Multi-scale analysis
The multi-scale analysis ( MRA , English: m ulti r esolution a nalysis ) or -approximation ( MSA , English: m ulti s cale a pproximation ) of the function space is a functional analytical basic construction of the wavelet theory, which describes the approximation properties of the discrete wavelet transformation . In particular, it explains the possibility and functionality of the fast wavelet transform algorithm .
definition
A multiscale analysis of space L² (R) consists of a series of nested subspaces
- ,
which fulfills both self-similarity conditions in time / space and scale / frequency as well as completeness and regularity conditions.
- Self-similarity in time requires that every subspace is invariant with shifts by integral multiples of . This means that for every function there is a function with .
- Self-similarity between different scales requires that all subspaces are time- scaled copies of one another, with the scaling or stretching factor being. This means that for every function there is a function with . For example , if the beam has a restricted beam , the beam is compressed by a factor of . In other words, the resolution (in the sense of points on a screen) of the l th subspace is higher than the resolution of the k th subspace.
- Regularity requires that the model subspace the linear hull (or even algebraically topologically closed) of the integer shifts one or a finite number of generating functions or is. These integer shifts should at least form a Riesz basis, but better a Hilbert basis of the subspace , from which a rapid decrease in the infinite of the generating functions follows. The latter is trivially fulfilled for functions with a compact carrier. The generating functions are called scaling functions or father wavelets . Often they are constructed as (piece-wise) continuous functions with a compact carrier .
- Completeness requires that these nested subspaces fill the entire space, that is, their union should be tight in ; also that they are not redundant, that is, their average may only contain the zero element.
Scaling function
In practically the most important case, that there is only one scaling function with compact carrier in the MRA and that this generates a Hilbert basis in the subspace , this fulfills a two-scale equation (in English literature: refinement equation )
- .
The sequence of numbers occurring there is called the scaling sequence or mask and must be a discrete low-pass filter , which in this case means that
- and
is fulfilled, or that the Fourier series
at the zero point the value 1 and at the location of a zero point has .
It is a basic task of wavelet design to determine conditions under which the desired properties of , such as continuity , differentiability etc. follow. Should be orthogonal, i.e. H. be perpendicular to all integer shifts of itself, so must
- and
for apply, using the Fourier series the condition is .
Usually these sequences are given as coefficient sequences of a Laurent polynomial , that is . The normalization is thus written as , the low-pass property as or for a , the orthogonality condition as .
Examples
- The Haar wavelet has a scaling mask
- The wavelet with order of the Daubechies family has the scaling mask
Nested subspaces
Let be an orthogonal scaling function. Then an affine function system and a sequence of scaling subspaces can be defined. So then applies and is an orthonormal basis of .
The wavelet sequence can now be defined with any odd number , where . The wavelet is thus defined as
and the wavelet subspaces as . This results in an orthogonal decomposition of the scaling spaces known as herringbone
and generally at .
The basic analytical requirement of an MRA is that the wavelet subspaces fully utilize the, that is, should be a dense subspace of .
literature
- Alfred Louis , Peter Maaß , Andreas Rieder : Wavelets: Theory and Applications . 2nd Edition. Teubner, 1998, ISBN 3-519-12094-1 .