# Chemometrics from basick to wavelet transform - Chau F.T

ISBN 0-471-20242-8

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2. Construct a wavelet function f (x) using Equations (4.19) and (4.20).

The MRSD provides us the techniques for constructing wavelet functions. However, a further question is what kind of wavelet functions are preferred. Most applications of wavelets exploit their ability to efficiently approximate or represent particular classes of signals with few nonzero wavelet coefficients. This applies not only for data compression but also for noise removal in chemical data analysis and fast calculations. The design of wavelet function f(x) must therefore be optimized to produce a maximum number of wavelet coefficients dj,k = {f(x), fj,k(x)) in (4.21) that are close to zero. A signal f(x) has few nonnegligible wavelet coefficients if most of the fine-scale wavelet coefficients are small. This depends mostly on the smoothness of signal f (x) and the properties of wavelet function f (x) itself. The following features are very important to a wavelet function with the mentioned optimal property:

Vanishing Moments

A wavelet function f (x) has p vanishing moments if

Why do we introduce the concept of vanishing moments? Simply speaking, if a wavelet function f (x) has larger vanishing moments, then the wavelet coefficients |cj,k| of a smooth function f(x) are much smaller on a larger scale j (or at finer resolution).

4.1.3. Basic Properties of Wavelet Function

wavelet function examples

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Compact Support

The compact support of a wavelet function f (x) is the maximal interval outside of which wavelet function has zero value. For example, the support of Haar wavelet function f (x) is [0, 1) [see Eq. (4.1)]. In general, the smaller the size of such compact, the fewer the highamplitude wavelet coefficients there are. There exists a relationship between the sizes of support of wavelet function and the corresponding scaling filter. If a scaling filter {hk | k = ... , -2, -1,0,1,2,...} has only nonzero values for N1 < k < N2, then the corresponding wavelet function f (x) has a support of size N2 - N1.

Regularity/Smoothness

The regularity of f (x) is a more complicated mathematical concept. It is related to the definition of Holder continuity and other factors. In this chapter we discuss only continuity and smoothness. For example, the Haar wavelet is an example of discontinous wavelets.

4.2. WAVELET FUNCTION EXAMPLES

In the last section we provided MRSD as a tool to construct some desired wavelet functions. Can any wavelet function be found from a MRSD? The simplest Haar provides us an example that can be constructed by the MRSD generated from a special scaling function: the basic unit step function 0(x) = 1[01)(x). In this section we are about to build more examples and to provide readers with several wavelet functions for their possible applications.

4.2.1. Meyer Wavelet

As mentioned in the rest of Section 4.1.2, one can construct a wavelet function in the Fourier transform. The Meyer wavelet function f (x) is the first example of a wavelet function given its Fourier transform, found by French mathematician Y. Meyer in the 1980s. The Meyer wavelet function is a frequency band-limited function whose Fourier transform has a compact support, namely, nonzero only in a finite interval of frequency variable o>. It is constructed by using the second strategy. The resultant scaling function 0(x) is defined by its Fourier transform. In Meyer’s method, only the Fourier transform f (a>) of the Meyer wavelet function f (x) is given. As the expression is very complicated, we omit it here. Readers can consult Daubechies’s book [10] for further details.

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fundamentals of wavelet transform

<>(x)

y(x)

Figure 4.4. Meyer scaling function 0 and wavelet f.

However, we have no explicit formulas for both the Meyer wavelet function f (x) and the Meyer scaling function 0(x), although the Fourier transforms of these two functions have been given. Figure 4.4 displays the corresponding Meyer scaling function 0(x) and Meyer wavelet function f (x). You are encouraged to plot these figures by yourselves with the aid of the MATLAB wavelet toolbox.

Neither the Meyer wavelet function f (x) nor the Meyer scaling function 0(x) has compact support, that is, f (x) = 0 and 0(x) = 0 for any x except for some points, but these functions do decrease to 0 when x at a very fast speed. Visually they appear as a small wave (see Fig. 4.4), so they can be considered as local waves.

These two functions have better regularity, that is, they are infinitely differentiable, and f (x) has an infinite number of vanishing moments; thus, for any integer m > 0, one has

/

+W

xm f (x)dx = 0

4.2.2. B-Spline (Battle-Lemarie) Wavelets

The first strategy is used to construct B-spline wavelet series, assuming a scaling function first. Let Bm(x) be the box spline of degree m, as in Chapter 2. Bm(x) can be computed by convolving the basic unit step function 1[01) [see Eq. (4.6)] with itself (m + 1) times. It has a compact support centered at 0 or 2, and is a piecewise polynomial function with (m - 1) times continuous differentiability. Its Fourier transform is

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