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Казанский национальный исследовательский технологический университет

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Chau Chemometrics From Basics to Wavelet Transform

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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
p = 2	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 φ(x ) whose integer translates are orthonormal. Assume that the scaling function φ(x ) generates a MRSD {Sj }−j =+∞∞ . 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 ) = k	+∞
f (x ) = k	ck φ(x − k )	(4.22)
	=−∞

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 ψ(x ). The algorithm is deﬁned as follows:

+∞
cj ,k = m hm−2k cj −1,m ,	(4.23)
=−∞
+∞
dj ,k = m gm−2k cj −1,m .	(4.24)
=−∞

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 =

{dj ,k |k = . . . , −2, −1, 0, 1, 2, . . . } are, respectively, the convolutions of

{cj −1,k \|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, g1, g0, g−1, 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 −1	(4.25)
dj = G cj −1	(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 ; dJ −1; . . . ; d1}.

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 − √

, h

3 − √

, h

3 + √

, h

1 + √

0 =

1 =

4√

and from Equation (4.20)

1 + √

, g

3 + √

, g

3 − √

, g

1 − √

−2 = −

4√

−1 =

4√

0 = −

4√

FWT

c1 d1

c2	d2
c3	d3

Figure 4.7. The structure of a three-level fast wavelet transform.

fast wavelet algorithm and packet algorithm

121

At the ﬁrst-level decomposition, we take j = 1 in the algorithms (4.23) and (4.24). By the algorithm we have

+∞

c1,−2 = m hm+4c0,m

=−∞

5 + 3√

4√

0,−4

0,−3

0,−2

0,−1

+∞

c1,−1 = m hm+2c0,m

=−∞

= h0c0,−2 + h1c0,−1 + h2c0,0 + h3c0,1 =

4√

+∞

c1,0 = m hm c0,m

=−∞

5 − 3√

0,0 +

0,1 +

0,2 +

0,3

4√

and while for other k , c1,k

= 0. Hence

. . . , 0, 0,

5 + 3√

, 0, 0, . . .

4√

Similarly, we have

+∞

d1,−1 = m gm+2c0,m

=−∞

−1 −

√

0,−3 +

0,−2 +

0,−1 =

4√

−2

0,−4

−1

+∞

d1,0 = m gm c0,m

=−∞

2√

= g−2c0,−2 + g−1c0,−1 + g0c0,0 + g1c0,1 =

4√

+∞

d1,1 = m gm−2c0,m

=−∞

1 −

√

0,0 +

0,1 +

0,3 =

−2

−1

0,2

4√

122

fundamentals of wavelet transform

and others d1,k

= 0. Hence

−1 − √

2√

1 − √

, 0, 0, . . .

1 =

. . . , 0, 0,

4√

Similar calculations produce

19 + 14√

22 − 10√

7 − 4√

2,−2

, c

2,−1 =

, c

2,0

4 − 9√

, d

2 + 10√

, d

2 − √

2,−1

2,0

2,1 =

91 + 73√

, c

82 − 62√

, c

19 − 11√

3,−2

3,−1 =

3,0

64√

37 − 55√

, d

58 − 22√

, d

5 − 3√

3,−1

3,0

= −

3,1 =

64√

For example, at three-level decomposition, the ﬁnal decomposition consists of the following discrete signals:

			. . . , 0, 0,	−1 − √						,			2√						,	1 − √											, 0, 0, . . .
d		=						3									3											3
	1			4√									4√							4√
					2												2							2
			. . . , 0, 0,	4 − 9√					,			2 + 10√											,		2 − √												, 0, 0, . . .
d	2	=					3														3															3

			16												16														16
			. . . , 0, 0,	37 − 55√										,				58 − 22√																,	5 − 3√							, 0, 0, . . .
d		=									3				−																	3								3
	3			64√
						2														64√						2										64√			2
			. . . , 0, 0,	91 + 73√										,		82			− 62√											,			19 − 11√										, 0, 0, . . .
c		=									3																3														3

	3			64√		2													64√			2														64√		2

4.3.2.Inverse Fast Wavelet Transform

Now we turn to the reconstruction problem. Suppose that we have an J -level decomposition {cJ ; dJ ; dJ −1; . . . ; d1} from a certain discrete signal c0, where cj = {cj ,k |k = . . . , −2, −1, 0, 1, 2, . . . } and dj = {dj ,k |k = . . . , −2, −1, 0, 1, 2, . . . }. The inverse wavelet transformation can be used to attain the aim of reconstructing a signal c0; that is, when we are given all the information provided by rulers on different scales, we can reconstruct the information of the whole length. The inverse wavelet is also a successive

fast wavelet algorithm and packet algorithm

123

procedure. In general we can get, at any level j , the inverse fast wavelet transform (IFWT)

+∞	+∞
cj ,k = m cj +1,m hk −2m + m dj +1,m gk −2m		(4.27)
=−∞	=−∞

which reconstructs the signal cj from cj +1 and dj +1 and can be recursively used to get the original signal c0 from j = J to j = 1.

Let us explain the meaning of reconstruction algorithm (4.27). Deﬁne a new signal sequence

c	cj +1,m	if l = 2m	for l	Z
ˆl =	0	if l = 2m	+ 1

which is created from the sequence {cj +1,m |m = . . . , −2, −1, 0, 1, 2, . . . } by inserting 0 values between its components. Consider the ﬁrst part of summation in the right-hand side (RHS) of (4.27). This part can be viewed as the discrete convolutions between the resulted signal cˆ = {cˆl |l = . . . , −2, −1, 0, 1, 2, . . . } and the ﬁlter H = {hk |k = . . . , −2, −1, 0, 1, 2, . . . }, that is, following an ‘‘upsampling’’ of factor 2 calculate the convolutions between the upsampled signal and the ﬁlter H = {hk |k = . . . , −2, −1, 0, 1, 2, . . . }. The second part of the summation in (4.27) has a similar explanation. The upsampling-convolution procedures described above are denoted by H and G, respectively; then the algorithm (4.27) is simply written as

cj = Hcj +1 + Gdj +1

Example 4.6. From Example 4.5, the two-level decomposed signals are

			. . . , 0, 0,		−1 − √					,			2√				,	1 −		√						, 0, 0, . . .
d		=						3								3								3
	1				4√							4						4√
						2															2
					4 − 9√							2 + 10√								,		2 −					√		, 0, 0, . . .
d	2	=	. . . , 0, 0,				3		,										3									3

				16								16								16
					19 + 14√									,	22 − 10√										,		7 − 4√
c	2	=	. . . , 0, 0,								3												3							3	, 0, 0, . . .
																											16
				16													16

In order to reconstruct the original signal c0, we have to get the smooth signal c1 at the level 1 ﬁrst.

124

fundamentals of wavelet transform

By the algorithm in Equation (4.27) we have

+∞

c1,−2 = m c2,m h−2−2m + m d2,m g−2−2m

=−∞

= c2,−2h2 + c2,−1h0 + d2,−1g0 + d2,0g−2

19 + 14√

3 + √

22 − 10√

1 − √

4√

4 − 9√

−3 + √

2 + 10√

−1 −

√

5 + 3√

4√

+∞

c1,−1 = m c2,m h−1−2m + m d2,m g−1−2m

=−∞

= c2,−2h3 + c2,−1h1 + d2,−1g1 + d2,0g−1

19 + 14√

1 + √

22 − 10√

3 − √

4√

4 − 9√

1 − √

2 + 10√

3 + √

4√

+∞

c1,0 = m c2,m h−2m + m d2,m g−2m

=−∞

= c2,−1h2 + c2,0h0 + d2,0g0 + d2,1g−2

22 − 10√

3 + √

7 − 4√

1 − √

4√

2 + 10√

−3 +

√

2 − √

−1 − √

5 − 3√

4√

Finally, similar calculations will result in the original signal c0. Take the component c0,0 as an example here:

+∞

c0,0 = m c1,m h−2m + m d1,m g−2m

=−∞

= c1,−1h2 + c1,0h0 + d1,0g0 + d1,1g−2

3 +

√

5 − 3√

1 −

√

4√

2√

−3 +

√

1 −

√

−1 −

√

+ 4√

4√

fast wavelet algorithm and packet algorithm

125

4.3.3. Finite Discrete Signal Handling with Wavelet Transform

The fast wavelet transforms (4.23) and (4.24) as well as the reconstruction algorithm (4.27) are deﬁned for inﬁnite signal sequences and the length of the ﬁlters H = {hk |k = . . . , −2, −1, 0, 1, 2, . . . } and G = {gk |k = . . . , −2, −1, 0, 1, 2, . . . } may also be inﬁnite, although most of the known wavelet ﬁlters have an ﬁnite support. However, any signals obtained from equipment is of ﬁnite length. Hence there exists a problem when dealing with a ﬁnite discrete signal.

For the sake of simplicity or in practice, we will assume that the support of ﬁlters H = {hk |k = . . . , −2, −1, 0, 1, 2, . . . } and G = {gk |k = . . . , −2, −1, 0, 1, 2, . . . } is ﬁnite, denoted by L. This is reasonable. Actually, most wavelet functions have ﬁlters with ﬁnite lengths. The ﬁlters of many other wavelet functions have rapidly decreasing properties; thus, when |k | is sufﬁciently large, one has hk ≈ 0 and gk ≈ 0. Moreover, without any confusion the indices of ﬁlter components are supposed to be from 0 to L − 1; thus other components hk and gk are zeros for k = 0, 1, . . . , L − 1. As a consequence of this assumption, one can write

H = {hk }Lk−=10 = {h0, h1, . . . , hL−1} and

G = {gk }Lk−=10 = {g0, g1, . . . , gL−1}

Generally the observed signal c0 is assumed to be of ﬁnite length. Without loss of generality, such a signal with ﬁnite length N is denoted by

c0 = {c0,k }Nk =−01 = {c0,0, c0,1, . . . , c0,N−1}

Let us consider the decomposition algorithm (4.23), with an analysis similar to that in Equation (4.24). In Section 4.3.1 we interpreted this algorithm as convolution between the signal c0 and the ﬁlter H = {h−k |k = . . . , −2, −1, 0, 1, 2, . . . } = {. . . , h2, h1, h0, h−1, h−2, . . . } in the case of level j = 1. In other words, each component in signal c1 is the inner product (the sum of products of components by components) of signal c0 and ﬁlter series H = {hk |k = . . . , −2, −1, 0, 1, 2, . . . } but with two-step shifting (right-hand forward) of components of H once. For example, we have

	L−1
c1,0 =	k	(4.28)
c1,0 =	hk c0,k	(4.28)
	=
	0
and
L−1
k		(4.29)
c1,1 =	hk c0,k +2	(4.29)
	=0

126 fundamentals of wavelet transform

and so on. The component c1,0 in (4.28) can be interpreted as average of vector c0 = (c0,0, c0,1, . . . , c0,N−1) of length N with weights H = {h0, h1, . . . , hL−1} (assume that N > L), and the component c1,1 in (4.29) is the average of vector c0 = (c0,0, c0,1, . . . , c0,N−1) but generated by moving window of weights H two steps righthand forward (for a discussion of moving-window ﬁltering, see Chapter 2). Other components in c1 can be computed similarly. However, there are some difﬁculties while computing

c1,−1 by

	L−1
c1,−1 =	k	(4.30)
c1,−1 =	hk c0,k −2	(4.30)
	=0

which is needed to reconstruct c0,0 by the reconstruction algorithm (4.27). In fact, we should at least have c1,−L/2, . . . , c1,−2 and c1,−1 to reconstruct c0,0, assuming that L is even. The difﬁculty in implementing (4.30) is that we have no information about c0,−2 and c0,−1. In order to obtain c1,−L/2, information about c0,−L, . . . , c0,−2 and c0,−1 should be provided. The similar situation occurs at the other end of c0.

The basic method to overcome the difﬁculty is to extend the signal c0 with a length L (in fact, L − 2 is sufﬁcient) at both sides. Let us still denote the extended signal by c0

c0 = {c˜0,−L, . . . , c˜0,−1, c0,0, c0,1, . . . , c0,N−1, c˜0,N , . . . , c˜0,N−1+L}

where the c˜0,k values are components to be added. It is interesting to notice that whatever the extension c˜0,k terms are, the decomposition performed using scheme (4.27) is perfect and the original signal can be recovered using inverse fast wavelet transform (4.27) from

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

and

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

where N is assumed to be even. This assumption is ﬁxed in the following discussion.

At the second level of fast wavelet transform, signal c1 = {c1,−L/2, . . . , c1,−1, c1,0, c1,1, . . . , c1,N/2−1} should be similarly extended in order to obtain perfect reconstruction at level 2 and so on. The length of cj and dj depends on the length of ﬁlter, the length of original signal c0, and the number of levels.

In actual applications, there exist many sophisticated methods for side extension. Typical methods for extending a signal include zero padding, symmetrization, extrapolation, and periodic extension.

fast wavelet algorithm and packet algorithm

127

Zero Padding. This method assumes that the signal is zero outside the original support. For example, if the original ﬁnite signal is

c0 = {1, −2, 3, 2, 1, 3}

with length 6, then one of possible zero-padded signals is

czero0 = {0, 0, 0, 0, 1, −2, 3, 2, 1, 3, 0, 0, 0, 0}.

However, the obvious disadvantage of zero padding is that discontinuities are artiﬁcially created at the sides.

Symmetrization. This method assumes that signals can be recovered outside their original support by symmetric boundary-value replication. For example, the signal

c0 = {1, −2, 3, 2, 1, 3}

may be extended as

csym0 = {2, 3, −2, 1, 1, −2, 3, 2, 1, 3, 3, 1, 2, 3}

This is the default mode of the wavelet transform in wavelet toolbox 2 of MATLAB. Symmetrization has the disadvantage of artiﬁcially creating discontinuities of the ﬁrst derivative at the border, but this method works well in general for images.

Extrapolation. To create some values beyond the ﬁnite signal c0, one can employ extrapolation methods. One such method is polynomial extrapolation. Let us consider linear extrapolation. For example, in order to obtain c0,−1, ﬁrst deﬁne a line that intersects points (0, c0,0) and (1, c0,1) and then compute c0,−1 such that the point ( − 1, c0,−1) is on the line. Hence we have

c0,−1 = 2c0,0 − c0,1

In other words, c0,0 is the mean value of c0,−1 and c0,1. In the similar calculation c0,−2, c0,−3 as well as c0,N , the value of c0,N+1 can be determined step by step. Hence in this procedure signal

c0 = {1, −2, 3, 2, 1, 3}

may be extended as

cline0 = {13, 10, 7, 4, 1, −2, 3, 2, 1, 3, 5, 7, 9, 11}

128	fundamentals of wavelet transform

Periodization. This method assumes that signal has a period of its length N. The signal may be extended by repeating its series of components. For example, under periodization, signal

c0 = {1, −2, 3, 2, 1, 3}

may be extended as

cper0 = {3, 2, 1, 3, 1, −2, 3, 2, 1, 3, 1, −2, 3, 2}

in which the ﬁrst extended components [3,2,1,3] are repeated by the last four components of the original signal and the last extended components [1,−2,3,2] are repeated from the ﬁrst four components of the original signal. If there is a large difference between the ﬁrst and last components of the original signal, it is obvious that discontinuity at extended points will be introduced. One method to deal with this problem is to implement translation--rotation transformation (TRT)

k		− c0,0)
c0,TRTk = c0,k − c0,0 − N − 1	(c0,N−1	− c0,0)	(4.31)

where k (=0, 1, . . . , N − 1) is the index and N is the length of the signal.

After the TRT treatment, both cTRT and cTRT		have the same value 0. Then
0,0	0,N−1

cTRT is extended periodically. By direct computation, the signal

c0 = {1, −2, 3, 2, 1, 3}

is converted into

cTRT0 = {0, −3.4, 1.2, −0.2, −1.6, 0}

by TRT through Equation (4.31), then periodically extended as

cper0 = {1.2, −0.2, −1.6, 0, 0, −3.4, 1.2, −0.2, −1.6, 0, 0, −3.4, 1.2, −0.2}

Conclusion. In the following discussion we assume that the periodic extension is chosen. Under periodic extension we can prove that both approximation signal c1 and detail signal d1 given by (4.23) and (4.24) respectively are also periodic with a period of N/2. It means that we only have to store N/2 components for c1 and d1, respectively. Denote

by c1 = {c1,0, c1,1, . . . , c1,N/2−1} and d1 = {d1,0, d1,1, . . . , d1,N/2−1} at the

ﬁrst-level wavelet transform. Generally, at any level j , the approximation cj

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{cj −1,k \|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	=