Preface......Page 3
Contents......Page 12
Examples & Exercises......Page 15
1 Sound & Fourier Series......Page 22
1.1 Sound and Digital Sound: Loudness and Frequency......Page 24
1.1.1 The Frequency of a Sound......Page 26
1.1.2 Working with Digital Sound on a Computer......Page 27
1.2 Fourier Series: Basic Concepts......Page 32
1.2.1 Fourier Series for Symmetric and Antisymmetric Functions......Page 40
1.3 Complex Fourier Series......Page 42
1.4 Some Properties of Fourier Series......Page 49
1.4.1 Rate of Convergence for Fourier Series......Page 51
1.4.2 Differentiating Fourier Series......Page 52
1.5 Filters on Periodic Functions......Page 56
1.6 Convergence of Fourier Series*......Page 59
1.6.1 Interpretation of the Filters Corresponding to the Fejer and Dirichlet Kernels......Page 63
1.7 The MP3 Standard......Page 65
1.8 Summary......Page 67
2.1 Discrete Fourier Analysis and the Discrete Fourier Transform......Page 69
2.1.1 Properties of the DFT......Page 76
2.2 Connection Between the DFT and Fourier Series: Sampling and the Sampling Theorem......Page 80
2.3 The Fast Fourier Transform (FFT)......Page 86
2.3.1 Reduction in the Number of Arithmetic Operations......Page 90
2.3.2 The FFT When N Is Non-prime......Page 93
2.4 The Discrete Cosine Transform (DCT)......Page 99
2.4.1 Cosine Matrices......Page 103
2.5 Efficient Implementations of the DCT......Page 107
2.5.1 Efficient Implementations of the IDCT......Page 109
2.5.2 Reduction in the Number of Arithmetic Operations......Page 110
2.6 Summary......Page 113
3.1 Discrete Time Filters on Periodic Vectors......Page 116
3.2 General Discrete Time Filters......Page 125
3.2.1 A Second Approach to Finite Input......Page 134
3.2.2 Connection Between Convolution and Circular Convolution......Page 135
3.3 Low-Pass and High-Pass Filters......Page 139
3.4 IIR Filters......Page 149
3.5 Symmetric Filters and the DCT......Page 155
3.5.1 Implementations of Symmetric Filters......Page 156
3.6 Relations to Signal Processing......Page 159
3.7 Summary......Page 161
4.1 Motivation for Wavelets, and a Wavelet Based on Piecewise Constant Functions......Page 162
4.1.1 Function Approximation Property......Page 167
4.1.2 Detail Spaces and Wavelets......Page 169
4.2 Implementation of the DWT......Page 177
4.3 A Wavelet Based on Piecewise Linear Functions......Page 186
4.3.1 Detail Spaces and Wavelets......Page 188
4.3.2 Multiresolution Analysis: A Generalization......Page 191
4.4 Alternative Wavelet Based on Piecewise Linear Functions......Page 197
4.5 Summary......Page 206
5.1 DWT and IDWT in Terms of Filters......Page 208
5.1.1 Difference Between MRA Matrices and Filters......Page 212
5.2 Dual Filter Bank Transform and Dual Wavelets......Page 222
5.3 Symmetric Extensions......Page 229
5.4 A Generalization of the Filter Representation, and Its Use in Audio Coding......Page 235
5.4.1 Forward Filter Bank Transform in the MP3 Standard......Page 237
5.4.2 Reverse Filter Bank Transform in the MP3 Standard......Page 240
5.5 Summary......Page 244
6 Constructing interesting Wavelets......Page 247
6.1 From Filters to Scaling Functions and Mother Wavelets......Page 248
6.2.1 Symmetric Filters......Page 260
6.2.2 Orthonormal Wavelets......Page 263
6.2.3 The Proof of Bezout's Theorem......Page 265
6.3 A Design Strategy Suitable for Lossless Compression......Page 267
6.3.1 The Spline 5/3 Wavelet......Page 268
6.4 A Design Strategy Suitable for Lossy Compression......Page 269
6.5 Orthonormal Wavelets......Page 272
6.6 Summary......Page 274
7 Polyphase Representation of Filter Bank Transforms......Page 277
7.1 The Polyphase Representation and Perfect Reconstruction......Page 279
7.2 The Polyphase Representation and the Lifting Factorization......Page 283
7.2.2 The Piecewise Linear Wavelet......Page 286
7.2.3 The Spline 5/3 Wavelet......Page 287
7.2.5 Orthonormal Wavelets......Page 288
7.3 Polyphase Representations of Cosine Modulated Filter Banks and the MP3 Standard......Page 293
7.3.1 Polyphase Representation of the Forward Filter Bank Transform......Page 294
7.3.2 Polyphase Representation of the Reverse Filter Bank Transform......Page 296
7.3.3 Perfect Reconstruction......Page 297
7.3.4 The MP3 Standard Does Not Give Perfect Reconstruction......Page 300
7.4 Summary......Page 303
8.1 What Is an Image?......Page 305
8.2 Some Simple Operations on Images with Python......Page 308
8.3 Filter-Based Operations on Images......Page 314
8.3.1 Tensor Product Notation for Operations on Images......Page 316
8.4 Change of Coordinates in Tensor Products......Page 329
8.5 Summary......Page 333
9.1 Tensor Product of Function Spaces......Page 335
9.2 Tensor Product of Function Spaces in a Wavelet Setting......Page 337
9.2.1 Interpretation......Page 341
9.3 Experiments with Images Using Wavelets......Page 345
9.4 An Application to the FBI Standard for Compression of Fingerprint Images......Page 355
9.5 Summary......Page 360
A.1 Matrices......Page 361
A.2 Block Matrices......Page 362
A.3 Vector Spaces......Page 363
A.4 Inner Products and Orthogonality......Page 364
A.5 Coordinates and Change of Coordinates......Page 366
A.6 Eigenvectors and Eigenvalues......Page 367
A.7 Positive Definite Matrices......Page 369
A.8 Singular Value Decomposition......Page 370
Nomenclature......Page 371
Refs......Page 373
Index......Page 377