Shapes and diffeomorphisms 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 large‒deformation 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 the later 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 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