Discrete Wavelet Transform

Dr. D. Sundararajan, Department Head of Electrical and Electronics Engineering, Adhiyamaan College of Engineering, India.Dr. Sundararajan obtained his PhD in Electrical Engineering at Concordia University, Montreal, Canada in 1988. As the principle inventor of the latest family of DFT algorithms, he has written three books, three Patents (which have been granted by US, Canada and Britain), and several papers in IEEE Transactions and in the Proceedings of IEEE Conference.

"Doubtless, this nice book will stimulate the practical education in the theory of DWT and its applications." (Zentralblatt MATH, 2016)

• Preface xi List of Abbreviations xiii1 Introduction 11.1 The Organization of This Book 22 Signals 52.1 Signal Classifications 52.1.1 Periodic and Aperiodic Signals 52.1.2 Even and Odd Signals 62.1.3 Energy Signals 72.1.4 Causal and Noncausal Signals 92.2 Basic Signals 92.2.1 Unit-Impulse Signal 92.2.2 Unit-Step Signal 102.2.3 The Sinusoid 102.3 The Sampling Theorem and the Aliasing Effect 122.4 Signal Operations 132.4.1 Time Shifting 132.4.2 Time Reversal 142.4.3 Time Scaling 142.5 Summary 17Exercises 173 Convolution and Correlation 213.1 Convolution 213.1.1 The Linear Convolution 213.1.2 Properties of Convolution 243.1.3 The Periodic Convolution 253.1.4 The Border Problem 253.1.5 Convolution in the DWT 263.2 Correlation 283.2.1 The Linear Correlation 283.2.2 Correlation and Fourier Analysis 293.2.3 Correlation in the DWT 303.3 Summary 31Exercises 314 Fourier Analysis of Discrete Signals 374.1 Transform Analysis 374.2 The Discrete Fourier Transform 384.2.1 Parseval’s Theorem 434.3 The Discrete-Time Fourier Transform 444.3.1 Convolution 484.3.2 Convolution in the DWT 484.3.3 Correlation 504.3.4 Correlation in the DWT 504.3.5 Time Expansion 524.3.6 Sampling Theorem 524.3.7 Parseval’s Theorem 544.4 Approximation of the DTFT 554.5 The Fourier Transform 564.6 Summary 56Exercises 575 Thez-Transform 595.1 The z-Transform 595.2 Properties of the z-Transform 605.2.1 Linearity 605.2.2 Time Shift of a Sequence 615.2.3 Convolution 615.3 Summary 62Exercises 626 Finite Impulse Response Filters 636.1 Characterization 636.1.1 Ideal Lowpass Filters 646.1.2 Ideal Highpass Filters 656.1.3 Ideal Bandpass Filters 666.2 Linear Phase Response 666.2.1 Even-Symmetric FIR Filters with Odd Number of Coefficients 676.2.2 Even-Symmetric FIR Filters with Even Number of Coefficients 686.3 Summary 69Exercises 697 Multirate Digital Signal Processing 717.1 Decimation 727.1.1 Downsampling in the Frequency-Domain 727.1.2 Downsampling Followed by Filtering 757.2 Interpolation 777.2.1 Upsampling in the Frequency-Domain 777.2.2 Filtering Followed by Upsampling 787.3 Two-Channel Filter Bank 797.3.1 Perfect Reconstruction Conditions 817.4 Polyphase Form of the Two-Channel Filter Bank 847.4.1 Decimation 847.4.2 Interpolation 877.4.3 Polyphase Form of the Filter Bank 917.5 Summary 94Exercises 948 The Haar Discrete Wavelet Transform 978.1 Introduction 978.1.1 Signal Representation 978.1.2 The Wavelet Transform Concept 988.1.3 Fourier and Wavelet Transform Analyses 988.1.4 Time-Frequency Domain 998.2 The Haar Discrete Wavelet Transform 1008.2.1 The Haar DWT and the 2-Point DFT 1028.2.2 The Haar Transform Matrix 1038.3 The Time-Frequency Plane 1078.4 Wavelets from the Filter Coefficients 1118.4.1 Two Scale Relations 1168.5 The 2-D Haar Discrete Wavelet Transform 1188.6 Discontinuity Detection 1268.7 Summary 127Exercises 1289 Orthogonal Filter Banks 1319.1 Haar Filter 1329.2 Daubechies Filter 1359.3 Orthogonality Conditions 1469.3.1 Characteristics of Daubechies Lowpass Filters 1499.4 Coiflet Filter 1509.5 Summary 154Exercises 15510 Biorthogonal Filter Banks 15910.1 Biorthogonal Filters 15910.2 5/3 Spline Filter 16310.2.1 Daubechies Formulation 17010.3 4/4 Spline Filter 17010.3.1 Daubechies Formulation 17710.4 CDF 9/7 Filter 17810.5 Summary 183Exercises 18411 Implementation of the Discrete Wavelet Transform 18911.1 Implementation of the DWT with Haar Filters 19011.1.1 1-Level Haar DWT 19011.1.2 2-Level Haar DWT 19111.1.3 1-Level Haar 2-D DWT 19311.1.4 The Signal-Flow Graph of the Fast Haar DWT Algorithms 19411.1.5 Haar DWT in Place 19611.2 Symmetrical Extension of the Data 19811.3 Implementation of the DWT with the D4 Filter 20011.4 Implementation of the DWT with Symmetrical Filters 20311.4.1 5/3 Spline Filter 20311.4.2 CDF 9/7 Filter 20511.4.3 4/4 Spline Filter 20811.5 Implementation of the DWT using Factorized Polyphase Matrix 21011.5.1 Haar Filter 21111.5.2 D4 Filter 21311.5.3 5/3 Spline Filter 21611.6 Summary 219Exercises 21912 The Discrete Wavelet Packet Transform 22312.1 The Discrete Wavelet Packet Transform 22312.1.1 Number of Representations 22612.2 Best Representation 22712.2.1 Cost Functions 23012.3 Summary 233Exercises 23313 The Discrete Stationary Wavelet Transform 23513.1 The Discrete Stationary Wavelet Transform 23513.1.1 The SWT 23513.1.2 The ISWT 23613.1.3 Algorithms for Computing the SWT and the ISWT 23813.1.4 2-D SWT 24313.2 Summary 244Exercises 24414 The Dual-Tree Discrete Wavelet Transform 24714.1 The Dual-Tree Discrete Wavelet Transform 24814.1.1 Parseval’s Theorem 24814.2 The Scaling and Wavelet Functions 25214.3 Computation of the DTDWT 25314.4 Summary 262Exercises 26315 Image Compression 26515.1 Lossy Image Compression 26615.1.1 Transformation 26615.1.2 Quantization 26815.1.3 Coding 27015.1.4 Compression Algorithm 27315.1.5 Image Reconstruction 27715.2 Lossless Image Compression 28415.3 Recent Trends in Image Compression 28915.3.1 The JPEG2000 Image Compression Standard 29015.4 Summary 290Exercises 29116 Denoising 29516.1 Denoising 29516.1.1 Soft Thresholding 29616.1.2 Statistical Measures 29716.2 VisuShrink Denoising Algorithm 29816.3 Summary 303Exercises 303Bibliography 305Answers to Selected Exercises 307Index 319