Fundamentals of Wavelets. Theory, Algorithms, and Applications

Cover Page......Page 1
WILEY SERIES IN MICROWAVE AND OPTICAL ENGINEERING......Page 2
Title Page......Page 8
ISBN 9780470484135......Page 9
2 Mathematical Preliminary 6......Page 11
4 Time-Frequency Analysis 61......Page 12
6 Construction of Wavelets 114......Page 13
7 DWT and Filter Bank Algorithms 147......Page 14
9 Digital Signal Processing Applications 239......Page 15
10 Wavelets in Boundary Value Problems 308......Page 16
Index 353......Page 17
Preface to the Second Edition......Page 18
Preface to the First Edition......Page 19
1 What Is This Book All About?......Page 21
2.1 LINEAR SPACES......Page 26
2.2 VECTORS AND VECTOR SPACES......Page 28
2.3.2 Orthogonality and Biorthogonality......Page 31
2.4 LOCAL BASIS AND RIESZ BASIS......Page 34
2.4.2 Shannon Basis......Page 35
2.5.1 Example 1......Page 37
2.6 APPROXIMATION BY ORTHOGONAL PROJECTION......Page 38
2.7 MATRIX ALGEBRA AND LINEAR TRANSFORMATION......Page 39
2.7.1 Elements of Matrix Algebra......Page 40
2.7.2 Eigenmatrix......Page 41
2.7.3 Linear Transformation......Page 42
2.7.4 Change of Basis......Page 43
2.7.5 Hermitian Matrix, Unitary Matrix, and Orthogonal Transformation......Page 44
2.8.1 Sampling of Signal......Page 45
2.8.3 Convolution......Page 46
2.8.4 z - Transform......Page 48
2.8.5 Region of Convergence......Page 49
2.9 EXERCISES......Page 51
2.10 REFERENCES......Page 53
3.1 FOURIER SERIES......Page 54
3.2.1 Rectified Sine Wave......Page 56
3.2.2 Comb Function and the Fourier Series Kernel KN(t)......Page 57
3.3 FOURIER TRANSFORM......Page 59
3.4.2 Time Shifting and Time Scaling......Page 61
3.4.4 Moments......Page 62
3.4.6 Parseval\'s Theorem\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0......Page 63
3.5.1 The Rectangular Pulse......Page 64
3.5.2 The Triangular Pulse......Page 65
3.5.3 The Gaussian Function......Page 66
3.6 POISSON\'S SUM AND PARTITION OF UNITY\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0......Page 67
3.6.1 Partition of Unity......Page 69
3.7 SAMPLING THEOREM......Page 71
3.8 PARTIAL SUM AND GIBB\'S PHENOMENON\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0\0......Page 73
3.9.1 Discrete Fourier Basis and Discrete Fourier Series......Page 74
3.9.2 Discrete Time Fourier Transform ( DTFT )......Page 76
3.10 DISCRETE FOURIER TRANSFORM ( DFT )......Page 78
3.11 EXERCISES......Page 79
3.12 REFERENCES......Page 80
4 Time Frequency Analysis......Page 81
4.1 WINDOW FUNCTION......Page 83
4.2 SHORT TIME FOURIER TRANSFORM......Page 84
4.2.1 Inversion Formula......Page 85
4.2.3 Time Frequency Window......Page 86
4.2.4 Properties of STFT......Page 87
4.3 DISCRETE SHORT TIME FOURIER TRANSFORM......Page 88
4.4 DISCRETE GABOR REPRESENTATION......Page 90
4.5 CONTINUOUS WAVELET TRANSFORM......Page 91
4.5.1 Inverse Wavelet Transform......Page 93
4.5.2 Time Frequency Window......Page 94
4.6 DISCRETE WAVELET TRANSFORM......Page 96
4.7 WAVELET SERIES......Page 97
4.8 INTERPRETATIONS OF THE TIME FREQUENCY PLOT......Page 98
4.9 WIGNER VILLE DISTRIBUTION......Page 100
4.9.1 Gaussian Modulated Chirp......Page 101
4.9.2 Sinusoidal Modulated Chirp......Page 102
4.10 PROPERTIES OF WIGNER VILLE DISTRIBUTION......Page 103
4.10.2 Marginal Properties......Page 105
4.11 QUADRATIC SUPERPOSITION PRINCIPLE......Page 106
4.12 AMBIGUITY FUNCTION......Page 108
4.13 EXERCISES......Page 109
4.14.1 Short Time Fourier Transform......Page 110
4.14.2 Wigner Ville Distribution......Page 111
4.15 REFERENCES......Page 113
5 Multiresolution Analysis......Page 114
5.1 MULTIRESOLUTION SPACES......Page 115
5.2 ORTHOGONAL, BIORTHOGONAL, AND SEMIORTHOGONAL DECOMPOSITION......Page 117
5.3 TWO SCALE RELATIONS......Page 121
5.4 DECOMPOSITION RELATION......Page 122
5.5 SPLINE FUNCTIONS AND PROPERTIES......Page 123
5.5.1 Properties of Splines......Page 127
5.6 MAPPING A FUNCTION INTO MRA SPACE......Page 128
5.6.2 Cubic Splines (m=4)......Page 129
5.7 EXERCISES......Page 130
5.8 COMPUTER PROGRAMS 5.8.1 B Splines......Page 132
5.9 REFERENCES......Page 133
6 Construction of Wavelets......Page 134
6.1.1 Relationship between the Two Scale Sequences......Page 135
6.1.2 Relationship between Reconstruction and Decomposition Sequences......Page 137
6.2.1 Expression for {g0[k]}......Page 139
6.2.2 Remarks......Page 141
6.3 CONSTRUCTION OF ORTHONORMAL WAVELETS......Page 143
6.4.1 Shannon Scaling Function......Page 144
6.4.2 Meyer Scaling Function......Page 146
6.4.3 Battle Lemarié Scaling Function......Page 149
6.4.4 Daubechies Scaling Function......Page 150
6.5 CONSTRUCTION OF BIORTHOGONAL WAVELETS......Page 156
6.6.1 Iteration Method......Page 158
6.6.2 Spectral Method......Page 159
6.6.3 Eigenvalue Method......Page 160
6.7 EXERCISES......Page 161
6.8.1 Daubechies Wavelet......Page 164
6.8.2 Iteration Method......Page 165
6.9 REFERENCES......Page 166
7.1 DECIMATION AND INTERPOLATION......Page 167
7.1.1 Decimation......Page 168
7.1.2 Interpolation......Page 171
7.1.4 Interpolation Followed by Convolution......Page 174
7.2 SIGNAL REPRESENTATION IN THE APPROXIMATION SUBSPACE......Page 175
7.3 WAVELET DECOMPOSITION ALGORITHM......Page 177
7.4 RECONSTRUCTION ALGORITHM......Page 179
7.5 CHANGE OF BASES......Page 181
7.6.1 Change of Basis for Spline Functions......Page 184
7.6.2 Change of Basis for Spline Wavelets......Page 188
7.7 EXAMPLES......Page 190
7.8 TWO CHANNEL PERFECT RECONSTRUCTION FILTER BANK......Page 192
7.8.1 Spectral Domain Analysis of a Two Channel PR Filter Bank......Page 196
7.8.2 Time Domain Analysis......Page 204
7.9.2 Filter Bank in the Polyphase Domain......Page 209
7.10 COMMENTS ON DWT AND PR FILTER BANKS......Page 211
7.11 EXERCISES......Page 212
7.12.1 Decomposition and Reconstruction Algorithm......Page 213
7.13 REFERENCES......Page 215
8 Special Topics in Wavelets and Algorithms......Page 217
8.1.1 Finer Time Resolution......Page 218
8.1.2 Finer Scale Resolution......Page 221
8.1.3 Function Mapping into the Interoctave Approximation Subspaces......Page 224
8.1.4 Examples......Page 227
8.2 RIDGELET TRANSFORM......Page 240
8.3 CURVELET TRANSFORM......Page 242
8.4 COMPLEX WAVELETS......Page 244
8.4.1 Linear Phase Biorthogonal Approach......Page 247
8.4.3 Common Factor Approach......Page 248
8.5 LIFTING WAVELET TRANSFORM......Page 249
8.5.1 Linear Spline Wavelet......Page 253
8.5.3 Linear Interpolative Subdivision......Page 254
8.6 REFERENCES......Page 257
9 Digital Signal Processing Applications......Page 259
9.1 WAVELET PACKET......Page 260
9.2 WAVELET PACKET ALGORITHMS......Page 263
9.3.2 Soft Thresholding......Page 266
9.3.4 Implementation......Page 267
9.4 INTERFERENCE SUPPRESSION......Page 268
9.4.1 Best Basis Selection......Page 269
9.5.1 Pattern Recognition of Acoustic Signals......Page 272
9.5.2 Wavelets, Wavelet Packets, and FFT Features......Page 274
9.6.1 Two Dimensional Wavelets......Page 276
9.6.2 Two Dimensional Wavelet Packets......Page 278
9.6.3 Two Dimensional Wavelet Algorithm......Page 279
9.6.4 Wavelet Packet Algorithm......Page 282
9.7 EDGE DETECTION......Page 284
9.7.2 Laplacian of Gaussian Edge Detector......Page 285
9.7.4 Wavelet Edge Detector......Page 286
9.8.1 Basics of Data Compression......Page 287
9.8.2 Wavelet Tree Coder......Page 291
9.8.4 EZW Example......Page 292
9.8.5 Spatial Oriented Tree ( SOT )......Page 295
9.9 MICROCALCIFICATION CLUSTER DETECTION......Page 297
9.9.3 Wavelet Decomposition of the Subimages......Page 298
9.9.4 Wavelet Coefficient Domain Processing......Page 299
9.9.5 Histogram Thresholding and Dark Pixel Removal......Page 301
9.9.7 Results......Page 302
9.10.1 OFDM Multicarrier Communication Systems......Page 304
9.10.2 Wavelet Packet – Based MCCS......Page 305
9.11 THREE DIMENSIONAL MEDICAL IMAGE VISUALIZATION......Page 307
9.11.1 Three Dimensional Wavelets and Algorithms......Page 308
9.11.2 Rendering Techniques......Page 310
9.11.4 Summary......Page 311
9.12 GEOPHYSICAL APPLICATIONS......Page 313
9.12.1 Boundary Value Problems and Inversion......Page 314
9.12.2 Well Log Analysis......Page 315
9.12.3 Reservoir Data Analysis......Page 316
9.12.4 Downhole Pressure Gauge Data Analysis......Page 318
9.13.1 Two Dimensional Wavelet Algorithms......Page 319
9.13.2 Wavelet Packet Algorithms......Page 323
9.14 REFERENCES......Page 325
10 Wavelets in Boundary Value Problems......Page 328
10.1 INTEGRAL EQUATIONS......Page 329
10.2 METHOD OF MOMENTS......Page 333
10.3.1 Use of Fast Wavelet Algorithm......Page 334
10.3.2 Direct Application of Wavelets......Page 335
10.3.3 Wavelets in Spectral Domain......Page 337
10.4 WAVELETS ON THE BOUNDED INTERVAL......Page 342
10.5 SPARSITY AND ERROR CONSIDERATIONS......Page 344
10.6 NUMERICAL EXAMPLES......Page 347
10.7 SEMIORTHOGONAL VERSUS ORTHOGONAL WAVELETS......Page 354
10.8 DIFFERENTIAL EQUATIONS......Page 355
10.8.1 Multigrid Method......Page 356
10.8.2 Multiresolution Time Domain ( MRTD ) Method......Page 357
10.8.3 Haar MRTD Derivation......Page 358
10.8.4 Subcell Modeling in MRTD......Page 361
10.8.5 Examples......Page 363
10.9 EXPRESSIONS FOR SPLINES AND WAVELETS......Page 366
10.10 REFERENCES......Page 368
Index......Page 373
Back Page......Page 380