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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