Geometric Curve Evolution and Image Processing - Frédéric Cao

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Résumé :
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 at this point. This book is a rigorous and self-contained exposition of the techniques of motion by curvature. The approach is axiomatic and formulated in terms of geometric invariance with respect to the position of the observer. This is translated into mathematical terms, and the author develops the approach of Olver, Sapiro and Tannenbaum, which classifies all curve evolution equations. He then draws a complete parallel with another axiomatic approach using level-set methods: this leads to generalized curvature motions. Finally, novel, and very accurate, numerical schemes are proposed allowing one to compute the solution of highly degenerate evolution equations in a completely invariant way. The convergence of this scheme is also proved....

Biographie:
Jean-Michel Morel (PhD 1980) is Professor of Applied Mathematics at the Ecole Normale Sup?rieure de Cachan since 1997. He started his career in 1979 as assistant professor in Marseille Luminy, then moved in 1984 to University Paris Dauphine where he was promoted professor in 1992. His research is focused since 1990 on the mathematical analysis of image analysis and processing. He has received the Philip Morris Prize of Mathematics (1991), the Junior Institut Universitaire de France (1992), Prix Sciences et D?fense (1996), Classe exceptionnelle 2001 and three IEEE best paper awards (ICIP 97, ICASSP 05, and CVPR 05.) He has coauthored with S. Solimini a book on Variational Methods in Image Segmentation (Birkh?user 1994) and with Agn?s Desolneux and Lionel Moisan a book on Computation Gestalt Theory (Springer, 2008). He has advised 31 PhD's and is associate editor of five journals in applied mathematics and image analysis. He is also the editor in charge of Applied Mathematics for Springer Lecture Notes of Mathematics. Since 2002, Jean-Michel Morel belongs to the ISI list of highly cited mathematicians (http://isihighlycited.com/). Fr?d?ric Sur is assistant professor of mathematics and computer science at Ecole des Mines de Nancy and conducts his research at Loria laboratory. His research interest is computer vision, from low-level features definition to structure and motion problems. He defended his PhD thesis in applied mathematics and image analysis in 2004 at Ecole Normale Sup?rieure de Cachan. Then he held a postdoctoral position on statistical learning theory at Loria. Jos?-Luis Lisani received the Ph.D.degree on applied mathematics by the Universities of Paris-Dauphine (France) and of Illes Balears (Spain) in 2001. He is currently assistant professor of applied mathematics at the University of Illes Balears (Spain) and collaborates with the CMLA laboratory at ENS Cachan (France) and Cognitech Inc. (USA). His current researchinterests include image and video processing with applications to video-surveillance, database indexing and 3D images analysis. Pablo Mus? received the Ph.D. degree in applied mathematics from Ecole Normale Sup?rieure de Cachan, France, in 2004. From 2005 to 2006 he was with Cognitech, Inc., Pasadena, CA, USA, where he worked on computer vision and image processing applications. Then he held a postdoctoral position on applied mathematics at the Seismological Laboratory, California Institute of Technology. Since 2008, he has been an assistant professor of electrical engineering at the Facultad de Ingenier?a, Universidad de la Rep?blica, Uruguay. His research interests include signal and image processing and analysis. Fr?d?ric Cao is researcher in applied mathematics and computer vision, with INRIA, France. He obtained a PhD in 2000, then successively worked for the French Defence Agency, INRIA (National Institute of research in Computer Science), Cognitech Inc. He is currently with DxO Labs, as Research Director. His topics of research are shape recognition, motion analysis or digital photography. He is author of a book on geometric curve evolution and image processing, published as a volume of Lecture Notes in Mathematics, Springer....

Sommaire:
Preface.- Part I. The curve smoothing problem: 1. Curve evolution and image processing; 2. Rudimentary bases of curve geometry.- Part II. Theoretical curve evolution: 3. Geometric curve shortening flow; 4. Curve evolution and level sets.- Part III. Numerical curve evolution: 5. Classical numerical methods for curve evolution; 6. A geometrical scheme for curve evolution.- Conclusion and perspectives.- A. Proof of Thm. 4.3.4.- References.- Index.