1 Introduction: Why Wavelets? 1
2 Vectors and Matrices 15
2.1 Vectors, Inner Products, and Norms 16
Problems 21
2.2 Basic Matrix Theory 23
Problems 38
2.3 Block Matrix Arithmetic 40
Problems 48
2.4 Convolution and Filters 51
Problems 65
3 An Introduction to Digital Images 69
3.1 The Basics of Grayscale Digital Images 70
Problems 88
Computer Lab 91
3.2 Color Images and Color Spaces 91
Problems 103
Computer Lab 106
3.3 Huffman Coding 106
Problems 113
3.4 Qualitative and Quantitative Measures 114
Problems 120
Computer Labs 123
4 The Haar Wavelet Transformation 125
4.1 Constructing the Haar Wavelet Transformation 127
Problems 137
Computer Lab 140
4.2 Iterating the Process 140
Problems 146
Computer Lab 147
4.3 The Two-Dimensional Haar Wavelet Transformation 147
Problems 159
Computer Lab 161
4.4 Applications: Image Compression and Edge Detection 161
Problems 177
Computer Labs 181
5 Daubechies Wavelet Transformations 183
5.1 Daubechies Filter of Length 4 185
Problems 196
Computer Lab 203
5.2 Daubechies Filter of Length 6 203
Problems 212
Computer Lab 215
5.3 Daubechies Filters of Even Length 215
Problems 225
Computer Lab 228
6 Wavelet Shrinkage: An Application to Denoising 231
6.1 An Overview of Wavelet Shrinkage 232
Problems 237
Computer Lab 238
6.2 VisuShrink 238
Problems 245
Computer Lab 246
6.3 SureShrink 246
Problems 257
Computer Labs 260
7 Biorthogonal Wavelet Transformations 261
7.1 The (5; 3) Biorthogonal Spline Filter Pair 262
Problems 273
Computer Lab 278
7.2 The (8; 4) Biorthogonal Spline Filter Pair 278
Problems 283
Computer Lab 288
7.3 Symmetry and Boundary Effects 288
Problems 307
Computer Lab 311
7.4 Image Compression and Image Pansharpening 312
Computer Lab 320
8 Complex Numbers and Fourier Series 321
8.1 The Complex Plane and Arithmetic 322
Problems 332
8.2 Fourier Series 334
Problems 344
8.3 Filters and Convolution in the Fourier Domain 349
Problems 360
9 Filter Construction in the Fourier Domain 365
9.1 Filter Construction 366
Problems 377
9.2 Daubechies Filters 378
Problems 382
9.3 Coiflet Filters 382
Problems 395
9.4 Biorthogonal Spline Filter Pairs 400
Problems 410
Computer Lab 413
9.5 The Cohen–Daubechies–Feauveau 9/7 Filter 414
Problems 423
Computer Lab 426
10 Wavelet Packets 427
10.1 The Wavelet Packet Transform 428
Problems 435
10.2 Cost Functions and the Best Basis Algorithm 436
Problems 444
10.3 The FBI Fingerprint Compression Specification 446
Computer Lab 460
11 Lifting 461
11.1 The LeGall Wavelet Transform 462
Problems 471
Computer Lab 473
11.2 Z–Transforms and Laurent Polynomials 474
Problems 484
11.3 A General Construction of the Lifting Method 486
Problems 499
11.4 The Lifting Method – Examples 504
Problems 517
12 The JPEG2000 Image Compression Standard 525
12.1 An Overview of JPEG 526
Problems 532
12.2 The Basic JPEG2000 Algorithm 533
Problems 539
12.3 Examples 540
A Basic Statistics 547
A.1 Descriptive Statistics 547
Problems 549
A.2 Sample Spaces, Probability, and Random Variables 550
Problems 553
A.3 Continuous Distributions 553
Problems 559
A.4 Expectation 559
Problems 565
A.5 Two Special Distributions 566
Problems 568
