. We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, in-place calculation of the wavelet transform. Several examples are included. Key words. wavelet, multiresolution, second generation wavelet, lifting scheme AMS subject classifications. 42C15 1. Introduction. Wavelets form a versatile tool for representing general functions or data sets. Essentially we can think of them as data building blocks. Their fundamental property is that they allow for representations which are efficient and which can be computed fast. In other words, wavelets are capable of quickly capturing the essence of a data set with only a small set of coefficients. This is based on the fact that most data sets have correlation both in time (or space) and frequenc...

We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spline, pseudo-spline tight frames and symmetric biframes with short supports and high approximation orders. Several explicit examples are discussed. The connection of these frames with multiresolution analysis guarantees the existence of fast implementation algorithms, which we discuss briefly as well.

Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrix-valued filter banks leading to wavelet bases, multi-wavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar 2-channel wavelet systems. After reviewing this recently developed theory, we examine the use of multiwavelets in a filter bank setting for discrete-time signal and image processing. Multiwavelets di#er from scalar wavelet systems in requiring two or more input streams to the multiwavelet filter bank. We describe two methods (repeated row and approximation /deapproximation) for obtaining such a vector input stream from a one-dimensional signal. Algorithms for symmetric extension of signals at boundaries are then developed, and naturally integrated with approximation-based preprocessing. We describe an additional algorithm for multiwavelet processing of two-dimensional signals, two rows at a time, and develop a new family of multiwavelets (the constr...

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 shift-invariant 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...

CONTENTS Page DEDICATION PAGE : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ii ACKNOWLEDGMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iii LIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : v Chapter I. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 Preliminaries : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 Synopsis of Main Theorems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 II. MULTIWAVELETS AND SCALING VECTORS : : : : : : : : : : : : : : : : : : : : 11 Orthonormal Scaling Vector of Legendre Polynomials : : : : : : : :

Abstract. Orthogonal polynomials are used to construct families of C 0 and C 1 orthogonal, compactly supported spline multiwavelets. These families are indexed by an integer which represents the order of approximation. We indicate how to obtain from these multiwavelet bases for L 2 [0,1] and present a C 2 example.

Starting from any two compactly supported d-refinable function vectors in L 2 (R) multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L 2 (R) and they achieve the best possible orders of vanishing moments. When all the components of the two real-valued d-refinable function vectors are either symmetric or antisymmetric with their symmetry centers differing by half integers, such 2rd wavelet functions, which generate a pair of dual d-wavelet frames, can be real-valued and be either symmetric or antisymmetric with the same symmetry center. Wavelet frames from any d-refinable function vector are also considered. This paper generalizes the work in [5, 12, 13] on constructing dual wavelet frames from scalar refinable functions to the multiwavelet case. Examples are provided to illustrate the construction in this paper.

. This article discusses modern techniques for non-uniform sampling and reconstruction of functions in shift-invariant spaces. The reconstruction of a function or signal or image f from its non-uniform samples f(x j ) is a common task in many applications in data or signal or image processing. The non-uniformity of the sampling set is often a fact of life and prevents the use of the standard methods from Fourier analysis. The non-uniform sampling problem is usually treated in the context of band-limited functions and with tools from complex analysis. However, many applied problems impose di#erent a priori constraints on the type of function. These constraints are taken into consideration by investigating the problem in general shift-invariant spaces rather than for band-limited functions only. This generalization requires a new set of techniques and ideas. While some of the tools are implicitly used in certain questions in approximation theory, wavelet theory and frame theory, they hav...

. A two parameter family of multiresolution analyses of L 2 (R 2 ) each generated by three orthogonal, continuous, compactly supported scaling functions is constructed using fractal interpolation surfaces. The scaling functions remain orthogonal when restricted to certain triangles making them useful for problems with bounded domains. x1. Introduction In the usual construction of a wavelet using a multiresolution analysis (MRA) (V p ) there is a single function OE called the scaling function whose set of integer translates forms a Riesz basis for V 0 . In contrast, multiwavelets are constructed from an MRA that is generated by a finite set of scaling functions OE 1 ; : : : ; OE r . Definition 1. If e 1 ; : : : e n are independent vectors in R n and N is an integer greater than 1, then a nested sequence of closed linear subspaces (V p ) in L 2 (R n ) is called an MRA of multiplicity r with dilation N and lattice e 1 Z \Theta \Delta \Delta \Delta \Theta e nZ if a) ...

We consider r-vectors of spline functions with compact support and stable integer translates which satisfy a refinement equation with a finite number of terms. It is seen that for r = 1 or 2, the choice of such vectors is very limited. However for r &ge 3 there is sufficient flexibility, even for simple uniform knots, to allow the satisfaction of further useful properties. Mention is made of biorthogonality and orthogonality properties. Moreover it is shown that for each r ≥ 3...