Spline and Spline Wavelet Methods with Applications to Signal and Image Processing: Volume I: Periodic Splines

Preface......Page 6
Contents......Page 17
  1.1.1 One-Dimensional Periodic Signals......Page 24
  1.1.2 Two-Dimensional Periodic Signals......Page 26
 1.2 Periodic Discrete-Time Periodic Signals......Page 27
  1.2.1 One-Dimensional Periodic Discrete-Time Signals......Page 28
  1.2.2 Two-Dimensional Periodic Discrete-Time Signals......Page 30
 References......Page 31
  2.1.1 Definition of Periodic Filters......Page 32
  2.1.2 Multirate p-Filtering......Page 33
  2.2.1 Filter Banks......Page 35
  2.2.2 Characterization of p-Filter Banks......Page 36
 Reference......Page 37
 3.1 Periodic Discrete-Continuous Circular Convolution  and Zak Transform......Page 38
 3.2 A Leading Example: Periodic Polynomial Splines......Page 41
  3.2.1 Periodic B-Splines......Page 42
  3.2.2 Spaces of Periodic Splines......Page 44
 3.3 Periodic Discrete--Discrete Convolution  and Zak Transform......Page 47
  3.4.2 Periodic Discrete B-Splines......Page 51
  3.4.3 Spaces of Periodic Discrete Splines......Page 52
 References......Page 56
  4.1.1 Orthogonal Periodic Exponential Splines......Page 58
  4.1.2 Normalized Periodic Exponential Splines......Page 62
  4.1.3 Representation of Periodic Splines by Exponential  Splines Bases......Page 63
  4.1.4 Generators of Periodic Splines Spaces......Page 67
  4.1.5 Remarks on Orthonormal Bases of Splines Spaces  and SHA Spectra......Page 76
 4.2 SHA in Two-Dimensional Spline Spaces......Page 79
 References......Page 81
 5.1 One-Dimensional Smoothing Splines......Page 82
  5.1.1 Solution of the Unconstrained Minimization Problem......Page 83
  5.1.2 Solution of the Constrained Minimization Problem......Page 85
  5.1.3 Generators of Smoothing Splines......Page 87
 5.2 Two-Dimensional Smoothing Splines......Page 89
 References......Page 91
6 Calculation of Splines Values by Subdivision......Page 92
 6.1 Interpolatory Subdivision......Page 93
  6.2.1 Spline Spaces on Different Dyadic Resolution Scales......Page 94
  6.2.2 Insertion Rule......Page 96
  6.2.3 Periodic Spline Filters for Binary Subdivision......Page 97
  6.2.4 Computation of Periodic Splines at Dyadic Rational Points......Page 99
  6.3.1 Super-Resolution Spline Spaces (Triadic Scale)......Page 100
  6.3.3 Periodic Spline Filters for Ternary Subdivision......Page 102
  6.3.4 Computation of Periodic Splines at Triadic Rational Points......Page 105
  6.3.5 Practical Implementation......Page 107
 6.4 Two-dimensional Spline Subdivision......Page 108
  6.5.1 Upsampling Discrete-Time signals......Page 109
  6.5.2 Upsampling Digital Images......Page 112
 References......Page 118
 7.1 Spline Algorithms for Deconvolution......Page 119
  7.1.1 Solution of One-Dimensional Convolution Equations......Page 120
  7.1.2 Solution of Two-Dimensional Convolution Equations......Page 128
 7.2 Inversion of Heat Equation......Page 135
  7.2.1 Inversion of One-Dimensional Heat Equation......Page 139
  7.2.2 Inversion of Two-Dimensional Heat Equation......Page 143
 References......Page 153
8 Periodic Spline Wavelets and Wavelet  Packets......Page 154
 8.1 Spline Spaces of Different Resolution Scales......Page 155
  8.2.1 Two-Scale Relations in Spline Spaces......Page 157
  8.2.2 Orthogonal Complements to Spline Spaces......Page 158
  8.2.3 Refined Split of the Spline Space into Orthogonal  Subspaces......Page 160
  8.3.1 Transform Matrices......Page 163
  8.3.2 One-Level Transforms......Page 164
  8.3.3 Transforms to Coarser Levels......Page 172
  8.4.1 Definition of Spline Wavelet Packets......Page 178
  8.4.2 The SHA Spectra of Spline Wavelet Packets......Page 179
  8.4.3 Wavelet Packets Bases......Page 182
  8.4.4 Best Basis......Page 185
  8.4.5 Reconstruction of a Spline from Wavelet Packet Bases......Page 190
  8.5.1 2D Wavelet Packets Bases......Page 191
  8.5.2 2D Wavelet Packets Transforms......Page 193
  8.5.3 Best Basis......Page 201
 References......Page 203
  9.1.1 Summary of One-Level Wavelet Transform of a Spline......Page 204
  9.1.2 One-Level Wavelet Transform of a Signal......Page 207
  9.1.3 Multilevel Wavelet Packet Transforms of a Signal......Page 212
  9.1.4 Bases of Discrete-Time Wavelet Packets......Page 215
  9.2.1 One-Level 2D Wavelet Transform......Page 225
  9.2.2 Multilevel Wavelet Packet Transform......Page 227
  9.2.3 2D Wavelet Bases......Page 230
10 Deconvolution by Regularized  Matching Pursuit......Page 236
  10.1.1 Conventional MP......Page 237
  10.1.2 Outline of the Regularized MP for Deconvolution......Page 238
 10.2 Description of the RMP for Deconvolution......Page 239
  10.2.1 Dictionaries for RMP......Page 240
  10.2.2 Modelling Noise......Page 241
  10.2.3 Oblique Projection......Page 242
  10.2.4 Stopping Threshold and Reduction of Dictionaries......Page 246
  10.2.5 Pursuit......Page 248
  10.2.6 Computational Scheme......Page 249
  10.3.1 Dictionaries for Heat Equation RMP......Page 252
  10.3.2 Computational Scheme......Page 254
 10.4 Examples......Page 257
  10.4.1 Conclusions......Page 266
 References......Page 267
11 Block-Based Inversion of the Heat Equations......Page 268
  11.1.1 Preliminaries......Page 269
  11.1.2 Partial Solution of the Inversion Problem  in the Subspace 2rmathcalWm,l......Page 271
  11.1.3 Approximate Inversion of Heat Equation  in the Space 2rmathcalS......Page 276
  11.2.1 Preliminaries......Page 279
  11.2.2 Partial Solution of the Inversion Problem  in the Subspace 2rmathcalWm,l,tildel......Page 280
  11.2.3 Scheme for the Approximate Inversion  of the 2D Heat Equation......Page 285
 11.3 Numerical Examples......Page 288
 References......Page 294
 12.1 Outline of the Problem......Page 295
 12.2 Structure of the Recorded Data......Page 297
 12.3 Formulation of the Approach......Page 298
  12.4.1 Derivation of the Classes Signatures......Page 301
  12.4.2 Training the Classifiers......Page 305
  12.4.3 Identification of an Acoustic Signal......Page 307
 12.5 Examples......Page 308
  12.5.1 Structure of Presented Figures......Page 309
  12.5.2 Display of Results......Page 310
 References......Page 314
  13.1.1 Discrete Periodic Splines Spaces......Page 315
  13.1.2 Exponential Discrete Periodic Splines......Page 317
  13.1.3 Normalized Exponential Discrete Periodic Splines......Page 322
  13.1.4 Representation of Discrete Periodic Splines by Exponential Splines Basis......Page 324
  13.1.5 Generators of Periodic Discrete Splines\' Spaces......Page 327
  13.1.6 Smoothing Periodic Discrete Splines......Page 334
  13.2.1 Bases in Two-dimensional Discrete Spline Spaces......Page 339
  13.2.2 Outline of 2D DSHA Relations......Page 341
  13.2.3 Explicit Computation of a 2D Periodic Discrete Spline......Page 342
  13.2.4 2D Smoothing Periodic Discrete Splines......Page 344
 References......Page 350
 14.1 Two-Scale Relations......Page 351
  14.1.1 Orthogonal Complements of Discrete Spline Spaces......Page 352
  14.1.2 Refined Split of the Signal Space Π[N] into Orthogonal Subspaces......Page 353
  14.2.1 Transform Matrices......Page 355
  14.2.2 One-Level Transforms......Page 356
  14.2.3 Multilevel Wavelet Packet Transforms......Page 360
 14.3 Discrete Spline Wavelet Packets......Page 364
  14.3.1 The DFT Spectra of Spline Wavelet Packets......Page 366
  14.3.2 Discrete Splines Wavelet Packets Bases......Page 369
 14.4 2D Wavelet Packets......Page 372
  14.4.1 2D Discrete Spline Wavelet Packets Bases......Page 374
  14.4.2 2D Wavelet Packets Transforms......Page 376
  14.5.1 Outline of the RMP for Deconvolution......Page 382
  14.5.2 Computational Scheme......Page 385
  14.5.3 Numerical Examples......Page 388
 References......Page 396
15 Biorthogonal Wavelet Transforms......Page 397
  15.1.1 Matrix Expression of p-Filter Banks......Page 398
  15.1.2 Biorthogonal Bases Generated by PR p-Filter Banks......Page 400
  15.1.3 Multilevel Wavelet Transforms......Page 402
  15.1.4 Implementation of Multi-Level Wavelet Transforms......Page 406
 15.2 Compactly Supported Biorthogonal Wavelets......Page 408
  15.2.1 Biorthogonal Low-Pass FIR p-Filters......Page 409
  15.2.2 Examples......Page 410
  15.3.1 Restoration of Sampled Polynomials......Page 414
  15.3.2 Discrete Vanishing Moments......Page 415
  15.3.3 Examples......Page 416
 References......Page 417
  16.1.1 Lifting Steps......Page 418
  16.1.2 Filter Banks......Page 419
 16.2 Prediction Filters Derived from Polynomial Splines......Page 422
  16.2.1 Periodic Interpolating Splines......Page 423
  16.2.2 Prediction p-filters......Page 424
  16.2.3 Approximation Properties of Spline p-filters......Page 425
  16.2.4 Perfect Reconstruction p-filter Banks......Page 426
  16.2.5 Examples of p-filters Derived from Polynomial Splines......Page 427
  16.3.1 Summary for the Discrete Splines of Span 2......Page 431
  16.3.2 Filter Banks......Page 432
 References......Page 435
17 Wavelet Frames Generated by Spline Based p-Filter Banks......Page 436
  17.1.1 Oversampled p-Filter Banks with Downsampling  Factor of 2......Page 437
  17.1.2 Frames in the Space of Periodic Signals......Page 439
  17.2.1 Interpolating p-Filter Banks for Frame Generation......Page 445
  17.2.2 Tight and Semi-Tight Frames......Page 448
  17.2.3 Interpolating Three-Channel p-Filter Banks Using Spline Filters......Page 449
  17.3.1 Four-Channel Perfect Reconstruction p-Filter Banks......Page 457
  17.3.2 Low-Pass p-Filters......Page 462
 17.4 Four-Channel p-Filter Banks Using Spline Filters......Page 464
  17.4.1 Examples of p-Filter Banks With p-FIR p-Filters......Page 466
  17.4.2 Four-Channel p-Filter Banks With IIR p-Filters......Page 474
 References......Page 482
 18.1 Outline of the Restoration Scheme......Page 484
 18.2 Numerical Examples......Page 485
  18.2.1 Restoration Experiments with the ``Window\'\' Image......Page 486
  18.2.2 Restoration Experiments with the ``Barbara\'\' Image......Page 489
  18.2.3 Restoration Experiments with the ``Boats\'\' Image......Page 492
  18.2.5 Restoration Experiments with the ``Fingerprint\'\' Image......Page 494
 18.3 Comments on the Experiments......Page 495
 References......Page 497
 Guide to SplineSoftP......Page 498
 Glossary......Page 508
Index......Page 511