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.