Multiresolution analysis: Difference between revisions

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A '''Multiresolutionmultiresolution Analysisanalysis (MRA)''' or '''Multiscalemultiscale Approximationapproximation (MSA)''' is the desingdesign 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 equationsequation]]s (the ''[[ironing method]]'') and the [[pyramid methodsmethod]]s of [[image processing]] as introduced in 1981/83 by Peter J. Burt undand Edward H. Adelson.

== Definition ==

A ''Multiresolutionmultiresolution Analysisanalysis'' of the space ''L²(IR)'' consists of a [[sequence]] of nested subspaces

:<math>\dots\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 completeness and regularity relations.

* ''Self-similarity'' in ''time'' demands that each subspace ''V_k'' is invariant under shifts by [[integer]] multiples[[multiple]]s of ''2^-k''. I.e.That is, for each <math>f\in V_k,\; m\in\mathbb Z</math> there is a <math>g\in V_k</math> with <math>\forall x\in\mathbb R:\;f(x)=g(x+m2^{-k})</math>.

* ''Self-similarity'' in ''scale'' demands that all subspaces <math>V_k\subset V_l,\; k<l,</math> are time-scaled versions of each other, with scaling resp.respectively dilation factor ''2^l-k''. I.e., for each <math>f\in V_k</math> there is a <math>g\in V_l</math> with <math>\forall x\in\mathbb R:\;g(x)=f(2^{l-k}x)</math>. If f has limited support, then the support of g gets smaller, the resolution of the ''l''-th subspace is higher then the resolution of the ''k''-th subspace.

* ''Regularity'' demands that the model subspace ''V₀'' be generated as the linear hull (algebraically or even topologically closed) of the integer shifts of one or a finite number of generating functions <math>\phi</math> or <math>\phi_1,\dots,\phi_r</math>. Those integer shifts should at least form a frame for the subspace <math>V_0\subset L^2(\R)</math>, which imposes certain conditions on the decay at infinity. The generating functions are also known as '''scaling functions''' or '''father wavelets'''. In most cases one demands of those functions to be (piecewise) continuous with compact support.

* ''Completeness'' demands that those nested subspaces fill the whole space, i.e., their union should be dense in ''L&sup2;(IR)'', and that they are not too redundant, i.e., their intersection should only contain the zero element.

== Important conclusions ==