Description
Over the past 15 years, there has been a growing need in the medical image computing community for principled methods to process nonlinear geometric data. Riemannian geometry has emerged as one of the most powerful mathematical and computational frameworks for analyzing such data.Riemannian Geometric Statistics in Medical Image Analysis is a complete reference on statistics on Riemannian manifolds and more general nonlinear spaces with applications in medical image analysis. It provides an introduction to the core methodology followed by a presentation of state-of-the-art methods.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. 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 methodologyContent includes:- The foundations of Riemannian geometric methods for statistics on manifolds with emphasis on concepts rather than on proofs- Applications of statistics on manifolds and shape spaces in medical image computing- Diffeomorphic deformations and their applicationsAs the methods described apply to domains such as signal processing (radar signal processing and brain computer interaction), computer vision (object and face recognition), and other domains where statistics of geometric features appear, this book is suitable for researchers and graduate students in medical imaging, engineering and computer science.- A complete reference covering both the foundations and state-of-the-art methods- Edited and authored by leading researchers in the field- Contains theory, examples, applications, and algorithms- Gives an overview of current research challenges and future applications
Table of Contents
Part 1 Foundations of geometric statistics1. Introduction to differential and Riemannian geometry2. Statistics on manifolds3. Manifold-valued image processing with SPD matrices4. Riemannian geometry on shapes and diffeomorphisms5. Beyond Riemannian geometryPart 2 Statistics on manifolds and shape spaces6. Object shape representation via skeletal models (s-reps) and statistical analysis7. Efficient recursive estimation of the Riemannian barycenter on the hypersphere and the special orthogonal group with applications8. Statistics on stratified spaces9. Bias on estimation in quotient space and correction methods10. Probabilistic approaches to geometric statistics11. On shape analysis of functional dataPart 3 Deformations, diffeomorphisms and their applications12. Fidelity metrics between curves and surfaces: currents, varifolds, and normal cycles13. A discretize–optimize approach for LDDMM registration14. Spatially adaptive metrics for diffeomorphic image matching in LDDMM15. Low-dimensional shape analysis in the space of diffeomorphisms16. Diffeomorphic density registration
