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Multiresolution analysis

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Multiple-scale_analysis" title="Multiple-scale analysis">Multiple-scale analysis</a>.</div>

A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant <a href="Discrete_wavelet_transform" title="Discrete wavelet transform">discrete wavelet transforms</a> (DWT) and the justification for the <a href="Algorithm" title="Algorithm">algorithm</a> of the <a href="Fast_wavelet_transform" title="Fast wavelet transform">fast wavelet transform</a> (FWT). It was introduced in this context in 1988/89 by <a href="Stephane_Mallat" class="mw-redirect" title="Stephane Mallat">Stephane Mallat</a> and <a href="Yves_Meyer" title="Yves Meyer">Yves Meyer</a> and has predecessors in the <a href="Microlocal_analysis" title="Microlocal analysis">microlocal analysis</a> in the theory of <a href="Differential_equation" title="Differential equation">differential equations</a> (the ironing method) and the <a href="Pyramid_(image_processing)" title="Pyramid (image processing)">pyramid methods</a> of <a href="Image_processing" class="mw-redirect" title="Image processing">image processing</a> as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and <a rel="nofollow" class="external text" href="http://www-prima.inrialpes.fr/Prima/Homepages/jlc/jlc.html">James L. Crowley</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

A multiresolution analysis of the <a href="Lp_space" title="Lp space">Lebesgue space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} )}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} )}</annotation>

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</math><img src="./8722fb232f689925a4baa0e4ba478e43ee346672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} )}" loading="lazy"> consists of a <a href="Sequence" title="Sequence">sequence</a> of nested <a href="Linear_subspace" title="Linear subspace">subspaces</a>

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<annotation encoding="application/x-tex">{\displaystyle \{0\}\subset \dots \subset V_{1}\subset V_{0}\subset V_{-1}\subset \dots \subset V_{-n}\subset V_{-(n+1)}\subset \dots \subset L^{2}(\mathbb {R} )}</annotation>

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</math><img src="./8caa08502be6cc2aa5ddb3c4544d707e94957529.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:65.259ex; height:3.509ex;" alt="{\displaystyle \{0\}\subset \dots \subset V_{1}\subset V_{0}\subset V_{-1}\subset \dots \subset V_{-n}\subset V_{-(n+1)}\subset \dots \subset L^{2}(\mathbb {R} )}" loading="lazy"></dd></dl></dd></dl>

that satisfies certain <a href="Self-similarity" title="Self-similarity">self-similarity</a> relations in time-space and scale-frequency, as well as <a href="Complete_metric_space" title="Complete metric space">completeness</a> and regularity relations.

<ul><li>Self-similarity in time demands that each subspace Vk is invariant under shifts by <a href="Integer" title="Integer">integer</a> <a href="Multiple_(mathematics)" title="Multiple (mathematics)">multiples</a> of 2k. That is, for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in V_{k},\;m\in \mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle f\in V_{k},\;m\in \mathbb {Z} }</annotation>

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</math><img src="./814af0e779b631226db2c556b53c3f18e40481da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.674ex; height:2.509ex;" alt="{\displaystyle f\in V_{k},\;m\in \mathbb {Z} }" loading="lazy"> the function g defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)=f(x-m2^{k})}">

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<annotation encoding="application/x-tex">{\displaystyle g(x)=f(x-m2^{k})}</annotation>

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</math><img src="./6dd8cd624e48d5eb1c556bb4ed8c34cee8d38613.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.903ex; height:3.176ex;" alt="{\displaystyle g(x)=f(x-m2^{k})}" loading="lazy"> also contained in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{k}}</annotation>

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</math><img src="./f43bfe96795a33589c12e1500b843f6268d35f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.444ex; height:2.509ex;" alt="{\displaystyle V_{k}}" loading="lazy">.</li>

<li>Self-similarity in scale demands that all subspaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{k}\subset V_{l},\;k>l,}">

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<annotation encoding="application/x-tex">{\displaystyle V_{k}\subset V_{l},\;k>l,}</annotation>

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</math><img src="./838f6c34aa7220eedef0873706b2628489e6bbee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.949ex; height:2.509ex;" alt="{\displaystyle V_{k}\subset V_{l},\;k>l,}" loading="lazy"> are time-scaled versions of each other, with <a href="Scaling_(geometry)" title="Scaling (geometry)">scaling</a> respectively <a href="Dilation_(metric_space)" title="Dilation (metric space)">dilation</a> factor 2k-l. I.e., for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in V_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle f\in V_{k}}</annotation>

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</math><img src="./3c8987aa5a6243b12a22faed7f93027b1d684bcb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.563ex; height:2.509ex;" alt="{\displaystyle f\in V_{k}}" loading="lazy"> there is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in V_{l}}">

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<annotation encoding="application/x-tex">{\displaystyle g\in V_{l}}</annotation>

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</math><img src="./80c1bac662c1d0e9bb9b89ffc605da829f740d08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.034ex; height:2.509ex;" alt="{\displaystyle g\in V_{l}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\in \mathbb {R} :\;g(x)=f(2^{k-l}x)}">

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<annotation encoding="application/x-tex">{\displaystyle \forall x\in \mathbb {R} :\;g(x)=f(2^{k-l}x)}</annotation>

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</math><img src="./5d2354be922d13286abe68f268bbd3c85c69b1da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.514ex; height:3.176ex;" alt="{\displaystyle \forall x\in \mathbb {R} :\;g(x)=f(2^{k-l}x)}" loading="lazy">.</li>

<li>In the sequence of subspaces, for k>l the space resolution 2l of the l-th subspace is higher than the resolution 2k of the k-th subspace.</li>

<li>Regularity demands that the model <a href="Linear_subspace" title="Linear subspace">subspace</a> V0 be generated as the <a href="Linear_hull" class="mw-redirect" title="Linear hull">linear hull</a> (<a href="Algebraic_closure" title="Algebraic closure">algebraically</a> or even <a href="Topologically_closed" class="mw-redirect" title="Topologically closed">topologically closed</a>) of the integer shifts of one or a finite number of generating functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

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</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{1},\dots ,\phi _{r}}">

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<annotation encoding="application/x-tex">{\displaystyle \phi _{1},\dots ,\phi _{r}}</annotation>

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</math><img src="./05ce2e827fed97f0e5c6c56bca61e1fe064b8891.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.977ex; height:2.509ex;" alt="{\displaystyle \phi _{1},\dots ,\phi _{r}}" loading="lazy">. Those integer shifts should at least form a frame for the subspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}\subset L^{2}(\mathbb {R} )}">

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<annotation encoding="application/x-tex">{\displaystyle V_{0}\subset L^{2}(\mathbb {R} )}</annotation>

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</math><img src="./593a2c2805d4e1eb04c26140ca30eaacd2e4f73f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.632ex; height:3.176ex;" alt="{\displaystyle V_{0}\subset L^{2}(\mathbb {R} )}" loading="lazy">, which imposes certain conditions on the decay at <a href="Infinity" title="Infinity">infinity</a>. The generating functions are also known as <a href="Wavelet#Scaling_function" title="Wavelet">scaling functions</a> or <a href="Father_wavelets" class="mw-redirect" title="Father wavelets">father wavelets</a>. In most cases one demands of those functions to be <a href="Piecewise_continuous" class="mw-redirect" title="Piecewise continuous">piecewise continuous</a> with <a href="Compact_support" class="mw-redirect" title="Compact support">compact support</a>.</li>

<li>Completeness demands that those nested subspaces fill the whole space, i.e., their union should be <a href="Dense_set" title="Dense set">dense</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} )}">

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<annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} )}</annotation>

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</math><img src="./8722fb232f689925a4baa0e4ba478e43ee346672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} )}" loading="lazy">, and that they are not too redundant, i.e., their <a href="Intersection" title="Intersection">intersection</a> should only contain the <a href="Zero_element" title="Zero element">zero element</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Important_conclusions">Important conclusions</h2></div>

In the case of one continuous (or at least with bounded variation) compactly supported scaling function with orthogonal shifts, one may make a number of deductions. The proof of existence of this class of functions is due to <a href="Ingrid_Daubechies" title="Ingrid Daubechies">Ingrid Daubechies</a>.

Assuming the scaling function has compact support, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}\subset V_{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{0}\subset V_{-1}}</annotation>

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</math><img src="./b4a92d6d41dc405d7a2168ad6b00fc7d6a670158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.196ex; height:2.509ex;" alt="{\displaystyle V_{0}\subset V_{-1}}" loading="lazy"> implies that there is a finite sequence of coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}=2\langle \phi (x),\phi (2x-k)\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle a_{k}=2\langle \phi (x),\phi (2x-k)\rangle }</annotation>

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</math><img src="./dc1ed5ab3039865a89ea165d90dea2ac5a09a677.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.686ex; height:2.843ex;" alt="{\displaystyle a_{k}=2\langle \phi (x),\phi (2x-k)\rangle }" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |k|\leq N}">

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<annotation encoding="application/x-tex">{\displaystyle |k|\leq N}</annotation>

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</math><img src="./07c42bc885fcd1b9c0fb5077e93994f115b91961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.667ex; height:2.843ex;" alt="{\displaystyle |k|\leq N}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}=0}">

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<annotation encoding="application/x-tex">{\displaystyle a_{k}=0}</annotation>

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</math><img src="./361df98baa29ed00306879f899d6c6531c94028d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.579ex; height:2.509ex;" alt="{\displaystyle a_{k}=0}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |k|>N}">

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<annotation encoding="application/x-tex">{\displaystyle |k|>N}</annotation>

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</math><img src="./0ae5a41622e23b5345b8d2c1ca7d5a638637524a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.667ex; height:2.843ex;" alt="{\displaystyle |k|>N}" loading="lazy">, such that

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<annotation encoding="application/x-tex">{\displaystyle \phi (x)=\sum _{k=-N}^{N}a_{k}\phi (2x-k).}</annotation>

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</math><img src="./d095fa28d9db58baed99e5d9d90365d958e82944.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.587ex; height:7.509ex;" alt="{\displaystyle \phi (x)=\sum _{k=-N}^{N}a_{k}\phi (2x-k).}" loading="lazy"></dd></dl>

Defining another function, known as mother wavelet or just the wavelet

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<annotation encoding="application/x-tex">{\displaystyle \psi (x):=\sum _{k=-N}^{N}(-1)^{k}a_{1-k}\phi (2x-k),}</annotation>

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</math><img src="./e770638543a1491c12b939704962ed631b6d6762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.943ex; height:7.509ex;" alt="{\displaystyle \psi (x):=\sum _{k=-N}^{N}(-1)^{k}a_{1-k}\phi (2x-k),}" loading="lazy"></dd></dl>

one can show that the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}\subset V_{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{0}\subset V_{-1}}</annotation>

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</math><img src="./bbdee7b7a5bbd6bd7bffa8da4953816281c36ab3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.034ex; height:2.509ex;" alt="{\displaystyle W_{0}\subset V_{-1}}" loading="lazy">, which is defined as the (closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{0}}</annotation>

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</math><img src="./7ae15ff9b845587dc4e1816f59c3fed0e71a132f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{0}}" loading="lazy"> inside <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{-1}}</annotation>

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</math><img src="./4d9cabee0b9798d239405ad25c33589a3066a309.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.688ex; height:2.509ex;" alt="{\displaystyle V_{-1}}" loading="lazy">.<a href="#cite_note-1">[1]</a> Or put differently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{-1}}</annotation>

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</math><img src="./4d9cabee0b9798d239405ad25c33589a3066a309.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.688ex; height:2.509ex;" alt="{\displaystyle V_{-1}}" loading="lazy"> is the <a href="Orthogonal_direct_sum" class="mw-redirect" title="Orthogonal direct sum">orthogonal sum</a> (denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oplus }">

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</math><img src="./8b16e2bdaefee9eed86d866e6eba3ac47c710f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \oplus }" loading="lazy">) of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{0}}</annotation>

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</math><img src="./7f541f57fd799ba5137a2e50a1a728dde4306c06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{0}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{0}}</annotation>

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</math><img src="./7ae15ff9b845587dc4e1816f59c3fed0e71a132f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle V_{0}}" loading="lazy">. By self-similarity, there are scaled versions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{k}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{k}}</annotation>

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</math><img src="./aea7ab451421a62d7de329faed9aa86419bebeab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.282ex; height:2.509ex;" alt="{\displaystyle W_{k}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle W_{0}}</annotation>

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</math><img src="./7f541f57fd799ba5137a2e50a1a728dde4306c06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{0}}" loading="lazy"> and by completeness one has

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<mtext>closure of </mtext>

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<annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} )={\mbox{closure of }}\bigoplus _{k\in \mathbb {Z} }W_{k},}</annotation>

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</math><img src="./a069b97c114cb7b377a40ff040a27432499c1f9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.467ex; height:5.676ex;" alt="{\displaystyle L^{2}(\mathbb {R} )={\mbox{closure of }}\bigoplus _{k\in \mathbb {Z} }W_{k},}" loading="lazy"></dd></dl>

thus the set

<msub>

</mrow>

</msub>

<msup>

<msqrt>

</msqrt>

</mrow>

</mrow>

</msup>

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\psi _{k,n}(x)={\sqrt {2}}^{-k}\psi (2^{-k}x-n):\;k,n\in \mathbb {Z} \}}</annotation>

</semantics>

</math><img src="./44d50d16bd0729e999e32c0ee63137468560cf84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.104ex; height:3.843ex;" alt="{\displaystyle \{\psi _{k,n}(x)={\sqrt {2}}^{-k}\psi (2^{-k}x-n):\;k,n\in \mathbb {Z} \}}" loading="lazy"></dd></dl>

is a countable complete <a href="Orthonormal_wavelet" class="mw-redirect" title="Orthonormal wavelet">orthonormal wavelet</a> basis in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(\mathbb {R} )}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{2}(\mathbb {R} )}</annotation>

</semantics>

</math><img src="./8722fb232f689925a4baa0e4ba478e43ee346672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.124ex; height:3.176ex;" alt="{\displaystyle L^{2}(\mathbb {R} )}" loading="lazy">.

<ul><li><a href="Multigrid_method" title="Multigrid method">Multigrid method</a></li>

<li><a href="Multiscale_modeling" title="Multiscale modeling">Multiscale modeling</a></li>

<li><a href="Scale_space" title="Scale space">Scale space</a></li>

<li><a href="Time%E2%80%93frequency_analysis" title="Time–frequency analysis">Time–frequency analysis</a></li>

<li><a href="Wavelet" title="Wavelet">Wavelet</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFMallat,_S.G." class="citation web cs1">Mallat, S.G. <a rel="nofollow" class="external text" href="http://www.cmap.polytechnique.fr/~mallat/book.html">"A Wavelet Tour of Signal Processing"</a>. www.di.ens.fr. Retrieved 2019-12-30.</cite>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFChui1992" class="citation book cs1">Chui, Charles K. (1992). An Introduction to Wavelets. San Diego: Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-585-47090-1</bdi>.</cite></li>

<li><cite id="CITEREFAkansuHaddad1992" class="citation book cs1"><a href="Ali_Akansu" title="Ali Akansu">Akansu, A.N.</a>; Haddad, R.A. (1992). Multiresolution signal decomposition: transforms, subbands, and wavelets. Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-047141-6</bdi>.</cite></li>

<li>Crowley, J. L., (1982). <a rel="nofollow" class="external text" href="http://www-prima.inrialpes.fr/Prima/Homepages/jlc/papers/Crowley-Thesis81.pdf">A Representations for Visual Information</a>, Doctoral Thesis, Carnegie-Mellon University, 1982.</li>

<li><cite id="CITEREFBurrusGopinathGuo1997" class="citation book cs1"><a href="C._Sidney_Burrus" title="C. Sidney Burrus">Burrus, C.S.</a>; Gopinath, R.A.; Guo, H. (1997). Introduction to Wavelets and Wavelet Transforms: A Primer. Prentice-Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-13-489600-9</bdi>.</cite></li>

<li><cite id="CITEREFMallat1999" class="citation book cs1">Mallat, S.G. (1999). <a rel="nofollow" class="external text" href="http://www.cmap.polytechnique.fr/~mallat/book.html">A Wavelet Tour of Signal Processing</a>. Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-466606-X</bdi>.</cite></li></ul></div><div><div class="zim-footer">

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