Multiresolution analysis

Multiresolution analysis (MRA, English: multiresolution analysis) or approximation (MSA, English: multi- scale approximation) of the functional area is a functional analytic basic construction of wavelet theory, which describes the approximation properties of the discrete wavelet transform. In particular, it explains the way and mode of operation of the algorithm of fast wavelet transformation.

Definition

A multi-scale analysis of the space L ² (R ) consists of a sequence of nested subspaces

Satisfying both self-similarity conditions in time / space, and scale / frequency, as well as completeness and regularity.

  • Self-similarity in time requires that each subspace Vk is invariant under shifts by integer multiples of 2k. 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 each other, the scaling or stretching factor 2k - l. This means that for every function there is a function with. For example, if the carrier f a bounded support, then bawled from g by a factor of 2k - l. In other words, the resolution (in terms of dots on a screen) of the I- th sub- space is higher than the resolution of the k-th sub-space.
  • Regularity requires that the model subspace V0 is the linear hull ( algebraically closed or even topologically ) is the integer shifts of one or a finite number of generating functions or. These integer shifts should at least be a Riesz basis, but better form a Hilbert basis of the subspace, resulting in a rapid decrease at infinity of the generating functions follows. The latter is trivially satisfied for functions with compact support. The generating functions are called scaling functions or Vaterwavelets. They are often constructed as a ( piecewise ) continuous functions with compact support.
  • Completeness requires that these nested subspaces fill the entire space, that is, their union should be sealed in; further, that they are not redundant, that is, their intersection must contain only the zero element.

Scaling function

In the practically important case that there is only one scaling function with compact support in the MRA, and this generates a Hilbert basis in the subspace V0, it satisfies a two - scale equation ( in eng literature. Refinement equation )

The numbers occurring there is scale or sequence mask and must be a discrete low-pass filter, which means in this case that

Is satisfied, or that the Fourier series

The zero point is set to 1 and at the location of π, a zero point A ( π ) = 0

It is a basic task of the wavelet designs, determine conditions that follow under which desired properties, such as continuity, differentiability, etc.. If orthogonal, that is, be perpendicular to all integer shifts of itself, it must

Apply for, by means of the Fourier series is the condition.

Typically, these sequences are given as coefficient sequences of a Laurent polynomial, that is. The normalization writes itself as the low-pass characteristic as as or for one, the orthogonality condition.

Examples

  • The Haar wavelet has a scaling form
  • The wavelet with order of the Daubechies family has the scaling form

Nested subspaces

Be an orthogonal scaling function. Then an affine function system and a sequence can be defined by scaling subspaces. This then applies and is an orthonormal basis of.

With any odd now the wavelet sequence can be defined, where. This means that the wavelet defined as

And the Waveletunterräume as. With these results in what is known as fishbone orthogonal decomposition of the scaling spaces

And generally in.

The basic analytical exposure to a MRA is that the full use of the Waveletunterräume, that is intended to be a dense subspace of.

Functional analysis Haar-Wavelet
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