Multivariate Approximation and Applications

296 Pages·2001·11.09 MB·English

by  N. Dyn

Multivariate Approximation
and Applications
Edited by
N. DYN
Tel Aviv University
D. LEVIATAN
Tel Aviv University
D. LEVIN
Tel Aviv University
A. PINKUS
Technion-Israel Institute of Technology
1 CAMBRIDGE
UNIVERSITY PRESS
PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
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CAMBRIDGE UNIVERSITY PRESS
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© Cambridge University Press 2001
This book is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.
First published 2001
Printed in the United Kingdom at the University Press, Cambridge
Typeface Computer Modern 10/12pt System IATgX [UPH]
A catalogue record for this book is available from the British Library
Library of Congress Cataloguing in Publication data
ISBN 0 521 80023 4 hardback
Contents
List of contributors Va9e v
Preface vii
1 Characterization and construction of radial basis functions
R. Schaback and H. Wendland 1
2 Approximation and interpolation with radial functions
M.D. Buhmann 25
3 Representing and analyzing scattered data on spheres
H.N. Mhaskar, F.J. Narcowich and J.D. Ward 44
4 A survey on .^-approximation orders from shift-invariant
spaces K. Jetter and G. Plonka 73
5 Introduction to shift-invariant spaces. Linear independence
A. Ron 112
6 Theory and algorithms for nonuniform spline wavelets
T. Lyche, K. M0rken and E. Quak 152
7 Applied and computational aspects of nonlinear wavelet
approximation A. Cohen 188
8 Subdivision, multiresolution and the construction of
scalable algorithms in computer graphics P. Schroder 213
9 Mathematical methods in reverse engineering J. Hoschek 252
Index 285
m

Contributors
M.D. Buhmann
Mathematical Institute, Justus-Liebig University, 35392 Gieflen, Germany
email: Martin. Buhmann8@inath. uni-giessen. de
A. Cohen
Laboratoire a"Analyse Numerique, Universite Pierre et Marie Curie, Paris, Prance
email: [email protected]
J. Hoschek
Department of Mathematics, Darmstadt University of Technology, 64289 Darm-
stadt, Germany
email: hoschek88mathematik.tu-darmstadt.de
K. Jetter
Institut fur Angewandte Mathematik und Statistik, Universitat Hohenheim, 70593
Stuttgart, Germany
email: kjetter89uni-hohenheim.de
T. Lyche
Department of Informatics, University of Oslo, P.O. Box 1080 Blindern, 0316 Oslo,
Norway
email: tom98ifi.uio.no
H.N. Mhaskar
Department of Mathematics, California State University, Los Angeles, CA 90032,
USA
email: hmhaska88calstatela.edu
K. M0rken
Department of Informatics, University of Oslo, P.O. Box 1080 Blindern, 0316 Oslo,
Norway
email: knutm88ifi.uio.no
F.J. Narcowich
Department of Mathematics, Texas A&M University, College Station, TX 77843,
USA
email: fnarc88math.tamu.edu
vi Contributors
G. Plonka
Fachbereich Mathematik, Universitdt Duisburg, 47048 Duisburg, Germany
email: plonka@@math. uni-duisburg. de
E. Quak
SINTEF Applied Mathematics, P.O. Box 124 Blindern, 0314 Oslo, Norway
email: Ewald. QuakQSmath. s intef . no
A. Ron
Computer Sciences Department, 1210 West Dayton, University of Wisconsin-
Madison, Madison, WI 57311, USA
email: [email protected]
R. Schaback
Institut fur Numerische und Angewandte Mathematik, Universitat Gottingen,
Lotzestrafie 16-18, 37083 Gottingen, Germany
email: schabackQQmath. uni-goett ingen. de
P. Schroder
Department of Computer Science, California Institute of Technology, Pasadena,
CA 91125, USA
email: psS9cs.caltech.ed
J.D. Ward
Department of Mathematics, Texas A&M University, College Station, TX 77843,
USA
email: [email protected]
H. Wendland
Institut fur Numerische und Angewandte Mathematik, Universitat Gottingen,
Lotzestrafie 16-18, 37083 Gottingen, Germany
email: wendland99math.uni-goettingen.de
Preface
Multivariate approximation theory is today an increasingly active research
area. It deals with a multitude of problems in areas such as wavelets, multi-
dimensional splines, and radial-basis functions, and applies them, for exam-
ple, to problems in computer aided geometric design, geometric modeling,
geodesic applications and image analysis. The field is both fascinating and
intellectually stimulating since much of the mathematics of the classical
univariate theory does not straightforwardly generalize to the multivariate
setting which models many real-world problems; so new tools have had to
be, and must continue to be, developed.
This advanced introduction to multivariate approximation and related
topics consists of nine chapters written by leading experts that survey many
of the new ideas and tools and their applications. Each chapter introduces a
particular topic, takes the reader to the forefront of research and ends with
a comprehensive list of references.
This book will serve as an ideal introduction for researchers and graduate
students who wish to learn about the subject and see how it may be applied.
A more detailed description of each chapter follows:
Chapter 1: Characterization and construction of radial basis functions, by
R. Schaback (Gottingen) and H. Wendland (Gottingen)
This chapter introduces characterizations of (conditional) positive defi-
niteness and shows how they apply to the theory of radial basis functions.
Complete proofs of the (conditional) positive definiteness of practically all
relevant basis functions are provided. Furthermore, it is shown how some of
these characterizations may lead to construction tools for positive definite
functions. Finally, a new construction technique is given which is based on
discrete methods which leads to non-radial, even non-translation invariant,
local basis functions.
viii Preface
Chapter 2: Approximation and interpolation with radial Junctions, by
M.D. Buhmann (Giessen)
This chapter provides a short, up-to-date survey of some of the recent
developments in the research of radial basis functions. Among these new
developments are results on convergence rates of interpolation with radial
basis functions, recent contributions concerning approximation on spheres,
and computations of interpolants with Krylov space methods.
Chapter 3: Representing and analyzing scattered data on spheres, by
H.N. Mhaskar (California State Univ. at Los Angeles), F.J. Narcowich (Texas
A & M) and J.D. Ward (Texas A k M)
Geophysical or meteorological data collected over the surface of the earth
via satellites or ground stations will invariably come from scattered sites.
There are two extremes in the problems one faces when handling such data.
The first is representing sparse data by fitting a surface to it. The second is
analyzing dense data to extract features of interest. In this chapter various
aspects of fitting surfaces to scattered data are reviewed. Analyzing data is a
more recent problem that is currently being addressed via various spherical
wavelet schemes, which are discussed along with multilevel schemes. Finally
quadrature methods, which arise in many of the wavelet schemes as well as
some interpolation methods, are touched upon.
Chapter 4: A survey on L2-approximation orders from shift-invariant
spaces, by K. Jetter (Hohenheim) and G. Plonka (Duisburg)
The aim of this chapter is to provide a self-contained introduction to
notions and results connected with the /^-approximation order of finitely
generated shift-invariant spaces. Special attention is given to the principal
shift-invariant case, where the shift-invariant space is generated from the
multi-integer translates of a single generator. This case is of special interest
because of its possible applications in wavelet methods. The general finitely
generated shift-invariant space case is considered subject to a stability con-
dition being satisfied, and the recent results on so-called superfunctions are
developed. For the case of a refinable system of generators, the sum rules
for the matrix mask and the zero condition for the mask symbol, as well as
invariance properties of the associated subdivision and transfer operators,
are discussed.
Chapter 5: Introduction to shift-invariant spaces. Linear independence,
by A. Ron (Madison)
Shift-invariant spaces play an increasingly important role in various areas
of mathematical analysis and its applications. They appear either implicitly
or explicitly in studies of wavelets, splines, radial basis function approxima-
tion, regular sampling, Gabor systems, uniform subdivision schemes, and
Preface ix
perhaps in some other areas. One must keep in mind, however, that the
shift-invariant system explored in one of the above-mentioned areas might
be very different from those investigated in others. The theory of shift-
invariant spaces attempts to provide a uniform platform for all these dif-
ferent investigations. The two main pillars of that theory are the study of
the approximation properties of shift-invariant spaces, and the study of gen-
erating sets for such spaces. Chapter 4 had already provided an excellent
up-to-date account of the first topic. The present chapter is devoted to the
second topic, and its goal is to provide the reader with an easy and friendly
introduction to the basic principles of that topic. The core of the presenta-
tion is devoted to the study of local principal shift-invariant spaces, while
the more general cases are treated as extensions of that basic setup.
Chapter 6: Theory and algorithms for nonuniform spline wavelets, by
T. Lyche (Oslo), K. M0rken (Oslo), E. Quak (Oslo)
This chapter discusses mutually orthogonal spline wavelet spaces on non-
uniform partitions of a bounded interval, addressing the existence, unique-
ness and construction of bases of minimally supported spline wavelets. The
relevant algorithms for decomposition and reconstruction are considered as
well as some questions related to stability. In addition, a brief review is
given of the bivariate case for tensor products and arbitrary triangulations.
The chapter concludes with a discussion of some special cases.
Chapter 7: Applied and computational aspects of nonlinear wavelet app-
roximation, by A. Cohen (Paris)
Nonlinear approximation is recently being applied to computational ap-
plications such as data compression, statistical estimation and adaptive
schemes for partial differential and integral equations, especially through
the development of wavelet-based methods. The goal of this chapter is to
provide a short survey of nonlinear wavelet approximation from the per-
spective of these applications, as well as to highlight some remaining open
questions.
Chapter 8: Subdivision, multiresolution and the construction of scalable
algorithms in computer graphics, by P. Schroder (Caltech)
Multiresolution representations are a critical tool in addressing complexity
issues (time and memory) for the large scenes typically found in computer
graphics applications. Many of these techniques are based on classical sub-
division techniques and their generalizations. In this chapter we review two
exemplary applications from this area: multiresolution surface editing and
semi-regular remeshing. The former is directed towards building algorithms
which are fast enough for interactive manipulation of complex surfaces of
arbitrary topology. The latter is concerned with constructing smooth pa-
x Preface
rameterizations for arbitrary topology surfaces as they typically arise from
3D scanning techniques. Remeshing such surfaces then allows the use of
classical subdivision ideas. The particular focus here is on the practical
aspects of making the well-understood mathematical machinery applicable
and accessible to the very general settings encountered in practice.
Chapter 9: Mathematical methods in reverse engineering, by J. Hoschek
(Darmstadt)
In many areas of industrial applications it is desirable to create a computer
model of existing objects for which no such model is available. This pro-
cess is called reverse engineering. Reverse engineering typically starts with
digitizing an existing object. These discrete data must then be converted
into smooth surface models. This chapter provides a survey of the practi-
cal algorithms including triangulation, segmentation, feature lines detection,
B-spline approximation and trimming.
No book of this sort can be compiled without the help of many people.
First and foremost there are the authors. We thank each of them for their
efforts, and for their willingness to humor the whims of the editors. Secondly
we thank Diana Yellin, our T^jX expert, for bringing the various manuscripts
into a uniform format, and for her typing, retyping, retyping and patience.
N. Dyn
D. Leviatan
D. Levin
A. Pinkus