Provides easy learning and understanding of DWT from a signal processing point of view Presents DWT from a digital signal processing point of view, in contrast to the usual mathematical approach, making it highly accessible Offers a comprehensive coverage of related topics, including convolution and correlation, Fourier transform, FIR filter, orthogonal and biorthogonal filters Organized systematically, starting from the fundamentals of signal processing to the more advanced topics of DWT and Discrete Wavelet Packet Transform. Written in a clear and concise manner with abundant examples, figures and detailed explanations Features a companion website that has several MATLAB programs for the implementation of the DWT with commonly used filters “This well-written textbook is an introduction to the theory of discrete wavelet transform (DWT) and its applications in digital signal and image processing.” -- Prof. Dr. Manfred Tasche - Institut für Mathematik, Uni Rostock Full review at https://zbmath.org/?q=an:06492561

Table of Contents:
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