Spline and Spline Wavelet Methods with Applications to Signal and Image Processing: Volume II: Non-Periodic Splines

Preface......Page 6
 References......Page 12
Contents......Page 17
Acronyms......Page 23
  1.1.1 One-Dimensional Signals......Page 24
  1.1.2 Two-Dimensional Signals......Page 27
  1.2.1 One-Dimensional Discrete-Time Signals......Page 28
  1.2.2 Two-Dimensional Discrete Signals......Page 30
 References......Page 31
  2.1.1 Filters......Page 32
  2.1.3 Polyphase Representation......Page 36
  2.1.4 Interpolating Filters......Page 40
 2.2 Bases and Frames Generated by Filter Banks......Page 43
  2.2.1 Examples of Filter Banks Implementation......Page 44
  2.2.2 Implementation of Causal---Anticausal IIR Filters  with a Rational Transfer Function......Page 48
 2.3 Discrete-Time Butterworth Filters......Page 50
 References......Page 53
  3.1.1 Mixed Discrete-Continuous Convolution......Page 54
  3.1.2 Zak Transform of Continuous-Time Signals......Page 55
  3.2.1 B-Splines......Page 58
  3.2.2 Spline Spaces......Page 60
 3.3 Discrete-Discrete Convolution and Zak Transform......Page 63
  3.3.1 A Leading Example: Discrete Splines......Page 66
 References......Page 72
  4.1.1 Decaying Polynomial Splines......Page 74
  4.1.2 Zak Transform of B-Spline (Exponential Spline)......Page 75
  4.1.3 Integral Representation of Splines......Page 77
 4.2 Generators of Spline Spaces......Page 81
  4.2.1 Examples of the Spline Space Generators......Page 82
 4.3 Polynomial Interpolating Splines......Page 91
  4.3.1 Example: Cubic Interpolating Spline......Page 92
  4.3.2 Exponential Decay of Generators of Spline Spaces......Page 94
  4.3.3 Minimal Norm Property of Even-Order Splines......Page 96
 4.4 Polynomial Smoothing Splines (Global)......Page 97
  4.4.1 Explicit Expression of Smoothing Splines......Page 98
  4.4.2 Computation of Smoothing Splines......Page 101
  4.4.3 Smoothing Generators for the Spline Spaces......Page 102
  4.4.4 Examples......Page 105
 References......Page 108
 5.1 Moments of B-Splines......Page 109
  5.2.1 Simplest (Variation-Diminishing) Splines......Page 112
  5.2.2 Quasi-interpolating Splines......Page 113
  5.2.3 Examples......Page 127
 5.3 Approximation Properties of Splines......Page 130
 References......Page 135
 6.1 Preliminaries: Divided Differences  and Interpolating Polynomials......Page 137
 6.3 Cubic Splines......Page 139
  6.3.1 Simplest Cubic Splines......Page 140
  6.3.2 Quasi-interpolating Cubic Splines......Page 141
 6.4 Examples......Page 145
 References......Page 147
 7.1 Interpolatory Subdivision for Non-periodic Splines......Page 149
 7.2 Binary Subdivision for Non-periodic Splines......Page 150
  7.2.1 Spline Filters for Binary Subdivision......Page 151
  7.2.2 Splines Computation at Dyadic Rational Points......Page 153
  7.2.3 Practical Implementation......Page 154
  7.3.1 Spline Filters for Ternary Subdivision......Page 157
  7.3.2 Splines Computation at Triadic Rational Points......Page 159
  7.3.3 Practical Implementation......Page 162
 7.4 Upsampling Examples......Page 164
 References......Page 166
8 Polynomial Spline-Wavelets......Page 167
  8.1.1  Two-scale Relations for Exponential Splines......Page 169
  8.1.2 Exponential Wavelets......Page 171
  8.2.1 One Step of the Spline Wavelet Transforms......Page 173
  8.2.2 Multiscale Spline Wavelet Transforms......Page 177
 8.3 Generators of the Wavelet Space......Page 178
  8.3.1 B-Wavelets......Page 179
  8.3.2 Generators and Dual Generators......Page 184
  8.3.3 Examples of the Wavelet-space Generators......Page 186
 References......Page 191
 9.1 Discrete Splines\' Spaces......Page 192
  9.2.1 Exponential Discrete Splines......Page 194
  9.2.2 Characteristic Functions of the Discrete Splines\' Spaces......Page 196
  9.2.3 Implications of the Integral Representation  of Discrete Splines......Page 198
  9.2.4 Computation of Discrete Splines......Page 199
  9.3.1 Generators and Their Duals......Page 202
  9.3.2 Examples of Generators......Page 204
 References......Page 209
 10.1 Discrete Splines\' Spaces......Page 210
 10.2 Discrete Exponential Wavelets......Page 212
 10.3 Generators of the Discrete Spline Wavelet Spaces......Page 214
  10.3.1 Discrete B-Wavelets......Page 215
  10.3.2 Generators and Dual Generators of Discrete Spline Wavelet Spaces......Page 217
  10.4.1 Matrix Representation of the Two-Scale Relations......Page 223
  10.4.2 Transform of Splines\' Coordinates......Page 224
  10.5.1 One Step of Discrete-Spline Wavelet Transform......Page 227
  10.5.3 Examples......Page 232
 References......Page 235
11 Biorthogonal Wavelet Transforms......Page 236
  11.1.1 Matrix Expression of Filter Banks......Page 237
  11.1.2 Biorthogonal Bases Generated by PR Filter Banks......Page 239
  11.1.3 Multilevel Discrete-Time Wavelet Transforms......Page 242
  11.2.1 Design of the Biorthogonal Filter Bank......Page 249
 References......Page 258
  12.1.1 Primal Scheme......Page 259
  12.1.2 Dual Scheme......Page 260
  12.1.3 Filter Banks......Page 261
  12.2.1 Prediction Filters Derived from Polynomial Splines......Page 265
  12.2.2 Filter Banks......Page 267
  12.2.3 Examples of Filters Originating from Splines......Page 272
  12.3.1 Summary for the Discrete Splines of Span 2......Page 281
  12.3.2 Prediction Filters......Page 283
 References......Page 293
 13.1 Spatial and Spectral Meaning of Wavelet Transform Coefficients......Page 294
 13.2 SPIHT Coding Scheme......Page 298
  13.3.1 Local Cosine Transform (LCT)......Page 300
  13.3.2 A Hybrid Algorithm for Data Compression......Page 301
 13.4 Numerical Examples......Page 305
  13.4.1 Seismic Compression......Page 306
  13.4.2 Fingerprints......Page 308
  13.4.3 Compression of Multimedia Images......Page 310
  13.4.4 Conclusions......Page 314
 References......Page 315
14 Wavelet Frames Generated by Perfect Reconstruction Filter Banks......Page 317
  14.1.1 Matrix Expression of Filter Banks  with the Downsampling Factor 2......Page 318
  14.1.2 Frames Generated by Filter Banks......Page 321
  14.2.1 Interpolating Filter Banks for Frame Generation......Page 326
  14.2.2 Frames Derived from Triangular Factorization......Page 328
  14.2.3 Design of Frames Using Spline Filters......Page 330
  14.2.4 2D Frame Transforms......Page 347
  14.3.1 Four-Channel Oversampled Filter Banks......Page 349
  14.3.2 Low-Pass Filters......Page 354
  14.4.1 Summary of the Filter Bank Design Scheme......Page 358
  14.4.2 Outline of the Frame Transforms\' Implementation......Page 359
  14.4.3 Examples of Filter Banks with FIR Filters......Page 361
  14.4.4 Four-Channel Filter Banks with IIR Filters......Page 368
 References......Page 378
15 Biorthogonal Multiwavelets Originated  from Hermite Splines......Page 380
  15.1.1 Cubic Hermite Splines......Page 381
  15.1.2 Multifilters......Page 382
 15.2 Lifting Scheme of Wavelet Transform  of Vector-Signals......Page 385
  15.3.1 Structure of Multifilter Banks......Page 387
  15.3.2 Approximation Properties of Multifilters......Page 389
 15.4 Lifting Algorithms for Pre/Post-processing Phases......Page 391
  15.4.1 An Orthogonal Scheme of Third Approximation  Order (Haar Algorithm)......Page 392
  15.4.2 Schemes of the Fifth Approximation Order......Page 393
  15.5.1 Bases of Zero Level......Page 397
  15.5.2 Bases of the First Level......Page 398
 15.6 Extension of the Multiwavelet Transforms  to Coarser Levels......Page 405
 15.7 Two-Dimensional Multiwavelet Transforms......Page 408
 References......Page 409
16 Multiwavelet Frames Originated  From Hermite Splines......Page 410
  16.1.1 Three-Channel Multifilter Banks......Page 411
  16.1.2 Bases of Zero Level......Page 412
  16.1.3 Analysis Filter Banks of the First Level......Page 413
  16.1.4 Synthesis Filter Banks of the First Level......Page 414
  16.2.1 Signal\'s Expansion Over the First-Level Multi-frame......Page 415
  16.2.2 Extension of the Multi-frame Transforms  to Coarser Levels......Page 416
  16.3.1 Design Scheme......Page 418
  16.3.2 Transfer Functions of the Multifilter Banks......Page 419
  16.3.3 Example: Framelets Originating From the Haar Pre-Post-processing Scheme (j=0):......Page 421
 References......Page 424
Appendix AGuide to SplineSoftN......Page 425
Glossary......Page 436
Index......Page 438