selflink">Daubechies</a></li> <li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li> <li><a href="/wiki/Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Predictive type</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a> <ul><li><a href="/wiki/Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li></ul></li> <li><a href="/wiki/Linear_predictive_coding" title="Linear predictive coding">LPC</a> <ul><li><a href="/wiki/Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li> <li><a href="/wiki/Code-excited_linear_prediction" title="Code-excited linear prediction">CELP</a></li> <li><a href="/wiki/Log_area_ratio" title="Log area ratio">LAR</a></li> <li><a href="/wiki/Line_spectral_pairs" title="Line spectral pairs">LSP</a></li> <li><a href="/wiki/Warped_linear_predictive_coding" title="Warped linear predictive coding">WLPC</a></li></ul></li> <li>Motion <ul><li><a href="/wiki/Motion_compensation" title="Motion compensation">Compensation</a></li> <li><a href="/wiki/Motion_estimation" title="Motion estimation">Estimation</a></li> <li><a href="/wiki/Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li> <li><a href="/wiki/Psychoacoustics" title="Psychoacoustics">Psychoacoustic</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Data_compression#Audio" title="Data compression">Audio</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit_rate" title="Bit rate">Bit rate</a> <ul><li><a href="/wiki/Average_bitrate" title="Average bitrate">ABR</a></li> <li><a href="/wiki/Constant_bitrate" title="Constant bitrate">CBR</a></li> <li><a href="/wiki/Variable_bitrate" title="Variable bitrate">VBR</a></li></ul></li> <li><a href="/wiki/Companding" title="Companding">Companding</a></li> <li><a href="/wiki/Convolution" title="Convolution">Convolution</a></li> <li><a href="/wiki/Dynamic_range" title="Dynamic range">Dynamic range</a></li> <li><a href="/wiki/Latency_(audio)" title="Latency (audio)">Latency</a></li> <li><a href="/wiki/Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon theorem</a></li> <li><a href="/wiki/Sampling_(signal_processing)" title="Sampling (signal processing)">Sampling</a></li> <li><a href="/wiki/Silence_compression" title="Silence compression">Silence compression</a></li> <li><a href="/wiki/Sound_quality" title="Sound quality">Sound quality</a></li> <li><a href="/wiki/Speech_coding" title="Speech coding">Speech coding</a></li> <li><a href="/wiki/Sub-band_coding" title="Sub-band coding">Sub-band coding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Audio_codec" title="Audio codec">Codec</a> parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/A-law_algorithm" title="A-law algorithm">A-law</a></li> <li><a href="/wiki/%CE%9C-law_algorithm" title="Μ-law algorithm">μ-law</a></li> <li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a> <ul><li><a href="/wiki/Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li> <li><a href="/wiki/Delta_modulation" title="Delta modulation">DM</a></li></ul></li> <li><a href="/wiki/Fourier_transform" title="Fourier transform">FT</a> <ul><li><a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a></li></ul></li> <li><a href="/wiki/Linear_predictive_coding" title="Linear predictive coding">LPC</a> <ul><li><a href="/wiki/Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li> <li><a href="/wiki/Code-excited_linear_prediction" title="Code-excited linear prediction">CELP</a></li> <li><a href="/wiki/Log_area_ratio" title="Log area ratio">LAR</a></li> <li><a href="/wiki/Line_spectral_pairs" title="Line spectral pairs">LSP</a></li> <li><a href="/wiki/Warped_linear_predictive_coding" title="Warped linear predictive coding">WLPC</a></li></ul></li> <li><a href="/wiki/Modified_discrete_cosine_transform" title="Modified discrete cosine transform">MDCT</a></li> <li><a href="/wiki/Psychoacoustics" title="Psychoacoustics">Psychoacoustic model</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Image_compression" title="Image compression">Image</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chroma_subsampling" title="Chroma subsampling">Chroma subsampling</a></li> <li><a href="/wiki/Coding_tree_unit" title="Coding tree unit">Coding tree unit</a></li> <li><a href="/wiki/Color_space" title="Color space">Color space</a></li> <li><a href="/wiki/Compression_artifact" title="Compression artifact">Compression artifact</a></li> <li><a href="/wiki/Image_resolution" title="Image resolution">Image resolution</a></li> <li><a href="/wiki/Macroblock" title="Macroblock">Macroblock</a></li> <li><a href="/wiki/Pixel" title="Pixel">Pixel</a></li> <li><a href="/wiki/Peak_signal-to-noise_ratio" title="Peak signal-to-noise ratio">PSNR</a></li> <li><a href="/wiki/Quantization_(image_processing)" title="Quantization (image processing)">Quantization</a></li> <li><a href="/wiki/Standard_test_image" title="Standard test image">Standard test image</a></li> <li><a href="/wiki/Texture_compression" title="Texture compression">Texture compression</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Methods</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chain_code" title="Chain code">Chain code</a></li> <li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">DCT</a></li> <li><a href="/wiki/Deflate" title="Deflate">Deflate</a></li> <li><a href="/wiki/Fractal_compression" title="Fractal compression">Fractal</a></li> <li><a href="/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem" class="mw-redirect" title="Karhunen–Loève theorem">KLT</a></li> <li><a href="/wiki/Pyramid_(image_processing)" title="Pyramid (image processing)">LP</a></li> <li><a href="/wiki/Run-length_encoding" title="Run-length encoding">RLE</a></li> <li><a href="/wiki/Wavelet_transform" title="Wavelet transform">Wavelet</a> <ul><li><a class="mw-selflink selflink">Daubechies</a></li> <li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li> <li><a href="/wiki/Embedded_zerotrees_of_wavelet_transforms" title="Embedded zerotrees of wavelet transforms">EZW</a></li> <li><a href="/wiki/Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Data_compression#Video" title="Data compression">Video</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bit_rate" title="Bit rate">Bit rate</a> <ul><li><a href="/wiki/Average_bitrate" title="Average bitrate">ABR</a></li> <li><a href="/wiki/Constant_bitrate" title="Constant bitrate">CBR</a></li> <li><a href="/wiki/Variable_bitrate" title="Variable bitrate">VBR</a></li></ul></li> <li><a href="/wiki/Display_resolution" title="Display resolution">Display resolution</a></li> <li><a href="/wiki/Film_frame" title="Film frame">Frame</a></li> <li><a href="/wiki/Frame_rate" title="Frame rate">Frame rate</a></li> <li><a href="/wiki/Video_compression_picture_types" title="Video compression picture types">Frame types</a></li> <li><a href="/wiki/Interlaced_video" title="Interlaced video">Interlace</a></li> <li><a href="/wiki/Video#Characteristics_of_video_streams" title="Video">Video characteristics</a></li> <li><a href="/wiki/Video_quality" title="Video quality">Video quality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7.0em;font-weight:normal;"><a href="/wiki/Video_codec" title="Video codec">Codec</a> parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete_cosine_transform" title="Discrete cosine transform">DCT</a></li> <li><a href="/wiki/Differential_pulse-code_modulation" title="Differential pulse-code modulation">DPCM</a></li> <li><a href="/wiki/Deblocking_filter" title="Deblocking filter">Deblocking filter</a></li> <li><a href="/wiki/Lapped_transform" title="Lapped transform">Lapped transform</a></li> <li>Motion <ul><li><a href="/wiki/Motion_compensation" title="Motion compensation">Compensation</a></li> <li><a href="/wiki/Motion_estimation" title="Motion estimation">Estimation</a></li> <li><a href="/wiki/Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li> <li><a href="/wiki/Wavelet_transform" title="Wavelet transform">Wavelet</a> <ul><li><a class="mw-selflink selflink">Daubechies</a></li> <li><a href="/wiki/Discrete_wavelet_transform" title="Discrete wavelet transform">DWT</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Information_theory" title="Information theory">Theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Compressed_data_structure" title="Compressed data structure">Compressed data structures</a> <ul><li><a href="/wiki/Compressed_suffix_array" title="Compressed suffix array">Compressed suffix array</a></li> <li><a href="/wiki/FM-index" title="FM-index">FM-index</a></li></ul></li> <li><a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">Entropy</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a> <ul><li><a href="/wiki/Timeline_of_information_theory" title="Timeline of information theory">Timeline</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Prefix_code" title="Prefix code">Prefix code</a></li> <li><a href="/wiki/Quantization_(signal_processing)" title="Quantization (signal processing)">Quantization</a></li> <li><a href="/wiki/Rate%E2%80%93distortion_theory" title="Rate–distortion theory">Rate–distortion</a></li> <li><a href="/wiki/Redundancy_(information_theory)" title="Redundancy (information theory)">Redundancy</a></li> <li><a href="/wiki/Data_compression_symmetry" title="Data compression symmetry">Symmetry</a></li> <li><a href="/wiki/Smallest_grammar_problem" title="Smallest grammar problem">Smallest grammar problem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Community</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hutter_Prize" title="Hutter Prize">Hutter Prize</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mark_Adler" title="Mark Adler">Mark Adler</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Template"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Symbol_template_class_pink.svg/16px-Symbol_template_class_pink.svg.png" 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