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 L

^{2} 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.

### Table of Contents

Preface

Acknowledgments

Problem Introduction and Motivation

Elastic Shape Analysis: Metrics and Representations

Computing Geometrical Quantities

Statistical Analysis of Shapes

Case Studies Using Human Body and Anatomical Shapes

Landmark-driven Elastic Shape Analysis

Bibliography

Authors' Biographies

### About the Author(s)

**Ian H. Jermyn**, Durham University

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**, The Ohio State University

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**, Murdoch University and the University of South Australia

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**, Florida State University

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.