Multiresolution analysis

A Multiresolution Analysis (MRA) or Multiscale Approximation (MSA) is the desing 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 und Edward H. Adelson.

Definition

A Multiresolution Analysis of the space L²(IR) consists of a sequence of nested subspaces

$\dots \subset V_{0}\subset V_{1}\subset \dots \subset V_{n}\subset V_{n+1}\subset \dots \subset L^{2}(\mathbb {R} )$

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. I.e., 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 $V_{k}\subset V_{l},\;k<l,$ are time-scaled versions of each other, with scaling resp. 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 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 $\phi$ or $\phi _{1},\dots ,\phi _{r}$. Those integer shifts should at least form a frame for the subspace $V_{0}\subset L^{2}(\mathbb {R} )$, 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

This is only for the case of one continuous (or at least with bounded variation) compactly supported scaling function with orthogonal shifts (the existence of those is due to Ingrid Daubechies).

Then there is, because of $V_{0}\subset V_{1}$, a finite sequence of coefficients $a_{k}=2<\phi (x),\phi (2x-k)>$, $a_{k}=0$ 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

$L^{2}(\mathbb {R} )={\mbox{closure of }}\bigoplus _{k\in \mathbb {Z} }W_{k}$,

thus the set

$\{\psi _{k,n}(x)={\sqrt {2}}^{k}\psi (2^{k}x-n):\;k,n\in \mathbb {Z} \}$

is a countable complete orthonormal system in $\mathbb {R}$.