About this Topic - Mathematical Tools for Shape Analysis and Description - page 6

1. ABOUT THIS TOPIC
• Professional scientists in industry who wish to understand, implement and use 3D shape
analysis techniques in their own specific field.
1.5 HOW THIS TOPIC IS ORGANIZED
In chapter 2 , shape analysis in a nutshell , we give an overview of shape analysis discussing problems
and application areas. We move then to the technical part, whose structure reflects our attempt
to synthesize the presentation into coherent groups of mathematical concepts, where the driv-
ing criteria is the perspective on shape analysis. When the shape is analyzed with a geometrical
point of view, we are interested in measuring properties, finding characterizations of features,
e.g., protrusions, possibly at different scales and resolutions. Geometry and topology provide
necessary background to build three kinds of descriptions (see chapter 3 ), which are elaborated
in chapter 4 where concepts of differential geometry are discussed together with examples of ap-
plications. Spectral methods applied to 3D meshes are introduced in chapter 5 to show how the
multi-resolution study of shapes may be addressed by the extraction and analysis of the eigen-
structures , as it is done for signal processing. From the analysis of a single shape, we move forward
to the analysis of pairs of shapes, by introducing concepts pertaining to the transformations be-
tween shapes in chapter 6 ; the chapter is rather theoretical and introduces a number of interesting
distances between spaces.
We skim through algebraic topology in chapter 7 to land on differential topology in chap-
ter 8 , where a consistent bunch of mathematics is presented. ese concepts are needed to un-
derstand a set of methodologies for analyzing shapes which are characterized by the usage of
real-valued functions to detect important features of the shape: Reeb graphs in chapter 9 , Morse
complexes in chapter 10 and topological persistence in chapter 11 .
In chapter 12 we introduce a number of concepts and methods where objects are not seen
solely as geometric shapes, but also as objects with colors or textures, objects with a functional
meaning, or objects that live in collections of many other models. We conclude the topic with
chapter 13 with an overview of existing resources in the shape analysis field.
In all chapters, besides chapter 3 , which is mostly theoretical and 13 , we have included
one or more sections, named concepts in action , which discusses applications of the mathematics
defined.
e background knowledge in mathematics that we rely on concerns:
• basic elements of analysis: e.g., functions, continuity, differentiability;
• basic elements of linear algebra: e.g., vectors, matrices, eigenvalues;
• basic elements of set theory and algebra: e.g., sets, groups.
References to specific textbooks where readers may find additional information on the
mathematics described are given in the chapters where they are expected to be more useful.