WWavelets & Multiresolution Analysis

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Wavelets 165

PART V

Wavelets & Multiresolution Analysis

A DDITIONAL REFERENCES : A. Cohen, Numerical Analysis of Wavelet Methods, North-Holland, (2003) S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, (1999) I. Daubechies, Ten Lectures on Wavelets, SIAM, (1992) www.wavelet.org


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Wavelets 166

WAVELETS OVERVIEW (I)

What is wrong with FOURIER ANALYSIS ???

All spatial information is hidden in the PHASES of the expansioncoefcients and therefore not readily available

Localized functions (bumps) tend to have a very complexrepresentation in Fourier space

Local modication of the function affects its whole Fourier transform If the dominant frequency changes in space, only average frequencies are

encoded in Fourier coefcients

Remedy need an analysis tool that will encode both SPACE (TIME ) andFREQUENCY information at the same time

Following the convention, will work with TIME (t ) and FREQUENCY ( ) ,rather than wavenumber ( k )


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WAVELETS OVERVIEW (II) From D ISCRETE FOURIER TRANSFORM to INTEGRAL FOURIER

T RANSFORM Consider the space L2

of squareintegrable functionsdened on ; if f L2

satises suitable decay conditions at (which??), the D ISCRETE FOURIER TRANSFORM can be replaced with theINTEGRAL FOURIER TRANSFORM

f

f

x

e i t dt

f

t

f

ei t d

Interestingly, the Fourier Transforms (both discrete and integral) areconstructed as superpositions of DIL ATIONS of the function w

x

eix

(wk

t

w

kt

)

Want to construct an integral transform using a basis function which isvery localized (a wavelet); we will therefore need: DILATIONS

TRANSLATIONS


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Wavelets 168

WAVELETS G ABOR TRANSFORM (I) The history begins with a WINDOWED FOURIER TRANSFORM known as the

G ABOR TRANSFORM (1946)

G b f

f

t

e i t

g

t b

dt

where the WINDOW FUNCTION is given by g

t

12

e

t 2

4 with 0

Note that the Fourier transform of a Gaussian function is another Gaussianfunction, i.e.,

e

i xeax2

dx a e

2

4 a

Note also that the window function has the following normalization

g t

b

db

g x dx 1

Therefore, for the Gabor transform we obtain

G b f db f

Thus, the set

G b f : b of Gabor transforms of f decomposes theFourier transforms f of f exactly to give its LOCAL spectral information


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Wavelets 169

WAVELETS G ABOR TRANSFORM (II)

The WIDTH of the window function can be characterized by employing thenotion of the STANDARD DE VIATION

g 1

g

2

x2 g2 x dx

1 2

Note that for 0 g Proof:

g

8

1

4 can be evaluated setting 0 and a

2

1 in theexpression for the Fourier transform of a Gaussian function

x

2 g2 x dx can be evaluated differentiating twice the Fourier transform of aGaussian function and again setting 0 and a

2

1

Instead of localizing the Fourier transform of f , the Gabor transform mayequivalently be regarded as windowing f with the WINDOW FUNCTION G b

G b f f G b

f

t

G b

t dt G b

t ei t

2

e

t 2

4


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Wavelets 170

WAVELETS G ABOR TRANSFORM (III) Using the Parseval identity and noting that

G b

e ib

e

2

we obtain for the Gabor transform

G b f f G b

12

f

G b

12

f

eib

e

2

d

e

ib

2

eib f

g1

4 d

e ib

2

G

1

4

f

b

The third line ( in red ) indicates that up to a multiplicative factor e ib

the WINDOWED FOURIER TRANSFORM of f with g at t b, the WINDOWED INVERSE FOURIER TRANSFORM of f with g1

4 at

ARE EQUAL

!


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Wavelets 171

WAVELETS U NCERTAINTY PRINCIPLE (I) Consider more general window functions w L2

which satisfy therequirement

tw

t

L2

It can be shown that

t

1

2w

t

L2

w L1

the Fourier transform w is continuous

w L2

Note, however, that in general x w

x

L2

, therefore w may not in generalbe a FREQUENCY WINDOW FUNCTION

If w L2

is chosen so that both w and w satisfy the above condition, thenthe window Fourier transform

G b f

f

t

e i t

w

t b

dt

f

W b

where W b

ei t

w

t

b

, is called a SHORT

TIME

FOURIER TRANSFORM


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Wavelets 173

WAVELETS U NCERTAINTY PRINCIPLE (III) Therefore by choosing w L2

, such that xw

x

L2

and x w

x

L2

, to dene a windowed Fourier transform

G b f

we obtainlocalization in a TIME FREQUENCY WINDOW

x

b

w x

b

w

w

w

with area equal to 4ww

In fact, there is a relation between possible time and frequency windowswhich is made precise in the following theorem

H EISENBERG UNCERTAINTY PRINCIPLE Let w L2

be chosen sothat xw

x

L2

and x w

x

L2

. Then

ww 1

2Furthermore, equality is attained if and only if

w

t

cei t g

t b

where c

0,

0, and a

b

.


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Wavelets 174

WAVELETS U NCERTAINTY PRINCIPLE (IV) Proof of the HEISENBERG UNCERTAINTY PRINCIPLE

Let us assume that the centers x and are zero (if they are not, then wecan modify w as w

t

e i t f

t

x

)

We observe that

2w2w

t

2

w

t

2 dt

2

w

2 d

w

22 w

22

t

2

w

t

2 dt

w

t

2 dt

w

42

Using the Schwarz inequality we get

2w2w

1

w

42

tw

t

w

t

dt

2

1

w

42

t 2

w

t

w

t

w

t

w

t

dt

2

14

w

42

t

w

t

2

dt

2


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Wavelets 175

WAVELETS U NCERTAINTY PRINCIPLE (V)

Proof of the HEISENBERG UNCERTAINTY PRINCIPLE continued Integrating by parts and noting that lim

t

0 t f

t

0 (since

t 1 2w t L2 seen earlier) we obtain

2w2w

14

w

42

w

t

2 dt

2

14

An equality will be obtained when the Schwarz inequality becomes anequality; this implies that there exists b such that

w

t

2btw

t

so that there exists an a such that w

t

ae bt 2

Thus the GABOR TRANSFORM has the smallest possible timefrequencywindow.

The above Heisenberg Uncertainty Principle has farreaching consequences.


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Wavelets 176

INTEGRAL WAVELET TRANSFORM (I)

The shorttime Fourier transform has a RIGID timefrequency window, inthe sense that its width ( w) is unchanged for all frequencies analyzed; this

turns out to be a limitation when studying functions with varying frequencycontent

The INTEGRAL WAVELET TRANSFORM provides a window which: automatically narrows when focusing on high frequencies,

automatically widens when focusing on low frequencies If L2

satises the admissibility condition

C

2

d

then is called a BASIC WAVELET . Relative to every basic wavelet . theINTEGRAL WAVELET TRANSFORM (IWT) in L2

is dened by

W f

a

b

a

12

f

x

x ba

dx

f L2

a

0

b


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Wavelets 177

INTEGRAL WAVELET TRANSFORM (II) Hereafter we will assume that t

t

L2

and

L2

, so that thebasic wavelet provides a time-frequency window with nite area

From the above assumption it also follows that is a continuous functionand therefore niteness of C implies

0

0

t

dt 0

Setting

b;a

t

a

12

t b

a

the IWT can be written as

W f

b

a

f

b;a

If the wavelet has the center and radius given by t and , respectively,then the function b;a has its center at b at

and radius equal to a Thus, the IWT provides local information about the function f in a time

window

b

at a b at

a

which narrows down as a 0.


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Wavelets 178

INTEGRAL WAVELET TRANSFORM (III) Consider the Fourier transform of a basic wavelet

12 b;a

a

12

2

e i t

t ba

dt

a

a

12

2 e i b

Suppose that has the center and radius . Dening

we obtain a window function with center at the origin and unchanged radius

Applying the Parseval identity to the denition of the IWT we obtain

W f

a

b

a

a

12

2

f

ei t

a

d

which, modulo multiplication by a constant factor and a linear frequencyshift, localizes information about the function f to the FREQUENCYWINDOW

a

1a

a

1a


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Wavelets 179

INTEGRAL WAVELET TRANSFORM (IV) Note that the ratio of the CENTER FREQUENCY

a to the BANDWIDTH2 a

center frequencybandwidth

2

is independent of the scaling a ; thus, the bandwidth grows with frequency inan adaptive fashion ( constantQ ltering )

Reconstruction of a function from its IWTLet be a basic wavelet, then f

g L2

0

W f

b

a

W g

b

a

db

daa 2

12

C

f

g

Furthermore, for any f L2

and x at which f is continuous

f

x

2

C

0

W f

b

a

b;a

x

db

da

a 2

Proof using the Parseval identity, integrating with respect to da

a2 andusing the denition of C Note the role of the ADMISSIBILITY condition for


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Wavelets 180

D ISCRETE WAVELET TRANSFORM (I)

Consider the IWT at a discrete set of samples a 2 j and b k 2

j forsome j

k

W f

k 2 j

12 j

f x 2 j 2 2 j x

k dx f j k

where j

k 2 j 2

2 j x k

must be chosen so that j

k form a Riesz basis in L2

(i.e, the linear spanof j

k with j k is dense in L2

)

If j

k with j k is a RIESZ BASIS , the the relation

j

k l m

j

lk

m j k l m

uniquely denes ANOTHER RIESZ BASIS l m known as the DUAL BASIS

Thus, every function f L2

has a unique representation

f

x

j

k

f

j

k

j k

x


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Wavelets 182

D ISCRETE WAVELET TRANSFORM (III) Consider a wavelet and the Riesz basis j

k it generates; for each j , letW j denote THE CLOSURE OF THE L INEAR SPAN of

j

k : k

, i.e.,

W j clos L2

j

k : k

Evidently, L2

can be decomposed as a DIRECT SUM of the spaces W j(dots over pluses indicate direct sums)

L2

j

W j W 1 W 0 W 1

and therefore every function f L2 has a unique decomposition f

x

g1

x

g0

x

g1

x

where g j W j , j

if is an ORTHOGONAL WAVELET , then the subspaces W j

L2

areMUTUALLY ORTHOGONAL W j W l , j l which means that

g j

gl

0

j

l

where g j W j and gl W l


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Wavelets 183

D ISCRETE WAVELET TRANSFORM (IV)

Therefore, in such case, the direct sum becomes an ORTHOGONAL SUM

L2

j

W j W 1 W 0 W 1

Thus, an orthogonal wavelet generates an ORTHOGONALDECOMPOSITION of the space L2

, as the functions g j are

UNIQUE

MUTUALLY ORTHOGONAL


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Wavelets 184

M ULTIRESOLUTION ANALYSIS (I) For every wavelet (not necessarily orthogonal) we can consider the

following space V j L2

, j

V j W j 2 W j 1

The subspaces V j have the following very interesting properties:

1. V 1 V 0 V 1

2. clos L2

j V j

L2

3.

j

V j

0

4. V j

1 V j W j, j

5. f

x

V j f

2 x

V j

1 , j

Note that

In contrast to the subspaces W j which satisfy W j W l

0

, j

l, thesequence of subspaces V j is NESTED (1

) Every f L2

can be approximated with A RBITRARY ACCURACY byits projections P j f on V j (2

)


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M ULTIRESOLUTION ANALYSIS (II)

If the reference subspace V 0 is generated by a single SCALING FUNCTION L2

in the sense that

V 0 clos L2

0 k : k

where j

k 2 j

2

2 j x k

then all the subspaces V j are also generated by the same as

V j clos L2

j

k : k

in the same way as the subspaces W j are generated by the wavelet

In the MULTIRESOLUTION ANALYSIS at a given scale ( j

1) the subspace V j represents the LARGE SCALE features of the function the subspaces W j represents the SMALL SCALE features (details) of the

function

W l 186


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Wavelets 186

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WWavelets & Multiresolution Analysis

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