Handbook of Variational Methods for Nonlinear Geometric Data

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: SO(1)3 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 Frechet 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 Francois 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 Boettcher 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 Schnoerr 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 Andre 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 Schroedinger 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 Frechet _-Means and Their Strong Laws 10.4 Procrustes Analysis Viewed Through Frechet Means 10.5 A CLT for Frechet _-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 Trouve 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, Mauricio 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 Huper and Krzysztof A. Krakowski and Fatima 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

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• Matemática para cientistas da computação
• Modelação matemática
• Análise numérica
• Processamento de imagens