Matrix-Valued Wavelets. Ahmet Alturk. A creative component submitted to the graduate faculty

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1 Matrix-Valued Wavelets by Ahmet Alturk A creative component submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Major: Mathematics Program of Study Committee: Fritz Keinert, Major Professor Jonathan D.H. Smith Khalid Boushaba Iowa State University Ames, Iowa 2004 Copyright c Ahmet Alturk, All rights reserved.

2 ii Graduate College Iowa State University This is to certify that the master creative component of Ahmet Alturk has met the creative component requirements of Iowa State University Major Professor For the Major Program

3 iii TABLE OF CONTENTS CHAPTER 1. INTRODUCTION CHAPTER 2. SCALAR WAVELET THEORY Refinable Functions Multiresolution Analysis and Wavelets Moments, Modulation Matrix and Polyphase Matrix The Discrete Wavelet Transform CHAPTER 3. MULTIWAVELET THEORY Refinable Function Vectors Multiresolution Analysis and Multiwavelets Moments, Modulation Matrix and Polyphase Matrix The Discrete Multiwavelet Transform CHAPTER 4. MATRIX-VALUED WAVELETS Refinable Matrix-Valued Functions Matrix-Valued Multiresolution Analysis and Matrix-Valued Wavelets The Discrete Multiwavelet Transform Connection with Scalar Wavelets and Multiwavelets ACKNOWLEDGEMENTS

4 1 CHAPTER 1. INTRODUCTION Wavelets are mathematical functions that partition data into frequency components, and then analyze each component with a resolution matched to its scale. They have advantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities and sharp spikes [3]. The main property of wavelets is their time-frequency localization. It is this property that makes wavelets so useful in many areas, such as music, video images, signal processing, etc. The aim of this creative component is to give an introduction to matrix-valued wavelets, introduced in [11] and explain the connections between classical or scalar wavelets, multiwavelets, and matrix-valued wavelets. Matrix-valued wavelets are intended to be used to decorrelate a matrix-valued signal not only in the time domain but also between the components of matrix for a fixed time [11]. Video images are examples of such matrixvalued signals. We begin by reviewing scalar wavelets. We give the main definitions and fundamental theorems on scalar wavelet theory such as, refinable function, multiresolution analysis, moments, modulation matrix, polyphase matrix, and discrete wavelet transformation. We then review multiwavelets. We give counterparts to the definitions and basic theorems of scalar wavelet theory. We also show the similarities and differences between multiwavelets and scalar wavelets. We finally introduce matrix-valued wavelets for the matrix-valued function space L 2 ( R, C N N ). We also demonstrate how matrix-valued wavelets can be constructed from scalar wavelets, which in turn give rise to multiwavelets.

5 2 CHAPTER 2. SCALAR WAVELET THEORY A wave is usually defined as an oscillating function of time or space, such as a sinusoid. A wavelet is a small wave, which has its energy concentrated both in time and in space to give a tool for the analysis of transient, nonstationary or time-varying phenomena [1]. The main ideas of classical scalar wavelet theory will be developed in this chapter. 2.1 Refinable Functions Definition 2.1 A refinable function is a function ϕ : R C which satisfies a two-scale refinement equation or recursion relation of the form ϕ(x) = 2 k 1 The h k C are called the recursion coefficients. ϕ is called orthogonal if ϕ(x), ϕ(x k) = δ 0k, k Z. Example 2.2 The hat function is given by k=k 0 h k ϕ(2x k). (2.1) 1 + x, if 1 x 0; φ(x) = 1 x, if 0 < x 1; 0, otherwise.

6 Figure 2.1 Hat Function Recursion Relation It satisfies φ(x) = 1 2 φ(2x + 1) + φ(2x) + 1 φ(2x 1) 2 = 2 ( φ(2x + 1) + φ(2x) φ(2x 1)), so it satisfies (2.1) with h 1 = 1 2 2, h 0 = 1 2, h 1 = Since φ(x), φ(x 1) = 0, it is not orthogonal. Example 2.3 The Haar function is the simple unit-width, unit-height pulse function ϕ(t) shown in Figure 2.2. It is clear that ϕ(2t) can be used to construct ϕ(t) by ϕ(t) = ϕ(2t) + ϕ(2t 1),

7 Figure 2.2 Haar Function Recursion Relation which means that the coefficients are h 0 = h 1 = 1 2. The Haar function is orthogonal. Theorem 2.4 A necessary condition for orthogonality is h k h k 2l = δ 0l. (2.2) k

8 5 Proof. δ 0l = ϕ(x), ϕ(x l) = 2 h k θ(2x k), 2 h m θ(2x 2l m) k m = 2 h k h m θ(2x k), θ(2x 2l m) k m = h k h mδ k,2l+m k m = h k h k 2l. k Observe that the orthogonality condition implies that an orthogonal ϕ has even number of recursion cofficients. Otherwise, if we assume h k0, h k1 are the first and last nonzero coefficients in (2.1), equation (2.2) implies h k0 h k1 = 0, which is a contradiction. Definition 2.5 Let f be an integrable function over R. The Fourier Transform of f is denoted by ˆf and defined as ˆf(ξ) = 1 2π f(x)e iξx dx. The Fourier transform can be extended to functions f L 2. Theorem 2.6 For any function ϕ L 2 ( R), the following conditions are equivalent. 1. The system {ϕ 0,k ϕ(x k), k Z} is orthonormal. 2. l= ˆϕ(ξ + 2lπ) 2 = 1 2π almost everywhere.

9 6 Proof. The Fourier transform of ϕ(x k) is ˆϕ 0,k (ξ) = e ikξ ˆϕ(ξ). By the general Parseval relation f, g = ˆf, ĝ for the Fourier transform, we have ϕ(x k), ϕ(x m) = ϕ(x), ϕ(x m + k) = ˆϕ(ξ), ( ϕ(x m + k) )ˆ(ξ) = = = e i(m k)ξ ˆϕ(ξ) 2 dξ 2π(l+1) l= 2πl 2π e i(m k)ξ 0 l= e i(m k)ξ ˆϕ(ξ) 2 dξ ˆϕ(ξ + 2πl) 2 dξ. Since the set {e ikξ, k Z} is complete in L 2 (0, 2π), ϕ(x k), ϕ(x m) = δ k,m l= ˆϕ(ξ + 2πl) 2 = 1 2π almost everywhere. Definition 2.7 The symbol of a refinable function is the trigonometric polynomial h(ξ) = 1 k 1 2 k=k 0 h k e ikξ. Lemma 2.8 The Fourier transform of the refinable function ϕ satisfies ˆϕ(ξ) = h( ξ 2 ) ˆϕ(ξ ). (2.3) 2

10 7 Proof. The Fourier transform of ϕ(x) = 2 k h k ϕ(2x k) is ˆϕ(ξ) = 1 2 h k e iξk 2 ˆϕ( ξ 2 ). In term of symbols, we have the desired equation k ˆϕ(ξ) = h( ξ 2 ) ˆϕ(ξ 2 ). Lemma 2.9 The orthogonality condition (2.2) is equivalent to h(ξ) 2 + h(ξ + π) 2 = 1 (2.4) Proof. h(ξ) 2 + h(ξ + π) 2 = 1 h m h 2 ne i(m n)ξ [1 + ( 1) m n ] m,n = l h m h m 2le 2ilξ = l δ 0,l e 2ilξ = 1. Definition 2.10 The cascade algorithm is fixed point iteration applied to the refinement equation. Let ϕ (0) be an initial guess, and define ϕ (n+1) (x) = 2 k h k ϕ (n) (2x k).

11 8 Lemma 2.11 If h satisfies (2.4), and the cascade algorithm converges, then ϕ is orthogonal and has compact support. Proof. Let ϕ (0) be orthogonal with compact support. Condition (2.4) insures that orthogonality is preserved at each step of the cascade algorithm, so the limit function is also orthogonal. Likewise, the support of ϕ (n) converges to the interval [k 0, k 1 ]. Details can be found in [2]. Definition 2.12 The transition operator for the symbol h(ξ) is defined by T f(ξ) = h( ξ 2 ) 2 f( ξ 2 ) + h(ξ 2 + π) 2 f( ξ 2 + π). It maps 2π periodic functions into 2π periodic functions. Lemma 2.13 The cascade algorithm converges if and only if T has a single eigenvalue 1, and all other eigenvalues are smaller than 1 in magnitude. Note: This condition can be checked by inspecting the eigenvalues of a finite matrix constructed from the recursion coefficients h k. Details can be found in [2]. Definition 2.14 A refinable function ϕ L 2 has stable shifts if there are constants 0 < A B < so that A k c k 2 k c kϕ(x k) 2 B k c k 2 for all sequences {c k } with k c k 2 <. If ϕ is orthogonal, it is automatically stable with A = B = 1. Definition 2.15 The refinable function ϕ satisfies the minimal regularity conditions if it is a compactly supported L 2 solution of the refinement equation (2.1) with nonzero integral, and has stable shifts.

12 9 Theorem 2.16 Assume ϕ satisfies the minimal regularity conditions. Then the following conditions hold: 1. h(0) = 1; 2. k ϕ(x k) = c, c 0 constant; 3. h(π) = 0. Proof. This is proved in [5]. For orthogonal ϕ, the constant c in (2) must have absolute value 1. ϕ is usually normalized so that this constant is 1. We assume from now on that ϕ satisfies the minimal regularity conditions. 2.2 Multiresolution Analysis and Wavelets Definition 2.17 Multiresolution Analysis An orthogonal multiresolution analysis (MRA) consists of a nested sequence of closed subspaces V j, j Z, of L 2 ( R) satisfying 1. V j V j+1 for all j Z; 2. f(x) V j f(2x) V j+1, for all j Z; 3. j Z V j = {0}; 4. j Z V j = L 2 ( R); 5. There exists a function ϕ V 0 such that { } ϕ(x k) : k Z is an orthonormal basis for V 0 [7] [8].

13 10 The function ϕ whose existence is asserted in (5) is called a scaling function of the given MRA. Condition (2) gives us the main property of an MRA. Each V n consists of the functions in V 0 compressed by a factor of 2 n. Thus, an orthonormal basis of V n is given by { } ϕ nk : k Z, where ϕ nk (x) = 2 n/2 ϕ(2 n x k). The factor 2 n/2 preserves the L 2 -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) = 2 k h k ϕ(2x k) (2.5) for some coefficients h k. That means that ϕ is refinable. When does the solution to a given refinement equation generate an orthogonal MRA? Properties (1), (2) are automatic. Property (5) is covered in Lemma It can be shown that property (3) is satisfied by any ϕ of compact support, and that a sufficient condition for property (4) is h(ξ) > 0 in a neighborhood of the origin. Together, we have the following theorem. Theorem 2.18 Let h(ξ) = 1 2 k1 k=k 0 h k e ikξ be such that the cascade algorithm converges, h(0) = 1,

14 11 h(ξ) 2 + h(ξ + π) 2 = 1. Then the solution ϕ(t) for the refinement equation exists and is a scaling function for an MRA {V j }. Proof. The proof can be found in [7]. Remark 2.19 It is possible to define non-orthogonal MRAs. However, there are several advantages to requiring that the scaling functions and wavelets be orthogonal. Orthogonal basis functions allow simple calculation of expansion coefficients and have a Parseval s theorem that allows a partitioning of the signal energy in the wavelet transform domain. Hence, we always assume that ϕ is orthogonal. The important features of the signal can better be described or parameterized not by using ϕ j,k (t) and increasing j to increase the size of the subspace spanned by the scaling functions, but by defining a different set of functions ψ j,k (t) that span the differences between the spaces V j. These functions are called wavelets. The orthogonal complement of V j in V j+1 is defined as W j. That is, all members of V j are orthogonal to all members of W j. It will be shown later that W 0 is spanned by the integer translates of a single function ψ, called mother wavelet. We require ϕ j,k (t), ψ j,l (t) = ϕ j,k (t)ψ j,l (t)dt = 0 for k, j, l Z. We may start at any V j, say at j = 0, V 0 V 1 V 2... L 2.

15 12 V2 V1 W0 W1 V0 Figure 2.3 Scaling Function and Wavelet Vector Spaces Now we define the wavelet spanned subspace W 0 such that V 1 = V 0 W 0, which extends to V 2 = V 0 W 0 W 1. In general, this gives L 2 = V 0 W 0 W 1 W 2... Figure 2.3 shows the nesting of the scaling function spaces V j for different scales j and how the wavelet spaces are disjoint differences (except for the zero element) or orthogonal complements. The scale of the inital space is arbitrary and could be chosen at a higher resolution of, say, j = 5; L 2 = V 5 W 5 W or at a lower resolution such as j = 5 to give L 2 = V 5 W 5 W 4 W 3...

16 13 or at even j = L 2 = W 2 W 1 W 0 W 1 W 2... to eliminate the scaling space altogether. We can also describe the relation of V 0 to the wavelet spaces as W W 1 = V 0. Since W 0 V 1, the mother wavelet ψ can be represented as ψ(t) = n g n 2ϕ(2t n) (2.6) where n Z, for some coefficients g n [1]. Like for the scaling function, we define the symbol of ψ as g(ξ) = 1 g k e ikξ. 2 The Fourier transform of (2.6) is ˆψ(ξ) = g( ξ 2 ) ˆϕ(ξ 2 ). k A necessary condition for orthogonality of ϕ and ψ is hk h k 2l = g k g k 2l = δ 0l, hk g k 2l = g k h k 2l = 0, or equivalently h(ξ) 2 + h(ξ + π) 2 = g(ξ) 2 + g(ξ + π) 2 = 1, h(ξ)g(ξ) + h(ξ + π)g(ξ + π) = g(ξ)h(ξ) + g(ξ + π)h(ξ + π) = 0. This is proved as in Theorem 2.4 or Lemma 2.9.

17 Figure Haar Wavelet Function Example 2.20 If we define ψ(x) = 1, if 0 x 1 2 ; 1, if 1 2 < 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. We now have constructed a set of functions ϕ k (t) and ψ j,k (t) that span all of L 2 ( R). Thus, any function g(t) L 2 ( R) can be written g(t) = c k ϕ k (t) + d j,k ψ j,k (t) k= j=0 k= as a series expansion in terms of the scaling function and wavelets [2]. Theorem 2.21 For any orthogonal MRA with scaling function ϕ 1. n W n is dense in L 2 ;

18 15 2. W k W n if k n; 3. f(x) W n f(2x) W n+1 for all n Z; 4. f(x) W n f(x 2 n k) W n for all n, k Z; 5. There exists a function ψ L 2 so that { } ψ(x k) : k Z forms an orthonormal basis of W 0, and { } ψ n,k : n, k Z forms an orthonormal basis of L 2 ; 6. Since ψ V 1, it can be represented as ψ(x) = 2 k g k ϕ(2x k) for some coefficients g k. The function ψ is called the wavelet function or mother wavelet. ϕ and ψ together form a wavelet. Proof. The proof can be found in [5]. Remark 2.22 If h k are the recursion coefficients of ϕ, then we can choose g k = ( 1) k h N k (2.7) where N is any odd number. The proof will be given at the end of this chapter. 2.3 Moments, Modulation Matrix and Polyphase Matrix Definition 2.23 The k th discrete moments of ϕ, ψ are defined by m k = 1 l k h l, 2 l

19 They are related to the symbols by 16 n k = 1 l k g l. 2 l m k = i k D k h(0), where D stands for differentiation. n k = i k D k g(0). Note that discrete moments are uniquely defined [1]. Definition 2.24 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 = 2πi k D k ˆϕ(0), ν k = 2πi k D k ˆψ(0). The continuous moment µ 0 is not determined by the refinement equation. For orthogonal wavelets, it must satisfy µ 0 = 1. Without loss of generality we assume µ 0 = 1. Theorem 2.25 The continuous and discrete moments are related by µ k = 2 k k ν k = 2 k t=0 k t=0 ( ) k m k t µ t, t ( ) k n k t µ t. t Starting with µ 0 = 1, all other continuous moments can be computed from these relations.

20 17 Definition 2.26 The matrix is called the modulation matrix. M(ξ) = h(ξ) h(ξ + π) g(ξ) g(ξ + π) Definition 2.27 A trigonometric polynomial matrix with the property U(ξ) U(ξ) = I. is called paraunitary. Remark 2.28 By Lemma 2.9, it is clear that the modulation matrix of an orthogonal refinable function is paraunitary. Definition 2.29 The polyphase symbols of the recursion coefficients are defined as h 0 (ξ) = k h 2k e ikξ, h 1 (ξ) = k h 2k+1 e ikξ. Note that they do not get a factor of 1 2 like the regular symbols. Definition 2.30 The matrix P (ξ) = h 0(ξ) g 0 (ξ) h 1 (ξ) g 1 (ξ) is called the polyphase matrix. It is also paraunitary.

21 18 Proof of Remark 2.12: Since we know that the modulation matrix of an orthogonal refinable function is paraunitary, we have M(ξ)M(ξ) = I. This implies M(ξ) = M(ξ) 1, (2.8) where M(ξ) 1 = 1 det(m(ξ)) g(ξ + π) h(ξ + π) g(ξ) h(ξ). By assumption, the entries of the modulation matrix M(ξ) are trigonometric polynomials, as are those of the M(ξ). However, this is only possible if det(m(ξ)) is monomial. Since det(m(ξ)) = h(ξ)g(ξ + π) g(ξ)h(ξ + π), det(m(ξ + π)) = det(m(ξ)). Hence, det(m(ξ)) must be of odd degree. Using paraunitary properties of M(ξ), we conclude det(m(ξ)) = αe i(2n+1)ξ, α = 1, n Z. By comparing the entries of (2.8), we obtain that g k = ( 1) k h N k.

22 The Discrete Wavelet Transform Definition 2.31 The orthogonal projection of an arbitrary function f L 2 onto V n is given by P n f = k f, ϕ nk ϕ nk. Since, by definition, the functions in V n have resolution (or scale) 2 n, P n f is called an approximation to f at resolution 2 n. Definition 2.32 The difference between the approximation at resolution 2 n and 2 n 1 is called the fine detail at resolution 2 n. In other words, Q n f(x) = P n+1 f(x) P n f(x). Q n is also an orthogonal projection. Assume that we have a function s V n s(x) = k s nkϕ nk (x) represented by its coefficient vector s n. We decompose s into its components in V n 1, W n 1 : s = P n 1 s + Q n 1 s = j s, ϕ n 1,j ϕ n 1,j + j s, ψ n 1,j ψ n 1,j = j s n 1,jϕ n 1,j + j d n 1,jψ n 1,j. Observe that the decomposed signal consists of two pieces s n 1, d n 1. Lemma 2.33 ϕ n 1,j, ϕ nk = h k 2j, ϕ n 1,j, ψ nk = g k 2j.

23 20 Proof. We will prove the first one: ϕ n 1,j, ϕ nk = = 2 n 1 2 ϕ(2 n 1 x j)2 n 2 ϕ(2 n x k)dx 2 n 2 h l ϕ(2 n x 2j l)2 n 2 ϕ(2 n x k)dx l = l h l δ 2j+l,k = h k 2j. The proof of the second one is similar. Using these formulas along with s nk = s, ϕ nk d nk = s, ψ nk we find s n 1,j and d n 1,j. s n 1,j = s, ϕ n 1,j = k s nkϕ nk, ϕ n 1,j = k s nk ϕ nk, ϕ n 1,j = k s nkh k 2j, or s n 1,j = k h k 2j s nk. Similarly, d n 1,j = k g k 2j s nk, and s nk = j [h k 2js n 1,j + g k 2jd n 1,j ].

24 21 Thus, we have the following algorithm. Algorithm: Assume we have the signal s n = {s nk }. Decomposition: s n 1,j = k h k 2j s nk, d n 1,j = k g k 2j s nk. Reconstruction: s nk = j [h k 2js n 1,j + g k 2jd n 1,j ].

25 22 CHAPTER 3. MULTIWAVELET THEORY 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. In addition, we consider a dilation factor of m rather than 2. With a dilation factor of m > 2, we still have one scaling function, but we get m 1 wavelets instead of one. 3.1 Refinable Function Vectors Definition 3.1 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) = m k 1 k=k 0 H k ϕ(mx k), k Z. (3.1) r is called the multiplicity of ϕ. The integer m 2 is the dilation factor. The recursion coefficients H k are r r matrices.

26 23 The refinable function vector ϕ is called orthogonal if ϕ(x), ϕ(x k) = ϕ(x)ϕ (x k)dx = δ 0k I. I is an r r identity matrix. The inner product is an r r matrix. Example 3.2 An example with multiplicity 2 and dilation factor 2 is the constantlinear refinable function vector, ϕ(x) = 1 3(2x 1) where x [0, 1]. It satisfies ϕ 1 (x) = ϕ 1 (2x) + ϕ 1 (2x 1), ϕ 2 (x) = [ 3 2 ϕ 1(2x) ϕ 2(2x)] + [ 2 ϕ 1(2x 1) ϕ 2(2x 1)]. This function vector is refinable with H 0 = , H 1 = It is orthogonal.

27 24 Theorem 3.3 A necessary condition for orthogonality is H k Hk ml = δ 0l I. (3.2) Proof. k δ 0l I = ϕ(x), ϕ(x l) = m k,n H k ϕ(mx k), ϕ(mx ml n) H n = k,n H k ϕ(y), ϕ(y + k ml n) H n = k,n H k δ 0,k ml n IH n = k H k H k ml. Here, the number of recursion coefficients need not be even, since H k0 Hk 1 = 0 does not imply that one or both of the matrices are zero. The Fourier transform of a function vector is defined element by element. Definition 3.4 The symbol of a refinable function vector is the trigonometric matrix polynomial H(ξ) = 1 k 1 m k=k 0 H k e ikξ. The Fourier transform of the refinement equation is ˆϕ(ξ) = H( ξ m ) ˆϕ( ξ m ). The proof is the same as in the scalar case. Theorem 3.5 For any function vector ϕ the following conditions are equivalent. 1. The system {ϕ 0,k ϕ(x k), k Z} is orthonormal.

28 25 2. l ˆϕ(ξ + 2lπ) ˆϕ (ξ + 2lπ) = 1 I almost everywhere. 2π Lemma 3.6 The orthogonality condition (3.3) is equivalent to Proof. m 1 k=0 m 1 k=0 H(ξ + 2πk m ) 2 = 1 m Here, we have used the formula m 1 k=0 e ik(u v) 2π m = H(ξ + 2πk m ) 2 = I. (3.3) = u,l = I. m 1 k=0 H u Hv e i(u v)ξ e ik(u v) 2π m u,v H u H u mle imlξ m, if u v = ml, l Z, 0, otherwise. As in the scalar case, condition (3.3) is sufficient if the cascade algorithm converges. Definition 3.7 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. Definition 3.8 The transition operator for the symbol H(ξ) is defined by T F (ξ) = m 1 k=0 H(ξ + 2π m k)f (ξ + 2π m k)h(ξ + 2π m k). This operator maps matrix-valued 2π periodic functions to matrix-valued 2π periodic functions. Theorem 3.9 Assume that H satisfies condition E(1) and the sum rules of order 1 with approximation vector y 0. Then the cascade algorithm converges for any starting function ϕ (0) which satisfies y 0 ϕ (0) (k) = c 0 iff the transition operator satisfies condition E(1). k

29 26 Proof. This is proved in [10]. As in the scalar case, it suffices to check condition E()1 for a certain matrix derived from the recursion coefficients H k. If the cascade algorithm converges, the limit function will automatically have compact support and nonzero integral if ϕ (0) has these properties. If H satisfies the orthogonality conditions (3.3) (or equivalently (3.6)), and ϕ (0) is orthogonal, the limit function will also be orthogonal. Definition 3.10 A refinable function vector ϕ L 2 has stable shifts if there are constants 0 < A B < so that A k c k 2 k c kϕ(x k) 2 B k c k 2 for any sequence of vectors {c k } with k c k 2 <. If ϕ is orthonormal, it is automatically stable with A = B = 1. Definition 3.11 The refinable function vector ϕ satisfies the minimal regularity conditions if ϕ has compact support, ϕ L 2, ϕ has stable shifts, and ϕ(x)dx 0. Theorem 3.12 Assume ϕ satisfies the minimal regularity conditions. Then the following conditions hold: 1. H(0) satisfies condition E(1). 2. There exists a vector y 0 0 so that 3. The same vector y 0 satisfies y0ϕ(x k) = 1. k y 0H( 2πk m ) = δ 0ky 0, k = 0, 1,... m 1.

30 27 4. The same vector y 0 satisfies k = 0, 1,... m 1. y 0 l H ml+k = 1 m y 0, Proof. This is proved in [9]. We assume from now on that ϕ satisfies the minimal regularity conditions. In particular, H(0) satisfies condition E(1). For later use, we state an additional theorem here. Definition 3.13 A collection of p function vectors ϕ 1,..., ϕ p L 2 has weakly stable shifts if there are constants 0 < A B < so that A k c k 2 p k c kϕ j (x k) 2 B k c k 2 j=1 for any sequence of vectors {c k } with k c k 2 <. Theorem 3.14 A necessary condition for the existence of exactly p linearly independent and weakly stable solutions of the refinement equation (3.1) is that H(0) satisfies condition E(p). Proof. The proof can be found in [4]. 3.2 Multiresolution Analysis and Multiwavelets Definition 3.15 Multiresolution Analysis and Multiwavelets An orthonormal multiresolution analysis (MRA) of L 2 is a doubly infinite nested sequence of subspaces of L 2... V 1 V 0 V 1 V 2...

31 28 with properties 1. V j V j+1 for all j Z; 2. f(x) V n f(mx) V n+1 for all n Z; 3. n V n = { 0 } ; 4. n V n is dense in L 2 ; 5. There exists a function vector ϕ L 2 so that { ϕl (x k) : l = 1, 2,..., r and k Z } forms an orthonormal basis of V 0. ϕ is called the multiscaling function. Condition (2) expresses the main property of an MRA: Each V n consists of the function in V 0 compressed by a factor of m n. Thus an orthonormal basis of V n is given by { ϕ nk : k Z }, where ϕ nk (x) = m n/2 ϕ(m n x k). The factor m n/2 preserves the L 2 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) = m k H k ϕ(mx k) for some coefficient matrices H k. It means ϕ is refinable. When does the solution to a given refinement equation generate an orthogonal MRA? Theorem 3.16 Let H(ξ) = 1 m k1 k=k 0 H k e ikξ be such that the cascade algorithm converges, H(0) satisfies condition E(1) and

32 29 H(ξ) satisfies the orthogonality condition (3.3). Then the solution ϕ(x) for the refinement equation exists and is a scaling function for an MRA {V j }. The proof is the same as in scalar case. As in the scalar wavelet case, we now define the orthogonal complement of V n in V n+1 as W n. However, W n is spanned by m 1 function vectors instead of one. In addition, these wavelets are orthogonal to each other and to the scaling function. Theorem 3.17 For any orthogonal MRA with multiscaling function ϕ 1. n W n is dense in L W k W n if k n. 3. f(x) W n f(mx) W n+1 for all n Z; 4. f(x) W n f(x m n k) W n for all n, k Z; 5. There exist function vectors ψ (s) L 2, s = 1,..., m 1 orthogonal to ϕ and to each other, so that { ψ (s) j (x k) : s = 1,..., m 1, and j = 1,..., r where k Z } forms an orthonormal basis of W 0, and { (s) ψ nk,j : s = 1,..., m 1, and j = 1,..., r where n, k Z} forms an orthonormal basis of L Since each ψ (s) V 1, it can be represented as ψ (s) (x) = m k G (s) k ϕ(mx k) for some coefficients G (s) k.

33 30 The function vectors ψ (s) are called the multiwavelet functions, ϕ and ψ (s) together form a multiwavelet. Proof. This is proved in [5]. Remark 3.18 In the scalar case, we have an explicit formula for g k. This is not possible for multiwavelets, but it is always possible to find G (s). We will discuss this below. k 3.3 Moments, Modulation Matrix and Polyphase Matrix Definition 3.19 The k th discrete moments of ϕ, ψ (s) are defined by M k = 1 l n H l, m N (s) k = 1 l m l l n G (s) l s = 1,... m 1. Discrete moments are r r matrices. They are related to the symbols by Discrete moments are uniquely defined. M k = i k D k H(0), N (s) k = i k D k G (s) (0). Definition 3.20 The k th continuous moments of ϕ, ψ (s) are µ k = x k ϕ(x)dx, ν (s) k = x k ψ (s) (x)dx. Continuous moments are r-vectors.

34 31 They are related to the Fourier transforms of ϕ, ψ (s) by µ k = 2πi k D k ˆϕ(0), ν (s) k = 2πi k D k ˆψ(s) (0). The continuous moment µ 0 is the unique eigenvector of H(0) with eigenvalue 1, normalized so that µ 0 = 1. ( This follows from orthonormality of ϕ ). All other continuous moments can then be calculated from these relations. Theorem 3.21 The continuous and discrete moments are related by In particular, µ k = m k k ν (s) k = m k t=0 k t=0 ( ) k M k t µ t t, ( ) k N (s) k t t µ t. µ 0 = M 0 µ 0 = H(0)µ 0. Once µ 0 has been chosen, all other continuous moments are uniquely defined and can be computed from these relations. Proof. This is proved in [5]. Definition 3.22 The matrix M(ξ) = H(ξ) H(ξ + 2π (m 1)2π )... H(ξ + ) m m G (1) (ξ) G (1) (ξ + 2π m )... G(1) (ξ + (m 1)2π m ).. G (m 1) (ξ) G (m 1) (ξ + 2π m )... G(m 1) (ξ + (m 1)2π m ).. is called the modulation matrix.

35 32 Definition 3.23 The polyphase symbols of the recursion coefficients are defined as H l (ξ) = k H mk+le ikξ where l = 0, 1,..., m 1. Definition 3.24 The matrix P (ξ) = H 0 (ξ) H 1 (ξ)... H m 1 (ξ) G (1) 0 (ξ) G (1) 1 (ξ)... G (1) m 1(ξ).. G (m 1) 0 (ξ) G (m 1) 1 (ξ)... G (m 1) m 1.. (ξ) is called the polyphase matrix. Remark 3.25 The modulation matrix and the polyphase matrix of an orthogonal multiwavelet are both paraunitary. Comment on Remark 3.18 : The problem of finding the multiwavelet functions when the multiscaling function is given is equivalent to matrix completion problem. Let with We want to find G (s) j P H = (H 0, H 1,..., H m 1 ) P H P H = I. so that P given in Definition 3.24 is paraunitary. This matrix completion problem can always be solved. Basic methods and examples are described in [5] and [6].

36 The Discrete Multiwavelet Transform Assume that we have a function s V n s(x) = k s nkϕ nk (x) represented by its coefficient vector s n. We decompose s into its components in V n 1, W n 1 : s = P n 1 s + Q n 1 s = j s, ϕ n 1,j ϕ n 1,j + j s, ψ n 1,j ψ n 1,j = j s n 1,jϕ n 1,j + j m 1 t=1 d (t) n 1,j ψ(t) n 1,j, where s n 1,j = s, ϕ n 1,j, d (t) n 1,j = s, ψ(t) n 1,j. Lemma 3.26 ϕ n 1,j, ϕ nk = H k mj, ϕ n 1,j, ψ (t) nk = G(t) k mj, Proof. The proof is the same as in scalar case. Using these formulas along with s nk = s, ϕ nk, d (t) nk = s, ψ(t) nk,

37 34 we find s n 1,j and d (t) n 1,j. s n 1,j = s, ϕ n 1,j = k s nkϕ nk, ϕ n 1,j = k s nk ϕ nk, ϕ n 1,j = k s nkh k mj, or s n 1,j = k H k mj s nk. Similarly, d (t) n 1,j = k G (t) k mj s nk. Thus, we have the following algorithm. Algorithm: Assume we have the signal s n = {s nk }. Decomposition: s n 1,j = k H k mj s nk, d (t) n 1,j = k G (t) k mj s nk. The decomposed signal consists of m pieces s n 1, d (t) n 1 t = 1,..., m 1 since, in this case, we have m 1 wavelets rather than one as in scalar case. Reconstruction: s nk = j H k mjs n 1,j + j m 1 t=1 G (t) k mj d(t) n 1,j.

38 35 CHAPTER 4. MATRIX-VALUED WAVELETS We now introduce matrix-valued multiresolution analysis (MMRA) for matrix-valued signals, where the concepts of orthogonality and orthonormal bases are similar to the ones in Hilbert spaces. Associated with MMRA, we define matrix-valued scaling functions and matrix-valued wavelet functions. We will show that matrix-valued scaling functions and wavelet functions can be generated from some lowpass and bandpass filters with matrix forms, which are called Matrix Quadrature Mirror Filters (MQMF s), via matrix dilation equations similar to the traditional wavelet theory. We will see that certain linear combinations of scalar-valued wavelets yield multiwavelets [11]. 4.1 Refinable Matrix-Valued Functions Matrix-Valued Functions: The signal space L 2 ( R, C N N ) f 11 (t) f 12 (t)... f 1N (t) f 21 (t) f 22 (t)... f 2N (t) F (t) = t R, f. kl (t) L 2 ( R), k, l = 1, 2,..., N f N1 (t) f N2 (t)... f NN (t) is the space of matrix-valued L 2 functions. The space L 2 ([a, b], C N N ) is defined sim-

39 36 ilarly by replacing the real line R with an interval [a, b]. Remark 4.1 In [11], these are called vector-valued functions. That makes absolutely no sense, so we changed the name. We use the norm ( N ) 1/2 F (t) f kl (t) 2 dt R k,l=1 The integral F (t)dt and Fourier transform ˆF are defined element by element. For two matrix-valued functions F, G L 2 ( R, C N N ), F, G is defined by F, G F (t)g (t)dt. R For convenience, we still call the operation, an inner product, although it is not an inner product in the common sense. Definition 4.2 A refinable matrix-valued function is a function Φ L 2 ( R, C N N ), which satisfies a two-scale refinement equation of the form Φ(t) = m k H k Φ(mt k), k Z. (4.1) The recursion coefficients H k are N N constant matrices. Definition 4.3 A sequence Φ k (t) L 2 ( R, C N N ), k Z, is called an orthonormal set in L 2 ( R, C N N ) if Φ k, Φ l = δ kl I N. A sequence Φ k (t) L 2 ( R, C N N ), k Z, is called an orthonormal basis for L 2 ( R, C N N ) if it is an orthonormal set in L 2 ( R, C N N ), and moreover for any F L 2 ( R, C N N ) there exists a sequence of N N constant matrices F k such that

40 37 F (t) = k Z F kφ k (t), for t R. The multiplication F k Φ k (t) for each fixed t is N N matrix multiplication, and the convergence for the infinite summation is in the sense of the norm. defined above. Theorem 4.4 A necessary condition for orthogonality is H k Hk 2m = δ m I, (4.2) where I is the identity matrix of the size N N and m Z. k Proof. See proof of Theorem 3.3. Example 4.5 Let Φ k (t) = diag(e i(k 1+k)t, e i(k 2+k)t,..., e i(k N +k)t ), k Z, where k 1, k 2,..., k N are fixed integers. This produces an orthonormal basis for L 2 ([0, 2π], C N N ). It is obvious that if {Φ k (t)} k Z is an orthonormal basis for L 2 ( R, C N N ), then {UΦ k (t)} k Z or {Φ k (t)u} k Z are also orthonormal bases for L 2 ( R, C N N ), where U is an N N unitary matrix. Definition 4.6 The symbol of a refinable matrix-valued function is the trigonometric matrix polynomial H(ξ) = 1 H k e ikξ. m k Lemma 4.7 The Fourier transform of the matrix-valued scaling function Φ satisfies Φ(ξ) = H( ξ m )ˆΦ( ξ m ). Proof. This is easy to show directly as in Chapters 2, 3, and is also proved in [11].

41 38 Lemma 4.8 The orthonormality condition is equivalent to m 1 k=0 H(ξ + 2πk m ) 2 = I, ξ R. (4.3) Proof. The proof is essentially identical to the proof of Lemma 3.6. It can also be found in [11]. 4.2 Matrix-Valued Multiresolution Analysis and Matrix-Valued Wavelets Definition 4.9 An orthonormal matrix-valued multiresolution analysis (MMRA) of L 2 ( R, C N N ) is a nested sequence of closed subspaces V j, j Z of L 2 ( R, C N N ) such that 1. V j V j+1, j Z 2. f(x) V j f(mt) V j+1 for all j Z; 3. j Z V j = {0} where 0 is the zero matrix. 4. j Z V j is dense in L 2 ( R, C N N ); 5. There exists a function Φ V 0 such that its translations Φ k (t) Φ(t k), k Z form an orthonormal basis for V 0. We call Φ(t) a matrix-valued scaling function for the MMRA {V j }. Each column of Φ is a refinable function vector for the same recursion coefficients as Φ. To span all of L 2 ( R, C N N ), the columns must be linearly independent. Thus, by Theorem 3.14, H(0) must satisfy condition E(N), so H(0) = I.

42 39 As in the multiwavelet case, this implies existence of Φ, via the infinite matrix product ˆΦ(ξ) = H( ξ m )ˆΦ(0). k k=1 The key to establishing the other properties is again the convergence of cascade algorithm. It is shown in [4] that a necessary condition for the existence of a weakly stable set of refinable function vectors is that the transition operator satisfies condition E(N 2 ). We conjecture that this is also a sufficient condition. The details have not been worked out in the literature yet. The approach taken in [11] is to prove the orthogonality of Φ from the orthogonality conditions (4.2), (4.3) for H directly, under the rather restrictive condition H k = H k k Z or H k = H k k Z. They claim that this automatically implies properties (1), (4) of an MMRA, but this is not clear. Since Φ(t) V 0 V 1, there exist constant N N matrices H k, k Z such that Φ(t) = m k H k Φ(mt k). For any ξ R, let G (l) (ξ) l = 1,..., m 1 satisfy and m 1 k=0 m 1 k=0 G (l) (ξ + 2πk m )H (ξ + 2πk ) = 0, (4.4) m G (l) (ξ + 2πk m )G (ξ + 2πk m ) = I N. (4.5) The problem of finding such G (l) wavelets when the scaling function is given is equivalent to the same matrix completion problem that we did for the multiwavelet case. This

43 40 matrix completion problem can always be done. Let ˆΨ (l) (ξ) = G (l) ( ξ m )ˆΦ( ξ ). (4.6) m We have the following result on the existence of a matrix-valued wavelet function. Proposition 4.10 Let Ψ (l) (t) be a matrix-valued function with Fourier transform (4.6) Then its translations Ψ (l) k (t) Ψ(l) (t k), k Z, form an orthonormal basis for W 0 V 1 V 0. Proof. The proof for the case m = 2 can be found in [11]. It is not hard to extend to general m. If we use the orthonormality conditions for general m along with (4.5), we obtain the desired result. Proposition 4.10 implies that a matrix-valued multiresolution analysis gives us matrixvalued wavelet functions Ψ (l) (t) whose dilations and translations Ψ (l) (t) jk (t) m j/2 Ψ (l) (t)(m j t k), j, k Z form an orthonormal basis for L 2 ( R, C N N ).

44 The Discrete Multiwavelet Transform Assume that we have a matrix valued function S L 2 ( R, C N N ) and its projection S J in V J for a fixed integerj can be written as S J (t) = k S J,k Φ Jk (t) = j<j k D (t) j,k Ψ jk(t) where By (4.6) and (4.1), for j J, We have S j,k = S, Φ jk, B (l) j,k = S, Ψ(l) jk l = 1,..., m 1 S j 1,k = n S j,n H n 2k, D (l) j 1,k = n S j,n G (l) n 2k. Thus, we have the following algorithm: Algorithm: Assume we have the signal S J = {S nk } Decomposition: S j 1,k = n S j,n H n 2k, D (l) j 1,k = n D j,n G (l) n 2k. Reconstruction: S j,n = ( S j 1,k H n 2k + k k m 1 l=1 D (l) ) j 1,k

45 Connection with Scalar Wavelets and Multiwavelets A matrix-valued scaling function of size N N can be constructed from N scalar scaling function as follows: Assume ϕ (k) (x) = m j h k j ϕ k (mx k), then Φ(x) = m j H j Φ(mx j) is a matrix-valued scaling function, where Φ(x) = U T diag(ϕ k (x))u, k = 1,..., N, H j = U T diag(h (k) )U, k = 1,..., N. for any fixed orthogonal matrix U. The columns of Φ are then multiwavelets. Example 4.11 Let ϕ 1, ϕ 2 be scalar-valued scaling functions and U = , then Φ(t) = 1 φ 1(t) + 3φ 2 (t) 4 3(φ1 (t) φ 2 (t)) 3(φ1 (t) φ 2 (t)) 3φ 1 (t) + φ 2 (t) is a matrix-valued scaling function.

46 43 ACKNOWLEDGEMENTS I would like to express my sincere gratitude and appreciation to my adviser Prof. Fritz Keinert for his encouragement and patience when I was writing this creative component. I would also like to thank Prof. Jonathan D. H. Smith for his kind support and advice during my graduate studies. My special thanks also go to my committee member Prof. Khalid Boushaba for his guidance and encouragement.

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