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
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.
- Self-similarity in time demands that each subspace Vk is invariant under shifts by integer multiples of 2k. That is, for each
the function
g defined as
also contained in
.
- Self-similarity in scale demands that all subspaces
are time-scaled versions of each other, with
scaling respectively
dilation factor 2
k-l. I.e., for each
there is a
with
\forallx\in\R: g(x)=f(2k-lx)
.
- In the sequence of subspaces, for k>l the space resolution 2l of the l-th subspace is higher than the resolution 2k 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
or
. Those integer shifts should at least form a frame for the subspace
, 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
, 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
implies that there is a finite sequence of coefficients
ak=2\langle\phi(x),\phi(2x-k)\rangle
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
| 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
.
See also
References
- Book: Chui, Charles K.. An Introduction to Wavelets. 1992. Academic Press. San Diego. 0-585-47090-1.
- Book: Ali Akansu. A.N.. Akansu. R.A.. Haddad. Multiresolution signal decomposition: transforms, subbands, and wavelets. Academic Press. 1992. 978-0-12-047141-6.
- Crowley, J. L., (1982). A Representations for Visual Information, Doctoral Thesis, Carnegie-Mellon University, 1982.
- Book: C. Sidney Burrus. C.S.. Burrus. R.A.. Gopinath. H.. Guo. Introduction to Wavelets and Wavelet Transforms: A Primer. Prentice-Hall. 1997. 0-13-489600-9.
- Book: Mallat, S.G.. A Wavelet Tour of Signal Processing. Academic Press. 1999. 0-12-466606-X.
Notes and References
- Web site: A Wavelet Tour of Signal Processing. Mallat, S.G.. www.di.ens.fr. 2019-12-30.