1 wavelets examples 王隆仁. 2 contents o introduction o haar wavelets o general order b-spline...
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Wavelets
Examples
王隆仁
2
Contents Introduction Haar Wavelets General Order B-Spline Wavelets Linear B-Spline Wavelets Quadratic B-Spline Wavelets Cubic B-Spline Wavelets Daubechies Wavelets
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I. Introduction
Wavelets are basis functions in continuous
time.
A basis is a set of linearly independent functions
that can be used to produce all admissible functions
:
The special feature of the wavelet basis is that all
functions are constructed from a single
mother wavelet .
)(tjk
. )()(,
kjjkjk tctf
)(tjk)(t
(1)
4
A typical wavelet is compressed times and shifted times. Its formula is
The remarkable property that is achieved by many wavelets is orthogonality. The wavelets are orthogonal when their “inner products” are zero :
Orthogonality leads to a simple formula for each coefficient in the expansion for .
jk jk
. )2()( ktt jjk
. 0)()(
dttt JKjk
JKc )(tf
(2)
5
Multiply the expansion displayed in equation (1) by
and integrate :
All other terms in the sum disappear because of orthogonality. Equation (2) eliminates all integrals of times , except the one term that has j=J and k=K. That term produces . Then is the ratio of the two integrals in equation (3). That is,
)(tJK . )()()(
2
-dttcdtttf JKJKJK
jk JK 2)(tJK JKc
(3)
. )()( dtttfc JKJK
6
II. Haar Wavelets
2.1 Scaling functions Haar scaling function is defined by
and is shown in Fig. 1. Some examples of its translated and scaled versions are shown in Fig. 2-4.
The two-scale relation for Haar scaling function is
( )xfor x
otherwise
1 0 1
0
. )12( )2( )( xxx
7
0
1
-10 0.5 1 1.5 2 2.5
0
1
-10 0.5 1 1.5 2 2.5
0
1
-10 0.5 1 1.5 2 2.5
0
1
-10 0.5 1 1.5 2 2.5
Fig.1: Haar scaling function (x). Fig.2: Haar scaling function (x-1).
Fig.3: Haar scaling function (2x). Fig.4: Haar scaling function (2x-1).
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2.2 Wavelets The Haar wavelet (x) is given by
and is shown in Fig. 5. The two-scale relation for Haar wavelet is
( )x
for x
for x
otherwise
1 0
1 1
0
12
12
. )12( )2( )( xxx
9
Fig. 5: Haar Wavelet (x) .
0
1
-1
0 0.5 1 1.5 2 2.5
10
2.3 Decomposition relation Both of the two-scale relation together are called
the reconstruction relation.
The decomposition relation can be derived as
follows.
( )
( )
( )
( )
x
x
x
x
1 1
1 1
2
2 1
2 2
2
( )
( )
( )
( )
2
2 1
1 1
1 12
x
x
x
x
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3.1 Scaling functions
The m-th order B-Splines Nm is defined by
Note that the 1st order B-Spline N1(x) is the Haar scal
ing function.
0 10 1
)(1 otherwisexfor
xN
1
0 1
11
)(
)()(
dttxN
dttNNxN
m
mm
(4)
(5)
III. General Order B-Spline Wavelets
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The two-scale relation for B-spline scaling functions of general order m is
where the two-scale sequence {pk} for B-spline scal
ing functions are given by :
m
kmkm kxNpxN
0
)2()(
. 0for , 2 1 mkkmp m
k
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3.2 Wavelets The two-scale relation for B-spline wavelets for gen
eral order m is given by
where
23
0
)2()(m
kmkm kxNqx
m
lm
mkk lkNl
mq0
21 )1(2)1(
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3.3 Decomposition relation The decomposition relation for m-th order B-Spli
ne is
where
Zlkxbkxalxk
klkl ,)()()2( 22
Zlmllmk
kk cqa 2,212
112
1
Zlmllmk
kk cpb 2,212
112
1
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4.1 Scaling functions
Linear B-Spline N2(x) is derived from the recurren
ce (4) and (5) as the case m=2 for general B-Splines as follows and is shown in Fig.6 .
otherwise 0
21for 2 10for
)()(2 xxxx
xxN (6)
IV. Linear B-Spline Wavelets
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Then the functions in V1 subspace are
expressed explicitly as follows and is shown in Fig.7 .
otherwise 0
1for 22
for 2
)2(
)2(
2k
21
2k
21
2k
2k
2
xxk
xkx
kx
kxN
)2( kx
(7)
17
Fig. 6: Linear B-Spline N2(x) .
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Fig. 7: Linear B-Spline N2(2x-k) .
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Since the support of is [0, 2], its two-scale relation is in the form
By substituting the expressions (6) and (7) for each 1/2 interval between [0, 2] into (8), the coefficients {pk} are obtained and the two scale relation for Line
ar B-Spline is shown in Fig.8 and is given by
. )2()(2
0
k
k kxpx
)(x
(8)
. )22(2
1)12()2(
2
1)( xxxx
20
Fig. 8: Two-scale relation for N2 .
21
4.2 Wavelets The two-scale relation for Linear B-Spline wavelets
for general order m=2 is
where
4
022 )2()(
kk kxNqx
)1()(2)1(
)1(22)1(
44421
2
04
1
kNkNkN
lkNlq
k
l
kk
22
The term N4(k) is cubic B-spline and the recursion r
elation for general order B-spline is given by
This relation is useful to compute Nm(k) at some inte
ger values. Non-zero values of Nm(k) for some small
m are summarized in Table 1.
ZkkN k for , )( 1,2
)1(1
)(1
)( 11 k-Nm
km kN
m
kkN mmm
23
Table 1: Non-zero Nm(k) values for m = 2 ,…, 6 .
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Then the two-scale sequence {qk} for is computed as follows:
Thus the Linear B-Spline wavelets is
)(2 x
121
61
21
44421
4
21
21
44421
3
65
35
21
44421
2
21
21
44421
1
121
61
21
44421
0
))(()}3()4(2)5(){(
)1)(()}2()3(2)4(){(
))(()}1()2(2)3(){(
)1)(()}0()1(2)2(){(
))(()}1()0(2)1(){(
NNNq
NNNq
NNNq
NNNq
NNNq
)42()32(
)22()12()2()(
2121
221
265
221
2121
2
xNxN
xNxNxNx
25
Fig. 9: Linear B-Spline wavelet .)(2 x
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4.3 Decomposition relation
The decomposition sequences {ak} and {bk} are
written for Linear B-Spline (m=2) as
Noting that only three {pk} and five {qk} are non
-zero, i.e.,
and
Zlllk
kk cqa 4,23
112
1
Zlllk
kk cpb 4,23
112
1
},1,{},,{ 21
21
210 ppp
},,,,{},,,,{ 121
21
65
21
121
43210qqqqq
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5.1 Scaling functions
Quadratic B-spline N3(x) is shown in Fig.10 and gi
ven by
otherwise 032for )3(21for )(10for
)()(2
21
223
43
221
3xxxxxx
xxN
V. Quadratic B-Spline Wavelets
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Fig. 10: Quadratic B-Spline N3(x) .
29
Functions in V1 space are expressed as
otherwise 01for )32(
1for )2(for )2(
)2(
)2(
23
2k
2k2
21
2k
21
2k2
23
43
21
2k
2k2
21
3
xkxxkx
xkx
kx
kxN
)2( kx
30
The two-scale relation for quadratic B-Spline N3(x) i
s shown in Fig.11 and given as follow:
)32(4
1)22(
4
3
)12(4
3)2(
4
1)(
xx
xxx
31
Fig. 11: Two-scale relation for N3(x) .
32
5.2 Wavelets The quadratic B-spline wavelet is shown in Fig.12 a
nd the two-scale relation is given by
)72(480
1)62(
480
29
)52(480
147)42(
480
303
)32(480
303)22(
480
147
)12(480
29)2(
480
1)(
33
33
33
333
xNxN
xNxN
xNxN
xNxNx
33
Fig. 12: Quadratic B-Spline wavelet .)(3 x
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5.3 Decomposition relation
The decomposition sequences {ak} and {bk} are
written for Quadratic B-Spline (m=3) as
Noting that only four {pk} and eight {qk} are non
-zero, i.e., and
Zlllk
kk cqa 6,25
112
1
Zlllk
kk cpb 6,25
112
1
},,,{},,,{ 41
43
43
41
3210 pppp
},,,,,,,{},,,,,,,{ 4801
48029
480147
480303
480303
480147
48029
4801
76543210qqqqqqqq
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6.1 Scaling functions Cubic B-spline N4(x) shown in Fig.13 is given by
otherwise 0 43for )4(
32for )4460243(21for )412123(10for
)()(
361
2321
2361
361
4
xx
xxxxxxxxxx
xxN
VI. Cubic B-Spline Wavelets
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Fig. 13: Cubic B-Spline N4(x) .
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The two-scale relation for cubic B-Spline N4(x) is
and is shown in Fig.14.
)42(8
1
)32(8
4)22(
8
6
)12(8
4)2(
8
1)(
x
xx
xxx
38
Fig. 14: Two-scale relation for N4(x) .
39
6.2 Wavelets The Cubic B-Spline wavelet is shown in Fig.15
.
6.3 Decomposition relation The decomposition sequences for Cubic B-Spli
ne are :
Zlllk
kk cqa 8,27
112
1
Zlllk
kk cpb 8,27
112
1
40
Fig. 15: Cubic B-Spline wavelet .)(4 x
41
7.1 Scaling functions
Daubechies scaling function is defined by the fo
llowing two-scale relation :
)32(4
31)22(
4
33
)12(4
33)2(
4
31
)2()(
33
33
3
03
xx
xx
kxpx
DD
DD
kk
D
VII. Daubechies Wavelets
D3
42
That is, non-zero values of the two-scale sequence {pk} are :
Note that the coefficients {pk} have properties p0 +
p2 = p1 + p3 = 1 . Figure 16 and 17 show the Daube
chies scaling functions, N is the length of the coefficients.
4
31,
4
33,
4
33,
4
31},,,{ 3210 pppp
43
Fig. 16: Daubechies Scaling Functions, N=4,6,8,10.
44
Fig. 17: Daubechies Scaling Functions, N=12,16,20,40.
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7.2 Wavelets
The two-scale relation for the Daubechies wavelets i
s in the following form :
)12(4
31)2(
4
33
)12(4
33)22(
4
31
)2()(
33
33
1
233
xx
xx
kxqx
DD
DD
k
Dk
D
46
Therefore the non-zero values of the two-scale sequence {qk} are :
Figure 18 and 19 show the Daubechies wavelets, N is the length of the coefficients.
4
31,
4
33,
4
33,
4
31
},,,{
},,,{
0123
1012
pppp
qqqq
47
Fig. 18: Daubechies Wavelets, N=4,6,8,10.
48
Fig. 19: Daubechies Wavelets, N=12,16,20,40.
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