Handbook of Variational Methods for Nonlinear Geometric Data. (2020)

• Record Type: Book
• Further Information: Note: Philipp Grohs, Martin Holler, Andreas Weinmann.
• Editors: Grohs, Philipp; Holler, Martin; Weinmann, Andreas
• Contents: 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 Atr
• Publisher Details: Cham : Springer
• Publication Date: 2020
• Copyright Date: 2020
• Extent: 1 online resource (701 pages)
• Subjects: Mathematics; Computer mathematics; Optical data processing; Computers -- Data Processing; Computers -- Computer Graphics; Mathematics -- Applied; Maths for computer scientists; Image processing; Mathematical modelling; Mathematics -- Counting & Numeration; Numerical analysis; Computer science--Mathematics
• Languages: English
• ISBNs: 9783030313517
• Related ISBNs: 9783030313500
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