Wavelets made easy

Cover......Page 1
Wavelets Made Easy......Page 4
Contents......Page 8
Preface......Page 12
Outline......Page 14
PART A Algorithms for Wavelet Transforms......Page 16
1.0 INTRODUCTION......Page 17
1.1 SIMPLE APPROXIMATION......Page 18
EXERCISES......Page 21
1.2.1 The Basic Haar Wavelet Transform......Page 22
1.2.2 Significance of the Basic Haar Wavelet Transform......Page 24
1.2.3 Shifts and Dilations of the Basic Haar Transform......Page 25
EXERCISES......Page 27
1.3.1 Initialization......Page 28
1.3.2 The Ordered Fast Haar Wavelet Transform......Page 29
1.3.2.4 Results......Page 31
1.3.2.8 Third Sweep......Page 33
EXERCISES......Page 34
1.4 THE IN-PLACE FAST HAAR WAVELET TRANSFORM......Page 35
1.4.1.3 Replacement......Page 36
1.4.2.3 Second Sweep......Page 37
1.4.2.8 In-Place Fast Haar Wavelet Transform......Page 39
EXERCISES......Page 40
1.5 THE IN-PLACE FAST INVERSE HAAR WAVELET TRANSFORM......Page 42
1.6.1 Creek Water Temperature Analysis......Page 45
1.6.2 Financial Stock Index Event Detection......Page 47
EXERCISES......Page 49
2.0 INTRODUCTION......Page 50
2.1.1 Two-Dimensional Approximation with Step Functions......Page 51
2.1.2 Tensor Products of Functions......Page 53
2.1.3 The Basic Two-Dimensional Haar Wavelet Transform......Page 56
2.1.4 Two-Dimensional Fast Haar Wavelet Transform......Page 60
2.2.1.1 Random Noise......Page 63
2.2.1.2 Band-Specific Noise......Page 65
2.2.2 Data Compression......Page 66
EXERCISES......Page 71
2.2.3 Edge Detection......Page 72
2.3.1 Fast Reconstruction of Single Values......Page 74
EXERCISES......Page 77
2.4.1 Creek Water Temperature Compression......Page 79
EXERCISES......Page 80
2.4.2 Financial Stock Index Image Compression......Page 81
2.4.3 Two-Dimensional Diffusion Analysis......Page 82
2.4.4 Three-Dimensional Diffusion Analysis......Page 83
EXERCISES......Page 86
3.1 CALCULATION OF DAUBECHIES WAVELETS......Page 87
3.2 APPROXIMATION OF SAMPLES WITH DAUBECHIES WAVELETS......Page 96
3.2.1 Approximate Interpolation......Page 97
3.2.2 Approximate Averages......Page 98
3.3.1 Zigzag Edge Effects from Extensions by Zeros......Page 99
3.3.2 Medium Edge Effects from Mirror Reflections......Page 102
3.3.3 Small Edge Effects from Smooth Periodic Extensions......Page 104
EXERCISES......Page 108
3.4 THE FAST DAUBECHIES WAVELET TRANSFORM......Page 109
3.5 THE FAST INVERSE DAUBECHIES WAVELET TRANSFORM......Page 115
3.6 MULTIDIMENSIONAL DAUBECHIES WAVELET TRANSFORMS......Page 121
3.7.1 Hangman Creek Water Temperature Analysis......Page 124
EXERCISES......Page 125
3.7.2 Financial Stock Index Image Compression......Page 126
EXERCISES......Page 127
PART B Basic Fourier Analysis......Page 128
4.1.1 Number Fields......Page 129
4.1.2 Linear Spaces......Page 132
4.1.3 Linear Maps......Page 134
4.2 PROJECTIONS......Page 135
4.2.1 Inner Products......Page 136
4.2.2 Gram-Schmidt Orthogonalization......Page 141
4.2.3 Orthogonal Projections......Page 143
4.3.1 Application to Three-Dimensional Computer Graphics......Page 146
EXERCISES......Page 147
4.3.2 Application to Ordinary Least-Squares Regression......Page 148
EXERCISES......Page 149
4.3.3 Application to the Computation of Functions......Page 150
EXERCISES......Page 153
4.3.4 Applications to Wavelets......Page 154
EXERCISES......Page 157
5.1 THE DISCRETE FOURIER TRANSFORM (DFT)......Page 159
5.1.1 Definition and Inversion......Page 160
5.1.2 Unitary Operators......Page 167
EXERCISES......Page 168
5.2.4 Bit Reversal......Page 175
5.2.1 The Forward Fast Fourier Transform......Page 169
5.2.3 Interpolation by the Inverse Fast Fourier Transform......Page 173
EXERCISES......Page 174
5.3.1 Noise Reduction Through the Fast Fourier Transform......Page 177
5.3.2 Convolution and Fast Multiplication......Page 179
5.4 MULTIDIMENSIONAL DISCRETE AND FAST FOURIER TRANSFORMS......Page 183
6.0 INTRODUCTION......Page 187
6.1.1 Orthonormal Complex Trigonometric Functions......Page 188
6.1.2 Definition and Examples of Fourier Series......Page 189
6.1.3 Relation Between Series and Discrete Transforms......Page 194
6.1.4 Multidimensional Fourier Series......Page 195
6.2.1 The Gibbs-Wilbraham Phenomenon......Page 197
6.2.2 Piecewise Continuous Functions......Page 199
EXERCISES......Page 202
6.2.3 Convergence and Inversion of Fourier Series......Page 203
6.2.4 Convolutions and Dirac\'s Function .......Page 204
6.2.5 Uniform Convergence of Fourier Series......Page 206
6.3 PERIODIC FUNCTIONS......Page 212
PART C Computation and Designof Wavelets......Page 214
7.1.1 Definition and Examples of the Fourier Transform......Page 215
EXERCISES......Page 218
7.2 CONVOLUTIONS AND INVERSION OF THE FOURIER TRANSFORM......Page 219
7.3 APPROXIMATE IDENTITIES......Page 223
7.3.1 Weight Functions......Page 224
7.3.2 Approximate Identities......Page 225
7.3.3 Dirac Delta(.) Function......Page 229
7.4 FURTHER FEATURES OF THE FOURIER TRANSFORM......Page 230
7.4.1 Algebraic Features of the Fourier Transform......Page 231
7.4.2 Metric Features of the Fourier Transform......Page 233
7.4.3 Uniform Continuity of Fourier Transforms......Page 237
EXERCISES......Page 238
7.5 THE FOURIER TRANSFORM WITH SEVERAL VARIABLES......Page 239
7.6.1 Shannon\'s Sampling Theorem......Page 244
7.6.2 Heisenberg\'s Uncertainty Principle......Page 246
EXERCISES......Page 247
8.1 EXISTENCE, UNIQUENESS, AND CONSTRUCTION......Page 248
8.1.1 The Recursion Operator and Its Adjoint......Page 249
EXERCISES......Page 252
8.1.2 The Fourier Transform of the Recursion Operator......Page 253
8.1.3 Convergence of Iterations of the Recursion Operator......Page 255
EXERCISES......Page 262
8.2 ORTHOGONALITY OF DAUBECHIES WAVELETS......Page 263
8.3 MALLAT\'S FAST WAVELET ALGORITHM......Page 268
9.1 COMPUTATIONAL FEATURES OF DAUBECHIES WAVELETS......Page 272
9.1.1 Initial Values of Daubechies\' Scaling Function......Page 273
9.1.2 Computational Features of Daubechies\' Function......Page 276
9.1.3 Exact Representation of Polynomials by Wavelets......Page 283
9.2.1 Accuracy of Taylor Polynomials......Page 284
9.2.2 Accuracy of Signal Representations by Wavelets......Page 288
9.2.3 Approximate Interpolation by Daubechies\' Function......Page 291
EXERCISES......Page 293
PART D Directories......Page 294
Acknowledgments......Page 295
Collection of Symbols......Page 296
Bibliography......Page 298
Index......Page 301