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1

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

3

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).

8

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

11

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

13

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(

14

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

15

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) .

18

Fig. 7: Linear B-Spline N2(2x-k) .

19

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 .

24

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

26

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

27

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

28

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

34

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

35

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

36

Fig. 13: Cubic B-Spline N4(x) .

37

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.

45

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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