Variational and Level Set Methods in Image Segmentation
Image segmentation consists of dividing an image domain into disjoint regions according to a characterization of the image within or in-between the regions. Therefore, segmenting an image is to divide its domain into relevant components. The efficient sol
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Amar Mitiche ยท Ismail Ben Ayed
Variational and Level Set Methods in Image Segmentation
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Prof. Amar Mitiche Institut National de la Recherche Scientifique (INRS) 800, de la Gaucheti`ere Ouest Montreal, Quebec, Canada [email protected]
Dr. Ismail Ben Ayed Institut National de la Recherche Scientifique (INRS) 800, de la Gaucheti`ere Ouest Montreal, Quebec, Canada [email protected]
ISSN 1866-2609 e-ISSN 1866-2617 ISBN 978-3-642-15351-8 e-ISBN 978-3-642-15352-5 DOI 10.1007/978-3-642-15352-5 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010935288 c Springer-Verlag Berlin Heidelberg 2010 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Contents
1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2
INTRODUCTORY BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Euler-Lagrange equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Variable domain of integration . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Descent methods for unconstrained optimization . . . . . . . . . . . . . . . . 2.2.1 Real functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Integral functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Level sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Optical flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 The gradient equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 The Horn and Schunck formulation . . . . . . . . . . . . . . . . . . . . . 2.4.3 The Aubert, Kornprobst, and Deriche formulation . . . . . . . . . 2.4.4 Optical flow of
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