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Biasotti S., Falcidieno B., Giorgi D., Spagnuolo M. Mathematical Tools for Shape Analysis and Description

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Biasotti S., Falcidieno B., Giorgi D., Spagnuolo M. Mathematical Tools for Shape Analysis and Description
Morgan and Claypool, 2014. — 138 p. — (Synthesis Lectures in Computer Graphics and Animation). — ISBN-13 978-1627053631.
This book is a guide for researchers and practitioners to the new frontiers of 3D shape analysis and the complex mathematical tools most methods rely on. The target reader includes students, researchers and professionals with an undergraduate mathematics background, who wish to understand the mathematics behind shape analysis.
The authors begin with a quick review of basic concepts in geometry, topology, differential geometry, and proceed to advanced notions of algebraic topology, always keeping an eye on the application of the theory, through examples of shape analysis methods such as 3D segmentation, correspondence, and retrieval.
A number of research solutions in the field come from advances in pure and applied mathematics, as well as from the re-reading of classical theories and their adaptation to the discrete setting. In a world where disciplines (fortunately) have blurred boundaries, the authors believe that this guide will help to bridge the distance between theory and practice.
About this Book
Shape and Shape Analysis
Why Math for 3D Shape Analysis?
What this Book is and What it is Not
Expected Readers
How this Book is Organized .
3D Shape Analysis in a Nutshell
3D Shape Analysis: Problems and Solutions
Applications
Geometry, Topology, and Shape Representation
Metric and Metric Spaces
Geodesic Distance
Topological Spaces
Continuous and Smooth Functions Between Topological Spaces
Manifolds
Charts
Smooth Manifold
Orientability
Tangent Space
Riemannian Manifold
Differential Geometry and Shape Analysis
Geodesic Distances on Surfaces
Curvature on Surfaces
Spectral Methods for Shape Analysis
Laplace Operators
Heat Equation
Maps and Distances between Spaces
Space Transformations
Distances between Spaces
Algebraic Topology and Topology Invariants
Cell Decompositions
Homology
Differential Topology and Shape Analysis
Critical Points and Morse Functions
Topological Analysis rough (Lower) Level Sets
Homology of Manifolds
Reeb Graphs
Reeb Graph Definition
Reeb Graphs on 2- and 3-Manifolds
Concepts in Action .
Morse and Morse-Smale Complexes
Basic Concepts
Concepts in Action
Topological Persistence
Basic Concepts
Persistence Diagrams
Persistence Spaces
Concepts in Action
Beyond Geometry and Topology
3D Textured Shape Retrieval
Qualitative Organization of Collections of 3D Models
Recognition of Functional Parts of Man-made Objects
Resources
Software
3D Datasets and Benchmarks
Bibliography
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