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# Wavelets 1

Apr 05, 2018

## Documents

Nitasha Sharma
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Wavelet basicsHennie ter Morsche

1. Introduction

2. The continuous/discrete wavelet transform

3. Multi-resolution analysis

4. Scaling functions

5. The Fast Wavelet Transform

6. Examples

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1. Introduction

For a given univariate function f , the Fourier transform of f and the inverse are given by

f () = f ( t )ei t dt . f ( t ) =

12 f () e i t d .

Parseval: ( f , g ) = ( f , g)/ 2 , ( f , g ) =

f ( t ) g (t ) dt .

e ( t )

=ei t ,

0()

=(

0)

f ( 0) = ( f , e0) = ( f , 0 )

0 0.5 1 1.5 2 2.5 3 3.5 41

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

TIJD0 5 10 15

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

HERTZ

Figure 1: The frequency break and its amplitude-spectrum

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The short time Fourier transform

Given a Window function g

g

L2(IR) , g =1 g is real-valued.

The short time Fourier transform F (u , ) of a function f isde ned by

F (u , ) = f ( t )eiut g ( t ) dt , f ( t ) =

12

F (u , ) e iut g ( t ) d du ,

gu, ( t ) :=e iut g ( t ) , F (u , ) = ( f , gu , )

( f , gu, ) =1

2( f , gu, ) ( Parseval) .

gu, () =ei (u) g ( u ).Fixed window width in time and frequency.

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2. The continous/discrete Wavelet transform

The continuous Wavelet transform

Given in L2

(IR) .Introduce a family of functions a ,b (a > 0, bIR ) as follows

a ,b( t ) =1

a (( t b)/ a ) ( t IR), a ,b = .

The continuous wavelet transform F (a , b) of a function f isde ned by

F (a , b) = ( f , a ,b) =1

a f ( t )(( t b)/ a ) dt .( f , a ,b) =

12

( f , a ,b) Parseval .where

a ,b() = a e i b

( a ),

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The inverse wavelet transform

f ( t )

=C 1

0

1

a2

F (a , b) a ,b(t ) da db .

C = 0 |() |2

d .

Needed ( 0) =0, i.e.,

( t ) dt

=0.

This is the reason why the functions a ,b are called wavelets.

is called the Motherwavelet.

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Example : The Mexican hat (Morlet wavelet)

( t )

=

2

3 14 (1

t 2)et 2 / 2.

5 4 3 2 1 0 1 2 3 4 50.4

0.2

0

0.2

0.4

0.6

0.8

1

TIJD as2 1.5 1 0.5 0 0.5 1 1.5 2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

Hertz

Figure 2: The Mexican hat

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The wavelet transform of the frequency break using the Mexi-can hat

0 0.5 1 1.5 2 2.5 3 3.5 41

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

TIJD0 5 10 15

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

HERTZ

Figure 3: frequency break

a s

c h

a a

l

100 200 300 400 500 600 700 800 900 1000

2

4

16

32

64

128

Figure 4: Grey value picture of the waveletcof cinten

Horizontal b-axis contains 1000 samples on interval [0, 1].The vertical axis contains the a -values: 2 , 4, . . . , 128.

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The discrete wavelet transform

Sampling in the a -b plane.

a 0 > 1, b0 > 0a =a 0 , b =k a 0 b0, (k , ZZ ).The translation step is adapted to the scale

k , ( t ) =a/ 2

0 ( a 0 t k b0).Dyadic wavelets: a0 =2, b0 =1.

k , ( t ) =2 / 2 ( 2 t k ).( f , k , ) are called waveletcoef cients .

Discrete Wavelet transform: f ( f , k , )a. Problem of reconstruction:

f = k , ( f , k , ) k , .b. Problem of decomposition:

f = k , a k , k ,It would be nice if the functions k , constitute an orthonormalbasis of L2(IR) . (orthogonal wavelets)

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For orthogonal wavelets the reconstruction formula and the de-composition formula coincide.A biorthogonal wavelets system consists of two sets of waveletsgenerated by a mother wavelet and a dual wavelet

, for

which

( k , , m,n ) =k ,m ,n ,for all integer values k , , m en n .We assume that ( k , ) constitute a so called Riesz basis (nu-merically stable) of L2(IR) , i.e.

A ( f , f ) k ,

k , 2 B ( f , f )

for positive constants A en B , where f = k , k , k , .The reconstruction formula now reads

f

= k ,( f , k , ) k , .

Examples of biorthogonal wavelets are the bior family imple-mented in the MATLAB Toolbox

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3. Multi-resolution analysis

For a given function f , let

f = k =

( f , k , ) k , ,

Then

f =

= f .

f can be interpreted as that part of f which belongs to thescale .So, f = =f is a decomposition of f to different scalelevels .The function f belongs to the scale space W spanned by( k , ) with xed .

The space W 0 is spanned by the integer translates of the motherwavelet .

For integer n the function

gn( t ) =n1

= f ( t )

contains all the information of f up to scale level n 1.So gn

V n , where

V n =n1

=W .

It follows that V n =V n1W n1 (nZZ ) direct sum.10

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Properties of the sequence (V n )

a) V n1V n (n geheel ) ,

b)n

ZZ V n =L2(IR) ,

c)n

ZZ V n = {0},

d) f ( t )

V nf (2t )

V n+1 ,

e) f ( t )

V 0f ( t +1)V 0 .

If a sequence of subspaces (V n ) satis es the properties a) to e),then it is called a Multi-Resolution-Analysis (MRA) of L2(IR) .

If there exists a function such that V 0 is spanned by the in-

teger translates of , then is called a scaling function for theMRA.As a consequence one has that V n is spanned by k ,n , (n xed),

k ,n =2n/ 2 ( 2n t k )

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4. Scaling functions

Suf cient conditions for a compactly supported function tobe a scaling function for an MRA.

1. There exists a sequence of numbers ( pk ) , from which only a nite number differs from zero, such that

( t ) =

k = pk ( 2t k ) 2-scale relation .

2. The so-called Riesz function has no zeros on the unit circle.

Autocorrelation function of : () := ( t + ) ( t ) dt .Riesz function R( z) =

m=( m) zm .

3. Partition of the unity

k

( t k ) 1.

The Laurent polynomial P ( z) = 12 k pk zk is called the twoscale symbol of .

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Examples

B-splines of order m:

P ( z)

= z

+1

2

m

The Daubechies scaling function of order 2

P2( z) =12

1 + 34 +

3 + 34

z +3 3

4 z2 +

1 34

z3 .

For an orthonormal system one has

R( z) 1,|P ( z)|2 + |P ( z)|2 1 (| z| =1)

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Based on a given MRA with scaling function one may con-struct wavelets by rst completing the spaces V to a spaceV +1 by means of a space W , i.e.V +1 = V W in such away that there exists a function such that W is spanned by(( 2 t k )) .To satisfy V 1 =V 0W 0 the following conditions are necessaryand suf cient:

1. W 0V 1,

2. W 0 V 0 = {0},3. ( 2t )

V 0

W 0 and ( 2t 1)V 0W 0 .It follows that

( t ) =

k =qk ( 2t k ),

( 2t ) =

k

=

(a k ( t k ) +bk ( t k )) ( t IR),

( 2t 1) =

k =(ck ( t k ) +d k ( t k )) ( t IR).

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By introducing the Laurent series A( z) = k a k zk , B( z) =k bk z

k , C ( z) = k ck zk and D ( z) = k d k zk and the sym-bol Q ( z) = k qk zk for the wavelet , the application of theFourier-transform to the previous equations and the 2-scale re-lation for the scaling function nally lead to the following setof equations, which must hold for complex z with | z| =1.

A( z2) P ( z) + B( z2) Q ( z) =1/ 2, A( z2) P ( z) + B( z2) Q ( z) =1/ 2,C ( z2) P ( z) + D( z2) Q ( z) = z/ 2,C ( z2) P (

z)

+ D( z2) Q (

z)

= z/ 2,

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Now let (assuming the inverse exists)

P ( z) Q ( z)

P ( z) Q ( z)1

=H ( z) H ( z)G ( z) G ( z)

,

where

H ( z) =k

h k zk ,

G ( z) =k

gk zk .

Then

A( z2) = ( H ( z) + H ( z))/ 2, B( z2) = (G ( z) +G ( z))/ 2,C ( z2) = z ( H ( z) H ( z))/ 2, D( z2) = z (G ( z) G ( z))/ 2, .

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We now have

( 2t

k )

=

m=h 2m

k ( t

m)

+g2m

k ( t

m) ( t

IR).

It can be shown that the symbol P ( z) for the dual scaling andthe symbol Q ( z) for the dual wavelet will satisfy

P ( z) = H ( z1),Q ( z) =Q ( z1).

For orthogonal wavelets based on an orthogonal scaling func-tion one may choose

qk = (1)k p1k .

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5. The Fast Wavelet Transform

To obtain a wavelet decomposition of a function f in practice,one rst approximates f by a function from a space V n , which

is close to f . So let us assume that f itself belongs to V n . So

f =

k =a k ,nk ,n

Since V n =n1=W , one has

f =n1

=

k =d k , k ,

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V n =V n1W n1 implies

f =

k

=

a k ,n k ,n =

k

=

a k ,n1k ,n1 +

k

=

d k ,n1 k ,n1 .

Due to

k ,n =

m= 2 h 2mk m,n1 + 2 g2mk m,n1 .

we obtain

f =

k

=

a k ,n k ,n =

k

=

a k ,n 2 ( m

=

(h 2mk m,n1+g2mk m,n1)).

Our conclusion is

a m,n1 =

k = 2 h 2mk a k ,n , d m,n1 =

k = 2 g2mk a k ,n .

convolution and subsequently downsampling ( m 2 m) yieldsthe two sequences a (n1) = (a m,n1) en d (n1) = (d m,n1) .

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A repeated application of the previous operation leads to a de-composition of f to coarser levels, which can be expressed bythe following scheme and ltering proces.

a (n)-

@ @

@ Ra (n1)d (n1)

- @

@ @ R

a (n2)d (n2)

. . .-

@ @

@ Ra (n N )d (n N )

-

-

-

Lo_d

Hi_d

a (n)

-

-

a (n1)

d (n1)

?

?

-

-

Figure 5: Decomposition

Filter coef cients are 2 h k for the low pass lter and 2 gk for the high pass lter.

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ReconstructionIf a 1 and d 1 are given then we may reconstruct the approx-imation coef cients a .

f = f 1 +w 1

f =

k =a k , k ,

=

k =a k , 1k , 1 +

k =d k , 1 k , 1

=

k =

m=a

k , 11

2 p

m

2k +m,

+

k =

m=d k , 1

1 2 qm 2k +m, .

Hence,

k =a k , k ,

= k =

m=

1 2 a k , 1 pm2k +d k , 1qm2k m, .

Conclusion:

a k , =1

2

m=(a m, 1 pk 2m +d m, 1qk 2m ).

upsampling and subsequently convolution

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-a( 1)

-d ( 1)

6

6

-

-

Lo_r

Hi_r

?

6

-a( )

Figure 6: Reconstruction

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6. Examples

1. Haar wavelet

General characteristics:OrthogonalSupport width 1Filters length 2Number of vanishing moments for : 1Scaling function yes

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.5

1

0.5

0

0.5

1

1.5

Figure 7: Haar wavelet

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2. Daubechies family

General characteristics:

Order N

=1, . . .

OrthogonalSupport width 2 N 1Filters length 2 N Number of vanishing moments for N Scaling function yes

0 2 4 6 8 0.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2db4 : phi

0 2 4 6 8 1

0.5

0

0.5

1

1.5db4 : psi

Figure 8: Daubechies order 4

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3. Coi et family

General characteristics:

Order N

=1, . . . , 5

OrthogonalSupport width 6 N 1Filters length 6 N Symmetry near fromNumber of vanishing moments for 2 N

0 5 10 15 20 25 0.2

0

0.2

0.4

0.6

0.8

1

1.2coif4 : phi

0 5 10 15 20 25 1

0.5

0

0.5

1

1.5coif4 : psi

Figure 9: Coi et order 4

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Meyer wavelet

General characteristics:

OrthogonalCompact support noEffective support [-8, 8]Symmetry yesScaling function yes

10 5 0 5 10 0.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2Meyer scaling function

10 5 0 5 10 1

0.5

0

0.5

1

1.5Meyer wavelet function

Figure 10: Meyer

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