Boundary functions for wavelets and their properties

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1 Graduate Theses and Dissertations Iowa State University Capstones, Theses and Dissertations 009 Boundary functions for wavelets and their properties Ahmet Alturk Iowa State University Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Alturk, Ahmet, "Boundary functions for wavelets and their properties" (009) Graduate Theses and Dissertations This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository For more information, please contact

2 Boundary functions for wavelets and their properties by Ahmet Altürk A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Applied Mathematics Program of Study Committee: Fritz Keinert, Major Professor Jonathan D H Smith Khalid Boushaba Ryan Martin Sung-Yell Song Iowa State University Ames, Iowa 009 Copyright c Ahmet Altürk, 009 All rights reserved

3 ii TABLE OF CONTENTS LIST OF TABLES iv LIST OF FIGURES v ABSTRACT vi CHAPTER 1 Introduction 1 11 Overview of the Dissertation CHAPTER Wavelet Theory 4 1 Introduction 4 Refinable Functions and Scalar Wavelets 4 3 Cascade Algorithm and Point Values 8 4 Multiresolution Approximations 9 5 Moments and Approximation Order 13 6 Discrete Wavelet Transform 15 7 Refinable Function Vectors and Multiwavelets 18 CHAPTER 3 Refinable Boundary Functions 5 31 Introduction 5 3 Endpoint Functions 3 33 Continuity for Scalar Boundary Functions Continuity for Multiboundary Functions Approximation Order 51 CHAPTER 4 Wavelets on Finite Intervals Introduction 54

4 iii 4 Data Extension Approach Boundary Function Approach Matrix Completion Approach Madych Approach for Scalar Wavelets A New Approach to Multiwavelet Endpoint Modification 60 CHAPTER 5 Elaborated Examples Introduction 70 5 D 4 Scalar Wavelet CL() Multiwavelet DGHM Multiwavelet CL(3) Multiwavelet Comparison of two approaches 90 CHAPTER 6 Concluding Remarks 9 61 Applications 9 6 Future Research 9 APPENDIX A Linear Algebra Results 94 BIBLIOGRAPHY 96 ACKNOWLEDGEMENTS 99

5 iv LIST OF TABLES Table 31 Recursion formulas 8

6 v LIST OF FIGURES Figure 1 Haar scaling function recursion relation 5 Figure Hat function recursion relation 6 Figure 3 D 4 scaling function 7 Figure 4 Two band filter bank 18 Figure 5 DGHM scaling functions 3 Figure 31 CL() at level 0 31 Figure 3 CL() at level -1 3 Figure 51 Daubechies D 4 first solution 7 Figure 5 Daubechies D 4 second solution 73 Figure 53 CL first solution 76 Figure 54 CL second solution 77 Figure 55 DGHM first solution 80 Figure 56 CL3 first solution 84 Figure 57 CL3 second solution 86 Figure 58 CL3 third solution 88 Figure 59 CL3 fourth solution 89 Figure 510 CL3 regular, but not a linear combination of boundary-crossing functions 90

7 vi ABSTRACT Wavelets and wavelet transforms have been studied extensively since the 1980s It has been shown that the Discrete Wavelet Transform (DWT) can be applied to an enormous number of applications in virtually every branch of science The DWT is designed for decomposing and reconstructing infinitely long signals In practice, we can only deal with finitely long signals This raises the very important question of handling boundaries What should be done near the ends? Several approaches have been proposed to deal with this problem In this dissertation, we consider the boundary function approach The idea is to alter the DWT by constructing appropriate boundary functions at each end so that finite length signals can be analyzed accurately This dissertation contains two sets of new results One is about smoothness and approximation order properties of boundary functions The other is about finding boundary functions The results have been elaborated upon for some specific wavelets

8 1 CHAPTER 1 Introduction In most basic terms, wavelets are basis functions (signals) that are local in both time and frequency Locality in time means that wavelets are zero or nearly zero outside of a finite time interval Locality in frequency implies that wavelets live in certain frequency bands They are primarily used for decomposing and reconstructing functions As a result of their applicability to a variety of research areas such as signal analysis, numerical analysis, physics, music, and many more, the interest in them has grown enormously over the years [11], [1], [18], [7], [9] Different approaches to wavelet theory in one- or higher-dimensional setting are available in the literature [11], [], [4], [5] We start with a refinable function ϕ, which is the solution to the following two-scale refinement equation ϕ(x) = k 1 k=k 0 h k ϕ(x k) (11) Under some mild conditions, ϕ produces a multiresolution approximation and a wavelet function ψ [] Translations and dilations of ψ yield the basis functions That is, { } ψ nk (x) := n/ ψ( n x k), n, k Z forms an orthonormal basis of L ( R) [10] Details are given in section 4 Similarly, starting with a refinable function vector ϕ, which is the solution to the two-scale matrix refinement equation ϕ(x) = k 1 k=k 0 H k ϕ(x k), k Z, one can obtain multiwavelets Multiwavelets are desirable and useful tools because they can have important properties such as orthogonality, approximation order, smoothness, symmetry or antisymmetry, short support, and continuity simultaneously [15], [3], [6]

9 Mathematically, analysis on wavelets is represented by the wavelet transform The Continuous Wavelet Transform maps a function of time f into a function of two arguments (time and frequency) and is defined as W f (a, b) = 1 a f(t)ψ ( t b a )dt, where a > 0, b R, and ψ is the wavelet function [1] In words, it is the integral over all time of the function multiplied by the complex conjugate of dilated and shifted wavelets The Discrete Wavelet Transform is obtained by discretizing (dyadic sampling) the timescale parameter in the wavelet transform This is a remarkable transform in that it is easy to implement simple algorithms are available, and they do not use the functions ϕ and ψ explicitly The Discrete Wavelet Transform acts on infinitely long signals For finite signals, a problem occurs near the boundaries By using some specific data extension approaches, one can extend a finite signal so that DWT can act on the extended signal [6], [3] However, the usual approaches all suffer from various flaws: loss of orthogonality or approximation order, introducing discontinuities, making the decomposed signal longer, etc 11 Overview of the Dissertation In this dissertation, we modify the algorithm for the Discrete Wavelet Transform by introducing boundary functions so that it can be applied to finite signals In preparation for this, in chapter 3, we follow two approaches to construct boundary functions The first approach is based on a recursion relation The second approach focuses on a linear combination of functions that cross boundaries We first investigate how these two approaches are connected By this comparison we show that not every refinable boundary function is a combination of boundary-crossing functions or vice versa We then obtain conditions under which these two approaches are equivalent Because a certain degree of approximation and continuity are desirable for wavelets, we show under what conditions continuous boundary functions with approximation order 1 can be obtained In particular, we give an explicit set of

10 3 coefficients that produces the same continuous boundary function with approximation order 1 for both approaches when there is only a single boundary function We also demonstrate by an example that this is not the case when we have a boundary function vector more than one boundary function (see chapter 5) The work contained in this chapter will appear in [], [3] In chapter 4, the Madych approach [1] for obtaining boundary functions for scalar wavelets is investigated We show that under certain extra conditions it also works for multiwavelets We also construct a more general approach to multiwavelet endpoint modification, which works in all cases In addition, this construction is unique up to the choice of arbitrary orthogonal matrices The tradeoff here is that not every orthogonal matrix produces a useful set of coefficients that are used for constructing boundary functions To overcome this, we search for orthogonal matrices that produce some useful set of coefficients in the sense that they produce boundary functions with certain properties The choice of orthogonal matrix is based on the results from chapter 3 In fact, the necessity of choosing suitable matrices is what motivated the work in chapter 3 The results obtained in this chapter will appear in [], [3] In chapter 5, the results obtained in previous chapters are applied to some specific wavelets The explicit boundary functions with certain properties for the specific wavelets are obtained as well We also show by an example that one can construct a regular boundary function that cannot be written as linear combinations of boundary-crossing functions In chapter 6, we discuss possible applications and direction for future research

11 4 CHAPTER Wavelet Theory 1 Introduction This chapter begins by stating the basic definitions and theorems in scalar wavelet theory It then presents basic properties of multiwavelets The DWT, which will be used throughout the dissertation, for both scalar wavalets and multiwavelets is also introduced Refinable Functions and Scalar Wavelets Definition 1 A function ϕ : R C is called a refinable function if it satisfies a two-scale refinement equation or recursion relation of the form ϕ(x) = k 1 k=k 0 h k ϕ(x k) (1) This equation is also referred to as dilation equation The h k C are called the recursion coefficients It is possible to consider refinement equations with an infinite number of recursion coefficients, as long as they satisfy suitable decay conditions However, we shall assume that there are only finitely many recursion coefficients Definition The inner product of two functions φ and ϕ is defined by φ(x), ϕ(x) = φ(x)ϕ (x)dx, where the stands for complex conjugation Definition 3 ϕ is called orthogonal if ϕ(x), ϕ(x k) = δ 0k, k Z

12 5 Haar recursion relation Figure 1 Haar scaling function recursion relation Two refinable functions ϕ and ϕ are called biorthogonal if ϕ(x), ϕ(x k) = δ 0k, k Z ϕ is called a dual of ϕ Example 1 The Haar function is the simple unit-width, unit-height pulse function It is clear that ϕ(x) can be reconstructed from ϕ(x) and ϕ(x 1) That is, ϕ(x) = ϕ(x) + ϕ(x 1), (see figure 1), which means that the coefficients are h 0 = h 1 = 1 The Haar function is orthogonal Example The hat function is given by 1 + x, if 1 x 0; ϕ(x) = 1 x, if 0 < x 1; 0, otherwise

13 6 Hat function recursion relation Figure Hat function recursion relation It satisfies ϕ(x) = 1 ϕ(x + 1) + ϕ(x) + 1 ϕ(x 1) = ( 1 1 ϕ(x + 1) + ϕ(x) + 1 ϕ(x 1)), (see figure ), so the recursion coefficients are h 1 = 1, h 0 = 1, h 1 = 1 Since ϕ(x), ϕ(x 1) 0, it is not orthogonal Example 3 The scaling function of the Daubechies wavelet D 4 has recursion coefficients h 0 = , h 1 = , h = 3 3 4, h 3 = It is orthogonal [11], (see figure 3) Theorem 4 A necessary condition for orthogonality is This is proved in [0] h k h k l = δ 0l () k As a corollary of this theorem, one can deduce that an orthogonal ϕ has an even number of recursion coefficients

14 Figure 3 D 4 scaling function Many refinable functions cannot be represented in closed form However, we are able to compute the point values and analyze other properties, such as smoothness and regularity But before delving into the details, we first analyze and express the above relations in the frequency domain Definition 5 Let f be an integrable function over R The Fourier Transform of f is denoted by ˆf and is defined by ˆf(ξ) = 1 π f(x)e iξx dx Definition 6 The symbol of a refinable function is the trigonometric polynomial h(ξ) = 1 h k e ikξ Taking the Fourier transform of the refinement equation (1), we obtain k ˆφ(ξ) = h( ξ ) ˆφ( ξ ) (3) Equation (3) is the refinement equation in the frequency domain Theorem 7 A necessary condition for orthogonality in the frequency domain is The proof can be found in [11] and [0] h(ξ) + h(ξ + π) = 1 (4)

15 8 3 Cascade Algorithm and Point Values Definition 8 The cascade algorithm is a fixed point iteration applied to the refinement equation The n th iteration with a chosen initial function ϕ (0) is given by ϕ (n+1) (x) = k h k ϕ (n) (x k) Convergence of the cascade algorithm is part of the sufficient conditions in many settings [11] The following lemma is an example Lemma 9 If the recursion coefficients satisfy () or equivalently, if the symbol h satisfies (4), and the cascade algorithm converges, then ϕ is orthogonal and has compact support This is proved in [11] The cascade algorithm can be used to find approximate point values of ϕ(x) The exact point values of ϕ can also often be found by solving an eigenvalue problem While it may not work for some cases, it usually works for continuous ϕ It is easy to show that if ϕ satisfies the recursion equation (1), it has support [k 0, k 1 ] If we let ϕ be the vector of point values of ϕ at the integers in [k 0, k 1 ], ϕ(k 0 ) ϕ =, ϕ(k 1 ) the refinement equation admits the form ϕ = T ϕ, (5) which is an eigenvalue problem Note: For notational convenience, vectors are set in boldface type

16 9 Example 4 Consider the scaling function of the Daubechies wavelet D 4 The equation (5) takes the following form h T ϕ = h h 1 h h 3 h h h 3 ϕ(0) ϕ(1) ϕ() ϕ(3) = ϕ(0) ϕ(1) ϕ() ϕ(3) The solution, normalized to k ϕ(k) = 1, is ϕ(0) = 0, ϕ(1) = 1 + 3, ϕ() = 1 3, ϕ(3) = 0 Then, one can use the refinement equation to find the values of ϕ at any dyadic rational Remark 10 If we iterate (3), we obtain [ ] ˆϕ(ξ) = h( ξ k ) ˆϕ(0) k A solution to the refinement equation exists under assumptions that ˆϕ(0) 0 and that the infinite product converges This approach is not very effective to find ϕ, but it is useful for smoothness estimates and existence proofs 4 Multiresolution Approximations Multiresolution Approximation Multiresolution approximation (MRA) is one of the main concepts in wavelet theory Any multiresolution approximation provides us with not only the wavelet function but also a way for understanding and deriving orthonormal wavelet bases Basic definitions and results are stated for orthonormal MRA throughout this section Definition 11 A multiresolution approximation of L ( R) is a chain of closed subspaces V n, n Z, V 1 V 0 V 1 L ( R) satisfying

17 10 1 V n V n+1 for all n Z; f(x) V n f(x) V n+1, for all n Z; 3 f(x) V n = f(x n k) V n, for all k Z; 4 n Z V n = {0}; 5 n Z V n = L ( R); 6 There exists a function ϕ L such that { } ϕ(x k) : k Z is an orthonormal basis for V 0 [] MRA The function ϕ, whose existence is asserted in (6), is called a scaling function of the given Condition () gives us the main property of an MRA Each V n consists of the functions in V 0 compressed by a factor of n Thus, an orthonormal basis of V n is given by { } ϕ nk (x) := n/ ϕ( n x k), k Z The factor n/ preserves the L -norm Since V 0 V 1, ϕ can be written in terms of the basis of V 1 as ϕ(x) = k h k ϕ 1,k (x) = k h k ϕ(x k), (6) for some coefficients h k That means that ϕ is refinable Theorem 1 Assume that h(0) = 1, h(ξ) + h(ξ + π) = 1, and that the cascade algorithm converges Then the solution ϕ(x) for the refinement equation exists and is a scaling function for an MRA

18 11 This is proved in [] The orthogonal projection P n of a function f L into V n is defined by P n f = k Z f, ϕ n,k ϕ n,k This is also called an approximation to f at scale n It is the best approximation of f in V n The key point of MRA can be expressed not by using P n f and increasing n, but by using the difference between approximations to f at successive scales n and n 1 More precisely, let Q n = P n+1 P n Q n is also an orthogonal projection onto a closed subspace W n W n is the orthogonal complement of V n in V n+1 ; V n+1 = V n W n This implies that Hence, V n = n 1 k= W k L ( R) = W n n Z Mallat [] showed that an orthonormal basis of W 0 is generated from integer translates of single function ψ(x) L ( R), called wavelet function or mother wavelet, which is essentially determined by the scaling function Since W 0 V 1, the wavelet function ψ can be represented as ψ(t) = g n ϕ(t n) (7) n Z for some coefficients g n, [7] Since W n is comprised of functions in W 0 compressed by a factor of n, { } ψ nk (x) := n/ ψ( n x k), k Z, forms an orthonormal basis of W n Moreover,

19 produces an orthonormal basis for L ( R) [10] 1 { } ψ nk, n, k Z This implies that an arbitrary function f L ( R) can be written as and also as f(x) = k f(x) = n,k f, ψ nk ψ nk, (8) f, ϕ jk ϕ jk + k f, ψ nk ψ nk (9) n j Equation (8) or (9) is called a wavelet decomposition of f Example 5 If we define ψ(x) = 1, if 0 x 1 ; 1, if 1 < x 1; 0, elsewhere, then ψ is an orthonormal wavelet for the MRA generated by the Haar scaling function This is called the Haar Wavelet For a given orthogonal MRA, the scaling function ϕ(x) and the wavelet function ψ(x) must satisfy the following properties under assumptions that ϕ(x) L 1 ( R) L ( R) and ψ(x) L 1 ( R) The following statements are all proved in [31] Theorem 13 R ϕ(x)dx = 1 Corollary 14 R ψ(x)dx = 0 The necessary conditions can be written in terms of recursion coefficients as follows 1 k h k = ; k g k = 0; 3 k h kh k l = k g kgk l = δ 0l; 4 k g kh k l = 0 for all l Z; 5 k h m k h n k + k g m k g n k = δ nm

20 13 5 Moments and Approximation Order Definition 15 The k th discrete moments of ϕ and ψ are defined by They are related to the symbols by m k = 1 l k h l, n k = 1 l k g l l l m k = i k D k h(0), n k = i k D k g(0), where D stands for differentiation Discrete moments are uniquely defined [7] Definition 16 The k th continuous moments of ϕ and ψ are defined by µ k = x k ϕ(x)dx, ν k = x k ψ(x)dx They are related to the Fourier transforms of ϕ and ψ by µ k = πi k D k ˆϕ(0), ν k = πi k D k ˆψ(0) The continuous moment µ 0 is not determined by the refinement equation It depends on the scaling of ϕ Theorem 17 The continuous and discrete moments are related by µ k = k k ν k = k t=0 k t=0 ( ) k m k t µ t, t ( ) k n k t µ t t Starting with given µ 0, all other continuous moments can be computed from these relations [0]

21 14 Definition 18 A function ψ has p vanishing moments if ψ(x)x n dx = 0, n = 0, 1,, p 1 R Definition 19 A scaling function has approximation order p if all polynomials of degree less than p can be expressed as a linear combination of its integer shifts That means the wavelet transform of any polynomial of degree less than p will have all its wavelet coefficients zero Definition 0 The recursion coefficients h k satisfy the sum rules of order p if ( 1) k k m h k = 0, m = 0, 1,, p 1 k It is important to know that the sum rules of order p can be equivalently stated as the symbol h has a zero of order p at ξ = π There are actually more equivalent statements, all of which are combined in the following fundamental theorem Theorem 1 Assume that the recursion coefficients h k satisfy the sum rules of order 1 Then the following statements are equivalent 1 ϕ has approximation order p {h k } satisfies the sum rules of order p 3 h (m) (π) = 0, m = 0, 1,, p 1, and so it can be factored as h(ξ) = (1 + e iξ ) p t(ξ), where t is some trigonometric polynomial 4 D n ˆϕ(kπ) = 0, k Z {0}, n = 1,,, p 1 This is proved in [8], [19] Example 6 The Haar wavelet has 1 vanishing moment and so only constant functions can be reproduced by the scaling functions The symbol is h(ξ) = 1 (1 + e iξ )

22 15 6 Discrete Wavelet Transform From now on we concentrate on orthogonal wavelets with compact support As we defined in section 4, V n+1 = V n W n = V n 1 W n 1 W n = V 0 W 0 W 1 W n 1 The series expansion for a given function s(x) V n is either s(x) = P n s(x) = k = k s, ϕ nk ϕ nk (x) s nk ϕ nk(x) or s(x) = P n 1 s(x) + Q n 1 s(x) = j s, ϕ n 1,j ϕ n 1,j (x) + j s, ψ n 1,j ψ n 1,j (x) = j s n 1,jϕ n 1,j (x) + d n 1,jψ n 1,j (x) where s nk = s, ϕ nk, d nk = s, ψ nk Note that the decomposed signal consists of two pieces s n 1 and d n 1 Lemma ϕ n 1,j, ϕ nk = h k j, ψ n 1,j, ϕ nk = g k j

23 16 Proof ϕ n 1,j, ϕ nk = = = l n 1 ϕ( n 1 x j) n ϕ ( n x k)dx n 1 h l h l ϕ( n x j l) n ϕ ( n x k)dx l n ϕ( n x j l)ϕ ( n x k)dx = l h l δ j+l,k = h k j The second identity is proved in a similar manner Algorithm: Discrete Wavelet Transform Assume we have the signal s n = {s nk } Decomposition: s n 1,j = k d n 1,j = k h k j s nk, g k j s nk Reconstruction: s nk = j [h k j s n 1,j + g k j d n 1,j] The DWT can also be interpreted as a product of an infinite matrix and a vector The decomposition step for the s-coefficients and the d-coefficients, respectively, turns out to be s n 1, 1 h 1 h 0 h 1 h s n, 1 s n 1,0 = h 1 h 0 h 1 h s n,0, s n 1,1 h 1 h 0 h 1 h s n,1

24 17 and d n 1, 1 d n 1,0 d n 1,1 = g 1 g 0 g 1 g g 1 g 0 g 1 g g 1 g 0 g 1 g s n, 1 s n,0 s n,1 The notation becomes easier if we interleave s- and d- coefficients: s n 1, 1 d n 1, 1 s n 1,0 d n 1,0 s n 1,1 d n 1,1 }{{} (sd) n 1 = h 1 h 0 h 1 h g 1 g 0 g 1 g h 1 h 0 h 1 h g 1 g 0 g 1 g h 1 h 0 h 1 h g 1 g 0 g 1 g }{{} T s n, 1 s n,0 s n,1 }{{} s n, where T is an infinite block Toeplitz matrix Decomposition: (sd) n 1 = T s n (10) Reconstruction: s n = T (sd) n 1 (11) In engineering terms, decomposition and reconstruction amount to the filtering scheme in figure 4 Here, the and denote the upsampling and the downsampling operators, respectively stands for the convolution operator Digital signal processors use the sequences {h k } and {g n } as filter coefficients, which are also referred to as lowpass filter and highpass filter, respectively

25 18 s n s n 1 d n 1 h g h g s n 1 d n 1 s n Figure 4 Two band filter bank 7 Refinable Function Vectors and Multiwavelets We now consider the case where there are several functions, grouped together into a function vector, which are jointly refinable The recursion coefficients in this case are no longer scalars Instead they are matrices, and the symbols are trigonometric matrix polynomials Definition 3 A refinable function vector is a vector valued function ϕ(x) = ϕ 1 (x) ϕ r (x), where ϕ k : R C, which satisfies a two scale matrix refinement equation of the form ϕ(x) = k 1 k=k 0 H k ϕ(x k), k Z r is called the multiplicity of ϕ The recursion coefficients H k are r r matrices The inner product of two function vectors φ and ϕ is defined by φ(x), ϕ(x) = φ(x)ϕ (x)dx

26 19 For a function vector, ϕ denotes the complex conjugate transpose, so this inner product is an r r matrix The refinable function vector ϕ is called orthogonal if ϕ(x), ϕ(x k) = δ 0k I, k Z I is an r r identity matrix Two refinable function vectors ϕ and ϕ are called biorthogonal if ϕ(x), ϕ(x k) = δ 0k I, k Z Example 7 An example with multiplicity is the constant-linear refinable function vector, ϕ(x) = 1 3(x 1), where x [0, 1] It satisfies ϕ 1 (x) = ϕ 1 (x) + ϕ 1 (x 1), ϕ (x) = [ 3 ϕ 1(x) ϕ (x)] + [ ϕ 1(x 1) + 1 ϕ (x 1)] This function vector is refinable with H 0 = , H 1 = It is orthogonal Theorem 4 A necessary condition for orthogonality is H k Hk l = δ 0lI (1) This is proved in [0] k

27 0 Definition 5 The symbol of a refinable function vector is the trigonometric matrix polynomial H(ξ) = 1 k 1 k=k 0 H k e ikξ The Fourier transform of the refinement equation is ˆϕ(ξ) = H( ξ )ˆϕ(ξ ) Lemma 6 The orthogonality condition (1) is equivalent to H(ξ)H(ξ) + H(ξ + π)h(ξ + π) = I A proof can be found in [0] As in the scalar case, these conditions are sufficient if the cascade algorithm converges Definition 7 A matrix A satisfies Condition E(p) if it has a nondegenerate p fold eigenvalue 1, and all other eigenvalues are smaller than 1 in magnitude Condition E means Condition E(1) Definition 8 A compactly supported refinable function vector ϕ has linearly independent shifts if for all sequences of vectors { c k }, k c k ϕ(x k) = 0 = c k = 0 for all k Definition 9 The refinable function vector ϕ satisfies the minimal regularity conditions if ϕ has compact support, ϕ L, ϕ has linearly independent shifts, and ϕ(x)dx 0 Theorem 30 Let ϕ have minimal regularity Then the following conditions hold: 1 H(0) satisfies condition E There exists a vector y 0 0 so that y0ϕ(x k) = 1 k

28 1 3 The same vector y 0 satisfies y 0H(πk) = δ 0k y 0, k = 0, 1 4 The same vector y 0 satisfies k = 0, 1 y 0 l H l+k = 1 y 0, This is proved in [3] Definition 31 Multiresolution Approximation and Multiwavelets A Multiresolution Approximation (MRA) of L is a doubly infinite nested sequence of subspaces of L V 1 V 0 V 1 V with properties 1 V n V n+1 for all n Z; f(x) V n f(x) V n+1 for all n Z; 3 f(x) V n f(x n k) V n for all k Z; 4 n V n = { 0 } ; 5 n V n = L ( R); 6 There exists a function vector ϕ L so that { ϕl (x k) : l = 1,,, r and k Z } forms an orthonormal basis of V 0 ϕ is called the multiscaling function Condition () expresses the main property of an MRA: Each V n consists of the function in V 0 compressed by a factor of n Thus, an orthonormal basis of V n is given by { ϕnk (x) := n/ ϕ( n x k), k Z }

29 The factor n/ preserves the L norm Since V 0 V 1, ϕ can be written in terms of the basis of V 1 as ϕ(x) = k H k ϕ 1k (x) = k H k ϕ(x k) for some coefficient matrices H k It means ϕ is refinable The orthogonal projection P n of a function f L into V n is defined by P n f = k Z f, ϕ n,k ϕ n,k, which is also called an approximation to f at scale n As in the scalar case, we define Q n to be the difference between approximations to f at successive scales n and n 1 : Q n = P n+1 P n Q n is also an orthogonal projection onto a closed subspace W n W n is the orthogonal complement of V n in V n+1 ; This implies that V n+1 = V n W n V n = n 1 k= It is known that there exists a function vector ψ L so that its integer translates form an orthonormal basis of W 0 Since W 0 V 1, the function vector ψ can be represented as W k ψ(x) = k Z G k ϕ(x k) for some coefficients G k The function vector ψ is called the multiwavelet function ϕ and ψ together form a multiwavelet Example 8 The orthogonal scaling function vector of the DGHM wavelet [14], (see figure 5), has recursion coefficients H 0 = , H 1 = ,

30 Figure 5 DGHM scaling functions H = The associated wavelet coefficients are G 0 = G = , H 3 =, G 1 =, G 3 = From now on, we assume that ϕ satisfies the minimal regularity conditions In particular, H(0) satisfies condition E, Definition 3 The k th discrete moments of ϕ and ψ are defined by M k = 1 l n H l, N k = 1 l n G l Discrete moments are r r matrices They are related to the symbols by l l M k = i k D k H(0),

31 4 N k = i k D k G(0) Discrete moments are uniquely defined Definition 33 The k th continuous moments of ϕ, ψ are µ k = x k ϕ(x)dx, ν k = x k ψ(x)dx Continuous moments are r vectors They are related to the Fourier transforms of ϕ, ψ by µ k = πi k D k ˆϕ(0), ν k = πi k D k ˆψ(0) The continuous moment µ 0 is only defined up to a constant multiple by the refinement equation, depending on the scaling of ϕ Theorem 34 The continuous and discrete moments are related by In particular, µ k = k k ν (s) k = k t=0 k t=0 ( ) k M k t µ t t, ( ) k N (s) t k t µ t µ 0 = M 0 µ 0 = H(0)µ 0 Once µ 0 is chosen, all other continuous moments can be computed from these relations [0]

32 5 CHAPTER 3 Refinable Boundary Functions 31 Introduction The motivation in standard wavelet analysis was to construct orthonormal bases for families of functions defined on R In practice, we often only deal with the functions on a closed subset [A, B] of the real line In this section we focus on functions on [A, B] We begin with internal scaling functions supported inside the given interval, and introduce appropriate boundary functions at each end so that this new family of functions obeys the multiresolution hierarchy and gives the accuracy that we want to achieve It is shown in [13] that one can form an orthogonal MRA for a closed interval [A, B] by first fixing an orthogonal MRA on the line given by the scaling function ϕ and the wavelet ψ and then constructing the spaces V n [A, B] The spaces V n [A, B] are spanned by scaling functions supported near the left end (left boundary functions), scaling functions supported near the right end (right boundary functions), and interior scaling functions This construction yields an MRA of L ([A, B]) The wavelet spaces W n [A, B] are constructed by the relation V n+1 [A, B] = V n [A, B] W n [A, B] See [9], [13], [4] for a complete treatment In our setup, we consider continuous orthogonal multiwavelets of multiplicity r with recursion coefficients H 0, H 1,, H N, G 0, G 1,, G N, so that the support is a subset of the interval [0, N] We assume that the finite interval is [0, M] with M large enough so that left and right ends don t interfere with each other We expect that there are M multiscaling functions at level zero (in V 0 ) or n M multiscaling functions at level n (in V n ) The interior multiscaling functions are those whose support fits completely in [0, M] We

33 6 therefore have (M N + 1) interior multiscaling functions, numbered ϕ 0 to ϕ M N+1, at level 0 Some shifts of ϕ(x) cross the boundaries These functions are called boundary-crossing multiscaling functions These are ϕ N+1 through ϕ 1 at the left end and ϕ M N+1 through ϕ M 1 at the right end Let L be the number of individual left boundary functions denoted by ϕ L j (x), j = L,, 1 Likewise, let R be the number of individual right boundary functions denoted by ϕ R j (x), j = M N + 1,, M N + R Thus, we have (M N + 1)r + L + R = Mr at level 0, which implies that L + R = (N 1)r (31) For orthogonal scalar wavelets, N is odd, so we can always take L = R However, this is not necessarily true for multiwavelets We will show that the choice of L and R for multiwavelets depends on the recursion coefficients H k From now on, we will consider only the left boundary functions The analogous analysis can be done at the right end as well Let L, not necessarily equal to the multiplicity r, be the number of left boundary functions supported inside [0, N 1] at level zero We stack them into a single vector ϕ L so that we have the following: ϕ L (x) = ϕ L 0 (x) = ϕ L 0, 1 (x) ϕ L 0, L (x) = ϕ L 1 (x) ϕ L L (x) We want the boundary functions to be orthogonal to each other and to the interior functions The interior function vectors are denoted by ϕ j (x) = ϕ(x j), j = 0, 1,

34 7 At level n, ϕ L n(x) = n ϕ L ( n x), ϕ nj (x) = n ϕ( n x j) To maintain the multiresolution hierarchy, we require the boundary function vector ϕ L (x) to satisfy the following recursion relation: N ϕ L n 1(x) = Aϕ L n(x) + B l ϕ nl (x), (3) l=0 where A = ( ) a jk = a 1, 1 a 1, L, ϕ L n 1,j, ϕ L n,k = a jk, a L, 1 a L, L L L B = ( B 0 B 1 B N ) L (N 1)r, ϕ L n 1, ϕ n,k = B k, B j = b j,1, j = 0, 1,, N b j,l L r For n = 1, the recursion relation yields ϕ L (x) = Aϕ L (x) + N B l ϕ(x l) As we shall show in the following, this representation yields the decomposition and the reconstruction algorithm Table 31 gives the recursion formulas in the interior and at the left end l=0

35 8 Formulas in Interior Formulas at Left End ϕ n,j (x) = n ϕ( n x j) ϕ L n,j (x) = n ϕ L j (n x) (no shift) ϕ n 1,j = l H l j ϕ n,l ϕ L n 1,j = l a j,l ϕ L n,l + l b l,j ϕ n,l ϕ n 1,j, ϕ n,k = H k j ϕ L n 1,j, ϕl n,k = a jk ϕ n,k, ϕ n 1,j = H k j ϕ L n,k, ϕl n 1,j = a jk j, k 0 j, k = L,, 1 ϕ L n 1, ϕ n,k = B k k 0 Table 31 Recursion formulas Sample Calculations: ϕ L n 1,j, ϕ L n,k = l = l a j,l ϕ L n,l + l a j,l δ l,k + 0 b l,j ϕ n,l, ϕ L n,k = a j,k ϕ n,k, ϕ L n 1,j = ϕ n,k, l a j,l ϕ L n,l + l b l,j ϕ n,l = 0 + l b l,j δ l,k = b k,j

36 9 Decomposition: Let s V n [0, M] s = P n s = s, ϕ L L }{{ n ϕn + } (s L k n ) = P n 1 s + Q n 1 s s, ϕ n,k ϕ n,k + s, ϕ }{{} (s n,k ) R n ϕ (s R n ) }{{} R n = (s L n 1) ϕ L n 1 + j (s n 1,j ) ϕ n 1,j + (s R n 1) ϕ R n 1 +(d L n 1) ψ L n 1 + j (d n 1,j ) ψ n 1,j + (d R n 1) ψ R n 1 Thus, (s L n 1) = P n s, ϕ L n 1 = (s L n) ϕ L n, ϕ L n 1 + k (s n,k ) ϕ n,k, ϕ L n 1 + (s R n 1) ϕ R n, ϕ L n 1 = (s L n) A + k (s n,k ) B k + 0 If we take the transpose of both sides, we wind up with s L n 1 = As L n + k B k s n,k (33) Likewise, if ψ L n 1 = Cϕ L n + l D l ϕ n,l, then d L n 1 = Cs L n + k D k s n,k (34) As we expressed earlier, notation becomes simpler if we interleave s and d coefficients at level n 1 : s n = s L n s n,k, where sl n = s L n, L s R n s L n, 1

37 30 If then (sd) n 1 = s L n 1 d L n 1 s n 1,0 d n 1,0 s R n 1 d R n 1, (sd) n 1 = T n s n, (35) s n = T n(sd) n 1, (36) where Here T k = H k H k+1 G k G k+1 where k = 1,, K L 0 L 1 L K T 0 T 1 T K T T n = 0 T 1 T K T 0 T 1 T K R 0 R 1 R K r r, L k = B k D k B k 1 D k 1 L r, L 0 = A C L L In the discrete setting, the inner products in Table 31 go directly into the matrix representation of the transformation Example 31 Consider the standard orthogonal Chui-Lian CL() multiwavelet [8] It has multiplicity and is supported on [0, ] Recursion coefficients are H 0, H 1, H For simplicity, we consider the interval [0,4] Then, we expect to have 4 multifunctions (scaling,

38 31 functions) or 8 single functions at level 0 As shown in figure 31, there are 3 interior multifunctions or 6 single functions Thus, we expect 1 single function at each end At level 1, wavelets come in to play As shown in figure 3, there is 1 interior multifunction, and 1 single function at each end It adds up to 4 single functions for the scaling functions Likewise, there are 4 single functions for the wavelet Then, we have a total of 8 functions as expected The following is the decomposition of a signal with this transform s L 1, 1 d L 1, 1 s 1,0 d 1,0 s R 1,1 d R 1,1 A B C D H = 0 H 1 H 0 0 G 0 G 1 G E F G H s L 0, 1 s 0,0 s 0,1 s 0, s R 0,3, where L 0 = A C, L 1 = B D, R 0 = E G, R 1 = F H 1 1 ϕ L 0, 1 ϕ 0,0 ϕ 0,1 ϕ 0, ϕ R 0, Figure 31 CL() at level 0

39 3 ϕ L 1, 1 ϕ 1,0 ϕ R 1, Figure 3 CL() at level -1 3 Endpoint Functions As we pointed out earlier, it suffices to consider the left boundary functions Recall that the recursion formula at the left end yields ϕ L n 1(x) = Aϕ L n,l (x) + l B l ϕ n,l (x) (37) A is of size L L, and each B l is of size L r Definition 31 We will call ϕ L refinable if it satisfies such a recursion relation We will call ϕ L a regular boundary function if it is refinable, continuous, has approximation order 1, and ϕ L (0) 0 It is also quite natural to expect that boundary functions are linear combinations of boundary crossing functions We then have ϕ L (x) = k C k ϕ k (x) (38) Each C k is of size L r A reasonable direction for analysis is to explore how the equations (37) and (38) are connected To be more precise, we would like to know how to find C = (, C k, ) if we are given A and B = (, B l, ) and vice versa We also want to know if all C (A and B) lead to useful A and B (C) We note that we are only interested in regular boundary functions, because

40 33 the A and B coefficients are needed for the wavelet transform algorithms Since we want to be able to represent functions that are not zero at the endpoints, the condition ϕ L (0) 0 is added We are on the point of making our analysis at the left end We devote this section to investigate the connection between (37) and (38) The recursion relation at the left end for n = 1 is The interior recursion for n = 1 is ϕ L (x) = Aϕ L (x) + N B l ϕ(x l) (39) ϕ(x) = l=0 N H k ϕ(x k) (310) k=0 Assume that the left endpoint functions are linear combinations of boundary crossing functions That is, ϕ L (x) = 1 k= N+1 Then, the left hand-side of (39) can be written as ϕ L (x) = = 1 k= N+1 1 k= N+1 = l C k ϕ(x k), for x 0 (311) C k ϕ(x k) C k [ N H l ϕ(x k l)] l=0 C k H l ϕ(x k l) k (31) = s ( k C k H s k )ϕ(x s) = k ( l C l H k l )ϕ(x k) and the right-hand side of (39) yields Aϕ L (x) + N l=0 B l ϕ n,l (x) = A 1 k= N+1 Comparing (31) and (313), we obtain the following identities C k ϕ(x k) + N B k ϕ(x k) (313) C l H k l = AC k, k = K,, 1, l = N + 1,, 1, (314) l C l H k l = B k,, k = 0,, N, l = N + 1,, 1 (315) l k=0

41 34 In matrix notation, (314) and (315) are equivalent to CY = AC, (316) CZ = B, (317) where (Y Z) = (H l j ) = H N 1 H N 0 0 H N 3 H N H N 1 H N 0 H 4 H N H 0 H 1 H H 3 H N 1 H N The order of both Y and Z is (N 1)r (N 1)r The first question is whether it is always possible to find C given A and B, or vice versa Recall that the Kronecker product of two matrices A = (a ij ) m n and B = (b ij ) p q is defined by A B = a 11 B a 1 B a 1n B a 1 B a B a n B a m1 B a m B a mn B mp nq and the vec operator transforms a matrix into a vector by stacking the columns of the matrix one below the other For example, let D be a matrix of order m n with the j th column defined as d j Then vec(d) = d 1 d d n mn 1 There is an important identity that relates the Kronecker product and the vec operator:, vec(p QR) = (R T P )vec(q), (318) where P, Q, and R are matrices of order m n, n r, and r s, respectively This is proved in [16]

42 35 We now focus on the identity CY = AC If ς= rowspan(c), then rowspan(ac) ς This implies that ς is a left invariant subspace of Y, so the rows of C must be a linear combinations of left eigenvectors of Y If there is a single boundary function, this is actually a straight eigenvalue problem: c Y = ac, c = C and a = A An alternative approach uses the identity (318) CY = AC I L CY = ACI (N 1)r = (Y T I L )vec(c) = (I (N 1)r A)vec(C) = [(Y T I L ) (I (N 1)r A)]c = 0, where c = vec(c) and I L is the L L identity matrix If (Y T I L ) (I (N 1)r A) is a nonsingular matrix, then c = 0 Therefore, we require (Y T I L ) (I (N 1)r A) to be a singular matrix and need to determine its nullspace, which is again an eigenvalue problem Both approaches show that the answer to the question we raised at the beginning is no An arbitrary C does not in general lead to A and B, nor do A and B always lead to C Example 3 Consider the standard orthogonal Chui-Lian CL() multiwavelet In this case, N =, K = 1, Y = H 1, Z = H, A = (a), C = c, B = b The identities CY = AC and CZ = B provide two sets of solutions: and a = [ ] /4, c = 0 1, b = [ 14 a = [ ] /, c = 1 0, and b = 8 [ ], (319) ] (30) [ ] [ ] What we have shown here is that given C either as 1 0 or 0 1 or (a multiple of these), there exist corresponding A and B so that boundary function satisfies the recursion relation

43 36 (39) Similarly, given A and B, there can be found a corresponding C so that boundary function can be obtained from a linear combination of boundary crossing functions We shall show that if there is only a single boundary function, the choice c = (µ 0, µ 0, ) always works However, we note that other choices of C do not lead to refinable functions 33 Continuity for Scalar Boundary Functions In this section, we will discuss and derive conditions for constructing continuous scalar boundary functions As a first example, we will investigate the case of scalar wavelet with four recursion coefficients For this case, we obtain necessary and sufficient conditions We also show that a similar analysis can be done for the multiwavelet case (see next section) Let us assume that we have a continuous scalar wavelet with four recursion coefficients and compactly supported on [0, 3] There is one left boundary scaling function In addition, we also assume that the boundary scaling function has support on [0, ] Then the recursion relation for the boundary scaling function yields ϕ L (x) = [a 1 ϕ L (x) + b 0 ϕ(x) + b 1 ϕ(x 1)] (31) In what follows, we show that under some conditions on a 1, b 0, and b 1, ϕ L is continuous The cascade algorithm applied to the equation (31) will determine the boundary function Assume x [1, ] Then ϕ L (x) = 0, so (31) becomes ϕ L (x) = [b 0 ϕ(x) + b 1 ϕ(x 1)] on [1, ] The cascade algorithm converges in a single step and determines ϕ L (x) on [1, ] The continuity of ϕ carries over to ϕ L In the next step, ϕ L (x) = a 1 [b 0 ϕ(4x) + b 1 ϕ(4x 1)] + [b 0 ϕ(x) + b 1 ϕ(x 1)] on [ 1, ] If we keep on applying the cascade algorithm, we determine ϕ L (x) on (0, ] At each step, ϕ L (x) is a linear combination of continuous functions, and therefore continuous Furthermore, ϕ L (x) is uniformly continuous on [ɛ, ] for every ɛ > 0 independent of choice of

44 37 a 1, b 0, b 1 simply because ϕ L (x) is continuous on the closed interval [ɛ, ] for every ɛ > 0 We therefore only need to explore the behavior of ϕ L (x) as x 0 For x = 0, equation (31) says ϕ L (0) = a 1 ϕ L (0), which can only possibly be nonzero and continuous at 0 if a 1 = 1 Otherwise, we will not get any desirable results, since either ϕ L (0) = 0, which is not useful in applications, or it is discontinuous at 0, possibly unbounded However, we will investigate lim x 0 ϕl (x) independent of the value of a 1 Take any 0 < x 0 1, so that x ϕ( x 0 ) = [h 0 ϕ(x 0 ) + h 1 ϕ(x 0 1) +h }{{} ϕ(x 0 ) +h }{{} 3 ϕ(x 0 3) ] }{{} =0 =0 =0 = h 0 ϕ(x 0 ), ϕ( x 0 4 ) =( h 0 ) ϕ(x 0 ), ϕ( x 0 8 ) =( h 0 ) 3 ϕ(x 0 ), (3) ϕ( n x 0 ) =( h 0 ) n ϕ(x 0 ) From (31), ϕ L ( x 0 ) = [a 1 ϕ L (x 0 ) + b 0 ϕ(x 0 ) + b 1 ϕ(x 0 1) ] }{{} =( a 1 )ϕ L (x 0 ) + ( b 0 )ϕ(x 0 ), ϕ L ( x 0 4 ) =( a 1 )ϕ L ( x 0 ) + b 0 ϕ( x 0 ) =( a 1 ) ϕ L (x 0 ) + [( a 1 )( b 0 ) + ( b 0 )( h 0 )]ϕ(x 0 ), ϕ L ( n x 0 ) =( a 1 ) n ϕ L (x 0 ) +( b 0 )( a 1 ) n 1 ϕ(x 0 ) =0 +( b 0 )[( a 1 ) n ( h 0 ) + + ( a 1 )( h 0 ) n + ( h 0 ) n 1 ]ϕ(x 0 ) (33)

45 38 We now consider several cases for (33) It follows from the continuity of ϕ, and the convergence of the cascade algorithm, that h0 < 1 Case 1: a 1 > 1 In this case, it is conceivable that the coefficients of both ϕ L (x 0 ) and ϕ(x 0 ) will cancel each other and lead a useful ϕ L for some particular x 0 However, this is unlikely to happen for all x 0 Thus, ϕ L will most likely become unbounded in this case We will not consider this further Case : a 1 = 1 (33) reads ϕ L ( n x 0 ) =ϕ L (x 0 ) + ( b 0 )[1 + ( h 0 ) + + ( h 0 ) n + ( h 0 ) n 1 ]ϕ(x 0 ) lim n ϕl ( n x 0 ) =ϕ lb (x 0 ) + ( b 0 ) 1 ( h0 ) n 1 ϕ(x 0 ) h 0 for every x 0 (0, 1] If it is continuous at 0, then ϕ L (0) = ϕ L (x) + b0 1 h 0 ϕ(x), x (0, 1] In particular, Thus, ϕ L (0) = ϕ L (1) + b0 1 h 0 ϕ(1) ϕ L ϕ L (0) b0 1 (x) = ϕ(x) if x [0, 1]; h 0 [b0 ϕ(x) + b 1 ϕ(x 1)] if x [1, ] (34) Case 3: a 1 < 1 This leads to ϕ L (0) = 0, continuous at zero, which is not useful in practice We have obtained a necessary condition: If ϕ L is continuous and ϕ L (0) 0, then a 1 = 1 We now show that b 0 b 1 = h h 3 is a sufficient condition To achieve this, we claim that the function defined by (34) satisfies (31) if b 0 b 1 = h h 3

46 39 If x is in [1, ], then the two identities for ϕ lb (x) are equal ϕ L (x) = [b 0 ϕ(x) + b 1 ϕ(x 1)] (from 34) ϕ L (x) = ϕ L (x) + [b }{{} 0 ϕ(x) + b 1 ϕ(x 1)] (from 31) =0 Let x be in [0, 1 ] Equation (34) leads to ϕ L (x) = ϕ L b0 (0) 1 ϕ(x) h 0 = ϕ L b0 (0) 1 [ h0 ϕ(x) ] h 0 b0 1 [ h1 ϕ(x 1) h 0 }{{} =0 = ϕ L b 0 h 0 (0) 1 ϕ(x) h 0 On the other hand, equation (31) leads to + h ϕ(x ) }{{} =0 + ] h 3 ϕ(x 3) }{{} =0 ϕ L (x) = ϕ L (x) + b 0 ϕ(x) + b 1 ϕ(x 1) }{{} =0 = ϕ L b0 (0) 1 ϕ(x) + b 0 ϕ(x) h 0 = ϕ L (0) + ( b0 b 0 1 ) ϕ(x) h 0 = ϕ L b 0 h 0 (0) 1 ϕ(x) h 0 The two identities that we obtained for ϕ L (x) are the same, so there is no need to impose any constraints

47 40 The important interval is [ 1, 1] Let x be in [ 1, 1] From equation (34), we get ϕ L (x) = ϕ L b0 (0) 1 ϕ(x) h 0 = ϕ L b0 (0) 1 [ h0 ϕ(x) + h 1 ϕ(x 1) ] h 0 After combining like terms, we have b0 1 [ h ϕ(x ) h 0 }{{} =0 + ] h 3 ϕ(x 3) }{{} =0 = ϕ L b0 (0) 1 [ ( h0 h0 ϕ(4x) + h 1 ϕ(4x 1) )] h 0 b0 1 [ ( h0 h ϕ(4x ) + h 3 ϕ(4x 3) )] h 0 b0 1 [ ( h1 h0 ϕ(4x ) + h 1 ϕ(4x 3) )] h 0 b0 1 [ ( h1 h ϕ(4x 4) h 0 }{{} =0 + )] h 3 ϕ(4x 5) }{{} =0 ϕ L (x) =ϕ L (0) b 0(h 0 ) 1 h 0 ϕ(4x) b 0h 0 h 1 1 h 0 ϕ(4x 1) b 0 h 0 h 1 + h 1 h 0 ϕ(4x ) b 0 (h 1 ) + h 0 h 3 1 h 0 ϕ(4x 3) On the other hand, equation (31) leads to ϕ L (x) =ϕ L (x) + b 0 ϕ(x) + b 1 ϕ(x 1) = b 0 ϕ(4x) + b 1 ϕ(4x 1) + b 0 ϕ(x) + b 1 ϕ(x 1) (35) (36) Use the recursion relation for ϕ(x) and ϕ(x 1) That is ϕ(x) = h 0 ϕ(4x) + h 1 ϕ(4x 1) + h ϕ(4x ) + h 3 ϕ(4x 3) (37) ϕ(x 1) = h 0 ϕ(4x ) + h 1 ϕ(4x 3) + h ϕ(4x 4) + h }{{} 3 ϕ(4x 5) }{{} =0 =0 (38)

48 41 Substitute (37) and (38) into (36) and combine like terms to get ϕ L (x) = b 0 ϕ(4x) + b 1 ϕ(4x 1) + b 0 [ h0 ϕ(4x) + h 1 ϕ(4x 1) + h ϕ(4x ) + h 3 ϕ(4x 3) ] + b 1 [ h0 ϕ(4x ) + h 1 ϕ(4x 3) ] = b 0 (1 + h 0 )ϕ(4x) + ( b 1 + h 1 b0 )ϕ(4x 1) + (b 0 h + b 1 h 0 )ϕ(4x ) + (b 0 h 3 + b 1 h 1 )ϕ(4x 3) Equate (35) and (39) This lead us to the following identity for x [ 1, 1] ϕ L (0) = [ b 0 (1 + h 0 ) + b 0(h 0 ) 1 ] ϕ(4x) h }{{ 0 } a + [ b 0h 0 h b 1 + ( h 1 )( b 0 ) ] ϕ(4x 1) h } 0 {{} b + [ b 0 h + b 1 h 0 + h 1 + h b 0 h 0 1 ] ϕ(4x ) h }{{ 0 } c + [ b 0 h 3 + b 1 h 1 + (h 1 ) + h 0 h 3 b 0 1 ] ϕ(4x 3) h }{{ 0 } d Notice that the right hand side of above identity is a constant For scalar scaling functions, ϕ(x j) = constant, j hence all coefficients of ϕ that appears in the identity must be the same If a = b, then (39) b0 (1 + h 0 ) + b 0(h 0 ) b0 (1 + h 0 )(1 h 0 ) + b 0 (h 0 ) = b 0 h 0 h 1 1 h 0 = b 0h 0 h 1 1 h 0 + b 1 + ( h 1 )( b 0 ) + b 1 (1 h 0 ) + ( h 1 )( b 0 )(1 h 0 ) b 0 =b 1 (1 h 0 ) + ( h 1 )(b 0 ) b 0 (1 h 1 ) =b 1 (1 h 0 ) (330)

49 4 We use the fact that h k = k k h k+1 = 1 In particular for the case that we are considering this means h 0 + h = 1, h 1 + h 3 = 1 Hence, the equation (330) becomes If a = c, then b0 (1 + h 0 ) + b 0(h 0 ) b 0 h 3 = b 1 h (331) 1 h 0 = b 0 h + b 1 h 0 + b 0 h 0 h 1 + h 1 h 0 b0 = b 1 h 0 (1 h 0 ) + b 0 + h + b 0 h 0 h 1 b 0 ( h 0 h 0 h 1 ) = b 1 h 0 h b 0 h 0 (1 h 1 ) = b 1 h 0 h b 0 h 0 h 3 = b 1 h 0 h If a = d, then b0 (1 + h 0 ) + b 0(h 0 ) 1 h 0 = b 0 h 3 + b 1 h 1 + b 0 (h 1 ) + h 0 h 3 1 h 0 b0 (1 (h 1 ) ) = b 0 h 3 + b 1 h 1 (1 h 0 ) b 0 ( h 0 h 0 h 1 ) = b 1 h 0 h b 0 h 3 (1 + h 1 ) = b 0 h 3 + b 1 h 1 h b 0 h 1 h 3 = b 1 h 1 h All equalities lead to the same sufficient condition Assuming this condition is satisfied, we obtain ϕ L ϕ L (0) b 0 h (x) = ϕ(x) if x [0, 1]; [b0 ϕ(x) + b 1 ϕ(x 1)] if x [1, ], (33)

50 43 and ϕ L (0) = ϕ L (1) + b 0 h ϕ(1) = [ b 1 + b 0 h ]ϕ(1) + b 0 ϕ() = b 1 [1 + 1 h3 ]ϕ(1) + b 0 ϕ() 34 Continuity for Multiboundary Functions We now do the same analysis for multiwavelets as we did for scalar wavelets Assume that we have a continuous multiwavelet with 4 recursion coefficients and compactly supported on [0, 3] In addition, we also assume that the boundary scaling function has support on [0, ] We want the boundary function with compact support [0, ] to satisfy the dilation equation Hence, the recursion relation has the following form: ϕ L (x) = Aϕ L (x) + B 0 ϕ(x) + B 1 ϕ(x 1) (333) The cascade algorithm shows that ϕ L is continuous on (0, ], as in the scalar case ϕ L (x) = [B 0 ϕ(x) + B 1 ϕ(x 1)] on [1, ], ϕ L (x) = AB 0 ϕ(4x) + AB 1 ϕ(4x 1) + B 0 ϕ(x) + B 1 ϕ(x 1)] on [1/, ] If we continue iterating the boundary function, we determine ϕ L (x) on (0, ] The continuity of boundary function follows from the continuity of interior functions As in the scalar case, we focus on the behavior of the boundary function as x 0 If x = 0, ϕ L (0) = Aϕ L (0) Since we do not want ϕ L (0) = 0, this implies that ( 1, ϕ L (0) ) is an eigen-pair of A Consider ϕ(x) = [ H 0 ϕ(x) + H 1 ϕ(x 1) + H ϕ(x ) + H 3 ϕ(x 3) ]

51 44 Let x 0 be fixed in (0, 1], then ϕ( x 0 ) = [ H 0 ϕ(x 0 ) + H 1 ϕ(x 0 1) + H ϕ(x 0 ) + H 3 ϕ(x 0 3) ] =[ H 0 ]ϕ(x 0 ), ϕ( x 0 4 ) =[ H 0 ] ϕ(x 0 ), ϕ( x 0 8 ) =[ H 0 ] 3 ϕ(x 0 ), ϕ( n x 0 ) =[ H 0 ] n ϕ(x 0 ), ϕ L ( x 0 ) = Aϕ L (x 0 ) + B 0 ϕ(x 0 ) + B 1 ϕ(x 0 1) = Aϕ L (x 0 ) + B 0 ϕ(x 0 ), ϕ L ( x 0 4 ) = Aϕ L ( x 0 ) + B 0 ϕ( x 0 ) = A [ Aϕ L (x 0 ) + B 0 ϕ(x 0 ) ] + B 0 ϕ( x 0 ) = ( A) ϕ L (x 0 ) + A B 0 ϕ(x 0 ) + B 0 H0 ϕ(x 0 ) = ( A) ϕ L (x 0 ) + ( A B 0 + B 0 H0 )ϕ(x 0 ), ϕ L ( n x 0 ) = ( A) n ϕ L (x 0 ) + [ ( A) n 1 B 0 + ( A) n ( B 0 )( H 0 ) + (334) + + ( A)( B 0 )( H 0 ) n + ( B 0 )( H 0 ) n 1] ϕ(x 0 ) More precisely, ϕ L ( n x 0 ) = [ A] n ϕ L (x 0 ) + M n 1 ϕ(x 0 ), (335) where M n 1 = ( A) n 1 B 0 + ( A) n B 0 H0 + + B 0 ( H 0 ) n 1 We now investigate what happens to this sum as n

52 45 Case 1:ρ( A) > 1 In this case, the cascade algorithm will blow up near 0 for most initial guesses Thus, ϕ L will most likely become unbounded in this case We will not consider this further since we do not expect that there are useful solutions Case :ρ( A) = 1 This case will produce some interesting results as shown below Case 3:ρ( A) < 1 In this case, the cascade algorithm will converge to a continuous function and ϕ L (0) = 0 This is not useful for applications Recall that a matrix A satisfies Condition E(p) if it has a non-degenerate p fold eigenvalue 1, and all other eigenvalues are smaller than 1 in magnitude That is, λ 1 = λ = = λ p = 1 and λ j < 1, for j > p Notation: r j and l j are right and left eigenvectors of A, respectively Without loss of generality we assume that r 1, r,, r p are the eigenvectors of A corresponding to eigenvalue 1 V = (r 1,, r L ), Λ = I p λ p λ L, V 1 = l 1 l L, Γ = (r 1,, r p ) l 1 l p Theorem 3 Assume that A satisfies condition E(p) and the cascade algorithm for ϕ(x) converges for every initial guesses Then M n Γ( B 0 )(I H 0 ) 1

53 46 Proof By assumption, ( A)V = V Λ, A = V ΛV 1 Since ( A) n = V Λ n V 1, it is clear that ( A) n V I p 0 V 1 = (r 1,, r p ) 0 0 For j = 1,,, p, we have l j( A) = l j l 1 l p = Γ Thus, l jm n = l j[( A) n B 0 + ( A) n 1 B 0 H0 + + B 0 ( H 0 ) n ] = l j( B 0 )[I + H 0 + ( H 0 ) + + ( H 0 ) n ] l j( B 0 )(I H 0 ) 1 For j > p, we obtain l jm n = l j( B 0 )[λ n j + λ n 1 j ( H 0 ) + + ( H 0 ) n ] (336) We now use the fact that given ɛ > 0 and a matrix A, there is a norm such that A ρ(a)+ɛ [17] The convergence of cascade algorithm implies that there is a norm such that H 0 < 1 Let m = max( H 0, λ p+1,, λ L ) Taking the norm of the identity (336), we get l jm n l j n B 0 λ j l ( H 0 ) n l l=0 l j B 0 (n + 1)m n 0