Curves and Surfaces (eBook)

Pierre-Jean Laurent, Alain Le Mehaute, Larry L. Schumaker (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
532 Seiten
Elsevier Science (Verlag)
978-1-4832-6387-8 (ISBN)

Infos

Curves and Surfaces provides information pertinent to the fundamental aspects of approximation theory with emphasis on approximation of images, surface compression, wavelets, and tomography. This book covers a variety of topics, including error estimates for multiquadratic interpolation, spline manifolds, and vector spline approximation. Organized into 77 chapters, this book begins with an overview of the method, based on a local Taylor expansion of the final curve, for computing the parameter values. This text then presents a vector approximation based on general spline function theory. Other chapters consider a nonparametric technique for estimating under random censorship the amplitude of a change point in change point hazard models. This book discusses as well the algorithm for ray tracing rational parametric surfaces based on inversion and implicitization. The final chapter deals with the results concerning the norm of the interpolation operator and error estimates for a square domain. This book is a valuable resource for mathematicians.

Inhalt

Front Cover 1
Curves and Surfaces 4
Copyright Page 5
Table of Contents 6
PREFACE 11
CONTRIBUTORS 12
Chapter 1. Parametrization for Data Approximation 20
§1. Introduction 20
§2. Local Taylor Expansion of the Curve 20
§3. Criteria 21
§4. Application to Approximation with B-Splines 22
References 23
Chapter 2. A Vector Spline Approximation With Application to Meteorology 24
§1. Introduction 24
§2. The Minimization Problem 25
§3. Solution of P 
26 
§4. Limit Problems 28
§5. Numerical Results 28
References 29
Chapter 3. Kernel Estimation in Change-Point Hazard Rate Models 30
§1. Introduction 30
§2. The Model 31
§3. Main Results 33
§4. Simulation Results and Concluding Remarks 34
References 35
Chapter 4. Spline Manifolds 36
§1. Introduction 36
§2. Basic 
38 
§3. A Fundamental Result 40
§4. Spline Manifolds 41
§5. P(D)-positive Vectorial Distributions 43
§6. Extension of P(D)-spline 44
References 44
Chapter 5. Use of Simulated Annealing to Construct Triangular Facet Surfaces 46
§1. Introduction 46
§2. Optimal Triangulations 47
§3. Locally Optimal Triangulations and Edge Swapping 47
§4. Simulated Annealing 48
§5. An Example 49
§6. Conclusions 51
References 51
Chapter 6. 
52 
§1. Geometric Framework 52
§2. Analytical Conditions 53
§3. Geometric Continuity Between Two Rectangular (SBR) 53
References 55
Chapter 7. Ray Tracing Rational Parametric Surfaces 56
§1. Introduction 56
§2. Implicitization 57
§3. Intersection Problem 59
§4. Discussion 61
References 61
Chapter 8. Energy-Based Segmentation of Sparse Range Data 62
Abstract 62
§1. Segmentation: Introduction and Background 62
§2. Definition of the Model of World Surfaces 63
§3. Experimentation 66
References 69
Chapter 9. Error Estimates for Multiquadric Interpolation 70
§1. Introduction and Statement of Main Result 70
§2. Proof of Theorem 1 72
References 77
Chapter 10. A Geometrical Analysis for a Data Compression of 3D Anatomical Structures 78
§1. Introduction 78
§2. Data Format 78
§3. Feature Extraction 79
§4. Data Compression 81
§5. Contour Reconstruction 82
§6. Conclusion 83
References 84
Chapter 11. Ck Continuity of (SBR) Surfaces 86
§1. Framework 86
§2. Rectangular (SBR) Surfaces 86
§3. Triangular (SBR) Surfaces 88
References 89
Chapter 12. A Note on Piecewise Monotonie Bivariate Interpolation 90
§1. Introduction 90
§2. Outline of the Algorithm 91
§3. Conclusions 93
References 93
Chapter 13. Real-Time Signal Analysis with Quasi-Interpolatory Splines and Wavelets 94
§1. Introduction 94
§2. Spline Sampling of Digital Signals 95
§3. Wavelet Signal Decomposition 98
§4. Wavelet Signal Reconstruction 99
References 101
Chapter 14. Polynomial Expansions for Cardinal Interpolants and Orthonormal Wavelets 102
§1. Introduction 102
§2. The Case of Real 
103 
§3. The Case of Complex 
105 
§4. Construction of Quasi-Interpolation Operators 108
References 109
Chapter 15. Realtime Pipelined Spline Data Fitting for Sketched Curves 110
§1. Introduction 110
§2. Background 111
§3. Data Reduction 111
§4. Pipelining the Algorithm 114
§6. Remarks 120
References 120
Chapter 16. Remarks on Digital Terrain Modelling Accuracy 122
§1. On Terrain Modelling Accuracy 122
§2. The Displacement-Buckling Approach 123
§3. A Graphical Approach 124
References 125
Chapter 17. Convexity and Bernstein-Bézier Polynomials 126
§1. Introduction 126
§2. Some Prerequisites 128
§3. Planar Curves 131
§4. Convexity of Functional Bézier Surfaces 133
§5. Convexity of Piecewise Polynomial Surfaces 141
§6. Convexity of Parametric Bézier Patches 143
§7. Converse Theorems on Convexity 146
References 150
Chapter 18. How to Draw a Curve Using Geometrical Data 154
§1. Introduction 154
§2. Algebraic Regression 155
§3. Automatic Ordering 156
References 157
Chapter 19. The Generation of an Aerodynamical Propeller Using Partial Differential Equations 158
§1. Introduction 158
§2. The PDE Method 158
§3. Generating the Propeller 159
§4. Aerodynamics 160
§5. Influence of Parameters 161
References 161
Chapter 20. Fast Computation of Cross-Validated Robust Splines and Other Non-linear Smoothing Splines 162
§1. Introduction 162
§2. Choice of p by 
164 
§3. Monte-Carlo Computation of Trace Terms 165
§4. Numerical Experiments 165
References 167
Chapter 21. Szasz-Mirakyan Quasi-interpolants 168
§1. Introduction and Definitions 168
§2. Norms of the Left Quasi-interpolants 170
§3. Woronovskaya-type Relation 174
Acknowledgements 175
References 175
Chapter 22. Statistical Check On The Smoothing Parameter of a Method for Inversion of Fourier Series 176
References 179
Chapter 23. A General Method of Treating Degenerate Bézier Patches 180
§1. Introduction 180
§2. Computing Geometric Features 181
§3. Geometric Continuity Constraints over Degenerate Patches 182
References 183
Chapter 24. G1 Smooth Connection Between Rectangular and Triangular Bézier Patches at a Common Corner 184
§1. G1 Continuity Between Two Adjacent Bézier Patches 184
§2. G1 Continuity Around a Mixed N-patch Corner 185
References 187
Chapter 25. Regularity Conditions for a Class of Geometrically Continuous Curves and Surfaces 188
§1. Introduction 188
§2. Regularity Conditions for Bell-Shaped Functions 190
§3. Two Classes of Bell-Shaped Functions 192
References 195
Chapter 26. Splines and Digital Signal Processing 196
§1. Introduction 196
§2. B-Spline Digital Filters 197
References 199
Chapter 27. B-Rational Curves and Surfaces N-Rational Splines 200
§1. Introduction 200
§2. General Framework 200
§3. The (BR) Curves 201
§4. The N-rational Splines 202
§5. The (SBR) Surfaces 202
References 203
Chapter 28. Reparametrizations of Polynomial and Rational Curves 204
§1. Introduction 204
§2. Homographie Transformation 205
§3. Rational Quadratic Transformation 205
§4. Conclusion 207
References 207
Chapter 29. Numerical Stability of Geometric Algorithms 208
References 211
Chapter 30. Solving Implicit ODEs by Simplicial Methods 212
§1. Introduction 212
§2. PL Approximation of Implicit Manifolds 212
§3. Implicit Differential Equations and Singularities of Mappings 213
§4. A Simplicial Method to Solve Implicit ODEs 214
References 215
Chapter 31. On the Power of a posteriori Error Estimation for Numerical Integration and Function Approximation 216
§1. Introduction 216
§2. Proof of the Theorems 220
References 226
Chapter 32. Using the Refinement Equations for the Construction of Pre-Wavelets II: Powers of Two 228
§1. Introduction 228
§2. Multiresolution Analysis 230
§3. Symbol Calculus 238
§4. Stability 245
§5. Linear Independence 250
§6. Pre-Wavelet Decomposition 252
§7. Wavelet Decomposition 255
§8. Extensibility 260
References 263
Chapter 33. Elastica and Minimal-Energy Splines 266
§1. Elastica 266
§2. Minimal-energy Spline Segments 267
§3. Minimal-energy Splines 268
References 269
Chapter 34. A Distributed Algorithm for Surface/Plane Intersection 270
§1. Introduction 270
§2. The Subdivision Step 271
§3. The Intersection Step 271
§4. Experimental Examples 271
§5. Conclusions 273
References 273
Chapter 35. Construction of Exponential Tension B-splines of Arbitrary Order 274
§1. B-splines 274
§2. Conversion to a Bézier-like Form 275
References 277
Chapter 36. On the Almost Sure Limit of Probabilistic Recovery 278
References 286
Chapter 37. A New Curve Tracing Algorithm and Some Applications 288
§1. Introduction 288
§2. Curve Tracing Algorithm 289
§3. Applications 290
References 291
Chapter 38. Pseudo-Cubic Weighted Splines Can Be C2 or G2 292
§1. Introduction 292
§2. General Weighted Interpolating Spline 292
§3. Interpolating q-spline of 
293 
§4. Smoothing q-spline of 
294 
§5. Polar Representation of 
294 
References 295
Chapter 39. Composite Cr-Triangular Finite Elements of PS Type on a Three Direction Mesh 296
§1. Introduction 296
§2. PS Triangles of Class C2s 297
§3. PS Triangles of Class C2s+1 298
References 298
Chapter 40. Dynamic Segmentation: Finding the Edge With Snake Splines 300
§1. Introduction 300
§2. Modelling Curves Or Surfaces With Spline Functions 301
§3. Strength Fields and Potential Convolving 302
§4. Adaptative Evolution of the Snake-Spline 303
§5. Preliminary Results 304
§6. Conclusion 304
References 304
Chapter 41. Recent Developments in the Strang-Fix Theory for Approximation Orders 306
§1. Introduction 306
§2. The Strang-Fix Theory 308
§3. Functions with Rapid Decay 309
References 313
Chapter 42. Aligning Frames with the Tangent Curve of a B-spline Curve 314
§1. Introduction 314
§2. Linear Interpolation of Frames in 2D 315
§3. 5-spline Approximation of Frames in 2D 316
§4. Frame Splines in 3D 317
References 317
Chapter 43. Error Estimates for Interpolation by Generalized Splines 318
§1. Introduction 318
§2. Variational Formulation 319
§3. Error Estimates 322
§4. Miscellaneous Examples and Remarks 325
References 326
Chapter 44. Varying the Shape Parameters of Rational Continuity 328
§1. Introduction 328
§2. Rational Continuity 329
§3. Basis Functions 330
§4. Conclusion 333
References 333
Chapter 45. Detecting Cusps and Inflection Points in Curves 336
§1. Introduction 336
§2. Parametric Curves 337
§3. Necessary Condition for Cusps 337
§4. Necessary and Sufficient Condition for Cusps 338
§5. Proper Parametrizations 339
References 340
Chapter 46. Image-like Surfaces: Parallel Least Squares Approximation Methods 342
§1. Introduction 342
§2. Parallel Least Squares Methods 343
References 345
Chapter 47. Local Kriging Interpolation: Application to Scattered Data on the Sphere 346
§1. Introduction 346
§2. Local Kriging Interpolation or Approximation on the Sphere 346
§3. Conclusion 349
References 349
Chapter 48. Best Approximation of Circle Segments by Quadratic Bézier Curves 352
§1. Introduction 352
§2. Interpolatory Approximation 353
§3. General Quadratic Approximation 354
§4. Numerical Comparisons 357
Acknowledgements 357
References 357
Chapter 49. A Procedure for Determining Starting Points for a Surface/Surface Intersection Algorithm 358
§1. Introduction 358
§2. The Algorithm in Connection with the SSI Algorithm 359
§3. Correctness 360
§4. Efficiency 361
References 361
Chapter 50. Norms of Inverses for Matrices Associated with Scattered Data 362
§1. Introduction 362
References 368
Chapter 51. 2D Sampling in Tomography 370
§1. Introduction 370
§2. The Radon Inversion Formula 371
§3. Shannon's Sampling Theorem 372
§4. The Sampling Theorem of Petersen-Middleton 373
§5. The Sampling Theorem of Beurling 375
§6. Fourier Reconstruction 376
References 377
Chapter 52. Subdividing Multivariate Polynomials Over Simplices in Bernstein-Bézier Form Without de Casteljau Algorithm 380
§1. Introduction 380
§2. Three Alternatives for Subdividing Bernstein Polynomials 381
§3. Remarks 383
References 383
Chapter 53. Geometrically Smooth Interpolation by Triangular Bernstein-Bézier Patches With Coalescent Control Points 384
§1. Description of the Problem 384
§2. Degenerate Polynomial Patches 386
References 387
Chapter 54. Curve Fitting Using NURBS 388
§1. Introduction 388
§2. Fitting a Parametric Quadratic B-spline Curve 388
§3. Searching Conies 389
§4. NURBS Representation of Conies 390
§5. NURBS Representation of the Whole Curve 391
References 391
Chapter 55. Univariate Multiquadric Interpolation: Some Recent Results 392
§1. Introduction 392
§2. Conditions for O(h) Accuracy 395
§3. Conditions for 
399 
§4. A Property of the 
400 
§5. Discussion 402
References 402
Chapter 56. Periodic Spline Interpolation of Functions of Bounded Variation 404
§1. Introduction 404
§2. Statement of Results 405
§3. Proofs 406
References 407
Chapter 57. Least Squares Fitting by Linear Splines on Data Dependent Triangulations 408
§1. Introduction 408
§2. Choosing the Vertex Set 409
§3. Estimating the Vertex Values 409
§4. Data Dependent Triangulations 409
§5. Remarks 411
References 411
Chapter 58. How to Build Quasi-Interpolants: Application to Polyharmonic B-Splines 412
Introduction 412
§1. "Elementary" Quasi-interpolants 414
§2. "High Level Quasi-interpolants" 418
§3. Conclusion 422
References 423
Chapter 59. Algorithms for Local Convexity of Bézier Curves and Surfaces 424
§1. Introduction 424
§2. Bézier Curves 424
§3. Bézier Surfaces 426
§4. Conclusions 427
References 427
Chapter 60. Polynomial N-sided Patches 428
§1. Introduction 428
§2. Tangent Plane Continuity Between Two Patches 429
§3. Twist Compatibility 429
§4. Construction of Cross Derivatives 430
§5. Patch Equation 431
References 431
Chapter 61. Cubic Recursive Division With Bounded Curvature 432
§1. Introduction 432
§2. Definitions 433
§3. Cubic Constructions 434
§4. A New Construction 435
§5. Properties 435
§6. Limitations 435
References 435
Chapter 62. .-Convergence, A Criterion for Linear Approximation 436
§1. Definitions 436
§2. The Principle 437
§3. B-spline Approximation 437
§4. Tri-quadratic Interpolation 440
Chapter 63. Bernstein-Type Quasi-Interpolants 442
§1. Introduction 442
§2. Construction of Operators 443
§3. Some Applications 445
References 447
Chapter 64. Extension of the Problem of Best Interpolating Parametric Curves to L-Splines 448
§1. Introduction 448
§2. Existence 449
§3. Characterization 451
§4. Uniqueness 452
References 453
Chapter 65. Adaptive G1 Approximation of Range Data Using Triangular Patches 454
§1. Introduction 454
§2. Adaptive Triangulation 454
§3. A Triangular Gregory-Bézier Patch Model 456
§4. Adaptive G1 Approximation using Triangular Patches 456
References 457
Chapter 66. Universal Splines and Geometric Continuity 458
§1. Introduction 458
§2. Geometrie Continuity 459
§3. Universal Splines 460
§4. Constructing the Spline Control Points 462
§5. Constructing the Bézier Points 464
§6. Conclusion 465
References 465
Chapter 67. Procedural Construction of Patch-Boundary Curves 466
§1. The Construction Process 466
§2. Choice of Curve Parameters 467
§3. Opposite Edge Method 467
§4. Filling in Patches and Shape Parameters 469
References 469
Chapter 68. Efficient Computation of Multiple Knots Nonuniform Spline Functions 470
§1. Introduction 470
§2. The Derivative/summation Approach 471
§3. Generation of Q(u) from 
472 
References 473
Chapter 69. Chebyshev Approximation in IRn by 
474 
§1. Introduction 474
§2. Approximation by a Parameterized Curve 475
§3. Approximation by a Straight Line 475
§4. Approximation by Linear Manifolds 476
References 476
Chapter 70. A Building Method for Hierarchical Covering Spheres of a Given Set of Points 478
§1. The Covering Spheres 478
§2. Building the Spheres 479
§3. Uses 480
References 481
Chapter 71. The Variational Approach to Shape Preservation 482
§1. Introduction 482
§2. The Norm Minimizing Constrained Splines 483
§3. The Semi-Norm Case 488
§4. Numerical Algorithms 492
§5. Best Constrained Interpolation in Banach Spaces 493
References 495
Chapter 72. Spline Fitting Numerous Noisy Data With Discontinuities 498
§1. Introduction 498
§2. B-spline Fitting 499
§3. Knots Determination 500
§4. Discussion 501
References 501
Chapter 73. B-Spline Surfaces for Real-time Shape Design 502
§1. Introduction 502
§2. B-spline Surface Generation 503
§3. Implementation and Further Research 505
References 505
Chapter 74. Conversion of a Composite Trimmed Bézier Surface Into Composite Bézier Surfaces 506
§1. Introduction 506
§2. Definition of the Problem 506
§3. Conditions for a Solution 507
§4. The Algorithm 508
§5. Unsolved Problems 509
§6. Conclusions 510
References 510
Chapter 75. Multivariate Model Building With Additive Interaction and Tensor Product Thin Plate Splines 512
§1. Introduction 512
§2. Multiple Smoothing Parameters, Splines Based on W2m 514
§3. Thin Plate Splines 516
§4. Grand Models, Model Selection 518
§5. An Example 519
§6. Diagnostics 519
§7. Post Analysis Diagnostics, or Accuracy Statements 521
References 524
Chapter 76. Recursion Relations for 4 x 4 Determinants Related to Rational Cubic Bézier Curves 526
§1. Introduction 526
§2. The 4 x 4 Determinant Method 527
§3. The 4 x 4 Determinants before and after Subdivision 529
§4. Conclusion 531
References 531
Chapter 77. Lagrange Interpolation by Quadratic Splines On a Quadrilateral Domain of 
532 
§1. Introduction 532
§2. Notations and Problem 532
§3. Interpolation Operator and Error Estimates for a Square 533
Acknowledgements 535
References 535

Produkt-Details

• Erscheint lt. Verlag: 12.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• ISBN-10: 1-4832-6387-8 / 1483263878
• ISBN-13: 978-1-4832-6387-8 / 9781483263878