Spline and spline wavelet methods with applications to signal and image processing Volume III Selected Topics

Preface......Page 6
 References......Page 12
Acknowledgements......Page 14
Contents......Page 15
Acronyms......Page 21
  1.1.1 One-Dimensional ¶ Discrete-Time Signals......Page 23
  1.1.2 Two-Dimensional Periodic Discrete-Time Signals......Page 26
  1.2.1 Definition of Periodic Filters......Page 27
  1.2.2 Multirate p-Filtering......Page 28
  1.3.1 Modulation Matrices of p-Filter Banks......Page 30
  1.3.2 Polyphase Matrices of p-Filter Banks......Page 31
 References......Page 32
 2.1 Periodic B-Splines and Spline Spaces......Page 33
 2.2 Exponential Splines and Characteristic Sequences......Page 35
  2.3.1 Orthogonal Periodic Exponential Splines......Page 36
  2.3.2 Representation of Periodic Splines by Exponential Spline Bases......Page 37
 2.4 SHA in Two-Dimensional Spline Spaces......Page 41
 References......Page 42
  3.1.1 B-Splines and Discrete Spline Spaces......Page 43
  3.1.2 Exponential Discrete Splines and Characteristic Sequences......Page 45
  3.1.3 Interpolating and Quasi-interpolating Discrete Splines......Page 47
  3.1.4 Orthonormal Periodic Discrete Splines......Page 49
 3.2 Two-Dimensional Discrete Splines of Span 2......Page 52
  3.3.1 B-Splines and Discrete-Time Spline Spaces......Page 54
  3.3.2 Exponential Discrete-Time Splines and Characteristic Sequences......Page 57
  3.3.3 Orthonormal Periodic Discrete-Time Splines......Page 59
  3.3.4 Interpolating Discrete-Time Splines......Page 61
 References......Page 62
  4.1.1 Orthogonal Complement for Subspace  2rmathcalS[1]0......Page 63
  4.1.2 Orthogonal Complement for Subspace  pbarmathcalS[1]0......Page 66
  4.1.3 One-Level Wavelet Transform of a Signal......Page 67
  4.2.1 P-Filter Banks for the Second Decomposition Level......Page 70
  4.2.2 Two-Level Wavelet Transforms......Page 72
  4.2.3 Two-Level Wavelet Packet Transforms......Page 77
  4.2.4 Selection of Wavelet-Packet Basis......Page 81
  4.3.1 Discrete-Spline Wavelet Packets......Page 83
  4.3.2 Discrete-Time-Spline Wavelet Packets......Page 85
 4.4 Best Basis and Signal\'s Restoration from Wavelet Packet Bases......Page 86
 4.5 Illustrations......Page 88
 4.6 Waveband Spectra......Page 90
 References......Page 94
  5.1.1 Wavelets of the First Decomposition Level......Page 95
  5.1.2 Multi-level 2D Wavelet Transforms......Page 98
  5.2.1 Direct Wavelet Packet Transforms......Page 100
  5.2.2 Best Basis and Restoration of 2D Signals From Wavelet Packet Bases......Page 102
 6.1 Preliminaries......Page 105
 6.2 Local Cubic Splines......Page 108
  6.2.1 Simplest Cubic Splines......Page 109
  6.2.2 Quasi-interpolating Cubic Splines......Page 110
  6.2.3 Two-Dimensional Cubic Splines......Page 119
  6.3.1 Design of Quadratic Quasi-interpolating Splines......Page 122
  6.3.2 Extension of Splines to Boundaries and Extrapolation......Page 127
  6.3.3 Quadratic Splines on Uniform Grids......Page 129
  6.4.1 Cubic Splines......Page 132
  6.4.2 Quadratic Splines......Page 133
 References......Page 135
  7.1.1 One-Level Transform......Page 136
 7.2 Spline-Based Prediction and Updating Operators......Page 138
  7.2.1 Prediction and Updating Operators – Cubic Splines......Page 139
  7.2.2 Prediction and Updating Operators – Quadratic Splines......Page 150
 References......Page 153
  8.1.1 Lifting Steps......Page 154
  8.1.2 Filter Banks......Page 155
  8.2.2 Filter Banks......Page 157
  8.3.1 Preliminaries......Page 161
  8.3.2 Filter Banks......Page 163
 8.4 Discrete Vanishing Moments......Page 166
 References......Page 168
9 Wavelet Frames Generated by Spline-Based p-Filter Banks......Page 169
  9.1.1 Oversampled p-Filter Banks with Downsampling Factor 2......Page 170
  9.1.2 Frames in the Space of Periodic Signals......Page 171
  9.2.1 Four-Channel Perfect Reconstruction p-Filter Banks......Page 176
  9.2.2 Symmetric Tight Frames with Increased Redundancy......Page 181
  9.3.1 Examples of Six-Channel p-Filter Banks Generating Tight Frames......Page 182
  9.3.2 Examples of Eight-Channel FIR p-Filter Banks Generating Tight Frames......Page 186
  9.3.3 Four-Channel p-Filter Banks with IIR p-Filters Derived from Discrete Splines......Page 190
  9.3.4 Example of Image Restoration Using Tight Frames......Page 192
 References......Page 195
 10.1 Background......Page 196
  10.2.1 Optical Scheme......Page 197
 10.3 Compressed Sensing Model for Spectral Imaging......Page 199
  10.3.1 Discrete Mathematical Model......Page 200
  10.4.1 Rationale Behind the Method......Page 202
  10.4.2 Outline of Algorithm......Page 203
 10.5 Experimental Optical Arrangement and Calibration......Page 204
 10.6 Optical Experiments for Spectral Imaging......Page 206
 10.7 Comments......Page 213
 References......Page 214
 11.1 Background......Page 216
  11.2.1 Imaging Device......Page 217
  11.2.2 Outline of the Processing Scheme......Page 218
  11.2.3 Selection of Training Data......Page 220
  11.2.4 Preprocessing the Training Data......Page 221
  11.2.5 Selection of Characteristic Features......Page 223
  11.2.6 Dimension Reduction......Page 226
  11.2.7 Decision Units......Page 228
  11.2.8 Identification Phase......Page 229
 11.3 Experimental Results......Page 230
  11.5.1 Diffusion Maps......Page 232
  11.5.2 Geometric Harmonics......Page 234
 11.6 Appendix: Outline of the CART Algorithm......Page 235
 References......Page 237
 12.1 Introduction......Page 238
 12.2 Structure of the Recorded Acoustics Signals......Page 239
 12.3 Formulation of the Approach......Page 241
  12.3.1 Outline of the Approach......Page 242
  12.4.1 The Algorithm......Page 243
  12.4.2 Choice of the Analyzing Waveforms......Page 244
  12.4.3 Training the Algorithm......Page 245
  12.4.4 Identification of an Acoustic Signal......Page 247
 12.5 Experimental Results......Page 249
  12.5.1 Detection Experiments......Page 250
  12.5.2 Examples......Page 252
 12.6 Appendix: Random Search for a Near Optimal Footprint (RSNOFP) Scheme......Page 255
 References......Page 258
  13.1.1 Background......Page 259
  13.1.2 Outline of the Methodology......Page 261
  13.1.3 Time-Domain Meaning of Wavelet Packet Transform Coefficients......Page 262
 13.2 WPT-Based Methodology for the Fault Detection......Page 263
  13.2.1 Structure of the Recorded Data......Page 264
  13.2.2 Outline of the Research Methodology......Page 266
  13.3.1 Experiments with Bearing1......Page 267
  13.3.2 Experiments with Bearing2......Page 273
 13.4 Comments......Page 277
 13.5 Appendix: Envelope......Page 278
 References......Page 280
Appendix AGuide to SB_Soft1......Page 281
Glossary......Page 292
Bibliography......Page 295
Index......Page 300