Results 1  10
of
30
Biorthogonal Wavelet Expansions
 Constr. Approx
"... This paper is concerned with developing conditions on a given finite collection of compactly supported algebraically linearly independent refinable functions that insure the existence of biorthogonal systems of refinable functions with similar properties. In particular we address the close connectio ..."
Abstract

Cited by 59 (6 self)
 Add to MetaCart
This paper is concerned with developing conditions on a given finite collection of compactly supported algebraically linearly independent refinable functions that insure the existence of biorthogonal systems of refinable functions with similar properties. In particular we address the close connection of this issue with stationary subdivision schemes. Key Words: Finiteley generated shiftinvariant spaces, stationary subdivision schemes, matrix refinement relations, biorthogonal wavelets. AMS Subject Classification: 39B62, 41A63 1 Introduction During the past few years the construction of multivariate wavelets has received considerable attention. It is quite apparent that multivariate wavelets with good localazition properties in frequency and spatial domains which constitute an orthonormal basis of L 2 (IR s ) are hard to realize. On the other hand, it turns out that in many applications orthogonality is not really important whereas locality, in particular, compact support is very...
Vector cascade algorithms and refinable function vectors in Sobolev spaces
 J. Approx. Theory
, 2002
"... In this paper we shall study vector cascade algorithms and refinable function vectors with a general isotropic dilation matrix in Sobolev spaces. By investigating several properties of the initial function vectors in a vector cascade algorithm, we are able to take a relatively unified approach to st ..."
Abstract

Cited by 53 (34 self)
 Add to MetaCart
In this paper we shall study vector cascade algorithms and refinable function vectors with a general isotropic dilation matrix in Sobolev spaces. By investigating several properties of the initial function vectors in a vector cascade algorithm, we are able to take a relatively unified approach to study several questions such as convergence, rate of convergence and error estimate for a perturbed mask of a vector cascade algorithm in a Sobolev space W k p (R s)(1 � p � ∞, k ∈ N∪{0}). We shall characterize the convergence of a vector cascade algorithm in a Sobolev space in various ways. As a consequence, a simple characterization for refinable Hermite interpolants and a sharp error estimate for a perturbed mask of a vector cascade algorithm in a Sobolev space will be presented. The approach in this paper enables us to answer some unsolved questions in the literature on vector cascade algorithms and to comprehensively generalize and improve results on scalar cascade algorithms and scalar refinable functions to the vector case. Key words: vector cascade algorithm, vector subdivision scheme, refinable function vector, Hermite interpolant, initial function vector, error estimate, sum rules, smoothness.
Multidimensional Interpolatory Subdivision Schemes
"... : This paper presents a general construction of multidimensional interpolatory subdivision schemes. In particular, we provide a concrete method for the construction of bivariate interpolatory subdivision schemes of increasing smoothness by finding an appropriate mask to convolve with the mask of a t ..."
Abstract

Cited by 45 (11 self)
 Add to MetaCart
(Show Context)
: This paper presents a general construction of multidimensional interpolatory subdivision schemes. In particular, we provide a concrete method for the construction of bivariate interpolatory subdivision schemes of increasing smoothness by finding an appropriate mask to convolve with the mask of a threedirection box spline B r;r;r of equal multiplicities. The resulting mask for the interpolatory subdivision exhibits all the symmetries of the threedirection box spline and with this increased symmetry comes increased smoothness. Several examples are computed (for r = 2; : : : ; 8). Regularity criteria in terms of the refinement mask are established and applied to the examples to estimate their smoothness. AMS Subject Classification: Primary 45A05, 65D05, 65D15, 26B05 Secondary 41A15, 41A63 Keywords: interpolation, subdivision schemes, interpolatory subdivision schemes, box splines, wavelets y Research supported in part by NSERC Canada under Grant # A7687 1 1. Introduction and Metho...
Vector subdivision schemes and multiple wavelets
 Math. Comput
, 1998
"... Abstract. We consider solutions of a system of refinement equations written in the form φ = ∑ a(α)φ(2 ·−α), α∈Z where the vector of functions φ =(φ1,...,φr) T is in (Lp(R)) r and a is a finitely supported sequence of r × r matrices called the refinement mask. Associated with the mask a is a linear o ..."
Abstract

Cited by 42 (13 self)
 Add to MetaCart
Abstract. We consider solutions of a system of refinement equations written in the form φ = ∑ a(α)φ(2 ·−α), α∈Z where the vector of functions φ =(φ1,...,φr) T is in (Lp(R)) r and a is a finitely supported sequence of r × r matrices called the refinement mask. Associated with the mask a is a linear operator Qa defined on (Lp(R)) r by Qaf: = ∑ α∈Z a(α)f(2 ·−α). This paper is concerned with the convergence of the subdivision scheme associated with a, i.e., the convergence of the sequence (Qn a f)n=1,2,... in the Lpnorm. Our main result characterizes the convergence of a subdivision scheme associated with the mask a in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the L2convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations. Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry. 1.
Smoothness of multiple refinable functions and multiple wavelets
 SIAM J. Matrix Anal. Appl
, 1999
"... Abstract. We consider the smoothness of solutions of a system of refinement equations written in the form φ = a(α)φ(2 ·−α), α∈Z where the vector of functions φ =(φ1,...,φr) T is in (Lp(R)) r and a is a finitely supported sequence of r × r matrices called the refinement mask. We use the generalized L ..."
Abstract

Cited by 33 (8 self)
 Add to MetaCart
Abstract. We consider the smoothness of solutions of a system of refinement equations written in the form φ = a(α)φ(2 ·−α), α∈Z where the vector of functions φ =(φ1,...,φr) T is in (Lp(R)) r and a is a finitely supported sequence of r × r matrices called the refinement mask. We use the generalized Lipschitz space Lip ∗ (ν, Lp(R)), ν>0, to measure smoothness of a given function. Our method is to relate the optimal smoothness, νp(φ), to the pnorm joint spectral radius of the block matrices Aε, ε =0,1, given by Aε =(a(ε+2α−β))α,β, when restricted to a certain finite dimensional common invariant subspace V. Denoting the pnorm joint spectral radius by ρp(A0V,A1V), we show that νp(φ) ≥ 1/p − log 2 ρp(A0V,A1V) with equality when the shifts of φ1,...,φr are stable and the invariant subspace is generated by certain vectors induced by difference operators of sufficiently high order. This allows an effective use of matrix theory. Also the computational implementation of our method is simple. When p = 2, the optimal smoothness is also given in terms of the spectral radius of the transition matrix associated with the refinement mask. To illustrate the theory, we give a detailed analysis of two examples where the optimal smoothness can be given explicitly. We also apply our methods to the smoothness analysis of multiple wavelets. These examples clearly demonstrate the applicability and practical power of our approach.
Eigenvalues of (!2)H and Convergence of the Cascade Algorithm
 IEEE Trans. SP
, 1996
"... This paper is about the eigenvalues and eigenvectors of (# 2)H. The ordinary FIR filter H is convolution with a vector h = (h(0); : : : ; h(N )), the impulse response. The operator (# 2) downsamples the output y = h x, keeping the evennumbered components y(2n). Where H is represented by a consta ..."
Abstract

Cited by 32 (1 self)
 Add to MetaCart
(Show Context)
This paper is about the eigenvalues and eigenvectors of (# 2)H. The ordinary FIR filter H is convolution with a vector h = (h(0); : : : ; h(N )), the impulse response. The operator (# 2) downsamples the output y = h x, keeping the evennumbered components y(2n). Where H is represented by a constantdiagonal matrix  this is a Toeplitz matrix with h(k) on its kth diagonal  the oddnumbered rows are removed in (# 2)H. The result is a double shift between rows, yielding a block Toeplitz matrix with 1 \Theta 2 blocks. Iteration of the filter is governed by the eigenvalues. If the transfer function H(z) = P h(k)z \Gammak has a zero of order p at z = \Gamma1, corresponding to ! = ß, then (# 2)H has p special eigenvalues 1 2 ; 1 4 : : : ; \Gamma 1 2 \Delta p . We show how each additional "zero at ß" divides all eigenvalues by 2 and creates a new eigenvector for = 1 2 . This eigenvector solves the dilation equation OE(t) = 2 P h(k)OE(2t \Gamma k) at the integers t = n. ...
Convergence Of Multidimensional Cascade Algorithm
 Numer. Math
, 1998
"... . A necessary and sufficient condition on the spectrum of the restricted transition operator is given for the convergence in L 2 (R d ) of the multidimensional cascade algorithm for any starting function OE 0 whose shifts form a partition of unity. 1. Introduction This paper is a continuation o ..."
Abstract

Cited by 30 (12 self)
 Add to MetaCart
(Show Context)
. A necessary and sufficient condition on the spectrum of the restricted transition operator is given for the convergence in L 2 (R d ) of the multidimensional cascade algorithm for any starting function OE 0 whose shifts form a partition of unity. 1. Introduction This paper is a continuation of [8]. In [8] we obtained a complete characterization of stability and orthonormality of the shifts of a refinable function in terms of its refinenent mask. In this paper we present a complete characterization of the convergence in L 2 (R d ) of the multidimensional cascade algorithm with an arbitrary dilation matrix M in terms of the mask. For fixed integers d 1 and m 2; let M be a d \Theta d dilation matrix with j det(M)j = m: A dilation matrix is an integer matrix with all eigenvalues of modulus ? 1: Let ` 2 (Z d ); where Z d is the set of all multiintegers, be the space of all squaresummable sequences, and L 2 (R d ) the space of all squareintegrable functions. The...
Approximation by multiple refinable functions
 Canadian J. Math
, 1997
"... We consider the shiftinvariant space, S(Φ), generated by a set Φ = {φ1,..., φr} of compactly supported distributions on IR when the vector of distributions φ: = (φ1,..., φr) T satisfies a system of refinement equations expressed in matrix form as φ = � a(α)φ(2 · − α) α ∈ Z where a is a finitely su ..."
Abstract

Cited by 25 (8 self)
 Add to MetaCart
We consider the shiftinvariant space, S(Φ), generated by a set Φ = {φ1,..., φr} of compactly supported distributions on IR when the vector of distributions φ: = (φ1,..., φr) T satisfies a system of refinement equations expressed in matrix form as φ = � a(α)φ(2 · − α) α ∈ Z where a is a finitely supported sequence of r × r matrices of complex numbers. Such multiple refinable functions occur naturally in the study of multiple wavelets. The purpose of the present paper is to characterize the accuracy of Φ, the order of the polynomial space contained in S(Φ), strictly in terms of the refinement mask a. The accuracy determines the Lpapproximation order of S(Φ) when the functions in Φ belong to Lp(IR) (see, Jia [10]). The characterization is achieved in terms of the eigenvalues and eigenvectors of the subdivision operator associated with the mask a. In particular, they extend and improve the results of Heil, Strang and Strela [7], and of Plonka [16]. In addition, a counterexample is given to the statement of Strang and Strela [20] that the eigenvalues of the subdivision operator determine the accuracy. The results do not require the linear independence of the shifts of φ.
Construction of Multivariate Biorthogonal Wavelets by CBC Algorithm
 Adv. Comput. Math
, 1998
"... In applications, it is well known that short support, high vanishing moments and reasonable smoothness are the three most important properties of a biorthogonal wavelet. Based on our previous work on analysis and construction of optimal fundamental refinable functions and optimal biorthogonal wavele ..."
Abstract

Cited by 22 (10 self)
 Add to MetaCart
In applications, it is well known that short support, high vanishing moments and reasonable smoothness are the three most important properties of a biorthogonal wavelet. Based on our previous work on analysis and construction of optimal fundamental refinable functions and optimal biorthogonal wavelets, in this paper, we shall discuss the mutual relations among these three properties. For example, we shall see that any orthogonal scaling function, which is supported on [0; 2r \Gamma 1] s for some positive integer r and has accuracy order r, has Lp (1 p 1) smoothness not exceeding that of the univariate Daubechies orthogonal scaling function which is supported on [0; 2r \Gamma 1]. Similar results hold true for fundamental refinable functions and biorthogonal wavelets. Then, we shall discuss the relation between symmetry and the smoothness of a refinable function. Next, we discuss the coset by coset (CBC) algorithm reported in Han [29] to construct biorthogonal wavelets with arbitrar...