Multiresolution analysis explained

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.

Definition

L2(R)

consists of a sequence of nested subspaces

\{0\}...\subsetV1\subsetV0\subsetV-1\subset...\subsetV-n\subsetV-(n+1)\subset...\subsetL2(\R)

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

f\inVk,m\in\Z

the function g defined as

g(x)=f(x-m2k)

also contained in

Vk

.

Vk\subsetVl,k>l,

are time-scaled versions of each other, with scaling respectively dilation factor 2k-l. I.e., for each

f\inVk

there is a

g\inVl

with

\forallx\in\R:g(x)=f(2k-lx)

.

\phi

or

\phi1,...,\phir

. Those integer shifts should at least form a frame for the subspace

V0\subsetL2(\R)

, 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.

L2(\R)

, and that they are not too redundant, i.e., their intersection should only contain the zero element.

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

V0\subsetV-1

implies that there is a finite sequence of coefficients

ak=2\langle\phi(x),\phi(2x-k)\rangle

for

|k|\leqN

, and

ak=0

for

|k|>N

, such that
N
\phi(x)=\sum
k=-N

ak\phi(2x-k).

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

N
\psi(x):=\sum
k=-N

(-1)ka1-k\phi(2x-k),

one can show that the space

W0\subsetV-1

, which is defined as the (closed) linear hull of the mother wavelet's integer shifts, is the orthogonal complement to

V0

inside

V-1

.[1] Or put differently,

V-1

is the orthogonal sum (denoted by

) of

W0

and

V0

. By self-similarity, there are scaled versions

Wk

of

W0

and by completeness one has
2(\R)=closureofoplus
L
k\in\Z

Wk,

thus the set

\{\psik,n(x)=\sqrt2-k\psi(2-kx-n):k,n\in\Z\}

is a countable complete orthonormal wavelet basis in

L2(\R)

.

See also

References

Notes and References

  1. Web site: A Wavelet Tour of Signal Processing. Mallat, S.G.. www.di.ens.fr. 2019-12-30.