A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (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 equations (the ironing method) and the pyramid methods of image processing as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson and James L. Crowley.


A multiresolution analysis of the Lebesgue space consists of a sequence of nested subspaces

that satisfies certain self-similarity relations in time-space and scale-frequency, as well as completeness and regularity relations.

Important conclusions

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 Ingrid Daubechies.

Assuming the scaling function has compact support, then implies that there is a finite sequence of coefficients for , and for , such that

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

one can show that the space , which is defined as the (closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to inside .[1] Or put differently, is the orthogonal sum (denoted by ) of and . By self-similarity, there are scaled versions of and by completeness one has

thus the set

is a countable complete orthonormal wavelet basis in .

See also


  1. ^ Mallat, S.G. "A Wavelet Tour of Signal Processing". www.di.ens.fr. Retrieved 2019-12-30.