Introduction
Over the last two decades, there has been a growing need in the medical image computing community for principled methods to process nonlinear geometric data. Typical examples of data in this domain include organ shapes and deformations resulting from segmentation and registration in computational anatomy, and symmetric positive definite matrices in diffusion imaging. In this context, Riemannian geometry has gradually been established as one the most powerful mathematical and computational paradigms. This book aims at being an introduction to and a reference on Riemannian geometric statistics and its use in medical image analysis for researchers and graduate students. The book provides both descriptions of the core methodology and presentations of state-of-the-art methods used in the field. We wish to present this combination of foundational material and current research together with examples, applications, and algorithms in a volume that is edited and authored by the leading researchers in the field. In addition, we wish to provide an overview of current research challenges and future applications. Beyond medical image computing, the methods described in this book may also apply to other domains such as signal processing, computer vision, geometric deep learning, and other domains where statistics on geometric features appear. As such, the presented core methodology takes its place in the field of geometric statistics, the statistical analysis of data being elements of nonlinear geometric spaces. We hope that both the foundational material and the advanced techniques presented in the later parts of the book can be useful in domains outside medical imaging and present important applications of geometric statistics methodology. Part 1 of this edited volume describes the foundations of Riemannian geometric computing methods for statistics on manifolds. The book here emphasizes concepts rather than proofs with the goal of providing graduate students in computer science the mathematical background needed to start in this domain. Chapter 1 presents an introduction to differential, Riemannian and Lie group geometry, and chapter 2 covers statistics on manifolds. Chapters 3–5 present introductions to geometry of SPD matrices, shape analysis through the action of the diffeomorphism group, and geometry and statistical analysis beyond the Riemannian setting when an affine connection, not a metric, is available. Part 2 includes contributions from leading researchers in the field on applications of statistics on manifolds and shape spaces in medical image computing. In chapter 6, Stephen Pizer, Steve Marron, and coauthors describe shape representation via skeletal models and how this allows application of nonlinear statistical methods on shape spaces. Chapter 7 by Rudrasis Chakraborty and Baba Vemuri concerns estimation of the iterative Riemannian barycenter, a candidate for the generalization of the Euclidean mean value on selected manifolds. In chapter 8, Aasa Feragen and Tom Nye discuss the statistics on stratified spaces that generalize manifold by allowing variation of the topological structure. Estimation of templates in quotient spaces is the topic of chapter 9 by Nina Miolane, Loic Devilliers, and Xavier Pennec. Stefan Sommer discusses parametric statistics on manifolds using stochastic processes in chapter 10. In chapter 11, Ruiyi Zhang and Anuj Srivastava consider shape analysis of functional data using elastic metrics. Part 3 of the book focuses on diffeomorphic deformations and their applications in shape analysis. Nicolas Charon, Benjamin Charlier, Joan Glaunès, Pierre Roussillon, and Pietro Gori present currents, varifolds, and normal cycles for shape comparison in chapter 12. Numerical aspects of large deformation registration is discussed in chapter 13 by Thomas Polzin, Marc Niethammer, François-Xavier Vialard, and Jan Modersitzki. Francois-Xavier and Laurent Risser present spatially varying metrics for large deformation matching in chapter 14. Chapter 15 by Miaomiao Zhang, Polina Golland, William M. Wells, and Tom Fletcher presents a framework for low-dimensional representations of large deformations and its use in shape analysis. Finally, in chapter 16, Martin Bauer, Sarang Joshi, and Klas Modin study densities matching in the diffeomorphic setting. We are extremely grateful for this broad set of excellent contributions to the book by leading researchers in the field, and we hope that the book in its entirety will inspire new developments and research directions in this exciting intersection between applied mathematics and computer science. The editors February, 2019Introduction
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