Synopsis:
Statistical analysis of shapes of 3D objects is an important problem with a wide range of applications. This analysis is difficult for many reasons, including the fact that objects differ in both geometry and topology. In this manuscript, we narrow the problem by focusing on objects with fixed topology, say objects that are diffeomorphic to unit spheres, and develop tools for analyzing their geometries. The main challenges in this problem are to register points across objects and to perform analysis while being invariant to certain shape-preserving transformations.
We develop a comprehensive framework for analyzing shapes of spherical objects, i.e., objects that are embeddings of a unit sphere in ℝ, including tools for: quantifying shape differences, optimally deforming shapes into each other, summarizing shape samples, extracting principal modes of shape variability, and modeling shape variability associated with populations. An important strength of this framework is that it is elastic: it performs alignment, registration, and comparison in a single unified framework, while being invariant to shape-preserving transformations.
The approach is essentially Riemannian in the following sense. We specify natural mathematical representations of surfaces of interest, and impose Riemannian metrics that are invariant to the actions of the shape-preserving transformations. In particular, they are invariant to reparameterizations of surfaces. While these metrics are too complicated to allow broad usage in practical applications, we introduce a novel representation, termed square-root normal fields (SRNFs), that transform a particular invariant elastic metric into the standard L² metric. As a result, one can use standard techniques from functional data analysis for registering, comparing, and summarizing shapes. Specifically, this results in: pairwise registration of surfaces; computation of geodesic paths encoding optimal deformations; computation of Karcher means and covariances under the shape metric; tangent Principal Component Analysis (PCA) and extraction of dominant modes of variability; and finally, modeling of shape variability using wrapped normal densities.
These ideas are demonstrated using two case studies: the analysis of surfaces denoting human bodies in terms of shape and pose variability; and the clustering and classification of the shapes of subcortical brain structures for use in medical diagnosis.
This book develops these ideas without assuming advanced knowledge in differential geometry and statistics. We summarize some basic tools from differential geometry in the appendices, and introduce additional concepts and terminology as needed in the individual chapters.
Ian H. Jermyn received a B.A. Honours degree (First Class) in Physics from Oxford University, and a Ph.D. in Theoretical Physics from the University of Manchester, UK. After working as a postdoc at the International Centre for Theoretical Physics in Trieste, Italy, he studied for and received a Ph.D. in Computer Vision from the Computer Science department of the Courant Institute of Mathematical Sciences at New York University. He then joined the Ariana research group at INRIA Sophia Antipolis, France, first as a postdoctoral researcher, and then as a Senior Research Scientist. Since September 2010, he has been Associate Professor (Reader) in Statistics in the Department of Mathematical Sciences at Durham University. His research concerns statistical geometry: the statistical modeling of shape and geometric structure, particularly using random fields with complex interactions and Riemannian geometry. This work is motivated by problems of shape and texture modelling in image processing, computer vision, and computer graphics. Using a Bayesian approach, it has been extensively applied to different types of images, including biological and remote sensing imagery. He is also interested in information geometry as applied to inference. Sebastian Kurtek is currently an Assistant Professor in the Department of Statistics at The Ohio State University, which he joined in 2012. He received a B.S. degree in Mathematics from Tulane University in 2007, and M.S. and Ph.D. degrees in Biostatistics from Florida State University in 2009 and 2012, respectively. His main research interests include statistical shape analysis, functional data analysis, statistical image analysis, statistics on manifolds, medical imaging, and computational statistics. In particular, he is interested in the interplay between statistics and Riemannian geometry, and their role in developing solutions to various applied problems. He is a member of the American Statistical Association, Institute of Mathematical Statistics, and the IEEE. Hamid Laga received his Ph.D. degree in Computer Science from Tokyo Institute of Technology in 2006. He is currently an Associate Professor at Murdoch University (Australia) and an Adjunct Associate Professor with the Phenomics and Bioinformatics Research Centre (PBRC) of the University of South Australia (UniSA). His research interests span various fields of computer vision, computer graphics, and image processing, with a special focus on the 3D acquisition, modeling, and analysis of the shape of static and deformable 3D objects. Anuj Srivastava is a Professor of Statistics and a Distinguished Research Professor at the Florida State University. He obtained his Ph.D. degree in Electrical Engineering from Washington University in St. Louis in 1996 and was a visiting research associate at the Division of Applied Mathematics at Brown University during 1996-1997. He joined the Department of Statistics at the Florida State University in 1997 as an Assistant Professor, and was promoted to full Professor in 2007. He has held visiting positions at INRIA, France, University of Lille, France, and Durham University, UK. His areas of research interest include statistics on nonlinear manifolds, statistical image understanding, functional data analysis, and statistical shape theory. He has published more than 200 papers in refereed journals and proceedings of refereed international conferences. He has been an associate editor for leading journals in computer vision and image processing, including IEEE PAMI, IEEE TIP, JMIV, and CVIU. He is a fellow of IEEE, IAPR, and ASA. G rard Medioni received the Dipl me d'Ingenieur in Information at The cole Nationale Sup rieure es T l communications, in 1977, and the M.S. and Ph.D. degrees in Computer Science from the University of Southern California, in 1980 and 1983, respectively. He has been at USC since then, and is currently Professor of Computer Science and Electrical Engineering, co-director of the Institute for Robotics and Intelligent Systems (IRIS), and co-director of the USC Games Institute. He served as Chairman of the Computer Science Department from 2001 to 2007. Prior to this, he was President and CEO of I.C. Vision, in Los Angeles, California, and held positions of Associate Professor, from 1992-1999, Assistant Professor, from 1987-1992, and Research Assistant Professor, from 1983-1987, at the Departments of Computer Science and Electrical Engineering, at the University of Southern California. From 1979-1983, he was a Research Assistant in the Intelligent Systems Group at the University of Southern California. Prior to his academic career, he was a research engineer at Underwater Signal Processing Division at Thomson-CSF, in Cagnes sur Mer, France. From 2000 to 2001, while on sabbatical leave, he was Chief Technical Officer at Geometrix, Inc. in San Jose, California. Professor Medioni has made significant contributions to the field of computer vision. His research covers a broad spectrum of the field, such as edge detection, stereo and motion analysis, shape inference and description, and system integration. He has published 3 books, over 50 journal papers and 150 conference articles, and is the recipient of 8 international patents. Prof. Medioni is associate editor of the Image and Vision Computing Journal, associate editor of the Pattern Recognition and Image Analysis Journal, and associate editor of the International Journal of Image and Video Processing. Prof. Medioni served as program co-chair of the 1991 IEEE CVPR Conference in Hawaii, of the 1995 IEEE Symposium on Computer Vision in Miami, general co-chair of the1997 IEEE CVPR Conference in Puerto Rico, conference co-chair of the 1998 ICPR Conference in Australia, general co-chair of the 2001 IEEE CVPR Conference in Kauai, general co-chair of the 2007 IEEE CVPR Conference in Minneapolis, and general co-chair of the upcoming 2009 IEEE CVPR Conference in Miami. He is a Fellow of IAPR, a Fellow of the IEEE, and a Fellow of AAAI. Sven Dickinson received the B.A.Sc. degree in Systems Design Engineering from the University of Waterloo in 1983, and the M.S. and Ph.D. degrees in Computer Science from the University of Maryland, in 1988 and 1991, respectively. He is currently Professor of Computer Science at the University of Toronto, where he serves as Acting Chair. Prior to that, he served as Departmental Vice Chair, from 2003-2006, and as Associate Professor, from 2000-2007. From 1995-2000, he was an Assistant Professor of Computer Science at Rutgers University, where he also held a joint appointment in the Rutgers Center for Cognitive Science (RuCCS) and membership in the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS). From 1994-1995, he was a Research Assistant Professor in the Rutgers Center for Cognitive Science, and from 1991-1994, a Research Associate at the Artificial Intelligence Laboratory, University of Toronto. He has held affiliations with the MIT Media Laboratory (Visiting Scientist, 1992-1994), the University of Toronto (Visiting Assistant Professor, 1994 1997), and the Computer Vision Laboratory of the Center for Automation Research at the University of Maryland (Assistant Research Scientist, 1993-1994, Visiting Assistant Professor, 1994 1997). Prior to his academic career, he worked in the computer vision industry, designing image processing systems for Grinnell Systems Inc., San Jose, CA, 1983-1984, and optical character recognition systems for DEST, Inc., Milpitas, CA, 1984-1985. His research interests revolve around the problem of object recognition, in general, and generic object recognition, in particular. He has explored a multitude of generic shape representations, and their common representation as hierarchical graphs has led to his interest in inexact graph indexing and matching. His interest in shape representation and matching has also led to his research in object tracking, vision-based navigation, content based image retrieval, and the use of language to guide perceptual grouping, object recognition, and motion analysis. One of the focal points of his research is the problem of image abstraction, which he believes is critical in bridging the representational gap between exemplar-based and generic object recognition. He has published over 100 papers on these topics in refereed journals, conferences, and edited collections. In 1996, he received the NSF CAREER award for his work in generic object recognition, and in 2002, received the Government of Ontario Premiere's Research Excellence Award (PREA), also for his work in generic object recognition. He was co-chair of the 1997, 1999, 2004, and 2007 IEEE International Workshops on Generic Object Recognition (or Object Categorization), co chaired the DIMACS Workshop on Graph Theoretic Methods in Computer Vision in 1999, and co-chaired the First International Workshop on Shape Perception in Human and Computer Vision in 2008. From 1998-2002, he served as Associate Editor of the IEEE Transactions on Pattern Analysis and Machine Intelligence, in which he also co-edited a special issue on graph algorithms and computer vision, which appeared in 2001. He currently serves as Associate Editor for the journals: International Journal of Computer Vision; Image and Vision Computing; Pattern Recognition Letters; IET Computer Vision; and the Journal of Electronic Imaging.
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