Shapes and diffeomorphisms / Laurent Younes

This book is an attempt at providing a description of a large range of methods used to represent, detect and compare shapes (more generally, deformable objects) together with the mathematical background that they require.

Chapters 1–4 contain shape representation methods, including classical aspects of the differential geometry of curves and surfaces, aspects from other fields that have positively impacted the analysis of shape in practical applications, like invariant moments or medial axes. Discretization issues involved in the representation of these geometric objects are also discussed.

In Chapters 5–7, the author studies curve and surface evolution algorithms and how they relate to segmentation methods used to extract shapes from images, using active contours or deformable templates. In Chapters 8–9, basic concepts related to diffeomorphisms are introduced, discussing how to use ordinary differential equations associated to vector fields belonging to reproducing kernel Hilbert spaces to provide a computationally convenient framework.

Chapters 10–11 focus on the registration of deformable objects using diffeomorphism methods that optimize a matching functional combined with a regularization term that penalizes the distance of a diffeomorphism to the identity within the group.

The last two chapters (12 and 13) discuss metric aspects of shape analysis with focus on the relation between distances and group actions (global and infinitesimal points of view are presented). Kendall’s classic metric over configurations of labeled points is included, as well as a discussion of Riemannian metrics on plane curves. The final chapter is a presentation of the theory of metamorphosis.

In the appendices the author presents fundamental concepts needed to understand the rest of the book: elements of Hilbert space theory, differential and Riemannian geometry, ordinary differential equations, optimization algorithms, principal component analysis, and dynamic programming.

This book on applied mathematics is of interest to engineers and computer scientists having direct applications in the computerized analysis of medical images. This theory will as well lead to other interesting applications in the future. (zbMath)