Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2, Volume 20, surveys the contemporary developments relating to the analysis and learning of images, shapes and forms, covering mathematical models and quick computational techniques. Chapter cover Alternating Diffusion: A Geometric Approach for Sensor Fusion, Generating Structured TV-based Priors and Associated Primal-dual Methods, Graph-based Optimization Approaches for Machine Learning, Uncertainty Quantification and Networks, Extrinsic Shape Analysis from Boundary Representations, Efficient Numerical Methods for Gradient Flows and Phase-field Models, Recent Advances in Denoising of Manifold-Valued Images, Optimal Registration of Images, Surfaces and Shapes, and much more.
1. 1 Introduction - p. 2 2. 2 Preliminaries: Diffusion Maps - p. 3 3. 3 Alternating Diffusion - p. 5 4. 4 Self-adjoint operators for recovering hidden components - p. 17 5. 5 Applications - p. 23 6. Chapter 2: Intrinsic and extrinsic operators for shape analysis - p. 39 7. Abstract - p. 42 8. 1 Introduction - p. 42 9. 2 Preliminaries - p. 44 10. 3 Theoretical aspects and numerical analysis - p. 45 11. 4 Spectral shape analysis and applications - p. 58 12. 5 Relevant geometric operators - p. 76 13. 6 Summary and experiments - p. 92 14. 7 Conclusion and future work - p. 103 15. Acknowledgements - p. 104 16. Chapter 3: Operator-based representations of discrete tangent vector fields - p. 115 17. Abstract - p. 117 18. 1 Introduction - p. 118 19. 2 Smooth functional vector fields - p. 120 20. 3 Discrete functional vector fields - p. 128 21. 4 Divergence-based functional vector fields - p. 136 22. 5 Conclusion and future work - p. 146 23. Chapter 4: Active contour methods on arbitrary graphs based on partial differential equations - p. 147 24. Chapter 5: Fast operator-splitting algorithms for variational imaging models: Some recent developments - p. 190 25. Abstract - p. 192 26. 1 Introduction - p. 192 27. 2 Regularizers and associated variational models for image restoration - p. 193 28. 3 Basic results, notations and an introduction to operator-splitting methods - p. 198 29. 4 Operator splitting method for Euler elastica energy functional - p. 203 30. 5 An operator-splitting method for the Willmore energy-based variational model - p. 210 31. 6 An operator-splitting method for the L1-mean curvature variational model - p. 217 32. 7 Operator-splitting methods for the ROF model - p. 220 33. 8 Conclusion - p. 227 34. Acknowledgements - p. 227 35. Chapter 6: From active contours to minimal geodesic paths: New solutions to active contours problems by Eikonal equations - p. 232 36. Abstract - p. 234 37. 1 Introduction - p. 234 38. 2 Active contour models - p. 236 39. 3 Minimal paths for edge-based active contours problems - p. 243 40. 4 Minimal paths for alignment active contours - p. 247 41. 5 Orientation-lifted Randers minimal paths for Euler–Mumford elastica problem - p. 253 42. 6 Randers minimal paths for region-based active contours - p. 261 43. 7 Conclusion - p. 268 44. Acknowledgements - p. 269 45. Chapter 7: Computable invariants for curves and surfaces - p. 271 46. Abstract - p. 273 47. 1 Introduction - p. 274 48. 2 Scale invariant metric - p. 277 49. 3 Equi-affine invariant metric - p. 285 50. 4 Affine metric - p. 288 51. 5 Applications - p. 294 52. 6 Summary - p. 310 53. Acknowledgements - p. 310 54. Chapter 8: Solving PDEs on manifolds represented as point clouds and applications - p. 314 55. Chapter 9: Tighter continuous relaxations for MAP inference in discrete MRFs: A survey - p. 349 56. Abstract - p. 351 57. 1 LP relaxation - p. 354 58. 2 Cluster pursuit algorithms - p. 362 59. 3 Cycles in the graph - p. 369 60. 4 Tighter subgraph decompositions - p. 377 61. 5 Semidefinite programming-based relaxation - p. 380 62. 6 Characterizing tight relaxations - p. 394 63. 7 Conclusion - p. 396 64. Chapter 10: Lagrangian methods for composite optimization - p. 400 65. Abstract - p. 401 66. 1 Introduction - p. 402 67. 2 The Lagrangian framework - p. 404 68. 3 The convex setting - p. 414 69. 4 The nonconvex setting - p. 424 70. Acknowledgements - p. 433 71. Chapter 11: Generating structured nonsmooth priors and associated primal-dual methods - p. 436 72. Abstract - p. 437 73. 1 Introduction - p. 438 74. 2 Nonsmooth priors - p. 449 75. 3 Numerical algorithms - p. 462 76. 4 Bilevel optimization - p. 469 77. 5 Numerical examples - p. 480 78. Chapter 12: Graph-based optimization approaches for machine learning, uncertainty quantification and networks - p. 502 79. Chapter 13: Survey of fast algorithms for Euler's elastica-based image segmentation - p. 531 80. Chapter 14: Recent advances in denoising of manifold-valued images - p. 552 81. Chapter 15: Image and surface registration - p. 578 82. Chapter 16: Metric registration of curves and surfaces using optimal control - p. 611 83. Abstract - p. 613 84. 1 Introduction - p. 614 85. 2 Building metrics via submersions - p. 616 86. 3 Optimal control framework - p. 619 87. 4 Chordal metrics on shapes - p. 622 88. 5 Intrinsic metrics - p. 626 89. 6 Outer deformation metric models - p. 635 90. 7 A hybrid metric model - p. 639 91. 8 Conclusion - p. 641 92. Acknowledgements - p. 642 93. Chapter 17: Efficient and accurate structure preserving schemes for complex nonlinear systems - p. 646 94. Index - p. 671
Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2, Volume 20, surveys the contemporary developments relating to the analysis and learning of images, shapes and forms, covering mathematical models and quick computational techniques. Chapter cover Alternating Diffusion: A Geometric Approach for Sensor Fusion, Generating Structured TV-based Priors and Associated Primal-dual Methods, Graph-based Optimization Approaches for Machine Learning, Uncertainty Quantification and Networks, Extrinsic Shape Analysis from Boundary Representations, Efficient Numerical Methods for Gradient Flows and Phase-field Models, Recent Advances in Denoising of Manifold-Valued Images, Optimal Registration of Images, Surfaces and Shapes, and much more.
1. 1 Introduction - p. 2 2. 2 Preliminaries: Diffusion Maps - p. 3 3. 3 Alternating Diffusion - p. 5 4. 4 Self-adjoint operators for recovering hidden components - p. 17 5. 5 Applications - p. 23 6. Chapter 2: Intrinsic and extrinsic operators for shape analysis - p. 39 7. Abstract - p. 42 8. 1 Introduction - p. 42 9. 2 Preliminaries - p. 44 10. 3 Theoretical aspects and numerical analysis - p. 45 11. 4 Spectral shape analysis and applications - p. 58 12. 5 Relevant geometric operators - p. 76 13. 6 Summary and experiments - p. 92 14. 7 Conclusion and future work - p. 103 15. Acknowledgements - p. 104 16. Chapter 3: Operator-based representations of discrete tangent vector fields - p. 115 17. Abstract - p. 117 18. 1 Introduction - p. 118 19. 2 Smooth functional vector fields - p. 120 20. 3 Discrete functional vector fields - p. 128 21. 4 Divergence-based functional vector fields - p. 136 22. 5 Conclusion and future work - p. 146 23. Chapter 4: Active contour methods on arbitrary graphs based on partial differential equations - p. 147 24. Chapter 5: Fast operator-splitting algorithms for variational imaging models: Some recent developments - p. 190 25. Abstract - p. 192 26. 1 Introduction - p. 192 27. 2 Regularizers and associated variational models for image restoration - p. 193 28. 3 Basic results, notations and an introduction to operator-splitting methods - p. 198 29. 4 Operator splitting method for Euler elastica energy functional - p. 203 30. 5 An operator-splitting method for the Willmore energy-based variational model - p. 210 31. 6 An operator-splitting method for the L1-mean curvature variational model - p. 217 32. 7 Operator-splitting methods for the ROF model - p. 220 33. 8 Conclusion - p. 227 34. Acknowledgements - p. 227 35. Chapter 6: From active contours to minimal geodesic paths: New solutions to active contours problems by Eikonal equations - p. 232 36. Abstract - p. 234 37. 1 Introduction - p. 234 38. 2 Active contour models - p. 236 39. 3 Minimal paths for edge-based active contours problems - p. 243 40. 4 Minimal paths for alignment active contours - p. 247 41. 5 Orientation-lifted Randers minimal paths for Euler–Mumford elastica problem - p. 253 42. 6 Randers minimal paths for region-based active contours - p. 261 43. 7 Conclusion - p. 268 44. Acknowledgements - p. 269 45. Chapter 7: Computable invariants for curves and surfaces - p. 271 46. Abstract - p. 273 47. 1 Introduction - p. 274 48. 2 Scale invariant metric - p. 277 49. 3 Equi-affine invariant metric - p. 285 50. 4 Affine metric - p. 288 51. 5 Applications - p. 294 52. 6 Summary - p. 310 53. Acknowledgements - p. 310 54. Chapter 8: Solving PDEs on manifolds represented as point clouds and applications - p. 314 55. Chapter 9: Tighter continuous relaxations for MAP inference in discrete MRFs: A survey - p. 349 56. Abstract - p. 351 57. 1 LP relaxation - p. 354 58. 2 Cluster pursuit algorithms - p. 362 59. 3 Cycles in the graph - p. 369 60. 4 Tighter subgraph decompositions - p. 377 61. 5 Semidefinite programming-based relaxation - p. 380 62. 6 Characterizing tight relaxations - p. 394 63. 7 Conclusion - p. 396 64. Chapter 10: Lagrangian methods for composite optimization - p. 400 65. Abstract - p. 401 66. 1 Introduction - p. 402 67. 2 The Lagrangian framework - p. 404 68. 3 The convex setting - p. 414 69. 4 The nonconvex setting - p. 424 70. Acknowledgements - p. 433 71. Chapter 11: Generating structured nonsmooth priors and associated primal-dual methods - p. 436 72. Abstract - p. 437 73. 1 Introduction - p. 438 74. 2 Nonsmooth priors - p. 449 75. 3 Numerical algorithms - p. 462 76. 4 Bilevel optimization - p. 469 77. 5 Numerical examples - p. 480 78. Chapter 12: Graph-based optimization approaches for machine learning, uncertainty quantification and networks - p. 502 79. Chapter 13: Survey of fast algorithms for Euler's elastica-based image segmentation - p. 531 80. Chapter 14: Recent advances in denoising of manifold-valued images - p. 552 81. Chapter 15: Image and surface registration - p. 578 82. Chapter 16: Metric registration of curves and surfaces using optimal control - p. 611 83. Abstract - p. 613 84. 1 Introduction - p. 614 85. 2 Building metrics via submersions - p. 616 86. 3 Optimal control framework - p. 619 87. 4 Chordal metrics on shapes - p. 622 88. 5 Intrinsic metrics - p. 626 89. 6 Outer deformation metric models - p. 635 90. 7 A hybrid metric model - p. 639 91. 8 Conclusion - p. 641 92. Acknowledgements - p. 642 93. Chapter 17: Efficient and accurate structure preserving schemes for complex nonlinear systems - p. 646 94. Index - p. 671
