Riemannian Geometric Statistics in Medical Image Analysis

Riemannian Geometric Statistics in Medical Image Analysis

by Xavier Pennec, Stefan Sommer, Tom Fletcher
Released September 2019
Publisher(s): Academic Press
ISBN: 9780128147269

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Book 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 methodology

Content 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 applications

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.

  • 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

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Contributors
  6. Introduction
    1. Introduction
  7. Part 1: Foundations of geometric statistics
    1. 1: Introduction to differential and Riemannian geometry
      1. Abstract
      2. 1.1. Introduction
      3. 1.2. Manifolds
      4. 1.3. Riemannian manifolds
      5. 1.4. Elements of analysis in Riemannian manifolds
      6. 1.5. Lie groups and homogeneous manifolds
      7. 1.6. Elements of computing on Riemannian manifolds
      8. 1.7. Examples
      9. 1.8. Additional references
      10. References
    2. 2: Statistics on manifolds
      1. Abstract
      2. 2.1. Introduction
      3. 2.2. The Fréchet mean
      4. 2.3. Covariance and principal geodesic analysis
      5. 2.4. Regression models
      6. 2.5. Probabilistic models
      7. References
    3. 3: Manifold-valued image processing with SPD matrices
      1. Abstract
      2. Acknowledgements
      3. 3.1. Introduction
      4. 3.2. Exponential, logarithm, and square root of SPD matrices
      5. 3.3. Affine-invariant metrics
      6. 3.4. Basic statistical operations on SPD matrices
      7. 3.5. Manifold-valued image processing
      8. 3.6. Other metrics on SPD matrices
      9. 3.7. Applications in diffusion tensor imaging (DTI)
      10. 3.8. Learning brain variability from Sulcal lines
      11. References
    4. 4: Riemannian geometry on shapes and diffeomorphisms
      1. Abstract
      2. 4.1. Introduction
      3. 4.2. Shapes and actions
      4. 4.3. The diffeomorphism group in shape analysis
      5. 4.4. Riemannian metrics on shape spaces
      6. 4.5. Shape spaces
      7. 4.6. Statistics in LDDMM
      8. 4.7. Outer and inner shape metrics
      9. 4.8. Further reading
      10. References
    5. 5: Beyond Riemannian geometry
      1. Abstract
      2. 5.1. Introduction
      3. 5.2. Affine connection spaces
      4. 5.3. Canonical connections on Lie groups
      5. 5.4. Left, right, and biinvariant Riemannian metrics on a Lie group
      6. 5.5. Statistics on Lie groups as symmetric spaces
      7. 5.6. The stationary velocity fields (SVF) framework for diffeomorphisms
      8. 5.7. Parallel transport of SVF deformations
      9. 5.8. Historical notes and additional references
      10. References
  8. Part 2: Statistics on manifolds and shape spaces
    1. 6: Object shape representation via skeletal models (s-reps) and statistical analysis
      1. Abstract
      2. Acknowledgements
      3. 6.1. Introduction to skeletal models
      4. 6.2. Computing an s-rep from an image or object boundary
      5. 6.3. Skeletal interpolation
      6. 6.4. Skeletal fitting
      7. 6.5. Correspondence
      8. 6.6. Skeletal statistics
      9. 6.7. How to compare representations and statistical methods
      10. 6.8. Results of classification, hypothesis testing, and probability distribution estimation
      11. 6.9. The code and its performance
      12. 6.10. Weaknesses of the skeletal approach
      13. References
    2. 7: Efficient recursive estimation of the Riemannian barycenter on the hypersphere and the special orthogonal group with applications
      1. Abstract
      2. Acknowledgements
      3. 7.1. Introduction
      4. 7.2. Riemannian geometry of the hypersphere
      5. 7.3. Weak consistency of iFME on the sphere
      6. 7.4. Experimental results
      7. 7.5. Application to the classification of movement disorders
      8. 7.6. Riemannian geometry of the special orthogonal group
      9. 7.7. Weak consistency of iFME on so(n)
      10. 7.8. Experimental results
      11. 7.9. Conclusions
      12. References
    3. 8: Statistics on stratified spaces
      1. Abstract
      2. Acknowledgements
      3. 8.1. Introduction to stratified geometry
      4. 8.2. Least squares models
      5. 8.3. BHV tree space
      6. 8.4. The space of unlabeled trees
      7. 8.5. Beyond trees
      8. References
    4. 9: Bias on estimation in quotient space and correction methods
      1. Abstract
      2. Acknowledgement
      3. 9.1. Introduction
      4. 9.2. Shapes and quotient spaces
      5. 9.3. Template estimation
      6. 9.4. Asymptotic bias of template estimation
      7. 9.5. Applications to statistics on organ shapes
      8. 9.6. Bias correction methods
      9. 9.7. Conclusion
      10. References
    5. 10: Probabilistic approaches to geometric statistics
      1. Abstract
      2. 10.1. Introduction
      3. 10.2. Parametric probability distributions on manifolds
      4. 10.3. The Brownian motion
      5. 10.4. Fiber bundle geometry
      6. 10.5. Anisotropic normal distributions
      7. 10.6. Statistics with bundles
      8. 10.7. Parameter estimation
      9. 10.8. Advanced concepts
      10. 10.9. Conclusion
      11. 10.10. Further reading
      12. References
    6. 11: On shape analysis of functional data
      1. Abstract
      2. 11.1. Introduction
      3. 11.2. Registration problem and elastic approach
      4. 11.3. Shape space and geodesic paths
      5. 11.4. Statistical summaries and principal modes of shape variability
      6. 11.5. Summary and conclusion
      7. Appendix. Mathematical background
      8. References
  9. Part 3: Deformations, diffeomorphisms and their applications
    1. 12: Fidelity metrics between curves and surfaces: currents, varifolds, and normal cycles
      1. Abstract
      2. Acknowledgements
      3. 12.1. Introduction
      4. 12.2. General setting and notations
      5. 12.3. Currents
      6. 12.4. Varifolds
      7. 12.5. Normal cycles
      8. 12.6. Computational aspects
      9. 12.7. Conclusion
      10. References
    2. 13: A discretize–optimize approach for LDDMM registration
      1. Abstract
      2. 13.1. Introduction
      3. 13.2. Background and related work
      4. 13.3. Continuous mathematical models
      5. 13.4. Discretization of the energies
      6. 13.5. Discretization and solution of PDEs
      7. 13.6. Discretization in multiple dimensions
      8. 13.7. Multilevel registration and numerical optimization
      9. 13.8. Experiments and results
      10. 13.9. Discussion and conclusion
      11. References
    3. 14: Spatially adaptive metrics for diffeomorphic image matching in LDDMM
      1. Abstract
      2. 14.1. Introduction to LDDMM
      3. 14.2. Sum of kernels and semidirect product of groups
      4. 14.3. Sliding motion constraints
      5. 14.4. Left-invariant metrics
      6. 14.5. Open directions
      7. References
    4. 15: Low-dimensional shape analysis in the space of diffeomorphisms
      1. Abstract
      2. Acknowledgements
      3. 15.1. Introduction
      4. 15.2. Background
      5. 15.3. PPGA of diffeomorphisms
      6. 15.4. Inference
      7. 15.5. Evaluation
      8. 15.6. Results
      9. 15.7. Discussion and conclusion
      10. References
    5. 16: Diffeomorphic density registration
      1. Abstract
      2. Acknowledgements
      3. 16.1. Introduction
      4. 16.2. Diffeomorphisms and densities
      5. 16.3. Diffeomorphic density registration
      6. 16.4. Density registration in the LDDMM-framework
      7. 16.5. Optimal information transport
      8. 16.6. A gradient flow approach
      9. References
  10. Index

Product information

  • Title: Riemannian Geometric Statistics in Medical Image Analysis
  • Author(s): Xavier Pennec, Stefan Sommer, Tom Fletcher
  • Release date: September 2019
  • Publisher(s): Academic Press
  • ISBN: 9780128147269