Wavelet analysis : twenty years' developments : proceedings of the International Conference of Computational Harmonic Analysis : Hong Kong, China, 4-8 June 2001

318 Pages·2002·12.802 MB·English

by  International Conference of Computational Harmonic Analysis, Zhou D.-X.

Wavelet Analysis
Twenty Years' Developments
SERIES IN ANALYSIS
Series Editor: Professor Roderick Wong
City University of Hong Kong, Hong Kong, China
Published
Vol. 1 Wavelet Analysis
edited by Ding-Xuan Zhou
Proceedings of the International Conference of
Computational Harmonic Analysis
Wavelet Analysis
Twenty Years' Developments
Hong Kong, China 4-8 June 2001
Editor
Ding-Xuan Zhou
City University of Hong Kong, China
V fe World Scientific
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World Scientific Publishing Co. Pte. Ltd.
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WAVELET ANALYSIS
Twenty Years' Developments
Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd.
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PREFACE
Wavelet Analysis has been developing very fast within the last twenty years.
More and more people are working on this important research field, including
mathematicians from the areas of harmonic analysis, approximation theory,
and scientific computation, and engineers from the areas of signal processing,
image analysis, and computer graphics.
The International Conference of Computational Harmonic Analysis was
held in Hong Kong during the period of June 4-8, 2001. It was organized by
an International Organizing Committee formed by Charles K. Chui (Stanford
University and University of Missouri), Roderick Wong and Ding-Xuan Zhou
(City University of Hong Kong).
The purpose of the conference was to bring together mathematicians and
engineers interested in the computational aspects of harmonic analysis. The
central conference theme was wavelet analysis in the broadest sense, covering
time-frequency and time-scale analysis, filter banks, fast numerical compu
tations, spline methods, multiscale algorithms, approximation theory, signal
processing, and a great variety of applications. The conference was attended
by more than 130 participants from 21 countries. The program included 10
plenary lectures, 28 invited talks, and 48 contributed presentations.
Wavelet analysis is still in the happy position that these topics from mathe
matics are of great interest in applications. The search for better wavelets (and
more generally for better representations of signals and images) has certainly
not ended. Splines and orthogonal wavelets led to biorthogonal wavelets for
images, and then to ridgelets, beamlets, ... that can capture edges. Frames are
becoming popular, in spite of (or because of) their redundancy. After twenty
astonishing years, there is still time for new ideas and new basis functions!
The present proceedings contain sixteen papers from the lectures given
by plenary and invited speakers. These include expository articles surveying
various aspects of the twenty years' developments of wavelet analysis, and
original research papers reflecting the wide range of research topics of current
interest. We apologize that we are not able to include all the interesting lectures
because of the size limitation.
This conference was sponsored by the Croucher Foundation, Liu Bie Ju
Centre for Mathematical Sciences, City University of Hong Kong, Hong Kong
v
VI
Mathematical Society, Hong Kong Pei Hua Education Foundation Limited,
and K. C. Wong Education Foundation. We would like to thank all of these
organizations and institutions for their generous financial support to the con
ference. We are indebted to Professor Roderick Wong for his very strong
support. Special thanks are due to our host, City University of Hong Kong,
for providing the facilities, and our colleagues for their help with carrying out
the program. In particular, we appreciate the efficient secretarial work by Miss
Shirley Cheung from the Liu Bie Ju Centre for Mathematical Sciences, and the
efforts of our reviewers in the preparation of this volume.
Finally, thanks go to the scientific committee members and all 130 partic
ipants for making the conference a success.
Ding-Xuan Zhou
City University of Hong Kong
June, 2002
Contents
Preface v
Non-uniform Sampling: Exact Reconstruction from Non-uniformly
Distributed Weighted-averages 1
Akram Aldroubi and Hans G. Feichtinger
Squeezable Bases and Semi-regular Multiresolutions 9
Derek Bruff and Douglas P. Hardin
Multilevel Structure of NURBS and Formulation of NURBlets 23
Charles K. Chui and Jian-Ao Lian
Adaptive Wavelet Methods — Basic Concepts and Applications
to the Stokes Problem 39
Wolfgang Dahmen, Karsten Urban, and Jiirgen Vorloeper
Nonstationary Wavelets 81
S. Dekel and D. Leviatan
Spline-type Spaces in Gabor Analysis 100
Hans G. Feichtinger
Spectrum of Transition, Subdivision and Multiscale Operators 123
Xiaojie Gao, S. L. Lee, and Qiyu Sun
Biorthogonal Refinable Functions and Wavelets from Spaces
Generalising Splines 139
T. N. T. Goodman
The Initial Functions in a Cascade Algorithm 154
Bin Han
On the Self-affine Sets and the Scaling Functions 179
Xing-Gang He, Ka-Sing Lau, and Hui Rao
Cascade Algorithms in Wavelet Analysis 196
Rong-Qing Jia
vii
VIII
Methods for Constructing Nonseparable Compactly Supported
Orthonormal Wavelets 231
Ming-Jun Lai
On Some Quantum and Analytical Properties of Fractional
Fourier Transforms 252
Jianhong Shen
Block Tridiagonal Matrices and the Kalman Filter 266
Gilbert Strang
A Special Class of Wavelet Frame Functions 281
Wenchang Sun, Deyun Yang, and Xingwei Zhou
Advances in Wavelet Algorithms and Applications 289
Mladen Victor Wickerhauser
NON-UNIFORM SAMPLING: EXACT RECONSTRUCTION
FROM NON-UNIFORMLY DISTRIBUTED
WEIGHTED-AVERAGES
AKRAM ALDROUBI
Department of Mathematics
Vanderbilt University
Nashville, TN 37240, USA
E-mail: [email protected]
HANS G. FEICHTINGER
Department of Mathematics
University of Vienna
Strudlhofg. 4, A-1090, Vienna, AUSTRIA
E-mail: [email protected]
In this article, we discuss the problem of reconstructing a function / in a lattice-
invariant subspace of L£ (IRd) from a family of non-uniformly distributed weighted-
averages {(ftipxA '• j £ J} using an approximation-projection iterative algorithm.
1. Introduction
The central topic of this article is a summary of some sufficient conditions
under which a function / satisfying some a priori conditions, expressed in terms
of certain function spaces, can be completely reconstructed from a collection
of local averages of the form {(/,i]} ) = J f(x)ip (x)dx : j € J}. When
Xj Xj
{ipx ••>'• j S J} are Dirac delta distributions, the data are the exact sample
values of / at the sampling points Xj, and the problem has a long history
already, under various assumptions, such as band-limitedness, or membership
of / in some spline-type spaces.
However, in applications, one needs to consider the case when {ip : j £ J}
Xj
are not Dirac delta distributions but functions that reflect the characteristic
of the more general sampling devices. For this case, the set {(/, ip ) • j € J}
Xj
consists of weighted-average sample values of /. Another case that includes
the particular situation where the data are of the form {f(xj) : j € J} is
when {tpx. = fi. : j € J} is a set of non-negative bounded measures with
x
compact support. For example, grouped data are preferred over exact sampling
l
2
values due to the fact that they have a better signal-to-noise ratio (cf. [19] for
statistical background in this direction).
Obviously, to reconstruct a function from the discrete data, we have to make
some general assumptions about the function /. Clearly these assumptions
must be sufficiently flexible to accommodate a large number of possible models
for /. Also, the standard assumption of band-limitedness should be a special
case or a limit case of the model spaces. It turns out that lattice-invariant
spaces are a sufficiently large family of possible model spaces which appear to
be adequate for our problem. They can be described as:
Vfffl = I E c^(- -Lk) • ceA , (l)
where <j> is a suitable generator, L is a d x d non-singular matrix, 1 < p < oo,
and v is a weight that controls the growth or decay of the functions in the
space V^{(j>). The matrix L transforms the lattice Zd to the lattice A, and
when L is the identity matrix, we obtain the standard shift-invariant spaces.
That a combination of atoms <j> (e.g. radial symmetric ones) with suitable
lattices A (related to sphere packing) is a good alternative for the usual voxel
representation of volume data has been observed also in another context ([18]).
As a matter of fact such spaces can also be considered over LCA groups ([11]),
where applications to the theory of Gabor multipliers are given.
Let us therefore come back to our main problem: generally we are given
a set of data {(f,ip ) '• j G J} and only an approximate information about
Xi
the function /, e.g. we may know that / is continuous, and we may know
the generator <f> approximately, but we may not know the value of p or the
rate of growth or decay of the function /. Moreover, even if we had the exact
information about the model, the data are typically corrupted by noise. Thus
our goal is to describe algorithms with the following properties:
(1) In the ideal case, i.e., when the model for / is exact and the data are
not corrupted by noise, the algorithm must reconstruct the function /
exactly, and must do so "fast".
(2) In the non-ideal case, e.g. when our information about / is partial or
when the data are corrupted by noise, the algorithm must converge and
must be able to "guess" the missing information.
In this paper we describe a family of Approximation-Projection algorithms
(AP algorithms) for recovering the function / 6 V^{(f>) exactly from the family
of weighted-averages, i.e., {(/, ip ) : j G J}. Such AP algorithms perform ex
Xi
act reconstruction, as long as the data are obtained from a function within the