Preface xi
List of Abbreviations xiii
1 Introduction 1
1.1 The Organization of This Book 2
2 Signals 5
2.1 Signal Classifications 5
2.1.1 Periodic and Aperiodic Signals 5
2.1.2 Even and Odd Signals 6
2.1.3 Energy Signals 7
2.1.4 Causal and Noncausal Signals 9
2.2 Basic Signals 9
2.2.1 Unit-Impulse Signal 9
2.2.2 Unit-Step Signal 10
2.2.3 The Sinusoid 10
2.3 The Sampling Theorem and the Aliasing Effect 12
2.4 Signal Operations 13
2.4.1 Time Shifting 13
2.4.2 Time Reversal 14
2.4.3 Time Scaling 14
2.5 Summary 17
Exercises 17
3 Convolution and Correlation 21
3.1 Convolution 21
3.1.1 The Linear Convolution 21
3.1.2 Properties of Convolution 24
3.1.3 The Periodic Convolution 25
3.1.4 The Border Problem 25
3.1.5 Convolution in the DWT 26
3.2 Correlation 28
3.2.1 The Linear Correlation 28
3.2.2 Correlation and Fourier Analysis 29
3.2.3 Correlation in the DWT 30
3.3 Summary 31
Exercises 31
4 Fourier Analysis of Discrete Signals 37
4.1 Transform Analysis 37
4.2 The Discrete Fourier Transform 38
4.2.1 Parseval's Theorem 43
4.3 The Discrete-Time Fourier Transform 44
4.3.1 Convolution 48
4.3.2 Convolution in the DWT 48
4.3.3 Correlation 50
4.3.4 Correlation in the DWT 50
4.3.5 Time Expansion 52
4.3.6 Sampling Theorem 52
4.3.7 Parseval's Theorem 54
4.4 Approximation of the DTFT 55
4.5 The Fourier Transform 56
4.6 Summary 56
Exercises 57
5 Thez-Transform 59
5.1 The z-Transform 59
5.2 Properties of the z-Transform 60
5.2.1 Linearity 60
5.2.2 Time Shift of a Sequence 61
5.2.3 Convolution 61
5.3 Summary 62
Exercises 62
6 Finite Impulse Response Filters 63
6.1 Characterization 63
6.1.1 Ideal Lowpass Filters 64
6.1.2 Ideal Highpass Filters 65
6.1.3 Ideal Bandpass Filters 66
6.2 Linear Phase Response 66
6.2.1 Even-Symmetric FIR Filters with Odd Number of Coefficients 67
6.2.2 Even-Symmetric FIR Filters with Even Number of Coefficients 68
6.3 Summary 69
Exercises 69
7 Multirate Digital Signal Processing 71
7.1 Decimation 72
7.1.1 Downsampling in the Frequency-Domain 72
7.1.2 Downsampling Followed by Filtering 75
7.2 Interpolation 77
7.2.1 Upsampling in the Frequency-Domain 77
7.2.2 Filtering Followed by Upsampling 78
7.3 Two-Channel Filter Bank 79
7.3.1 Perfect Reconstruction Conditions 81
7.4 Polyphase Form of the Two-Channel Filter Bank 84
7.4.1 Decimation 84
7.4.2 Interpolation 87
7.4.3 Polyphase Form of the Filter Bank 91
7.5 Summary 94
Exercises 94
8 The Haar Discrete Wavelet Transform 97
8.1 Introduction 97
8.1.1 Signal Representation 97
8.1.2 The Wavelet Transform Concept 98
8.1.3 Fourier and Wavelet Transform Analyses 98
8.1.4 Time-Frequency Domain 99
8.2 The Haar Discrete Wavelet Transform 100
8.2.1 The Haar DWT and the 2-Point DFT 102
8.2.2 The Haar Transform Matrix 103
8.3 The Time-Frequency Plane 107
8.4 Wavelets from the Filter Coefficients 111
8.4.1 Two Scale Relations 116
8.5 The 2-D Haar Discrete Wavelet Transform 118
8.6 Discontinuity Detection 126
8.7 Summary 127
Exercises 128
9 Orthogonal Filter Banks 131
9.1 Haar Filter 132
9.2 Daubechies Filter 135
9.3 Orthogonality Conditions 146
9.3.1 Characteristics of Daubechies Lowpass Filters 149
9.4 Coiflet Filter 150
9.5 Summary 154
Exercises 155
10 Biorthogonal Filter Banks 159
10.1 Biorthogonal Filters 159
10.2 5/3 Spline Filter 163
10.2.1 Daubechies Formulation 170
10.3 4/4 Spline Filter 170
10.3.1 Daubechies Formulation 177
10.4 CDF 9/7 Filter 178
10.5 Summary 183
Exercises 184
11 Implementation of the Discrete Wavelet Transform 189
11.1 Implementation of the DWT with Haar Filters 190
11.1.1 1-Level Haar DWT 190
11.1.2 2-Level Haar DWT 191
11.1.3 1-Level Haar 2-D DWT 193
11.1.4 The Signal-Flow Graph of the Fast Haar DWT Algorithms 194
11.1.5 Haar DWT in Place 196
11.2 Symmetrical Extension of the Data 198
11.3 Implementation of the DWT with the D4 Filter 200
11.4 Implementation of the DWT with Symmetrical Filters 203
11.4.1 5/3 Spline Filter 203
11.4.2 CDF 9/7 Filter 205
11.4.3 4/4 Spline Filter 208
11.5 Implementation of the DWT using Factorized Polyphase Matrix 210
11.5.1 Haar Filter 211
11.5.2 D4 Filter 213
11.5.3 5/3 Spline Filter 216
11.6 Summary 219
Exercises 219
12 The Discrete Wavelet Packet Transform 223
12.1 The Discrete Wavelet Packet Transform 223
12.1.1 Number of Representations 226
12.2 Best Representation 227
12.2.1 Cost Functions 230
12.3 Summary 233
Exercises 233
13 The Discrete Stationary Wavelet Transform 235
13.1 The Discrete Stationary Wavelet Transform 235
13.1.1 The SWT 235
13.1.2 The ISWT 236
13.1.3 Algorithms for Computing the SWT and the ISWT 238
13.1.4 2-D SWT 243
13.2 Summary 244
Exercises 244
14 The Dual-Tree Discrete Wavelet Transform 247
14.1 The Dual-Tree Discrete Wavelet Transform 248
14.1.1 Parseval's Theorem 248
14.2 The Scaling and Wavelet Functions 252
14.3 Computation of the DTDWT 253
14.4 Summary 262
Exercises 263
15 Image Compression 265
15.1 Lossy Image Compression 266
15.1.1 Transformation 266
15.1.2 Quantization 268
15.1.3 Coding 270
15.1.4 Compression Algorithm 273
15.1.5 Image Reconstruction 277
15.2 Lossless Image Compression 284
15.3 Recent Trends in Image Compression 289
15.3.1 The JPEG2000 Image Compression Standard 290
15.4 Summary 290
Exercises 291
16 Denoising 295
16.1 Denoising 295
16.1.1 Soft Thresholding 296
16.1.2 Statistical Measures 297
16.2 VisuShrink Denoising Algorithm 298
16.3 Summary 303
Exercises 303
Bibliography 305
Answers to Selected Exercises 307
Index 319
