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

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which can be represented by the functions in S-1 [see Eq. (4.7)], then

S0 c 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 c S-1 would be expressed, for instance, in a statement such as 1 km is 1000 m. Generally, one has

?c S2 c S1 c S0 c S-1 c S-2 c - (4.8)

and

f(x) e S0 ^ f e Sj for any j = ... , -2, -1,0,1,2,... . (4.9)

Alternatively, we might note that 0O,O(x) = 0(x) e S-1 and

0o,o(x) = -= 0-1,0(x) + 12 0-1,1 (x) (4.10)

Example 4.3. Let us consider Example 4.2 again. With the aid of Equations (4.7) and (4.10), we can represent the signal f1(x) in Example 4.2 as follows:

,^1 . . 2 ^4 , , 5

f1(x) = 2 0-1,-2(x) + 2 0-1,-1(x) + 2 0-1,o(x) + 2 0-1,1(x)

2 +1 2 -1 , N 5 + 4 N 5 - 4 , N

= 2 0o,-1(x ) +---2 0o,-1(x ) +--2 0o,o(x ) +--2 0o,o(x)

= 1.500,-1 (x) + 4.50o,o(x) + O.50o,-1 (x) + O.50o,o(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 0o,_1(x) is the mean value of the original

108

fundamentals of wavelet transform

signal on the interval [-1,0] and the coefficient 0.5 of ^0,_i(x) is the differential or variant of the signal on the same interval. The coefficient 4.5 of 00,0(x) is the mean value of the original signal on the interval [0,1] and the coefficient 0.5 of f0t0(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

Haars wavelet ^(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? Haars 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_ 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 ^(x) e S-1 while ^(x) e 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 c L2(R) (for the example of measuring a length, the following relation means that we have a series of lengths at different scales, ... lightyear, kilometer, meter, ... )

{0} c

c S2 c Si c Sq c S-i c S-2 c

c L2(R)

(4.11)

introduction to wavelet transform and wavelet packet transform 109

with the following properties2

TO

lim Sj =11 Sj = L2(R)

j TO .

(4.12)

TO

H Sj = {0}

(4.13)

j= TO

f (x) e So ^ f (j e Sj

(4.14)

and there exists a function 0(x) belonging to L2(R) whose integer translates

is an orthonormal basis3 in L2(R). We also say that 4>(x) generates a multiresolution signal decomposition {Sj}.

The term fj(x) is the representation of f on the scale space Sj and contains all details of f(x) up to finer resolution level j. For example, if the most accurate ruler used is a millimeter ruler, then we obtain a length approximation only down to a millimeter. Property (4.12) says that the signal approximation fj(x) from Sj converges to an original signal f (x) when j ^ -to (the precision becomes finer and finer). You can imagine that fj is the most accurate length to the length f(x) in all measurable lengths Sj under possible rulers; on the other hand, when j ^ +to (the precision becomes coarser and coarser), Equation (4.13) implies that we lose all the details in the signal f (x). We cannot measure a length with bigger and bigger rulers. Property (4.14) means that Sj is the 2j scale version of S0 by changing scale and Sj is spanned by the scaled functions

that is, each element f (x) in Sj (j fixed) can be written in the following form:

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