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Chemometrics from basick to wavelet transform - Chau F.T

Chau F.T Chemometrics from basick to wavelet transform - Wiley publishing , 2004. - 333 p.
ISBN 0-471-20242-8
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Coiflets are wavelet functions with much more symmetry than Daubechies wavelets. A coiflet function f (x) has 2p vanishing moments, and the corresponding scaling function 0(x) has (2p - 1) vanishing moments. Both functions have a compact support of length (6p - 1).
The coefficients of scaling filters are listed in Table 4.3.
118 fundamentals of wavelet transform
Table 4.2. Daubechies Wavelet Coefficients
k hk k kk
p = 1 0 0.707106781 p = 6 0 0.111540743
1 0.707106781 1 0.494623890
p = 2 0 0.482962913 2 0.751133908
1 0.836516304 3 0.315250352
2 0.224143868 4 -0.226264693
3 -0.129409523 5 -0.129766868
p = 3 0 0.332670553 6 0.097501606
1 0.806891509 7 0.027522866
2 0.459877502 8 -0.031582039
3 -0.135011020 9 0.000553842
4 -0.085441274 10 0.004777258
5 0.035226292 11 -0.001077301
p = 4 0 0.230377813 p= 7 0 0.077852054
1 0.714846571 1 0.396539319
2 0.630880768 2 0.729132091
3 -0.027983769 3 0.469782287
4 -0.187034812 4 -0.143906004
5 0.030841382 5 -0.224036185
6 0.032883012 6 0.071309219
7 -0.010597402 7 0.080612609
p = 5 0 0.160102398 8 -0.038029937
1 0.603829270 9 -0.016574542
2 0.724308528 10 0.012550999
3 0.138428146 11 0.000429578
4 -0.242294887 12 -0.001801641
5 -0.032244870 13 0.000353714
6 0.077571494
7 -0.006241490
8 -0.012580752
9 0.003335725
For more details, see Daubechies’s book [10]. We suggest that you explore coiflets using the MATLAB wavelet toolbox.
4.3. FAST WAVELET ALGORITHM AND PACKET ALGORITHM
In this section we introduce the basic algorithm, derived by Mallat [14] for fast computation of the discrete wavelet transform. In practice, any signals from an equipment, such as chemical spectrum, are discrete. Thus the most useful transform should be in discrete version. The discrete wavelet algorithms can be easily derived from the scaling equations. The wavelet
fast wavelet algorithm and packet algorithm 119
Table 4.3. The Coefficients of Scaling Filters Corresponding to Coiflets
k hk k kk
p = 1 0 -0.015655728 p = 3 0 -0.000034600
1 -0.072732620 1 -0.000070983
2 0.384864847 2 0.000466217
3 0.852572020 3 0.001117519
4 0.337897662 4 -0.002574518
5 -0.072732612 5 -0.009007976
P = 2 0 -0.000720549 6 0.015880545
1 -0.001823209 7 0.034555028
2 0.005611435 8 -0.082301927
3 0.023680172 9 -0.071799822
4 -0.059434419 10 0.428483476
5 -0.076488599 11 0.793777223
6 0.417005184 12 0.405176902
7 0.812723635 13 -0.061123390
8 0.386110067 14 -0.065771911
9 -0.067372555 15 0.023452696
10 -0.041464937 16 0.007782596
11 0.016387336 17 -0.003793513
packet algorithm, a more general algorithm, will be also introduced in this section.
4.3.1. Fast Wavelet Transform
Consider a scaling function 0(x) whose integer translates are orthonormal. Assume that the scaling function 0(x) generates a MRSD {Sj}yl°+TO. For a given discrete signal c = {ck\k = ... , -2, -1,0,1,2,...}, let us associate c with a signal function in S0:
f(x) = J2 Ck4>(x - k) (4.22)
k=—to
Mallat developed an algorithm, called fast wavelet transform (FWT), to express the signal f(x) of Equation (4.22) in terms of the corresponding wavelet function f (x). The algorithm is defined as follows:
cj,k = ^ ' hm—2kcj-1,m, (4.23)
m=-<x)
dj,k = y ] 9m-2kcj-1,m. (4.24)
m=-TO
120
fundamentals of wavelet transform
In the language of signal processing, Equations (4.3) and (4.24) mean that the signals Cj = {Cj,k\k = ... , -2, -1,0,1,2,...} and dj = {djk\k = ... , -2, -1,0,1,2,...} are, respectively, the convolutions of {Cj-1k\k = ... , -2, -1,0,1,2,...} with the filters H* = {h-k\k = ... , -2, -1,0,1,2,...} = {... , h2, h1, h0, h-1, h-2,...} and G* = {g-k\k = ... , -2, -1,0,1,2, ...} = {... , g2, g-i, g0, g_-|, g-2,...} followed by ‘‘downsampling’’ of factor 2. Denote still by H* and G* such the convolution operators (with downsampling), respectively, then the decomposition algorithms (4.23) and (4.24) can be written as
Cj = H *Cj -i
dj = G*Cj-i
(4.25)
(4.26)
The whole decomposition process is started from c0 := c until J levels of decomposition where J is a given number of scales. A three-level decomposition process has been shown by Figure 4.7.
After a J-level decomposition process, the initial discrete signal c0 has been turned into a sequence of newly generated signals {cJ;
dj ; d
J -1 ;
; di}.
Example 4.5. In order to see how to implement the decomposition algorithm , consider a special discrete signal c0 = ... , 0 , 0 , 1, 2 , 2 , 1, 0 , 0 , ... such that c0, -2 = 1, c0, -1 = 2 , c0 ,0 = 2 , C0 ,1 = 1 and other c0, k = 0. Choose a Daubechies wavelet with the filter coefficients:
, 1-V3 , 3-V3 , 3 + V3 , 1+V3
h0 =-------, h1 =-----—, h2 =-----------, h3 =
4V2 ’ and from Equation (4.20)
4V2 ’
g-2 = —
1 + V3 4V2 ’
g-1 =
3 + V3 4V2 ’

g0 = -
3 -4V2 ’
g1 =
4\p? 1 - 43
4V2 '
FWT
0
c
d
c
2
2
d
c
3
3
d
c
Figure 4.7. The structure of a three-level fast wavelet transform.
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