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## Wavelet Analysis

The task of identifying the properties of an acquired signal starts with its given representation as an element in the reference (i.e. fiducial, central) representation space . One singles out the large overall features by projecting it onto the next subspace . This projection process suppresses the finer details of the signal. They are no longer present when the signal is represented as an element of . Using the pyramid algorithm one repeats this process iteratively. In this process one moves from the resolution of to the lower resolution of

To keep track of the finer details suppressed by this process, one introduces , the orthogonal complement of in :

Thus any signal represented in is the unique sum of the signal represented in and its suppressed detail which lies in :

In brief,

The o.n. bases for and , and hence the representations of the signal in these spaces, are known and expressed in terms of the scaling function . These bases determine a unique basis for whose purpose is to keep track of the suppressed of the signal . The process of constructing the -basis resembles that for and . One starts with a square-integrable function , the mother wavelet''. By applying translations and dilations to it, one obtains the desired o.n. basis for , the space of details at resolution . The crucial part of this endeavor is the construction of the mother wavelet from the scaling function of the MSA. The construction is done by means of the following theorem by Mallat:

Theorem 26.1 (Wavelet generation theorem)
1. Let

be the hierarchy of vector spaces which make up the MSA whose scaling function is and whose corresponding pyramid algorithm is based on the filtering function

2. Let be a function whose Fourier transform is given by

 where

then
I.

 (2123)

is an o.n. basis for and
II.

 (2124)

is an o.n. basis for .

The validation of this theorem is a three step process.

1. First of all notice that the set of functions, Eq.(2.123), being orthogonal,

is equivalent to the statement that

 (2125)

The reasoning is identical to that leading to Eq.(2.120).
2. Secondly note that , and hence

 with

implies that any basis element of or of is a linear combination of the basis elements of . Applying this fact to the case , one has

The corresponding Fourier transformed equations are

 (2126) (2127)

3. Thirdly note that the orthogonality condition, Eq.(2.125), when combined with Eq.(2.127), yields a normalization condition on analogous to Eq.(2.122) on page ,

This is not the only constraint that must satisfy. One must also take into account that is the orthogonal complement of in This fact, which is expressed by

is equivalent to

 (2128)

Inserting Eqs.(2.126) and (2.127) into Eq.(2.128), using the fact that and are -periodic,

and taking advantage of Eq.(2.120), one finds that the additional constraint on is

Thus the filter functions and satisfy

 (2129) and (2130) (2131)

A good way of remembering these constraints is that the matrix

is unitary. These constraints are useful if for no other reasons than that they (i) place the two sequences of Fourier coefficients and on certain quadratic surfaces in the Hilbert space and that they (ii) establish a tight relation between the 's and the 's. Indeed, from Eq.(2.129) one finds

This relation is not unique. Other possibilities are

Each side of this equation is a Fourier series in powers of . Equating equal powers one finds

With both and at hand, one solves the two Eqs.(2.126) and (2.127). The solutions are

The inverse Fourier transform of these solutions yields the sought after scaling function

 and the mother wavelet

Shifting and dilating this mother wavelet yields the o.n. basis functions, Eq.(2.123), for

Exercise 26.8 (ORTHOGONALITY OF THE DETAIL SPACES)
Validate conclusion # II. of the theorem on page , i.e. point out why, whenever , the functions in the space of details are orthogonal to the functions in the space of details .

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Next: Sturm-Liouville Theory Up: Multiresolution Analysis Previous: The Scaling Function as   Contents   Index
Ulrich Gerlach 2010-12-09