Interpolatory periodic scaling functions and wavelets

From this analysis of general periodic scaling functions andwavelets, the extended interpolatory theory is developed in Chapter 2.. Examples of interpolatory periodic scaling functions a

SCALING FUNCTIONS AND WAVELETS

KOK CHI WEE

(B.Sc.(Hons.), NUS)

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF MATHEMATICS

NATIONAL UNIVERSITY OF SINGAPORE

2004

It has been quite a tough year taking the Accelerated Master’s Degree gramme Being the last scholar on this programme is an honour to me but at thesame time, rather pressurizing Full of confidence in the beginning, I started tolose faith in completing the degree as months passed without any fruitful research.There was no anxiety initially as my priority was placed on the coursework andteaching assignment Then came the December holidays when much of my timewas spent either experimenting cake recipes or dozing off from reading research ma-terial It was a period when I simply was not in the mood for research This wasfollowed by another semester of coursework, teaching assignment and zero amount

Pro-of research this time round It was not until the recent three months Pro-of intenseresearch and hard work when almost all the stuff in this thesis was created Iattribute this feat to the following people

First of all, I express my gratitude to my supervisor, Associate Professor GohSay Song He trusted that I would eventually come up with something for thisthesis when I conveyed to him my lack of motivation Without his encouragementand help, I would have quitted long ago And of course, this thesis is ascribable tohis guidance

ii

Next, I would like to thank my instructors and fellow classmates of the culinaryclasses I have attended during this arduous period Their joyous company lightened

my tense mood

Not forgetting my friends, I would like to thank Weiguang, Minghui, Jinfeng,Jeffrey, Jiayi, Fengying, Baoling and Philip whom I played badminton with almostevery week The game sessions with them helped me vent my frustrations

Last but not least, I would like to show my appreciation to my family membersfor their utmost care and concern that pulled me through the difficulties Mythanks goes to — my sisters for sharing their titbits, comics and teddy bears; mydad for all the little things in life; and my mom for keeping my room clean andcooking the homely meals

Chi WeeJuly 2004

Acknowledgements ii

1.1 Preliminaries 11.2 Multiresolution of L2[0, 2π) and Polyphase Splines 31.3 Periodic Wavelets from Polyphase Splines 12

2.1 Introduction 312.2 Interpolatory Properties 33

2.3 Existence of Interpolatory Wavelets 41

iv

3 Examples of Interpolatory Trigonometric Scaling Functions and

Wavelets play a major role in many applications like data compression, signaland image processing, and resolution enhancement Many studies have been con-ducted on wavelets and here we consider interpolatory periodic wavelets, a class ofwavelets that provides certain advantage over other wavelets.

This thesis continues the study of interpolatory periodic wavelets done in thehonours thesis [7] Periodic multiresolutions with dilation 2 were considered in thehonours thesis and extension is being made by considering a general dilation Mthat is greater than or equal to 2 Chapter 1 contains a detailed study of periodicmultiresolutions with general dilation, together with their corresponding scalingfunctions and wavelets From this analysis of general periodic scaling functions andwavelets, the extended interpolatory theory is developed in Chapter 2 Examples

of interpolatory periodic scaling functions and wavelets are given in Chapter 3 Allresults in Chapters 2 and 3 are new

The paper [4] supplied part of the material in Chapter 1 on which the remainingnew material in the chapter is built upon Results in Section 1.2, which is aboutperiodic multiresolutions, are found in [4] Some of these have been formulated

or proved differently so as to follow the presentation sequence here In contrast,

vi

the development of periodic wavelets in Section 1.3 takes a different approach.This section contains necessary and sufficient conditions for the existence of variouswavelets, some of which are innovated from [4] and are cornerstones of the enhancedinterpolatory theory This section also encompasses new material on wavelets thatform Riesz bases for L2[0, 2π) For results which have analogous versions in thehonours thesis, their proofs are omitted to avoid duplication whenever there is nosignificant change in technique.

Besides the consideration of a general dilation, the generalization of the polating height of both scaling functions and wavelets add considerable intricacies

inter-to the original interpolainter-tory theory This is reflected in the comprehensive theorydeveloped in Chapter 2 Section 2.2 examines interpolatory properties of scalingfunctions and wavelets In addition, relations between the interpolating natureand the interpolating height of the scaling functions are established With the aid

of the innovated material from Chapter 1 as well as several linear algebra results,the discussion on the existence of interpolatory wavelets in the honours thesis iselevated in Section 2.3 to the much more general setting on hand Section 2.3 alsoincludes a formula for the interpolatory wavelets

Modified versions of the de la Vall´ee Poussin means of the Dirichlet kernelsfor dilation M are created independently and shown to form interpolatory peri-odic scaling functions in Section 3.2 In Section 3.3, a sharp observation of thesemodified functions leads to an extensive family of periodic scaling functions, some

of which possess interpolation or orthonormality properties It is demonstrated

in both sections that corresponding interpolatory periodic wavelets that form thonormal or Riesz bases for L2[0, 2π) can be constructed, using the theory inChapter 2 The examples in Chapter 3 complement the extended interpolatorytheory

or-Chapter 1

Periodic Scaling Functions and Wavelets

Let L2[0, 2π) be the space of all 2π-periodic square-integrable complex-valuedfunctions with inner product h·, ·i and norm k · k given by

hf, gi := 1

2π

Z 2π 0

be used in subsequent chapters, has no correspondence in [4] or [7]

The results in this section will be used later and the reader can refer to Chapter 1

of [7] for their proofs, with or without simple modifications

1

Proposition 1.1 Suppose {Vk}k>0 is a sequence of nested, finite-dimensional spaces of L2[0, 2π) and Wk is the orthogonal complement of Vk in Vk+1 for each

(b) Each pair f and g from S

Define ˆf (n) := hf, ein·i, n ∈ Z, as the Fourier coefficients of any function

f ∈ L2[0, 2π) With this notation, we have the well-known Parseval’s Identity

Proposition 1.2 (Parseval’s Identity) For every pair f and g from L2[0, 2π),

(a) The function f is real-valued a.e if and only if ˆf (n) = ˆf (−n), n ∈ Z

(b) The function f is even a.e if and only if ˆf (n) = ˆf (−n), n ∈ Z

Proposition 1.4 Let M be an integer greater than or equal to 2 If a ∈ l2(Z),then vk,j := P

p∈Za(j + Mkp)ei(j+M k p)· converges for k > 0, j = 0, 1, , Mk− 1,and

0 otherwise

1.2 Multiresolution of L2[0, 2π) and

Polyphase Splines

Let r and M be positive integers with M > 2 A periodic multiresolution (MR)

of L2[0, 2π) with multiplicity r and dilation M is a sequence of subspaces {Vk}k>0

of L2[0, 2π) that satisfies the following three conditions

MR1 For each k > 0, dim Vk = rMk and there exist functions φmk ∈ Vk, m =

1, 2, , r, such that {Tl

kφm

k : m = 1, 2, , r, l = 0, 1, , Mk− 1} is a basisfor Vk, where Tl

rMk(M − 1) and Wk ⊥ Wk0 when k 6= k0 As to be shown later, analogous to Vk,

For a function f ∈ L2[0, 2π), Parseval’s identity implies that P

n∈Z| ˆf (n)|2 =

kf k2 < ∞, that is, ˆf ∈ l2(Z) Thus for k > 0, j = 0, 1, , Mk− 1, P

p∈Zf (j +ˆ

Mkp)ei(j+Mkp)·converges by Proposition 1.4 Given {φmk : k > 0, m = 1, 2, , r} ⊂

L2[0, 2π), the polyphase splines of these functions, vm

k,j, k > 0, m = 1, 2, , r,

r = 1 and M = 2 The next few propositions are properties of polyphase splinesand their proofs follow analogously from those of orthogonal splines, which thereader can refer to Propositions 3.2 to 3.4 of [7].

Proposition 1.5 For k > 0, m, m0 = 1, 2, , r, and j, j0 = 0, 1, , Mk− 1,

Proposition 1.7 For k > 0 and m = 1, 2, , r,

The next step after introducing polyphase splines is to apply their properties

to derive a few important results concerning MRs of L2[0, 2π) in general

Applying Tkl, l = 0, 1, , Mk− 1, to (1.1) and using Proposition 1.6,

Fixing k > 0 and m ∈ {1, 2, , r}, and defining the following vector functions

Theorem 1.8 For k > 0, given {φm

k : m = 1, 2, , r} ⊂ L2[0, 2π) with responding polyphase splines {vm

cor-k,j : m = 1, 2, , r, j = 0, 1, , Mk− 1}, thefollowing statements are equivalent

(i) The collection {Tl

kφm

k : m = 1, 2, , r, l = 0, 1, , Mk− 1} is linearlyindependent

(ii) The collection {vm

k,j : m = 1, 2, , r, j = 0, 1, , Mk− 1} is linearly pendent

inde-(iii) For each j = 0, 1, , Mk− 1, {vm

k,j : m = 1, 2, , r} is linearly independent

(iv) For each j = 0, 1, , Mk− 1, detMk(j) > 0.

Proof The proof follows the scheme (i) ⇔ (ii) ⇔ (iii) ⇔ (iv)

(i)⇒(ii): Consider Pr

m=1

PM k −1 j=0 cm

j vm k,j = 0 for some cm

indepen-(cm0 cm1 · · · cmMk −1)√1

MkFMk = 01×Mk

for each m, giving cm

j = 0 for all pairs of m and j Hence the conclusion

(i)⇐(ii): This converse direction is similarly proved using (1.3)

(ii)⇒(iii): This direction is clearly true

(ii)⇐(iii): ConsiderPM k −1

j=0

Pr m=1cm

j vm k,j = 0 for some cm

The next theorem is a characterization of the orthonormality of {Tklφmk : m =

1, 2, , r, l = 0, 1, , Mk− 1}

Theorem 1.9 For k > 0, given {φm

k : m = 1, 2, , r} ⊂ L2[0, 2π) with responding polyphase splines {vm

cor-k,j : m = 1, 2, , r, j = 0, 1, , Mk− 1}, thefollowing statements are equivalent

(i) The collection {Tklφmk : m = 1, 2, , r, l = 0, 1, , Mk− 1} is orthonormal

(ii) The collection {vm

k,j : m = 1, 2, , r, j = 0, 1, , Mk− 1} is orthogonalwith kvmk,jk2 = M1k for each pair of m and j

(iii) For each j = 0, 1, , Mk− 1, {vm

k,j : m = 1, 2, , r} is orthogonal with

kvm

k,jk2 = M1k for each m

(iv) For each j = 0, 1, , Mk− 1, Mk(j) = M1kIr

Proof The proof is in the order (i) ⇔ (ii) ⇔ (iii) ⇔ (iv)

(i)⇔(ii): Statement (i) is equivalent to hTl

kφm

k, Tl0

kφm0

k i = δl,l0δm,m0 for m, m0 ∈{1, 2, , r} and l, l0 ∈ {0, 1, , Mk− 1} Using (1.2), it can be checked that

√

MkFMk.Thus an equivalent condition for (i) is that for m, m0 = 1, 2, , r,

√

MkFM∗k



hvm k,j, vk,jm00iM

k −1 j,j 0 =0

√

MkFMk = δm,m0IMk.Upon simplifying, this is the same as

hvm k,j, vmk,j00i = 1

Mkδm,m0δj,j0

for m, m0 = 1, 2, , r, and j, j0 = 0, 1, , Mk− 1, which is (ii)

(ii)⇔(iii): Statement (ii) clearly implies (iii) The converse holds due to tion 1.5

Proposi-(iii)⇔(iv): This is straightforward by definition of Mk(j) 

The identities (1.3) and (1.4) are also crucial as they give another basis for eachmultiresolution subspace besides the usual basis consisting of the shifts of scalingfunctions.

Theorem 1.10 For k > 0, suppose that Vk is a subspace of L2[0, 2π) with {φm

k :

m = 1, 2, , r} ⊆ Vk and let {vm

k,j : m = 1, 2, , r, j = 0, 1, , Mk− 1} be thecorresponding polyphase splines

(a) The set {Tl

k,jk2 = M1k for each pair of m and j

Proof Suppose that {Tl

kφm

k : m = 1, 2, , r, l = 0, 1, , Mk− 1} is a basis for

Vk From (1.4), each vk,jm is in Vk; Theorem 1.8 gives that {vmk,j : m = 1, 2, , r, j =

0, 1, , Mk− 1} is linearly independent; and there are same number of elements

Tklφk := (Tklφ1k, , Tklφrk)T, bφk(n) := ˆφ1k(n), , ˆφrk(n)
T, vk,j := (v1k,j, , vk,jr )Tfor k > 0, l, j = 0, 1, , Mk− 1, and n ∈ Z

Theorem 1.11 Suppose that Vk, k > 0, are subspaces of L2[0, 2π) satisfyingMR1 Let φm

k, m = 1, 2, , r, be the scaling functions and vm

k, m = 1, 2, , r, into the scaling vector φk, denoting Hk+1(l) :=

hmk+1(l, m0)
rm,m0 =1 for l = 0, 1, , Mk+1− 1, and generating a complex Mk+1periodic sequence using these Mk+1 matrices, we obtain (ii)

-Suppose (ii) holds From Hk+1, we define bHk+1 parallel to the finite Fouriertransform:

where the periodicity of bHk+1 is due to the fact that ωk+1Mk+1 = 1 Letting ˆhpk+1(j, q)

to be the (p, q)-entry of bHk+1(j) for p, q = 1, 2, , r, we have the usual finiteFourier transform

k(n), m = 1, 2, , r, in a column vector gives (iii)

Next assume (iii) holds Fixing j ∈ {0, 1, , Mk− 1} and m ∈ {1, 2, , r},

k,j, m = 1, 2, , r, in a column vector for j = 0, 1, , Mk− 1, gives

Remark 1.12 Applying the inverse finite Fourier transform on (1.7) leads to

Corollary 1.13 Assume, in addition to the hypothesis of Theorem 1.11, that Vk,

k > 0, satisfy MR2 Then for k > 0, there exists a bHk+1 ∈ S(Mk+1)r×r such that

Proof Fix k > 0 Using the same bHk+1 in Statement (iv) of Theorem 1.11, anarbitrary entry of Mk(j), j = 0, 1, , Mk− 1, can be expressed as

where Proposition 1.5 is invoked in the last equality Each inner product in thelast expression is actually the corresponding entry of bHk+1(j + lMk)Mk+1(j +

We end this section with a characterization of MR3 As the proof of thischaracterization follows closely to the case for r = 1, M = 2, which was discussed

in detail in [7], we shall just state the result Interested readers can verify the

proof for themselves by replacing every incidence of 2k by Mk in Lemmas 3.9, 3.10,3.11 and Theorem 3.12 of [7] as well as adjusting the proof of Theorem 3.12 of [7]slightly to cater to multiple scaling functions.

Theorem 1.14 Let Vk, k > 0, be subspaces of L2[0, 2π) satisfying MR1 and MR2,each subspace with scaling functions φm

k, m = 1, 2, , r Then[

k>0

Vk = L2[0, 2π) ⇐⇒ {n ∈ Z : ˆφmk(n) = 0 for all k > 0, m = 1, 2, , r} = ∅

For this section, we assume a MR of L2[0, 2π) with the notations in the previoussection Recall that each Wk is the orthogonal complement of Vk in Vk+1 and

a fixed k > 0

Consider any collection of r(M − 1) functions in Wk, labelled as ψkm, m =

1, 2, , r(M − 1) Let the corresponding polyphase splines be denoted by

1, 2, , r(M − 1), q = 1, 2, , r, and Gk+1(l) = M1k+1

PMk+1−1 j=0 Gbk+1(j)ωk+1lj , l ∈

Z As for the Fourier coefficients, we have for n ∈ Z,

The analogue of Corollary 1.13 is

for j = 0, 1, , Mk− 1, where bGk+1 is uniquely determined in the previous graph One of the reasons of going through the preceding paragraph is to illustratethat there is a one-to-one correspondence between {ψm

para-k : m = 1, 2, , r(M − 1)}and bGk+1 This comes into play as the criteria for the existence of wavelets, to beshown later, are based on the existence of bGk+1 that satisfies certain conditions.Once such bGk+1 is found, the wavelets {ψm

k : m = 1, 2, , r(M − 1)} can bededuced

Lemma 1.15 For j, j0 ∈ {0, 1, , Mk− 1} with j 6= j0,

hum k,j, vk,jm00i = 0for all m = 1, 2, , r(M − 1), m0 = 1, 2, , r In addition,

Proof The arguments are similar to those of Proposition 1.5 and Corollary 1.13

i=1civi = 0 Due to the linear independence of {vi : i =

1, 2, , p}, each ci = 0 Similarly, each dj = 0 Hence the lemma 

To simplify notations, let us define the following for j = 0, 1, , Mk− 1:

Xk(j) := Hbk+1(j) Hbk+1(j + Mk) Hbk+1(j + (M − 1)Mk), (1.12)

Yk(j) := Gbk+1(j) Gbk+1(j + Mk) ...

Studies on interpolatory scaling functions and wavelets in L2(R) in the literatureinclude [1], [5], [8], [14] and [16] Interpolatory periodic scaling functions andwavelets have been... [10] and [12], with the setting of the firstthree being r = 1, M = and the last being r = M = This chapter investigatesgeneral properties of interpolatory periodic scaling functions and wavelets. .. class="page_container" data-page="39">

the functions on both sides of the equations agree almost everywhere In order togenerate results about interpolatory scaling functions and their polyphase splines,which