Variational Methods in Image Segmentation - Jean-Michel Morel

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Résumé :
This book contains both a synthesis and mathematical analysis of a wide set of algorithms and theories whose aim is the automatic segmen? tation of digital images as well as the understanding of visual perception. A common formalism for these theories and algorithms is obtained in a variational form. Thank to this formalization, mathematical questions about the soundness of algorithms can be raised and answered. Perception theory has to deal with the complex interaction between regions and edges (or boundaries) in an image: in the variational seg? mentation energies, edge terms compete with region terms in a way which is supposed to impose regularity on both regions and boundaries. This fact was an experimental guess in perception phenomenology and computer vision until it was proposed as a mathematical conjecture by Mumford and Shah. The third part of the book presents a unified presentation of the evi? dences in favour of the conjecture. It is proved that the competition of one-dimensional and two-dimensional energy terms in a variational for? mulation cannot create fractal-like behaviour for the edges. The proof of regularity for the edges of a segmentation constantly involves con? cepts from geometric measure theory, which proves to be central in im? age processing theory. The second part of the book provides a fast and self-contained presentation of the classical theory of rectifiable sets (the edges) and unrectifiable sets (fractals).

Sommaire:
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....

Well-written and very user-friendly. Each chapter is introduced by a brief summary of its aims... A very readable introduction to the mathematical theory of image segmentation. --ZAA