1 - 10 of 13 Chapters
[In this chapter, we introduce a number of concepts, focusing on those which are common to most of the following chapters. Many more concepts will be introduced later on, to support the formalization of advanced techniques for shape analysis.]
[In this chapter, we focus on a specific type of shape: surfaces. Surfaces play a fundamental role in applications as they are used to model boundaries of 3D objects in computer graphics. We will draw our attention to surfaces with a geometric perspective: geometry is one of the oldest fields in...
[The decomposition of a shape into simpler pieces is the ultimate goal of most mathematical methods that aim at combining synthesis with saliency. This is the peculiar characteristic of spectral methods: the decomposition and expression of a complex function into a set of simpler ones. Probably,...
[This chapter deals with shape transformations: a transformation, or map, is any function ϕ mapping a set X to another set Y (or to the set X itself). The simplest examples are Euclidean transformations: rotation, translation, scaling. A more elaborate question concerns the effect of maps on...
[Algebraic topology studies both topology spaces and functions through algebraic entities, such as groups or homomorphisms, by analyzing the representations (formally known as functors) that transform a topological problem into an algebraic one, with the aim of simplifying it. Then, the focus of...
[Morse theory can be seen as the investigation of the relation between functions defined on a manifold and the shape of the manifold itself. The key feature in Morse theory is that information on the topology of the manifold is derived from the information about the critical points of real...
[Reeb graphs encode the evolution and the arrangement of the level sets of a real function defined on a shape. They were first defined by a French mathematician, Georges Reeb, in 1946  as topological constructs. In recent years, Reeb graphs have become popular in computer graphics as tools...
[The intuition behind Morse and Morse-Smale complexes was given by Maxwell :
“Hence each point of the earth’s surface has a line of slope, which begins at a certain summit and ends in a certain bottom. Districts whose lines of slope run to the same bottom are called basins or dales. Those...
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