Shapes and Diffeomorphisms

Shapes are complex objects to apprehend, as mathematical entities, in terms also suitable for computerized analysis and interpretation. This volume provides the background that is required for this purpose, including different approaches that can be used.

This book covers mathematical foundations and methods for the computerized analysis of shapes, providing the requisite background in geometry and functional analysis and introducing various algorithms and approaches to shape modeling, with a special focus on the interesting connections between shapes and their transformations by diffeomorphisms. A direct application is to computational anatomy, for which techniques such as largedeformation diffeomorphic metric mapping and metamorphosis, among others, are presented. The appendices detail a series of classical topics (Hilbert spaces, differential equations, Riemannian manifolds, optimal control). The intended audience is applied mathematicians and mathematically inclined engineers interested in the topic of shape analysis and its possible applications in computer vision or medical imaging. The first part can be used for an advanced undergraduate course on differential geometry with a focus on applications while thelater chapters are suitable for a graduate course on shape analysis through the action of diffeomorphisms. Several significant additions appear in the 2nd edition, most notably a new chapter on shape datasets, and a discussion of optimal control theory in an infinite-dimensional framework, which is then used to enrich the presentation of diffeomorphic matching.

Suitable for an advanced undergraduate course in the differential geometry of curves and surfaces, featuring applications that are rarely treated in standard texts Provides a graduate-level theoretical background in shape analysis and connects it with algorithms and statistical methods Offers a unique presentation of diffeomorphic registration methods, which has no equivalent in the current literature

Auteur

A former student of the Ecole Normale Supérieure in Paris, Laurent Younes received his Ph.D. from the University Paris Sud in 1989. Now a professor in the Department of Applied Mathematics and Statistics at Johns Hopkins University (which he joined in 2003), he was a junior, then senior researcher at CNRS in France from 1991 to 2003. His research is in stochastic modeling for imaging and biology, shape analysis and computational anatomy. He is a core faculty member of the Center for Imaging Science and of the Institute for Computational Medicine at JHU.

Contenu
Preface to the 2nd Edition.- Preface to the 1st Edition.- Parametrized Plane Curves.- Medial Axis.- Local Properties of Surfaces.- Computations on Triangulated Surfaces- Evolving Curves and Surfaces.- Deformable templates.- Ordinary Differential Equations and Groups of Diffeomorphisms.- Building Admissible Spaces.- Deformable Objects and Matching Functionals.- Diffeomorphic Matching.- Distances and Group Actions.- Metamorphosis.- Analyzing Shape Datasets.- Appendices: Elements from Functional Analysis.- Elements from Differential Geometry.- Ordinary Differential Equations.- Introduction to Optimization and Optimal Control Theory. - Principal Component Analysis.- Dynamic Programming.- References.- Index.