Multiresolution analysis

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 Crowley.

Definition

A multiresolution analysis of the space $L^2(\mathbb{R})$ consists of a sequence of nested subspaces

\dots\subset V_0\subset V_1\subset\dots\subset V_n\subset V_{n+1}\subset\dots\subset L^2(\R)

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 of 2^-k. That is, for each $f\in V_k,\; m\in\mathbb Z$ there is a $g\in V_k$ with $\forall x\in\mathbb R:\;f(x)=g(x+m2^{-k})$ .
Self-similarity in scale demands that all subspaces $V_k\subset V_l,\; k are time-scaled versions of each other, with scaling respectively dilation factor$ 2^l-k. I.e., for each $f\in V_k$ there is a $g\in V_l$ with $\forall x\in\mathbb R:\;g(x)=f(2^{l-k}x)$ . If f has limited support, then the support of g gets smaller, the resolution of the l-th subspace is higher than 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 $\phi$ or $\phi_1,\dots,\phi_r$ . Those integer shifts should at least form a frame for the subspace $V_0\subset L^2(\R)$ , 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.

Completeness demands that those nested subspaces fill the whole space, i.e., their union should be dense in L²(IR), 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.

There is, because of $V_0\subset V_1$ , a finite sequence of coefficients $a_k=2 \langle\phi(x),\phi(2x-k)\rangle$ , for $|k|\leq N$ and $a_k=0$ for $|k|>N$ , such that

\phi(x)=\sum_{k=-N}^N a_k\phi(2x-k)

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

\psi(x):=\sum_{k=-N}^N (-1)^k a_{1-k}\phi(2x-k)

one can see that the space $W_0\subset V_1$ , which is defined as the linear hull of its integer shifts, is the orthogonal complement to $W_0$ inside $V_1$ . Or put differently, $V_1$ is the orthogonal sum of $W_0$ and $V_0$ . By self-similarity, there are scaled versions $W_k$ of $W_0$ and by completeness one has

L^2(\mathbb R)=\mbox{closure of }\bigoplus_{k\in\Z}W_k

thus the set

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

is a countable complete orthonormal wavelet basis in $L^2(\R)$ .

References

Charles K. Chui, An Introduction to Wavelets, (1992), Academic Press, San Diego, ISBN 91-58831

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Definition

Important conclusions

See also

References