Discrete wavelet transform: a signal processing approach

Content: 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