Variation Reduction and Shape Preservation
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Nonlinear Smoothing and Multiresolution Analysis
Carl Rohwer
Birkhäuser Verlag Basel . Boston . Berlin
Author: Carl Rohwer Department of Mathematics University of Stellenbosch 7602 Stellenbosch South Africa
2000 Mathematics Subject Classification: Primary 00A69, 41A46, 42C40; Secondary 06, 47, 62, 65, 94
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de.
ISBN 3-7643-7229-X Birkhäuser Verlag, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2005 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced of chlorine-free pulp. TCF ∞ Printed in Germany ISBN-10: 3-7643-7229-X ISBN-13: 978-3-7643-7229-3 987654321
www.birkhauser.ch
Contents Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
1. Operators on Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2. Basic Rank Selectors, Pulses and Impulses . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3. LU LU -Smoothers, Signals and Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . .
21
4. LU LU -Intervals and Similar Smoothers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
5. Smoothing and Approximation with Signals . . . . . . . . . . . . . . . . . . . . . . . . .
43
6. Variation Reduction and Shape Preservation . . . . . . . . . . . . . . . . . . . . . . . .
51
7. Multiresolution Analysis of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
8. The Discrete Pulse Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
9. Fair Comparison with Linear Smoothers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
10. Interpretation and Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
127
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
131
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
135
Foreword This monograph is intended as a simple introduction to the so-call
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