Part I Processing geometric data
1 Geometric Finite Elements
Hanne Hardering and Oliver Sander
1.1 Introduction
1.2 Constructions of geometric finite elements
1.2.1 Projection-based finite elements
1.2.2 Geodesic finite elements
1.2.3 Geometric finite elements based on de Casteljau’s algorithm
1.2.4 Interpolation in normal coordinates
1.3 Discrete test functions and vector field interpolation
1.3.1 Algebraic representation of test functions
1.3.2 Test vector fields as discretizations of maps into the tangent bundle
1.4 A priori error theory
1.4.1 Sobolev spaces of maps into manifolds
1.4.2 Discretization of elliptic energy minimization problems
1.4.3 Approximation errors . .
1.5 Numerical examples
1.5.1 Harmonic maps into the sphere
1.5.2 Magnetic Skyrmions in the plane
1.5.3 Geometrically exact Cosserat plates
2 Non-smooth variational regularization for processing manifold-valued
data
M. Holler and A. Weinmann
2.1 Introduction
2.2 Total Variation Regularization of Manifold Valued Data
vii
viii Contents
2.2.1 Models
2.2.2 Algorithmic Realization
2.3 Higher Order Total Variation Approaches, Total GeneralizedVariation
2.3.1 Models
2.3.2 Algorithmic Realization
2.4 Mumford-Shah Regularization for Manifold Valued Data
2.4.1 Models
2.4.2 Algorithmic Realization
2.5 Dealing with Indirect Measurements: Variational Regularization
of Inverse Problems for Manifold Valued Data
2.5.1 Models
2.5.2 Algorithmic Realization
2.6 Wavelet Sparse Regularization of Manifold Valued Data
2.6.1 Model
2.6.2 Algorithmic Realization
3 Lifting methods for manifold-valued variational problems
Thomas Vogt, Evgeny Strekalovskiy, Daniel Cremers, Jan Lellmann
3.1 Introduction
3.1.1 Functional lifting in Euclidean spaces
3.1.2 Manifold-valued functional lifting
3.1.3 Further related work
3.2 Submanifolds of RN
3.2.1 Calculus of Variations on submanifolds
3.2.2 Finite elements on submanifolds
3.2.3 Relation to [47]
3.2.4 Full discretization and numerical implementation
3.3 Numerical Results
3.3.1 One-dimensional denoising on a Klein bottle
3.3.2 Three-dimensional manifolds: SO¹3º
3.3.3 Normals fields from digital elevation data
3.3.4 Denoising of high resolution InSAR data
3.4 Conclusion and Outlook
4 Geometric subdivision and multiscale transforms
Johannes Wallner
4.1 Computing averages in nonlinear geometries
The Fréchet mean
The exponential mapping
Averages defined in terms of the exponential mapping
4.2 Subdivision
4.2.1 Defining stationary subdivision
Linear subdivision rules and their nonlinear analogues
4.2.2 Convergence of subdivision processes
4.2.3 Probabilistic interpretation of subdivision in metric spaces
4.2.4 The convergence problem in manifolds
4.3 Smoothness analysis of subdivision rules
4.3.1 Derivatives of limits
4.3.2 Proximity inequalities
4.3.3 Subdivision of Hermite data
4.3.4 Subdivision with irregular combinatorics
4.4 Multiscale transforms
4.4.1 Definition of intrinsic multiscale transforms
4.4.2 Properties of multiscale transforms
Conclusion
5 Variational Methods for Discrete Geometric Functionals
Henrik Schumacher and Max Wardetzky
5.1 Introduction
5.2 Shape Space of Lipschitz Immersions
5.3 Notions of Convergence for Variational Problems
5.4 Practitioner’s Guide to Kuratowski Convergence of Minimizers
5.5 Convergence of Discrete Minimal Surfaces and Euler Elasticae
Part II Geometry as a tool
6 Variational methods for fluid-structure interactions
François Gay-Balmaz and Vakhtang Putkaradze
6.1 Introduction
6.2 Preliminaries on variational methods
6.2.1 Exact geometric rod theory via variational principles
6.3 Variational modeling for flexible tubes conveying fluids
6.3.1 Configuration manifold for flexible tubes conveying fluid
6.3.2 Definition of the Lagrangian
6.3.3 Variational principle and equations of motion
6.3.4 Incompressible fluids
6.3.5 Comparison with previous models
6.3.6 Conservation laws for gas motion and Rankine-Hugoniot conditions
6.4 Variational discretization for flexible tubes conveying fluids
6.4.1 Spatial discretization 6.4.2 Variational integrator in space and time
6.5 Further developments
7 Convex lifting-type methods for curvature regularization
Ulrich Böttcher and Benedikt Wirth
7.1 Introduction .
7.1.1 Curvature-dependent functionals and regularization
7.1.2 Convex relaxation of curvature regularization functionals
7.2 Lifting-type methods for curvature regularization .
7.2.1 Concepts for curve- (and surface-) lifting
7.2.2 The curvature varifold approach
7.2.3 The hyper-varifold approach
7.2.4 The Gauss graph current approach
7.2.5 The jump set calibration approach
7.3 Discretization strategies
7.3.1 Finite differences
7.3.2 Line measure segments
7.3.3 Raviart–Thomas Finite Elements on a staggered gri
7.3.4 Adaptive line measure segments
7.4 The jump set calibration approach in 3D
7.4.1 Regularization model
7.4.2 Derivation of Theorem 7.4.2
7.4.3 Adaptive discretization with surface measures
8 Assignment Flows
Christoph Schnörr
8.1 Introduction
8.2 The Assignment Flow for Supervised Data Labeling
8.2.1 Elements of Information Geometry
8.2.2 The Assignment Flow
8.3 Unsupervised Assignment Flow and Self-Assignment
8.3.1 Unsupervised Assignment Flow: Label Evolution
8.3.2 Self-Assignment Flow: Learning Labels from Data
8.4 Regularization Learning by Optimal Control
8.4.1 Linear Assignment Flow
8.4.2 Parameter Estimation and Prediction
8.5 Outlook
9 Geometric methods on low-rank matrix and tensor manifolds
André Uschmajew and Bard Vandereycken
9.1 Introduction
9.1.1 Aims and outline
9.2 The geometry of low-rank matrices
9.2.1 Singular value decomposition and low-rank approximation
9.2.2 Fixed rank manifold
9.2.3 Tangent space
9.2.4 Retraction
9.3 The geometry of the low-rank tensor train decomposition
9.3.1 The tensor train decomposition
9.3.2 TT-SVD and quasi optimal rank truncation
9.3.3 Manifold structure
9.3.4 Tangent space and retraction
9.3.5 Elementary operations and TT matrix format
9.4 Optimization problems
9.4.1 Riemannian optimization
9.4.2 Linear systems
9.4.3 Computational cost
9.4.4 Difference to iterative thresholding methods
9.4.5 Convergence
9.4.6 Eigenvalue problems
9.5 Initial value problems
9.5.1 Dynamical low-rank approximation
9.5.2 Approximation properties
9.5.3 Low-dimensional evolution equations
9.5.4 Projector-splitting integrator
9.6 Applications
9.6.1 Matrix equations
9.6.2 Schrödinger equation
9.6.3 Matrix and tensor completion
9.6.4 Stochastic and parametric equations
9.6.5 Transport equations
9.7 Conclusions
Part III Statistical methods and non-linear geometry
10 Statistical Methods Generalizing Principal Component Analysis to
Non-Euclidean Spaces
Stephan Huckemann and Benjamin Eltzner
10.1 Introduction
10.2 Some Euclidean Statistics Building on Mean and Covariance
10.3 Fréchet _-Means and Their Strong Laws
10.4 Procrustes Analysis Viewed Through Fréchet Means
10.5 A CLT for Fréchet _-Means
10.6 Geodesic Principal Component Analysis
10.7 Backward Nested Descriptors Analysis (BNDA)
10.8 Two Bootstrap Two-Sample Tests
10.9 Examples of BNDA
10.10 Outlook
11 Advances in Geometric Statistics for manifold dimension reduction
Xavier Pennec
11.1 Introduction
11.2 Means on manifolds
11.3 Statistics beyond the mean value: generalizing PCA.
11.3.1 Barycentric subspaces in manifolds
11.3.2 From PCA to barycentric subspace analysis
11.3.3 Sample-limited Lp barycentric subspace inference
11.4 Example applications of Barycentric subspace analysis
11.4.1 Example on synthetic data in a constant curvature space
11.4.2 A symmetric group-wise analysis of cardiac motion in 4D image sequences
12 Deep Variational Inference
Iddo Drori
12.1 Variational Inference
12.1.1 Score Gradient
12.1.2 Reparametrization Gradient
12.2 Variational Autoencoder
12.2.1 Autoencoder
12.2.2 Variational Autoencoder
12.3 Generative Flows
12.4 Geometric Variational Inference
Part IV Shapes spaces and the analysis of geometric data
13 Shape Analysis of Functional Data
Xiaoyang Guo, Anuj Srivastava
13.1 Introduction
13.2 Registration Problem and Elastic Framework
13.2.1 The Use of the L2 Norm and Its Limitations
13.2.2 Elastic Registration of Scalar Functions
13.2.3 Elastic Shape Analysis of Curves
13.3 Shape Summary Statistics, Principal Modes and Models
14 Statistical Analysis of Trajectories of Multi-Modality Data
Mengmeng Guo, Jingyong Su, Zhipeng Yang and Zhaohua Ding
14.1 Introduction and Background
14.2 Elastic Shape Analysis of Open Curves
14.3 Elastic Analysis of Trajectories
14.4 Joint Framework of Analyzing Shapes and Trajectories
14.4.1 Trajectories of Functions
14.4.2 Trajectories of Tensors
15 Geometric Metrics for Topological Representations
Anirudh Som, Karthikeyan Natesan Ramamurthy and Pavan Turaga
15.1 Introduction
15.2 Background and Definitions
15.3 Topological Feature Representations
15.4 Geometric Metrics for Representations
15.5 Applications
15.5.1 Time-series Analysis
15.5.2 Image Analysis
15.5.3 Shape Analysis .
16 On Geometric Invariants, Learning, and Recognition of Shapes and
Forms
Gautam Pai, Mor Joseph-Rivlin, Ron Kimmel and Nir Sochen
16.1 Introduction
16.2 Learning Geometric Invariant Signatures For Planar Curves
16.2.1 Geometric Invariants of Curves
16.2.2 Learning Geometric Invariant Signatures of Planar Curves
16.3 Geometric Moments for Advanced Deep Learning on Point Clouds
16.3.1 Geometric Moments as Class Identifiers
16.3.2 Raw Point Cloud Classification based on Moments
Performance Evaluation
17 Sub-Riemannian Methods in Shape Analysis
Laurent Younes and Barbara Gris and Alain Trouvé
17.1 Introduction
17.2 Shape Spaces, Groups of Diffeomorphisms and Shape Motion
17.2.1 Spaces of Plane Curves
17.2.2 Basic Sub-Riemannian Structure
17.2.3 Generalization
17.2.4 Pontryagin’s Maximum Principle
17.3 Approximating Distributions
17.3.1 Control Points
17.3.2 Scale Attributes
17.4 Deformation Modules
17.4.1 Definition
17.4.2 Basic deformation modules
17.4.3 Simple matching example
17.4.4 Population analysis
17.5 Constrained Evolution
Normal Streamlines
Multi-shapes
Atrophy Constraints
Part V Optimization algorithms and numerical methods
18 First order methods for optimization on Riemannian manifolds
Orizon P. Ferreira, Maurício S. Louzeiro and Leandro F. Prudente
18.1 Introduction
18.2 Notations and Basic Results.
18.3 Examples of convex functions on Riemannian manifolds
18.3.1 General examples .
18.3.2 Example in the Euclidean space with a new Riemannianmetric
18.3.3 Examples in the positive orthant with a new Riemannian
18.3.4 Examples in the cone of SPD matrices with a new Riemannian metric
Bibliographic notes and remarks
18.4 Gradient method for optimization
18.4.1 Asymptotic convergence analysis
18.4.2 Iteration-complexity analysis
Bibliographic notes and remarks
18.5 Subgradient method for optimization
18.5.1 Asymptotic convergence analysis
18.5.2 Iteration-complexity analysis
Bibliographic notes and remarks
18.6 Proximal point method for optimization
18.6.1 Asymptotic convergence analysis
18.6.2 Iteration-complexity analysis
Bibliographic notes and remarks
19 Recent Advances in Stochastic Riemannian Optimization
Reshad Hosseini and Suvrit Sra
19.1 Introduction
Additional Background and Summary
19.2 Key Definitions
19.3 Stochastic Gradient Descent on Manifolds
19.4 Accelerating Stochastic Gradient Descent
19.5 Analysis for G-Convex and Gradient Dominated Functions
19.6 Example applications
20 Averaging symmetric positive-definite matrices
Xinru Yuan, Wen Huang, P.-A. Absil and K. A. Gallivan
20.1 Introduction
20.2 ALM Properties
20.3 Geodesic Distance Based Averaging Techniques
20.3.1 Karcher Mean (L2 Riemannian mean)
20.3.2 Riemannian Median (L1 Riemannian mean)
20.3.3 Riemannian Minimax Center (L1 Riemannian mean)
20.4 Divergence-based Averaging Techniques
20.4.1 Divergences
20.4.2 Left, Right, and Symmetrized Means Using Divergences
20.4.3 Divergence-based Median and Minimax Center
20.5 Alternative Metrics on SPD Matrices
21 Rolling Maps and Nonlinear Data
Knut Hüper and Krzysztof A. Krakowski and Fátima Silva Leite
21.1 Introduction
21.2 Rolling Manifolds Along Affine Tangent Spaces
21.2.1 Mathematical Setting
21.2.2 Rolling Manifolds
21.2.3 Parallel Transport
21.3 Rolling to Solve Interpolation Problems on Manifolds
21.3.1 Formulation of the Problem
21.3.2 Motivation
21.3.3 Solving the Interpolation Problem
21.3.4 Examples
21.3.5 Implementation of the Algorithm on S2
21.4 Some Extensions
21.4.1 Rolling a Hypersurface .
21.4.2 The Case of an Ellipsoid
21.4.3 Related Work
Part VI Applications
22 Manifold-valued Data in Medical Imaging Applications
Maximilian Baust and Andreas Weinmann
22.1 Introduction
22.1.1 Motivation
22.1.2 General Model
22.1.3 Organization of the Chapter
22.2 Pose Signals and 3D Ultrasound Compounding
22.2.1 Problem-specific Manifold and Model
22.2.2 Numerical Approach
22.2.3 Experiments
22.2.4 Discussion
22.3 Diffusion Tensor Imaging
22.3.1 Problem-specific Manifold and Model
22.3.2 Algorithmic Approach
22.3.3 Experiments
22.3.4 Discussion
22.4 Geometry Processing and Medical Image Segmentation
22.4.1 Problem-specific Manifold, Basic Model and Algorithm
22.4.2 Experiments
22.4.3 Extensions
22.4.4 Discussion
23 The Riemannian and Affine Geometry of Facial Expression and
Action Recognition
Mohamed Daoudi, Juan-Carlos Alvarez Paiva and Anis Kacem
23.1 Landmark representation
23.1.1 Challenges
23.2 Static representation
23.3 Riemannian geometry of the space of Gram matrices
23.3.1 Mathematical preliminaries
23.3.2 Riemannian manifold of positive semi-definite matrices of fixed rank
23.3.3 Affine-invariant and spatial covariance information of Gram matrices
23.4 Gram matrix trajectories for temporal modeling of landmark sequences
23.4.1 Rate-invariant comparison of Gram matrix trajectories
23.5 Classification of Gram matrix trajectories
23.5.1 Pairwise proximity function SVM
23.6 Application to Facial Expression and Action Recognition
23.6.1 2D facial expression recognition
23.6.2 3D action recognition
23.7 Affine-invariant shape representation using barycentric coordinates554
23.7.1 Relationship with the conventional Grassmannian representation
23.8 Metric learning on barycentric representation for expression recognition in unconstrained environments
23.8.1 Experimental results
24 Biomedical Applications of Geometric Functional Data Analysis
James Matuk, Shariq Mohammed, Sebastian Kurtek and Karthik
Bharath
24.1 Introduction
24.2 Mathematical Representation: Riemannian Metrics and
Simplifying Transforms .
24.2.1 Probability Density Functions
24.2.2 Amplitude and Phase in Elastic Functional Data
24.2.3 Shapes of Open and Closed Curves
24.2.4 Shapes of Surfaces
24.3 Nonparametric Metric-based Statistics
24.3.1 Karcher Mean
24.3.2 Covariance Estimation and Principal Component Analysis
24.4 Biomedical Case Studies
24.4.1 Probability Density Functions
24.4.2 Amplitude and Phase in Elastic Functional Data
24.4.3 Shapes of Open and Closed Curves
24.4.4 Shapes of Surfaces
