MA5242 Wavelets Lecture 3 Discrete Wavelet Transform

Wayne M. Lawton

Department of Mathematics

National University of Singapore

2 Science Drive 2

Singapore 117543

Email matwml@nus.edu.sgTel (65) 6874-2749

Riesz RepresentationTheorem. If is a finite dimensional unitary space

VvVv ,),(

},,{ 1 dbb

there exists an antilinear isomorphism V

such that

Proof. Let

and define

d

j jj bb1

)()(

Then

d

j jj vbbv1

,)()),((

)),((),()(11

d

j jj

d

j jj bvbvbb

V

VV :VvVvv ,),),(()( *

be an ONB for

Adjoint Transformations

Theorem: Given unitary spaces

WwVvvwTTvw ,),,(),( *

and a linear

there exists a uniquetransformation

WV ,

linear transformation

WVT :VWT :*

with

(adjoint of T)

Proof. Define

VWT :' by composition

let

WTT WWW ,'

VWTT WV :1'* be the Riesz Rep. transformations and define

WWVV WV :,:

Problem Set 1

and that

1. Assume that

are ONB for unitary spaces

between the matrices

WVT : is linear. Derive the relationship

][],[ TT

WV ,},,,{},,,{ 11 mn wwvv

that represent

with respect to these bases.TT ,

3. Derive the Riesz Representation, Adjoint and matrix representations, and characterization for orthogonal transformations for Euclidean spaces.

2. Prove that a transformation VVT :is unitary iff 1* TT

General Discrete Wavelet Transform

105432

10

123210

105432

10

123210

000

00

dddddd

ddddddd

cccccc

ccccccc

N

N

Convolution Representation

xd

c

b

a

Lx

x

x

x

x

LW

Lb

b

b

La

a

a

)21(

)3(

)2(

)1(

)0(

2

)1(

)2(

)0(

)1(

)2(

)0(

where a,b,c,dare infinite

sequences that extend the finite

sequences

Orthogonality Conditions

Theorem. The wavelet transform matrix is unitary iff

122

0)()2()(

kN

jkkjdjd

1,...,0 Nkfor all

122

0)()2()(

kN

jkkjcjc

122

00)2()(

kN

jkjdjc

122

00)2()(

kN

jkjcjd

Laurent Polynomials

Definition: A Laurent polynomial is a function

CC }0{\}0{\,)()( CzzkczP

Zk

ka

)()( jcjc

that admits a representation

where c is a finitely supported sequence.

Theorem: For seq. a, b,

Definition: For a sequence c let

)()()( zPzPzP baba TzzPzP

aa ),()(

and define the unit circle }1||:{ zCzT

Conjugate Quadrature Filters

Definition: A sequence c that satisfies the quadratic equations necessary for a wavelet transform matrix to me unitary is called a Conjugate Quadrature FilterTheorem. A sequence c is a CQF iff it satisfies

Theorem: Prove that if c is a CQF and if d is related to c by the equation on the previous page then d is also a CQF and the WT is unitary

TzzPzP cc ,2|)(||)(| 22

Theorem: If c is a CQF then the WT is unitary if

)12()1()( 1 kNckd k

Problem Set 2

1. Derive the conditions for a WT to be unitary. 2. Prove the theorems about Laurent polynomials and the two theorems on the preceding page.

4. Prove that d on the previous page is the same as

}0{\),()( 12 CzzPzzPc

Nd

3. Prove that c, d form a unitary WT iff

)()(

)()()(

21

, zPzP

zPzPzM

dc

dcdc

(the modulation matrix) is unitary for all

.Tz

Moment Conditions

Definition. d has -1< p vanishing moments if

1,...,0,0)(

pkjjdZj

k

Theorem. If c,d gives a unitary WT then d has -1< p vanishing moments

)(zPd has a factor pz )1( )(zPc

iff

has a factor pz )1( iff

Moment Consequences

Theorem. If d has -1< p vanishing moments and issupported on the set {0,1,…,2N-1} then

0)()( kxdkb

can be represented by a polynomial having degree < N

Proof. 1110

)( pjejeejx

N

if the finite sequence d(k),d(k-1),…,d(k-2N+1)

nN

m

p

n n mkemdkb )()()(12

0

1

0

0 by the binomial theorem and vanishing moments.

Riesz-Fejer Spectral Factorization

Theorem. A Laurent polynomial N is on0iff there exists a LP P such that

T

TzzPzN ,|)(|)( 2

Proof. Let

TzzzzN mk ,)()( )(

where

}0{\C be the set of roots of N

)(m is the multiplicity of Since N is real-valued /1furthermore, since N is non-negative the are even hence paired, now choose P to containone root from each pair and the result easily follows.

)(m

Daubechies Wavelets

Theorem. If c is a CQF supported on 0,1,…,2N-1 and then cca

)(zPcsatisfies and is uniquely determined by the equations

has a factorpz )1(

0

0

1

)12(

)23(

)21(

Na

Na

Na

12N1)(2N

12N2N)(3

12N2N)(1

12N2N32N1

111

Furthermore, TzzPa ,0)(and c can be chosen by the R.-F. Theorem.

Problem Set 3

1. Prove all of the Theorems after Problem Set 2.