Handbook of numerical analysis: Volume 20 Processing, analyzing and learning of images, shapes and forms : Part 2
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| Other Authors: | , , |
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| Format: | Book |
| Language: | English |
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Amsterdam
North-Holland
[2019]
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xix, 684 Seiten Illustrationen, Diagramme (teilweise farbig) |
| ISBN: | 9780444641403 |
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| adam_text | Contents Contributors Preface 1. Diffusion operators for multimodal dataanalysis xv xix 1 Tal Shnitzer, Roy R. Lederman, Gi-Ren Liu, Ronen Talmon, and Hau-Tleng Wu 1 2 3 2. Introduction Preliminaries: Diffusion Maps Alternating Diffusion 3.1 Problem formulation: Metric spaces and probabilistic setting 3.2 Illustrative example 3.3 Algorithm 3.4 Common manifold interpretation 4 Self-adjoint operators for recoveringhidden components 4.1 Problem formulation 4.2 Operator definition and analysis 4.3 Discrete setting 5 Applications 5.1 Shape analysis 5.2 Foetal heart rate recovery 5.3 Sleep dynamics assessment References 2 3 5 5 5 10 12 17 18 19 22 23 23 25 27 38 Intrinsic and extrinsicoperators for shape analysis 41 Yu Wang and Justin Solomon 1 2 3 Introduction Preliminaries 2.1 Extrinsic and intrinsic geometry 2.2 Operators and spectra Theoretical aspects and numericalanalysis 3.1 Basics of linear operators 3.2 PDEs and Green s functions 3.3 Operator derivation and discretization 3.4 Operators and geometry 3.5 Inverse problems 42 44 44 45 45 ’ 45 49 52 55 56 v
v! Contents 4 Spectra! shape analysis and applications 4.1 Spectral analysis: From Euclidean space to manifold 4.2 Spectral data analysis 4.3 Spectral analysis: Point embedding, signature, and geometric descriptors 4.4 Shape analysis and geometry processing 4.5 Other aspects of spectral shape analysis 4.6 Numerical aspects 5 Relevant geometric operators 5.1 Identity operator, area form, and mass matrix 5.2 Laplace-Beltrami (intrinsic Laplacian) 5.3 Combinatorial and graph Laplacians 5.4 Restricted Laplacian 5.5 Scale invariant Laplacian 5.6 Affine and equi-affine invariant Laplacian 5.7 Anisotropic Laplacian 5.8 Hessian and normal-restricted Hessian: Afamily of linearized energies 5.9 Modified Dirichlet energy 5.10 Hamiltonian operator and Schrödinger operator 5.11 Curvature Laplacian 5.12 Concavity-aware Laplacian 5.13 Extrinsic and relative Dirac operators 5.14 Intrinsic Dirac operator D¡ 5.15 Volumetric (extrinsic) Laplacian 5.16 Hessian energy 5.17 Single layer potential operator and kernelmethod 5.18 Dirichlet-to-Neumann operator (Poincaré-Steklov operator) 5.19 Other extrinsic methods 6 Summary and experiments 6.1 Experiments 6.2 Eigenfunctions 6.3 Heat kernel signatures 6.4 Segmentation 6.5 Distance or dissimilarity 7 Conclusion and future work 7.1 Summary 7.2 Future work Acknowledgements References 3. Operator-based representations of discrete tangent vector fields 58 58 59 62 66 72 73 76 76 78 80 80 81 81 82 83 83 84 85 85 86 88 88 89 90 ֊ 91 92 92 93 93 96 97 100 103 103 104 104 105 117 Mirela Ben-Chen and Omri Azencot 1 Introduction 1.1 Organization 119 120
Contents 2 Smooth functional vector fields 2.1 Notation 2.2 Directional derivative of functions 2.3 Functional vector fields 2.4 Flow maps 2.5 Functional flow maps 2.6 Lie bracket 3 Discrete functional vector fields 3.1 Notation 3.2 Directional derivative of functions 3.3 Functional vector fields 3.4 Flow maps 3.5 Functional flow maps 3.6 Lie bracket 4 Divergence-based functional vector fields 4.1 Smooth DFVF 4.2 Discrete DFVF 4.3 Mixed Lie bracket operator 4.4 The Lie bracket as a linear transformation on vector fields 4.5 Integrating the Lie bracket operator 4.6 Outlook 5 Conclusion and future work References 4. Active contour methods on arbitrary graphs based on partial differential equations vii 120 120 121 121 122 123 126 128 128 131 131 133 133 135 136 136 136 137 142 144 145 146 146 149 Christos Sakaridis, Nikos Kolotouros, Kimon Drakopoulos, and Petros Maragos 1 2 3 Introduction Background and related work Active contours on graphs via geometricapproximations of gradient and curvature 3.1 Geometric gradient approximation on graphs 3.2 Geometric curvature approximation on graphs 3.3 Gaussian smoothing on graphs 4 Active contours on graphs using afinite element framework 4.1 Problem formulation and numerical approximation 4.2 Locally constrained contour evolution 5 Experimental results 6 Conclusion References 150 151 155 155 163 166 169 169 174 178 186 187
viii 5. Contents Fast operator-splitting algorithms for variational imaging models: Some recent developments 191 Roland Głowiński, Shousheng Luo, and Xue-Cheng Tai 1 2 3 introduction Regularizers and associated variational modelsfor image restoration 193 2.1 2.2 2.3 2.4 2.5 2.6 193 194 195 196 197 198 Basic results, notations and an introduction to operator-splitting methods 3.1 3.2 3.3 3.4 4 Basic results and notations The Lie and Marchuk-Yanenko operator-splitting schemes for the time discretization of initial value problems Time discretization of the initial value problem (17) by the Lie scheme Asymptotic properties of the Lie and Marchuk-Yanenko schemes Operator splitting method for Euler elasticaenergy functional 4.1 4.2 4.3 4.4 5 Generalities Total variation regularization The Euler elastica regularization t1-Mean curvature and t1-Gaussian curvature energies Willmore bending energy Summary Formulation of the problem solution methods On the solution of problem On the solution of problem On the solution of problem 5.2 5.3 5.4 198 198 200 201 202 203 and operator-splitting (44) (45) (46) An operator-splitting method for the Willmore energy-based variational model 5.1 192 Formulation of the problem and operator-splitting solution methods On the solution of problem (77) On the solution of problem (78) Estimating у 203 207 208 210 210 210 213 215 216 6 An operator-splitting method for the L’-meancurvature 7 variational model Operator-splitting methods for the ROF model 217 221 7.1 7.2 7.3 221 222 224 Generalities: synopsis A first operator-splitting method A second
operator-splitting method 8 Conclusion Acknowledgements References 227 227 227
Contents ix New solutions to active contours problems by Eikonal equations 233 Da Chen and Laurent D. Cohen 1 Introduction 1.1 Outline 2 Active contour models 2.1 Edge-based active contour model 2.2 The piecewise smooth Mumford-Shah model and the piecewise constant reduction model 3 Minimal paths for edge-based active contours problems 3.1 Cohen-Kimmel minimal path model 3.2 Finsler and Randers minimal paths 4 Minimal paths for alignment active contours 4.1 Randers alignment minimal paths 4.2 Riemannian alignment minimal paths 5 Orientation-lifted Randers minimal paths for Euler-Mumford elastica problem 5.1 Euler-Mumford elastica problem and its Finsler metric interpretation 5.2 Finsler elastica geodesic path for approximating the elastica curve 5.3 Data-driven Finsler elastica metric 6 Randers minimal paths for region-based active contours 6.1 Hybrid active contour model 6.2 A Randers metric interpretation to the hybrid energy 6.3 Practical implementations 6.4 Application to image segmentation 7 Conclusion Acknowledgements References 7. Computable invariants for curves and surfaces 234 235 236 236 240 243 243 245 247 247 251 253 254 255 257 261 261 263 264 265 268 269 269 273 Oshri Halimi, Dan Raviv, Yonathan Aflalo, and Ron Kimmel 1 2 3 Introduction Scale invariant metric 2.1 Scale invariant arc-length for planar curves 2.2 Scale invariant metric for implicitly defined planar curves 2.3 Scale invariant metric for surfaces 2.4 Approximating the scale invariant Laplace-Beltrami operator for surfaces Equi-affine invariant metric 3.1 Equi-affine invariant arc-length 3.2 Equi-
affine metric for surfaces 274 277 277 278 279 280 285 285 286
x Contents 4 8. Affine metric 4.1 Affine invariant arc-length 4.2 Affine metric for surfaces 4.3 Approximating the Gaussian curvature 5 Applications 5.1 Self functional maps: A song of shapes and operators 5.2 Object recognition 6 Summary Acknowledgements References 288 288 289 292 294 294 305 310 310 311 Solving PDEs on manifolds represented as point clouds and applications 315 Rongjie Lai and Hongkai Zhao 1 2 Introduction Solving PDEs on manifolds represented as point clouds 2.1 Moving least square methods 2.2 Local mesh method 3 Solving Fokker-Planck equation for dynamic system 3.1 Double well potential 3.2 Rugged Mueller potential 4 Solving PDEs on incomplete distance data 5 Geometric understanding of point clouds data 5.1 Construction of skeletons from point clouds 5.2 Construction of conformal mappings frompoint clouds 5.3 Nonrigid manifolds registration using LB eigenmap 6 Conclusion Acknowledgements References 9. Tighter continuous relaxations for MAP inference in discrete MRFs: A survey 316 318 318 320 324 327 329 331 337 338 339 340 343 345 345 351 Hariprasad Kannan, Nikos Komodakis, and Nikos Paragios 1 2 3 4 5 LP relaxation 1.1 Tightening the polytope Cluster pursuit algorithms Cycles in the graph 3.1 Searching for frustrated cycles 3.2 Efficient MAP inference in a cycle 3.3 Planar subproblems Tighter subgraph decompositions 4.1 Global higher-order cliques Semidefinite programming-based relaxation 5.1 Rounding schemes for SDP relaxation 5.2 Problems where SDP relaxation has helped 354 357 362 369 372 374 375 377 378 380 392 393
Contents xi 6 Characterizing tight relaxations 7 Conclusion References 10. Lagrangian methods for compositeoptimization 394 396 396 401 Shoham Sabach and Marc Teboulle 1 2 Introduction The Lagrangian framework 2.1 Lagrangian-based methods: Basic elements and mechanism 2.2 Proximal mappings and minimization 2.3 Application examples 3 The convex setting 3.1 Preliminaries on the convex model (CM) 3.2 Proximal method of multipliers and fundamental Lagrangian-based schemes 3.3 One scheme for all: A perturbed PMM and its global rate analysis 3.4 Special cases of the perturbed PMM: Fundamental schemes 4 The nonconvex setting 4.1 The nonconvex nonlinear composite optimization— Preliminaries 4.2 ALBUM—Adaptive Lagrangian-based multiplier method 4.3 A methodology for global analysis of Lagrangian-based methods 4.4 ALBUM in action: Global convergence of Lagrangian-based schemes Acknowledgements References 11. Generating structured nonsmoothpriors and associated primal-dual methods 402 404 404 407 411 414 414 416 418 422 424 425 427 429 431 433 433 437 Michael Hintermüller and Kostas Papafitsoros 1 2 3 4 Introduction 1.1 Context 1.2 Main contributions and organization of this chapter Nonsmooth priors 2.1 Total Variation 2.2 Total generalized variation 2.3 Dualization 2.4 Dualization of the variational regularization problems Numerical algorithms Bilevel optimization 4.1 Background 4.2 Bilevel optimization—A monolithic approach 438 438 449 449 449 453 455 461 462 469 469 473
Contents XII 5 Numerical examples Discrete operators for (Pjv) 5.2 Bilevel TV numerical experiments 5.3 Discrete operators for (Pţgv) 5.4 Bilevel TGV numerical experiments References 5.1 12. Graph-based optimization approaches for machine learning, uncertainty quantification and networks 480 481 485 488 491 494 503 Andrea L. Bertozzi and Ekaterina Merkurjev 1 2 3 Introduction Graph theory Recent methods for semisupervised and unsupervised data classification 3.1 Semisupervised learning and the Ginzburg-Landau graph model 3.2 The graph MBO scheme for data classification and image processing 3.3 Heat kernel pagerank method 3.4 Unsupervised learning and the Mumford-Shah model 3.5 Imposing volume constraints 4 Total variation methods for semisupervised andunsupervised data classification 5 Uncertainty quantification within the graphical framework 6 Networks 7 Conclusion Acknowledgements References 13. Survey of fast algorithms for Euler s elastica-based image segmentation 504 505 507 508 511 513 514 515 518 522 523 526 527 527 533 Sung Ha Kang, Xue-Cheng Tai, and Wei Zhu 1 2 14. Introduction Piecewise constant representation and interface problems to illusory contour with curvature term 3 Euler s elastica-based segmentation models andfast algorithms 4 Discussion References 534 535 540 547 548 Recent advances in denoising of manifold-valued images 553 R. Bergmann, F. Laus, J. Perseti, and G. Steidl 1 2 Introduction Preliminaries on Riemannian manifolds 2.1 General notation 2.2 Convexity and Hadamard manifolds 554 557 557 560
Contents 15. 3 4 Intrinsic variational restoration models Minimization algorithms 4.1 Subgradient descent 4.2 Half-quadratic minimization 4.3 Proximal point and Douglas-Rachfordalgorithm 5 Numerical examples 6 Conclusions References 561 564 564 565 567 572 574 574 Image and surface registration 579 Ke Chen, Lok Ming Lui, and Jan Modersitzki 1 2 Introduction Mathematical background 2.1 Continuous and discrete images 2.2 A mathematical framework for image registration 3 Distance measures 3.1 Volumetric differences 3.2 Feature-based differences 4 Regularization 4.1 Regularization by ansatz-spaces, parametric registration 4.2 Quadratic regularizer 4.3 Nonquadratic regularizer 4.4 Registration penalties and constraints 4.5 Penalties for locally invertible maps 4.6 Diffeomorphic registration 4.7 Registration by inverse consistent approach 5 Surface registration 5.1 Brief introduction to surface geometry 5.2 Parameterization-based approaches 5.3 Laplace-Beltrami eigenmap approaches 5.4 Metric approaches 5.5 Functional map approaches 5.6 Relationship between SR and IR 6 Numerical methods 7 Deep learning-based registration 8 Conclusions References 16. Metric registration of curvesand surfaces using optimal control 580 583 583 585 585 585 588 590 590 591 592 594 594 595 596 597 597 599 600 600 601 601 603 605 606 606 613 Martin Bauer, Nicolas Charon, and Laurent Younes 1 2 3 Introduction Building metrics via submersions Optimal control framework 614 616 619
Contents XIV 4 5 Chordal metrics on shapes 622 4.1 4.2 4.3 4.4 622 623 624 626 Motivation General principle Oriented varifold distances Numerical aspects Intrinsic metrics 5.1 5.2 5.3 5.4 5.5 5.6 Reparametrization-invariant metrics on parametrized shapes The metric on the space of unparametrized shapes The induced geodesic distance The geodesic equation An optimal control formulation of the geodesic problem on the space of unparametrized shapes Numerical aspects 6 Outer deformation metric models 7 A hybrid metric model 8 Conclusion Acknowledgements References 17. Efficient and accurate structure preserving schemes for complex nonlinear systems 626 627 629 630 631 632 633 635 639 641 642 642 647 Jie Sheri 1 2 3 Introduction The SAV approach 647 649 2.1 2.2 653 654 Suitable energy splitting Adaptive time stepping Several extensions of the SAV approach 655 3.1 3.2 3.3 3.4 3.5 655 658 659 661 Problems with global constraints L1 minimization via hyper regularization Free energies with highly nonlinear terms Coupling with other physical conservation laws Dissipative/conservative systems which are not driven by free energy 4 Conclusion Acknowledgements References Index 664 666 666 666 671
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| spelling | Handbook of numerical analysis Volume 20 Processing, analyzing and learning of images, shapes and forms : Part 2 general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions Amsterdam North-Holland [2019] xix, 684 Seiten Illustrationen, Diagramme (teilweise farbig) txt rdacontent n rdamedia nc rdacarrier Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 s Analysis (DE-588)4001865-9 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Kimmel, Ron 1963- (DE-588)117552008X edt Ciarlet, Philippe G. 1938- Sonstige (DE-588)143368362 oth Tai, Xue-Cheng (DE-588)1199863300 edt Lions, Jacques-Louis 1928-2001 Sonstige (DE-588)124055397 oth Du, Qiang 1964- (DE-588)1188249320 edt (DE-604)BV002745459 20 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031629526&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Handbook of numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd Analysis (DE-588)4001865-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4001865-9 (DE-588)4128130-5 |
| title | Handbook of numerical analysis |
| title_auth | Handbook of numerical analysis |
| title_exact_search | Handbook of numerical analysis |
| title_full | Handbook of numerical analysis Volume 20 Processing, analyzing and learning of images, shapes and forms : Part 2 general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions |
| title_fullStr | Handbook of numerical analysis Volume 20 Processing, analyzing and learning of images, shapes and forms : Part 2 general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions |
| title_full_unstemmed | Handbook of numerical analysis Volume 20 Processing, analyzing and learning of images, shapes and forms : Part 2 general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions |
| title_short | Handbook of numerical analysis |
| title_sort | handbook of numerical analysis processing analyzing and learning of images shapes and forms part 2 |
| topic | Numerische Mathematik (DE-588)4042805-9 gnd Analysis (DE-588)4001865-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Numerische Mathematik Analysis Numerisches Verfahren |
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