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 methodology
Content includes:
As 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.
1. Introduction - p. xvii 2. Part 1: Foundations of geometric statistics - p. xix 3. 1: Introduction to differential and Riemannian geometry - p. xix 4. 2: Statistics on manifolds - p. 37 5. 3: Manifold-valued image processing with SPD matrices - p. 74 6. 4: Riemannian geometry on shapes and diffeomorphisms - p. 134 7. 5: Beyond Riemannian geometry - p. 167 8. Part 2: Statistics on manifolds and shape spaces - p. 229 9. 6: Object shape representation via skeletal models (s-reps) and statistical analysis - p. 229 10. 7: Efficient recursive estimation of the Riemannian barycenter on the hypersphere and the special orthogonal group with applications - p. 271 11. 8: Statistics on stratified spaces - p. 297 12. 9: Bias on estimation in quotient space and correction methods - p. 342 13. 10: Probabilistic approaches to geometric statistics - p. 376 14. 11: On shape analysis of functional data - p. 416 15. Part 3: Deformations, diffeomorphisms and their applications - p. 438 16. 12: Fidelity metrics between curves and surfaces: currents, varifolds, and normal cycles - p. 438 17. 13: A discretize–optimize approach for LDDMM registration - p. 477 18. 14: Spatially adaptive metrics for diffeomorphic image matching in LDDMM - p. 532 19. 15: Low-dimensional shape analysis in the space of diffeomorphisms - p. 556 20. 16: Diffeomorphic density registration - p. 576 21. Index - p. 605
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 methodology
Content includes:
As 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.
1. Introduction - p. xvii 2. Part 1: Foundations of geometric statistics - p. xix 3. 1: Introduction to differential and Riemannian geometry - p. xix 4. 2: Statistics on manifolds - p. 37 5. 3: Manifold-valued image processing with SPD matrices - p. 74 6. 4: Riemannian geometry on shapes and diffeomorphisms - p. 134 7. 5: Beyond Riemannian geometry - p. 167 8. Part 2: Statistics on manifolds and shape spaces - p. 229 9. 6: Object shape representation via skeletal models (s-reps) and statistical analysis - p. 229 10. 7: Efficient recursive estimation of the Riemannian barycenter on the hypersphere and the special orthogonal group with applications - p. 271 11. 8: Statistics on stratified spaces - p. 297 12. 9: Bias on estimation in quotient space and correction methods - p. 342 13. 10: Probabilistic approaches to geometric statistics - p. 376 14. 11: On shape analysis of functional data - p. 416 15. Part 3: Deformations, diffeomorphisms and their applications - p. 438 16. 12: Fidelity metrics between curves and surfaces: currents, varifolds, and normal cycles - p. 438 17. 13: A discretize–optimize approach for LDDMM registration - p. 477 18. 14: Spatially adaptive metrics for diffeomorphic image matching in LDDMM - p. 532 19. 15: Low-dimensional shape analysis in the space of diffeomorphisms - p. 556 20. 16: Diffeomorphic density registration - p. 576 21. Index - p. 605
