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