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1 Wavelets Examples 王王王
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1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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Page 1: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

1

Wavelets

Examples

王隆仁

Page 2: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

2

Contents Introduction Haar Wavelets General Order B-Spline Wavelets Linear B-Spline Wavelets Quadratic B-Spline Wavelets Cubic B-Spline Wavelets Daubechies Wavelets

Page 3: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline 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)

Page 4: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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)

Page 5: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 6: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 7: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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).

Page 8: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 9: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

9

Fig. 5: Haar Wavelet (x) .

0

1

-1

0 0.5 1 1.5 2 2.5

Page 10: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 11: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 12: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic 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

Page 13: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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(

Page 14: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 15: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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

Page 16: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic 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)

Page 17: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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Fig. 6: Linear B-Spline N2(x) .

Page 18: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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

Page 19: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 20: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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Fig. 8: Two-scale relation for N2 .

Page 21: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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

Page 22: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 23: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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Table 1: Non-zero Nm(k) values for m = 2 ,…, 6 .

Page 24: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 25: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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Fig. 9: Linear B-Spline wavelet .)(2 x

Page 26: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 27: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 28: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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Fig. 10: Quadratic B-Spline N3(x) .

Page 29: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 30: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 31: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

31

Fig. 11: Two-scale relation for N3(x) .

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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

Page 33: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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Fig. 12: Quadratic B-Spline wavelet .)(3 x

Page 34: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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

Page 35: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 36: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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

Page 37: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 38: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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Fig. 14: Two-scale relation for N4(x) .

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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

Page 40: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

40

Fig. 15: Cubic B-Spline wavelet .)(4 x

Page 41: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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

Page 42: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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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

Page 43: 1 Wavelets Examples 王隆仁. 2 Contents o Introduction o Haar Wavelets o General Order B-Spline Wavelets o Linear B-Spline Wavelets o Quadratic B-Spline Wavelets.

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Fig. 16: Daubechies Scaling Functions, N=4,6,8,10.

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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

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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

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Fig. 18: Daubechies Wavelets, N=4,6,8,10.

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Fig. 19: Daubechies Wavelets, N=12,16,20,40.