Chau Chemometrics From Basics to Wavelet Transform

and detail dj from the fast wavelet transform are defined as

cj = {cj ,0, cj ,1, . . . , cj ,N/2j −1}

and

dj = {dj ,0, dj ,1, . . . , dj ,N/2j −1}

Under periodic extension both decomposition [Eqs. (4.23) and (4.24)] and reconstruction [Eq. (4.27)] are very simple. Let us take j = 1 as an example and that assume N and L are even numbers:

1. First, extend signal c0 = {c0,k }Nk =−01 into

cext0 = {c0,0, c0,1, . . . , c0,N−1, . . . , c0,0, c0,1, . . . , c0,L−3}

with length (N + L − 2).

2.Implement a moving filter (see Chapter 2) along with cext0 by filter H = {h0, h1, . . . , hL−1} but move two steps at each time, up to N/2 times. The result of such moving filter is taken as approximation signal

c1 = {c1,0, c1,1, . . . , c1,N/2−1}.

3.Implement another moving filter (see Chapter 2) along with cext0 by filter G = {g0, g1, . . . , gL−1} but move two steps at each time, up to N/2 times. The result of such moving filter is taken as detail signal

d1 = {d1,0, d1,1, . . . , d1,N/2−1}.

Example 4.7. Consider a signal c0 = {1, 2, 3, 4}. The length of this signal is N = 4. Compute the approximation and detail signals given by Daubechies

Thus the extended signal should be of length N + L − 2 = 6:

cext0 = {1, 2, 3, 4, 1, 2}

By the moving-filter algorithm the first component of approximation signal is given by [see Eq. (4.28)]

and then by moving the filter two steps, the second component of approximation signal is [see Eq. (4.29)]

Similarly, the moving-filter algorithm with filter G is employed to cext0 . It turns out that

along cext1 with H and G are

c2,0 = h0c1,0 + h1c1,1 + h2c1,2 + h3c1,3 = 5

and

√

d2,0 = g0c1,0 + g1c1,1 + g2c1,2 + g3c1,3 = − 3.

This gives the final approximation and detail signal with period 1.

Now we will discuss the reconstruction algorithm for a signal with finite length. Still assuming that periodic extension has been employed in the decomposition procedure, we can describe the whole algorithm as follows:

1.Given approximation c1 = {c1,0, c1,1, . . . , c1,N/2−1} and detail d1 = {d1,0, d1,1, . . . , d1,N/2−1} which are generated by fast wavelet trans-

form with filters H = {h0, h1, . . . , hL−2, hL−1} and G = {g0, g1, . . . ,

gL2 , gL−1}, we obtain the first inverse filters as H = {hL−1, hL−2, . . . , h1, h0} and G = {gL−1, gL−2, . . . , g1, g0}. Let L be even and denote

L = 2m.

2.Periodically extend both c1 and d1 from the front to cext1 and dext1 with length of N2 + L2 − 1. Insert 0s between the components of both cext1 and dext1 as well as before the first components and after the last components such that the resulting signals have length of N + L − 1.

The resultant signals are

c˜1 = {0, c1,N/2−L/2−1, 0, . . . , 0, c1,N/2−1, 0, c1,0, 0, . . . , 0, c1,N/2−1, 0}

and

˜ = { } d1 0, d1,N/2−L/2−1, 0, . . . , 0, d1,N/2−1, 0, d1,0, 0, . . . , 0, d1,N/2−1, 0

˜˜

3.Implement moving filters (see Chapter 2) along with c1 and d1 with H and G , respectively. Add two results of moving filters together to

give the reconstruction signal.

Example 4.8. From approximation signal

and detail signal

2 d1 = 0, √

2

at level 1, use inverse fast wavelet transform with Daubechies wavelet of the second order to reconstruct the original signal.

Note that N/2 = 2 and L = 4, the extended and inserted approximation and detail signals, are

Similarly, we can compute c0,1, c0,2, and c0,3, and so on.

On the basis of the periodic assumption for a signal, the length of approx-

imation signal cj +1 = {cj +1,0, cj +1,1, . . . , cj +1,N/2j +1−1} and detail signal dj +1 = {dj +1,0, dj +1,1, . . . , dj +1,N/2j +1−1} is half the of length of cj . Ordinarily the length of the signal is assumed to be power 2 in the standard fast

wavelet transform. For a signal of any length the following ‘‘keep one’’ scheme can be applied:

1.Assume that the length of a finite discrete signal c0 = {c0,0, c0,1, . . . , c0,N−1} is N. If N is even, then implement the fast wavelet transform once to get approximation signal c1 and detail signal d1 with

the structure of a three-levels wavelet packet decomposition. All parts in each line of this diagram are labeled by the level j , and the number p of parts from left to right and are represented as a decomposed signal wjp = {wjp,k |k = . . . , −2, −1, 0, 1, 2, . . . }. For example, w00 corresponds to an original signal c at level 0. The idea of this decomposition is to start from a scale-oriented decomposition and then to analyze the obtained signals on frequency subbands.

The computation scheme for wavelet packets is easy when using an orthogonal wavelet. Assume that φ(x ) is a scaling function of a given MRSD and ψ(x ) is the corresponding wavelet function associated with a scaling filter H = {hk |k = . . . , −2, −1, 0, 1, 2, . . . }, and a wavelet filter G = {gk |

corresponding to its abovementioned part as

+∞

wj2,kp = hm−2k wjp−1,m m=−∞

+∞

wj2,kp+1 = gm−2k wjp−1,m m=−∞

Thus, wj2p and wj2p+1 are, respectively, the approximation signal and detail signal of wjp−1 from one-step fast wavelet transform. In Figure 4.8, each part, except for the parts in the bottom line, covers two subparts that are obtained from themselves by fast wavelet transform in one-step. In general, suppose that an original signal w00 is of finite length, say, N; then each part wjp (p = 0, 1, . . . , 2j − 1) should be of finite length 2Nj .

At the reconstruction stage each part can be reconstructed from its subparts by inverse fast wavelet transform as

It is obvious that the overall diagram of wjp is redundant for reconstruction of the original signal c0 = w00. One question to be answered is how many resulting signals wjp should be stored in order to reconstruct perfectly the original signal w00 at the top of the diagram without any redundant information. In order to answer this question, first let us note that all the information contained in any parts of wavelet packet is conserved in its own two subparts because of the orthogonal fast wavelet transform. Hence, if any part in a wavelet packet diagram is chosen to be stored, then any subparts as well as all sub-subparts and its upper parts do not have to be kept. Figure 4.9

136 fundamentals of wavelet transform

−1, 0, 1, 2, . . . } as done in standard fast wavelet transform while the recon-

4.4.2. Biorthogonal Spline Wavelets

In this section we introduce biorthogonal wavelets with a minimum size support for a specified number of vanishing moments. These biorthogonal wavelets with symmetric or antisymmetric supports are constructed by Cohen and co-workers [8] based on spline MRSD. These biorthogonal wavelets are known as CDF (Cohen--Daubechies--Feauveau) biorthogonal wavelets or spline biorthogonal wavelets.

In the construction of orthogonal spline wavelets, the scaling function φ(x ) is defined through the orthogonalized version of Fourier transform of the B-spline function. Now this time let us directly choose the B-spline function of degree m − 1 as a scaling function φm (x ) whose Fourier transform is

where ν = 0 for m even and ν = 1 for m odd.

Cohen, Daubechies, and Feauveau gave the formulas for the biorthog-

onal filter H with minimal length.

Table 4.4 gives the filter coefficients for some m and m˜ . The resulting scaling functions and wavelet functions for m = 3 and m˜ = 9 are shown in Figure 4.10.

4.4.3.A Computing Example

is consistent with that of fast

Fast biorthogonal wavelet transform is the same as standard fast wavelet transform, while inverse fast biorthogonal wavelet transform is implemented with two dual wavelet filtersH andG. Yet, the computation structure wavelet transform. Thus the strategy intro-

duced in Section 4.3.3 for dealing with signals of finite length still be valid for the fast biorthogonal wavelet transform and reconstruction.

In this section, we give an example demonstrating how to implement such algorithms in the case of biorthogonal spline wavelet. Consider