Geometric Curve Evolution and Image Processing
In image processing, "motions by curvature" provide an efficient way to smooth curves representing the boundaries of objects. In such a motion, each point of the curve moves, at any instant, with a normal velocity equal to a function of the curvature
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Fr´ed´eric Cao
Geometric Curve Evolution and Image Processing
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Author Fr´ed´eric Cao IRISA/ INRIA Campus Universitaire de Beaulieu 35042 Rennes Cedex France e-mail: [email protected] http://www.irisa.fr/prive/fcao
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Mathematics Subject Classification (2000): 35K65, 35D05, 53A04, 68U10 ISSN 0075-8434 ISBN 3-540-00402-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science + Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10904251
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Preface
These lectures intend to give a self-contained exposure of some techniques for computing the evolution of plane curves. The motions of interest are the so-called motions by curvature. They mean that, at any instant, each point of the curve moves with a normal velocity equal to a function of the curvature at this point. This kind of evolution is of some interest in differential geometry, for instance in the problem of minimal surfaces. The interest is not only theoretical since the motions by curvature appear in the modeling of various phenomena as crystal growth, flame propagation and interfaces between phases. More recently, these equations have also appeared in the young field of image processing where they provide an efficient way to smooth curves representing the contours of the objects. This smoothing is a necessary step for image analysis as soon as the analysis uses some local characteristics of the contours. Indeed, natural images are very noisy and differential features are unreliable if one is not careful before computing them. A solution consists in smoothing the curves to eliminate the small oscillations without changing the global shape of the contours
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