Table of Contents
Part 1: Foundations of geometric statistics
1: Introduction to differential and Riemannian geometry
1.4. Elements of analysis in Riemannian manifolds
1.5. Lie groups and homogeneous manifolds
1.6. Elements of computing on Riemannian manifolds
2.3. Covariance and principal geodesic analysis
3: Manifold-valued image processing with SPD matrices
3.2. Exponential, logarithm, and square root of SPD matrices
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
4: Riemannian geometry on shapes and diffeomorphisms
4.3. The diffeomorphism group in shape analysis
4.4. Riemannian metrics on shape spaces
4.7. Outer and inner shape metrics
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
Part 2: Statistics on manifolds and shape spaces
6: Object shape representation via skeletal models (s-reps) and statistical analysis
6.1. Introduction to skeletal models
6.2. Computing an s-rep from an image or object boundary
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
7.2. Riemannian geometry of the hypersphere
7.3. Weak consistency of iFME on the sphere
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)
8: Statistics on stratified spaces
8.1. Introduction to stratified geometry
8.4. The space of unlabeled trees
9: Bias on estimation in quotient space and correction methods
9.2. Shapes and quotient spaces
9.4. Asymptotic bias of template estimation
9.5. Applications to statistics on organ shapes
10: Probabilistic approaches to geometric statistics
10.2. Parametric probability distributions on manifolds
10.5. Anisotropic normal distributions
11: On shape analysis of functional data
11.2. Registration problem and elastic approach
11.3. Shape space and geodesic paths
11.4. Statistical summaries and principal modes of shape variability
Part 3: Deformations, diffeomorphisms and their applications
12: Fidelity metrics between curves and surfaces: currents, varifolds, and normal cycles
12.2. General setting and notations
13: A discretize–optimize approach for LDDMM registration
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
13.9. Discussion and conclusion
14: Spatially adaptive metrics for diffeomorphic image matching in LDDMM
14.2. Sum of kernels and semidirect product of groups
14.3. Sliding motion constraints
15: Low-dimensional shape analysis in the space of diffeomorphisms
15.7. Discussion and conclusion
16: Diffeomorphic density registration
16.2. Diffeomorphisms and densities
16.3. Diffeomorphic density registration
16.4. Density registration in the LDDMM-framework
16.5. Optimal information transport