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Shapes are complex objects to apprehend, as mathematical entities, in terms that also are suitable for computerized analysis and interpretation. This volume provides the background that is required for this purpose, including different approaches that can be used to model shapes, and algorithms that are available to analyze them. It explores, in particular, the interesting connections between shapes and the objects that naturally act on them, diffeomorphisms. The book is, as far as possible, self-contained, with an appendix that describes a series of classical topics in mathematics (Hilbert spaces, differential equations, Riemannian manifolds) and sections that represent the state of the art in the analysis of shapes and their deformations. A direct application of what is presented in the book is a branch of the computerized analysis of medical images, called computational anatomy.



1. Parametrized Plane Curves

We start with some definitions. Definition 1.1. A (parametrized plane) curve is a continuous mapping m : I → ℝ2, where I = [a, b] is an interval. The curve m is closed if m(a) = m(b). A curve m is a Jordan curve if it is closed and m has no self-intersection: m(x) = m(y) only for x = y or {x, y} = {a, b}. The curve is piecewise C 1 if m has everywhere left and right derivatives, which coincide except at a finite number of points. The range of a curve m is the set m([a, b]). It will be denoted R m. Notice that we have defined curves as functions over bounded intervals. Their range must therefore be a compact subset of ℝ2 (this forbids, in particular, curves with unbounded branches).
Laurent Younes

2. Medial Axis

The medial axis [24] transform associates a skeleton-like structure to a shape, which encodes its geometry. The medial axis itself (or skeleton) is the center of discs of maximal radii inscribed in the shape. The medial axis transform stores in addition the maximal radii.
Laurent Younes

3. Moment-Based Representation

Moments are global descriptors which can be used to represent shapes [117, 199, 177]. They are defined as integrals, either along the boundary of the shape, or over its interior. To simplify the discussion, we introduce a common notation for both cases. Let m be a non-intersecting closed curve, and Ω m its interior. Let f be a function defined on R2. We define the following moments.
Laurent Younes

4. Local Properties of Surfaces

In this chapter, we start discussing representations that can be associated to three-dimensional shapes, where surfaces now replace curves. For this reason, we start with basic definitions and results on the theory of surfaces in ℝ3. Although some parts are redundant with the abstract discussion of submanifolds that is provided in Appendix B, we have chosen to provide a more elementary presentation here, very close to [64], to ensure that this important section can be read independently.
Laurent Younes

5. Isocontours and Isosurfaces

In this chapter, we start discussing methods for the extraction of shapes (curves or surfaces) from discrete image data. Most of the methods we will discuss will proceed by energy minimization and be implemented using curve evolution equations. Such equations will be discussed in Chapter 6. In this first, short, chapter, we discuss what is probably the simplest option that is available to extract a curve or a surface from an image, which is to define it implicitly based on the image values.
Laurent Younes

6. Evolving Curves and Surfaces

In this chapter, we discuss how to represent curve or surface evolution using partial differential equations. This connects to fundamental mathematical results, some of them beyond the scope of this book, but also has important practical implications, especially when implementing optimization algorithms over curves and surfaces. One important example will be active contours, which will be described in the next chapter.
Laurent Younes

7. Deformable templates

Deformable templates represent shapes as deformations of a given prototype, or template. Describing a shape therefore requires providing the following information: (1) A description of the template. (2) A description of the relation between the shape and the template. This has multiple interesting aspects. The first one is that the template needs to be specified only once, for a whole family of curves. Describing the variation usually results in a simpler representation, typically involving a small number of parameters. The conciseness of the description is important for detection or tracking algorithms in which the shape is a variable, since it reduces the number of degrees of freedom. Another aspect is that small-dimensional representations are more easily amenable to probabilistic modeling, leading, as we will see, to interesting statistical shape models.
Laurent Younes

8. Ordinary Differential Equations and Groups of Diffeomorphisms

This chapter introduces spaces of diffeomorphisms, and describe how ordinary differential equations provide a convenient way of generating deformations.
Laurent Younes

9. Building Admissible Spaces

We have defined in the previous chapter a category of admissible spaces V that drive the construction of groups of diffeomorphisms as flows associated to ordinary differential equations with velocities in V. We now show how such spaces can be explicitly constructed, focusing on Hilbert spaces. This construction is fundamental, because it is intimately related to computational methods involving flows of diffeomorphisms. We will in particular introduce the notion of reproducing kernels associated to an admissible space, which will provide our main computational tool. We introduce this in the next section.
Laurent Younes

10. Deformable Objects and Matching Functionals

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

11. Diffeomorphic Matching

A standard way to ensure the existence of a smooth solution of a matching problem is to add a penalty term in the matching functional. This term would complete (10.1) to form
$$E_{I,I'} (\varphi ) = p(\varphi ) + D(\varphi.I,I').$$
A large variety of methods have been designed, in approximation theory, statistics or signal processing for solving ill-posed problems. The simplest (and typical) form of penalty function is
$$p(\varphi ) = ||\varphi - id||_H^2 $$
for some Hilbert (or Banach) space of functions.
Laurent Younes

12. Distances and Group Actions

We start, in this chapter, a description of how metric comparison between deformable objects can be performed, and how it interacts with the registration methods that we have studied in the previous chapters. We start with a general discussion on the interaction between distances on a set and transformation groups acting on it.
Laurent Younes

13. Metamorphosis

The infinitesimal version of the construction of Section 12.1.2 provides a metric based on transformations in which the objects can change under the action of diffeomorphisms but also under independent variations. We shall refer to such metrics as metamorphoses [150, 205, 206, 115]. They will result in formulations that enable both object registration and metric comparison.
Laurent Younes


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