Multiresolution analysis: Difference between revisions

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{{Short description|designDesign method of discrete wavelet transforms}}

{{Distinguish|Multiple-scale analysis}}

A '''multiresolution analysis''' ('''MRA''') or '''multiscale approximation''' ('''MSA''') is the design method of most of the practically relevant [[discrete wavelet transform]]s (DWT) and the justification for the [[algorithm]] of the [[fast wavelet transform]] (FWT). It was introduced in this context in 1988/89 by [[Stephane Mallat]] and [[Yves Meyer]] and has predecessors in the [[microlocal analysis]] in the theory of [[differential equation]]s (the ''ironing method'') and the [[pyramid (image processing)|pyramid method]]s of [[image processing]] as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and [http://www-prima.inrialpes.fr/Prima/Homepages/jlc/jlc.html James L. Crowley].

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A multiresolution analysis of the [[Lp space|Lebesgue space]] <math>L^2(\mathbb{R})</math> consists of a [[sequence]] of nested [[linear subspace|subspaces]]

::<math>\{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(\R)</math>

that satisfies certain [[self-similarity]] relations in time-space and scale-frequency, as well as [[Complete metric space|completeness]] and regularity relations.

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

:<math>\psi(x):=\sum_{k=-N}^N (-1)^k a_{1-k}\phi(2x-k),</math>

one can show that the space <math>W_0\subset V_{-1}</math>, which is defined as the (closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to <math>V_0</math> inside <math>V_{-1}</math>.<ref>{{Cite web|url=http://www.cmap.polytechnique.fr/~mallat/book.html|title=A Wavelet Tour of Signal Processing|last=Mallat, S.G.|first=|date=|website=www.di.ens.fr|url-status=live|archive-url=|archive-date=|access-date=2019-12-30}}</ref> Or put differently, <math>V_{-1}</math> is the [[orthogonal direct sum|orthogonal sum]] (denoted by <math>\oplus</math>) of <math>W_0</math> and <math>V_0</math>. By self-similarity, there are scaled versions <math>W_k</math> of <math>W_0</math> and by completeness one has{{citation needed|date=April 2013}}

:<math>L^2(\R)=\mbox{closure of }\bigoplus_{k\in\Z}W_k,</math>

thus the set

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==See also==

*[[Multigrid method]]

*[[Multiscale modeling]]

*[[Scale space]]