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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Parametrized Plane Curves

Abstract
This chapter introduces basic concepts on the geometry of plane curves, including arc length, curvature and change of parameter. It also contains a presentation of implicit and non-local representations, and discusses special cases of invariant parametrizations, relative to specific groups of transformations.
Laurent Younes

Chapter 2. The Medial Axis

Abstract
The medial axis (or skeleton) of a shape is the set of centers of discs of maximal radii inscribed in the shape. It provides a skeleton-like structure, which, when the associated maximal radii are also stored (providing the medial axis transform) uniquely encodes the shape geometry. This chapter described some fundamental properties of this transforms introduces some extensions.
Laurent Younes

Chapter 3. Local Properties of Surfaces

Abstract
This chapter introduces basic definitions and properties on the theory of surfaces and provides a fundamental background for the study of shapes in three dimensions. It presents several results on the local differential geometry of such objects, including metric properties and integration. Some of these elements are special cases of the abstract discussion of submanifolds in Appendix B, but are given here a more elementary presentation to ensure that this important material can be read independently.
Laurent Younes

Chapter 4. Computations on Triangulated Surfaces

Abstract
Triangulated surfaces provide a three-dimensional generalization of polygons in two dimensions. Surfaces are usually stored on computers in the form of triangulated surfaces, and these are the kinds of objects that must be handled in practical applications.
Laurent Younes

Chapter 5. Evolving Curves and Surfaces

Abstract
In this chapter, we discuss how curve or surface evolution can be formulated using partial differential equations, and discuss some applications in curve smoothing and image segmentation.
Laurent Younes

Chapter 6. Deformable Templates

Abstract
Deformable templates represent shapes as transformations of a given prototype, or template, shifting the modeling effort from the shape space itself to the space of transformations. The models that are presented in this chapter are linear versions of this approach, and can be seen as precursors of the nonlinear constructions, based on diffeomorphism groups,, that will be the main focus of the rest of this book.
Laurent Younes

Chapter 7. Ordinary Differential Equations and Groups of Diffeomorphisms

Abstract
This chapter begins our discussion of diffeomorphic shape analysis with the introduction of spaces of diffeomorphisms, notably those associated with flows of ordinary differential equations. This chapter relies heavily on concepts presented in Appendix C that the reader may want consult at this point.
Laurent Younes

Chapter 8. Building Admissible Spaces

Abstract
This chapter presents several explicit constructions of admissible spaces, focusing on Hilbert spaces. We will in particular introduce the notion of reproducing kernels associated to an admissible space, which will provide a powerful computational tool. Several important properties associated with such kernels are also discussed.
Laurent Younes

Chapter 9. Deformable Objects and Matching Functionals

Abstract
In the previous two chapters, we introduced and studied basic tools related to deformations and their mathematical representation using diffeomorphisms. In this chapter, we start investigating relations between deformations and the objects they affect, which we will call deformable objects, and discuss the variations of matching functionals, which are cost functions that measure the quality of the registration between two deformable objects.
Laurent Younes

Chapter 10. Diffeomorphic Matching

Abstract
This chapter discusses variational registration methods, minimizing registration costs with a guarantee that the optimal solution is a diffeomorphism. The main focus will be on methods optimizing over flows associated with differential equations, using concepts introduced in the previous chapters.
Laurent Younes

Chapter 11. Distances and Group Actions

Abstract
In this chapter we discuss metric comparisons between deformable objects and their relation to the registration methods that we have studied in the previous chapters. We will start with a discussion on general metric spaces, and follow with local considerations involving Riemannian geometry.
Laurent Younes

Chapter 12. Metamorphosis

Abstract
This chapter introduces metamorphoses, which refer to Riemannian metrics based on transformations in which objects can change under the action of diffeomorphisms (or other transformation groups) but also under independent variations, allowing for more flexibility in the evolution.
Laurent Younes

Chapter 13. Analyzing Shape Datasets

Abstract
We present in this chapter some “shape analysis” methods, among those that are mainly used in practice, where the goal is to provide a low-dimensional description and to perform statistical validations of hypotheses for datasets in which each object is a shape. Most recent applications of this framework have taken place in medical imaging, in which the shapes are provided by anatomical regions segmented by MRI or computer tomography scans. The analysis of the anatomy derived from such images is called computational anatomy and has generated a huge literature. Beside this important range of applications, shape analysis can also be used in computed vision, or in biology, which was, for example, the main focus of D’Arcy-Thompson’s seminal treatise on Growth and Form. We here focus on methods that derive from the analysis of diffeomorphisms developed in the previous chapters, leading to “morphometric”, or “diffeomorphometric” analyses.
Laurent Younes

Backmatter

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