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. Table of Contents: Acknowledgments / Figure Credits / About this Book / 3D Shape Analysis in a Nutshell / Geometry, Topology, and Shape Representation / Differential Geometry and Shape Analysis / Spectral Methods for Shape Analysis / Maps and Distances between Spaces / Algebraic Topology and Topology Invariants / Differential Topology and Shape Analysis / Reeb Graphs / Morse and Morse-Smale Complexes / Topological Persistence / Beyond Geometry and Topology / Resources / Bibliography / Authors' Biographies

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Silvia Biasotti is a researcher at CNR-IMATI. She graduated in Mathematics in 1998 and got a Ph.D. in Mathematics and Applications in 2004 and a Ph.D. in Information and Communi cation Technologies in 2008, all from the University of Genoa. She is co-author of more than 80 reviewed papers on geometric modeling, shape analysis and computational topology. Since 2005 she has been responsible for the CNR activity Topology and Homology for analyzing dig ital shapes, and teaches the Master course on methods of analysis of discrete surfaces and their applications at the Department of Mathematics, University of Genoa. Bianca Falcidieno is a research director of the National Research Council of Italy at CNR IMATI. She has been leading and coordinating research at international level in Computational Mathematics, Computer Graphics, 3D Media and Knowledge Technologies, strongly interact ing with industrial application fields. Author of more than 200 scientific reviewed papers and books, she was the editor in chief of the International Journal of Shape Modeling and guest-editor of several special issues. Since 2011 she is Fellow of the EUROGRAPHICS Association and for the 80th CNR anniversary she was included in the 12 top-level female researchers in the CNR history. Daniela Giorgi graduated cum laude in Mathematics at the University of Bologna (2002), and got a Ph.D. in Computational Mathematics from the University of Padova (2006). Since then she has been a researcher at the National Research Council of Italy. She has considerable expertise in mathematics (geometry and topology), and in image and 3D shape analysis. She has authored about 40 peer-reviewed publications on computational geometry and topology for shape analy sis, and has been participating in several international projects on related topics. She has been teaching BS and Master (Mathematics and its Applications) courses, and has been a lecturer at international schools Michela Spagnuolo got a Laurea Degree cum laude in Applied Mathematics from the University of Genova and a Ph.D. in Computer Science Engineering at INSA, Lyon. She is currently a research director at CNR-IMATI. She authored more than 130 reviewed papers, edited a book on 3D shape analysis, and was a guest-editor of several special issues. She is an associate editor of Computers & Graphics, and e Visual Computer. She is a member of the steering committee of Shape Modelling International, and of the EG workshops on 3D Object Retrieval. Since 2014 she is Fellow of the EUROGRAPHICS Association. Her interests include computational topology, shape analysis, shape similarity and matching