Chau Chemometrics From Basics to Wavelet Transform

conventional techniques, wavelet decomposition produces a family of hierarchically organized decompositions. The selection of a suitable level for the hierarchy will depend on the signal and experience. Often, selection of the level is based on a desired lowpass cutoff frequency.

Wavelet transform (WT) has attracted interest in applied mathematics for signal and image processing [7,10,15]. In contrast to some existing popular methods, especially the FFT, this new mathematical technique has been demonstrated to be fast in computation with localization and quick decay properties. Since 1989, WT has been proposed for signal processing in chemical studies owing to its efficiency, large number of basis functions available, and high speed in data treatment. It has been applied successfully in flow injection analysis, high-performance liquid chromatography, infrared spectrometry, mass spectrometry, nuclear magnetic resonance spectrometry, UV--visible spectrometry, and voltammetry for data compression and smoothing. More than a hundred papers have been published on applying WT in chemical studies [1--6,11--13].

4.1. INTRODUCTION TO WAVELET TRANSFORM AND

WAVELET PACKET TRANSFORM

As you have noted at the beginning of this chapter, a set or collection of templates should be provided or constructed when we want to analyze an object. Before we introduce the concept of wavelet and multiscale analysis, let us consider the following simple example.

Let us consider measuring a length or distance. First what we need is some standard rules or templates. In the metric system, these templates are units such as kilometer, meter, centimeter, millimeter, and micrometer. Using these templates, we can express (measure or analyze) a length as, for instance, 23 m, 43 cm, and 6 mm. In this procedure, we decompose a length into a (linear) combination of templates; thus we have an idea about the length. Of course, you can choose another template system to measure a length, for example, the imperial unit system. Which system should be chosen is a purpose-based task.

In signal processing, the conventional analyzing tool is Fourier transform. In this technique, a signal to be analyzed is decomposed into a linear combination of standard sine waves at different frequencies, namely, Fourier templates. The benefit from this technique is that we can obtain frequency information in the signal. As the sine wave is a global wave signal, the frequency information obtained will be global. However, sometimes one may want to determine the local frequencies, in which case it will not

introduction to wavelet transform and wavelet packet transform 103

4.1.1.A Simple Example: Haar Wavelet

From this subsection, we restrict ourselves in one-dimensional (1D) signal. Generally, any signal or physical phenomenon with finite energy can be represented by a square integrable function; that is, we say a signal f (x ) is of finite energy if the signal function satisfies the square-integrable condition

+∞

f (x )2dx < ∞

−∞

Mathematically, the set or collection of all such signal functions is denoted by L2(R). Thus f (x ) L2(R) means that the signal function f (x ) is square integrable.

In order to introduce Mallat’s MRSD algorithm, we first consider Haar basis decomposition of signals here. For simplicity, we denote by Z the set of (positive and negative) integers, that is, Z = {0, ±1, ±2, ±3, . . . }. In 1910, Haar proposed the following simple piecewise constant function

The Haar function ψ(x ) is shown in Figure 4.3.

For each pair of integers j , k Z, Haar constructed a templates at scales 2j and translation k

For any given (positive or negative) integers j and k , the template

√

ψ (x ) is a piecewise constant function with value −(1/ 2j ) at the interval

j ,k √

[2j k , 2j k +2j −1), (1/ 2j ) at the interval [2j k +2j −1, 2j (k +1)) and 0 outside the interval [2j k , 2j (k + 1)]. j is called the scaling level, or simply level, and k is called the location.

introduction to wavelet transform and wavelet packet transform 105

First consider the set of finite energy functions L2(R). Let us definite the inner product of two finite-energy functions f (x ) and g(x ) from L2(R) as

+∞

−∞

As we know, the inner product f (x ), g(x ) can be used for measuring the similarity of two signals, f (x ) and g(x ). The larger the magnitude of inner product is, the greater the similarity is. If the inner product of two functions is 0, then we call these two functions ‘‘orthogonal to each other.’’ If function f (x ) is orthogonal to a function g(x ), then they are said to be dissimilar.

The inner product of a function f (x ) with itself is called the energy of the function. Now, by computing integral, you can easily prove that each Haar template ψj ,k (x ) defined in Equation (4.2) has energy 1 and the inner product of any two different Haar templates is 0 (i.e., the different Haar templates are dissimilar). With these two properties, we can say that the Haar collection H consists of an orthonormal basis of L2(R), called the Haar wavelet basis. Haar further proved that any finite-energy function f (x ) can be decomposed into a linear combination form of

which means that the Haar family is complete or sufficient to represent any finite energy signals.

For the example of measuring a length, Equation (4.4) tells us the length is how many kilometers and how many meters and how many millimeters, and so on. In general, here we can say that the expansion coefficientsf (x ), ψj ,k (x ) of ψj ,k (x ) in Equation (4.4) gives us information on how the signal f (x ) of, for instance, a spectrum or a chromatagram is similar to the template signal ψj ,k (x ), or we can determine by f (x ), ψj ,k (x ) how much of the template ψj ,k (x ) a spectrum or a chromatagram has.

As the integral of ψj ,k (x ) over (−∞, +∞) is zero, each term in summation (4.4) can be interpreted as detailed information on the scale j and the location k . If we put all the detailed information on the scales greater than j together, which is done by truncating finer resolution details from the decomposition (4.4), the new resultant approximation signal

can be created. For the Haar wavelet basis H = {ψj ,k (x )|j , k = . . . , −2, −1, 0, 1, 2, . . . }, fj (x ) is piecewise constant on each interval [2j k , 2j (k +1))

with width 2j . When j → −∞, piecewise function fj (x ) approaches the actual signal f (x ). For each integer j , denote by Sj the set of all such piecewise functions fj (x ).

For example, in the case that j = 0, S0 consists of all piecewise functions over intervals [k , k + 1). Define a basic unit step function

Then it is easy to prove that each function f0(x ) in S0 can be written as a linear combination of translated versions of the basic unit step function φ(x ):

∞

f0(x ) = ck φ(x − k )

k =−∞

In the wavelet community, φ(x ) is called a scaling function. In much the same way as we defined the Haar wavelet function ψ(x ), we can define a family of dilations/scales and translates for scaling function φ(x ):

Example 4.2. Let f1(x ) be a piecewise constant function on half intervals defined as

introduction to wavelet transform and wavelet packet transform 107

As φ0,0(x ) is a function in collection S0 and the functions φ−1,0(x ) and φ−1,1 belong to the collection S−1, note that

This equation states that the function φ0,0(x ) can be represented as the linear combination of the functions in S−1. On the other hand, each function in S0 is a form of linear combinations of φ0,k (k = . . . , −2, −1, 0, 1, 2, . . . ), which can be represented by the functions in S−1 [see Eq. (4.7)], then S0 S−1, which means that any function in S0 can be represented a form of linear combinations of the functions in S−1.

In the example of measuring length, the relation S0 S−1 would be expressed, for instance, in a statement such as 1 km is 1000 m. Generally,

Equations (4.7) and (4.10), we can represent the signal f1(x ) in Example 4.2 as follows:

= 1.5φ0,−1(x ) + 4.5φ0,0(x ) + 0.5ψ0,−1(x ) + 0.5ψ0,0(x )

We call this relation Haar wavelet expansion/representation. The significance of the Haar wavelet representation can be easily explained by this example. The coefficient 1.5 of φ0,−1(x ) is the mean value of the original

signal on the interval [−1, 0] and the coefficient 0.5 of ψ0,−1(x ) is the differential or variant of the signal on the same interval. The coefficient 4.5 of φ0,0(x ) is the mean value of the original signal on the interval [0,1] and the coefficient 0.5 of ψ0,0(x ) is the differential or variant of the signal on the same interval.

As explained above, we have obtained local average/mean information (like the result given by a moving-average filter) and differential/derivative information (e.g., the result provided by Savitsky--Golay filter) by using Haar wavelet expansion.

4.1.2.Multiresolution Signal Decomposition

Haar’s wavelet ψ(x ) defined by Equation (4.1) is one simple example of wavelets. It has a simple structure and compact support; however, it is not continuous, as can be seen from Figure 4.3. Are there any other wavelets with better properties, say, continuity with even more smoothness, compact support (i.e., the template function is zero outside a finite interval), and symmetry? How can we find them? Haar’s example provides us with a nested approximation structure (4.8) based on Haar wavelet basis decomposition (4.7). In terms of the approximation, the collection Sj −1 can provide a more accurate approximation to a signal than that by Sj . You can imagine, in the example of measuring a length, that the meter ruler is more accurate than a kilometer ruler and a millimeter ruler is more accurate than a meter ruler, and so forth.

It is very important to note that the Haar wavelet function ψ(x ) S−1 while ψ(x ) S0. This fact implies that the Haar wavelet function could be found in the complement of S0 in S−1. This process can be expressed exactly in terms of a multiresolution signal decomposition (MRSD) first noted by Mallat [14] and used by Daubechies [9,10] to construct a class of new wavelet functions. Although it is very difficult to construct a wavelet function directly, an approximation nested series {Sj } with some properties such as (4.8) and (4.9) could be easily provided at sometimes. Then a wavelet function might be constructed from such an approximation nested series.

A multiresolution signal decomposition of L2(R) is a nested series of closed subspaces Sj L2(R) (for the example of measuring a length, the following relation means that we have a series of lengths at different scales,