Multiscale Transforms with Application to Image Processing (Signals and Communication Technology)

This book provides an introduction to image processing, an overview of the transforms which are most widely used in the field of image processing, and an introduction to the application of multiscale transforms in image processing. The book is divided into three parts, with the first part offering the reader a basic introduction to image processing. The second part of the book starts with a chapter on Fourier analysis and Fourier transforms, wavelet analysis, and ends with a chapter on new multiscale transforms. The final part of the book deals with all of the most important applications of multiscale transforms in image processing. The chapters consist of both tutorial and highly advanced material, and as such the book is intended to be a reference text for graduate students and researchers to obtain state-of-the-art knowledge on specific applications. The technique of solving problems in the transform domain is common in applied mathematics and widely used in research and industry, but is a somewhat neglected subject within the undergraduate curriculum. It is hoped that faculty can use this book to create a course that can be offered early in the curriculum and fill this void. Also, the book is intended to be used as a reference manual for scientists who are engaged in image processing research, developers of image processing hardware and software systems, and practising engineers and scientists who use image processing as a tool in their applications. Preface 6 Contents 8 About the Authors 13 Part I Introduction to Image Processing 15 1 Fundamentals of Digital Image Processing 16 1.1 Image Acquisition of Digital Camera 16 1.1.1 Introduction 16 1.2 Sampling 17 References 23 Part II Multiscale Transform 25 2 Fourier Analysis and Fourier Transform 26 2.1 Overview 26 2.2 Fourier Series 27 2.2.1 Periodic Functions 27 2.2.2 Frequency and Amplitude 27 2.2.3 Phase 29 2.2.4 Fourier Series of Periodic Functions 30 2.2.5 Complex Form of Fourier Series 31 2.3 Fourier Transform 32 2.3.1 2D-Fourier Transform 35 2.3.2 Properties of Fourier Transform 35 2.4 Discrete Fourier Transform 37 2.4.1 1D-Discrete Fourier Transform 38 2.4.2 Inverse 1D-Discrete Fourier Transform 41 2.4.3 2D-Discrete Fourier Transform and 2D-Inverse Discrete Fourier Transform 42 2.4.4 Properties of 2D-Discrete Fourier Transform 43 2.5 Fast Fourier Transform 45 2.6 The Discrete Cosine Transform 50 2.6.1 1D-Discrete Cosine Transform 50 2.6.2 2D-Discrete Cosine Transform 51 2.7 Heisenberg Uncertainty Principle 52 2.8 Windowed Fourier Transform or Short-Time Fourier Transform 52 2.8.1 1D and 2D Short-Time Fourier Transform 52 2.8.2 Drawback of Short-Time Fourier Transform 53 2.9 Other Spectral Transforms 53 References 54 3 Wavelets and Wavelet Transform 55 3.1 Overview 55 3.2 Wavelets 56 3.3 Multiresolution Analysis 58 3.4 Wavelet Transform 63 3.4.1 The Wavelet Series Expansions 63 3.4.2 Discrete Wavelet Transform 64 3.4.3 Motivation: From MRA to Discrete Wavelet Transform 65 3.4.4 The Quadrature Mirror Filter Conditions 67 3.5 The Fast Wavelet Transform 72 3.6 Why Use Wavelet Transforms 75 3.7 Two-Dimensional Wavelets 76 3.8 2D-discrete Wavelet Transform 77 3.9 Continuous Wavelet Transform 79 3.9.1 1D Continuous Wavelet Transform 79 3.9.2 2D Continuous Wavelet Transform 79 3.10 Undecimated Wavelet Transform or Stationary Wavelet Transform 80 3.11 Biorthogonal Wavelet Transform 80 3.11.1 Linear Independence and Biorthogonality 80 3.11.2 Dual MRA 82 3.11.3 Discrete Transform for Biorthogonal Wavelets 83 3.12 Scarcity of Wavelet Transform 86 3.13 Complex Wavelet Transform 88 3.14 Dual-Tree Complex Wavelet Transform 89 3.15 Quaternion Wavelet and Quaternion Wavelet Transform 93 3.15.1 2D Hilbert Transform 94 3.15.2 Quaternion Algebra 95 3.15.3 Quaternion Multiresolution Analysis 99 References 100 4 New Multiscale Constructions 103 4.1 Overview 103 4.2 Ridgelet Transform 104 4.2.1 The Continuous Ridgelet Transform 104 4.2.2 Discrete Ridgelet Transform 108 4.2.3 The Orthonormal Finite Ridgelet Transform 110 4.2.4 The Fast Slant Stack Ridgelet Transform 110 4.2.5 Local Ridgelet Transform 111 4.2.6 Sparse Representation by Ridgelets 111 4.3 Curvelets 112 4.3.1 The First Generation Curvelet Transform 112 4.3.2 Sparse Representation by First Generation Curvelets 113 4.3.3 The Second-Generation Curvelet Transform 114 4.3.4 Sparse Representation by Second Generation Curvelets 115 4.4 Contourlet 116 4.5 Contourlet Transform 117 4.5.1 Multiscale Decomposition 118 4.5.2 Directional Decomposition 119 4.5.3 The Discrete Contourlet Transform 120 4.6 Shearlet 122 4.7 Shearlet Transform 125 4.7.1 Continuous Shearlet Transform 125 4.7.2 Discrete Shearlet Transform 126 4.7.3 Cone-Adapted Continuous Shearlet Transform 128 4.7.4 Cone-Adapted Discrete Shearlet Transform 131 4.7.5 Compactly Supported Shearlets 133 4.7.6 Sparse Representation by Shearlets 135 References 136 Part III Application of Multiscale Transforms to Image Processing 140 5 Image Restoration 141 5.1 Model of Image Degradation and Restoration Process 141 5.2 Image Quality Assessments Metrics 142 5.3 Image Denoising 144 5.4 Noise Models 144 5.4.1 Additive Noise Model 145 5.4.2 Multiplicative Noise Model 145 5.5 Types of Noise 145 5.5.1 Amplifier (Gaussian) Noise 145 5.5.2 Rayleigh Noise 146 5.5.3 Uniform Noise 146 5.5.4 Impulsive (Salt and Pepper) Noise 147 5.5.5 Exponential Noise 147 5.5.6 Speckle Noise 147 5.6 Image Deblurring 148 5.6.1 Gaussian Blur 149 5.6.2 Motion Blur 149 5.6.3 Rectangular Blur 149 5.6.4 Defocus Blur 150 5.7 Superresolution 150 5.8 Classification of Image Restoration Algorithms 150 5.8.1 Spatial Filtering 151 5.8.2 Frequency Domain Filtering 154 5.8.3 Direct Inverse Filtering 159 5.8.4 Constraint Least-Square Filter 159 5.8.5 IBD (Iterative Blind Deconvolution) 160 5.8.6 NAS-RIF (Nonnegative and Support Constraints Recursive Inverse Filtering) 160 5.8.7 Superresolution Restoration Algorithm Based on Gradient Adaptive Interpolation 160 5.8.8 Deconvolution Using a Sparse Prior 161 5.8.9 Block-Matching 161 5.8.10 LPA-ICI Algorithm 161 5.8.11 Deconvolution Using Regularized Filter (DRF) 161 5.8.12 Lucy-Richardson Algorithm 162 5.8.13 Neural Network Approach 162 5.9 Application of Multiscale Transform in Image Restoration 162 5.9.1 Image Restoration Using Wavelet Transform 163 5.9.2 Image Restoration Using Complex Wavelet Transform 177 5.9.3 Image Restoration Using Quaternion Wavelet Transform 180 5.9.4 Image Restoration Using Ridgelet Transform 182 5.9.5 Image Restoration Using Curvelet Transform 185 5.9.6 Image Restoration Using Contourlet Transform 189 5.9.7 Image Restoration Using Shearlet Transform 194 References 197 6 Image Enhancement 207 6.1 Overview 207 6.2 Spatial Domain Image Enhancement Techniques 208 6.2.1 Gray Level Transformation 209 6.2.2 Piecewise-Linear Transformation Functions 210 6.2.3 Histogram Processing 211 6.2.4 Spatial Filtering 212 6.3 Frequency Domain Image Enhancement Techniques 213 6.3.1 Smoothing Filters 213 6.3.2 Sharpening Filters 215 6.3.3 Homomorphic Filtering 216 6.4 Colour Image Enhancement 217 6.5 Application of Multiscale Transforms in Image Enhancement 218 6.5.1 Image Enhancement Using Fourier Transform 220 6.5.2 Image Enhancement Using Wavelet Transform 222 6.5.3 Image Enhancement Using Complex Wavelet Transform 227 6.5.4 Image Enhancement Using Curvelet Transform 230 6.5.5 Image Enhancement Using Contourlet Transform 231 6.5.6 Image Enhancement Using Shearlet Transform 233 References 236 Appendix A Real and Complex Number System 240 Appendix B Vector Space 243 Appendix C Linear Transformation, Matrices 245 Appendix D Inner Product Space and Orthonormal Basis 247 Appendix E Functions and Convergence 250 E.1 Functions 250 E.2 Convergence of Functions 252 Index 255 Front Matter ....Pages i-xiv Front Matter ....Pages 1-1 Fundamentals of Digital Image Processing (Aparna Vyas, Soohwan Yu, Joonki Paik)....Pages 3-11 Front Matter ....Pages 13-13 Fourier Analysis and Fourier Transform (Aparna Vyas, Soohwan Yu, Joonki Paik)....Pages 15-43 Wavelets and Wavelet Transform (Aparna Vyas, Soohwan Yu, Joonki Paik)....Pages 45-92 New Multiscale Constructions (Aparna Vyas, Soohwan Yu, Joonki Paik)....Pages 93-129 Front Matter ....Pages 131-131 Image Restoration (Aparna Vyas, Soohwan Yu, Joonki Paik)....Pages 133-198 Image Enhancement (Aparna Vyas, Soohwan Yu, Joonki Paik)....Pages 199-231 Back Matter ....Pages 233-254