Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity
This book presents the state of the art in sparse and multiscale image and signal processing, covering linear multiscale transforms, such as wavelet, ridgelet, or curvelet transforms, and non-linear multiscale transforms based on the median and mathematical morphology operators. Recent concepts of sparsity and morphological diversity are described and exploited for various problems such as denoising, inverse problem regularization, sparse signal decomposition, blind source separation, and...
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This book presents the state of the art in sparse and multiscale image and signal processing, covering linear multiscale transforms, such as wavelet, ridgelet, or curvelettransforms, and non-linear multiscale transforms based on the median and mathematical morphology operators. Recent concepts of sparsity and morphological diversity are described and exploited for various problems such as denoising, inverse problem regularization, sparse signal decomposition, blind source separation, and compressed sensing.This book weds theory and practice in examining applications in areas such as astronomy, biology, physics, digital media, and forensics. A final chapter explores a paradigm shift in signal processing, showing that previous limits to information sampling and extraction can be overcome in very significant ways.MATLAB and IDL code accompany these methods and applications to reproduce the experiments and illustrate the reasoning and methodology of the research available for download at the associated Web site.
Acronyms ixNotation xiiiPreface xv1 Introduction to the World of Sparsity 11.1 Sparse Representation 11.2 From Fourier to Wavelets 51.3 From Wavelets to Overcomplete Representations 61.4 Novel Applications of the Wavelet and Curvelet Transforms 81.5 Summary 152 The Wavelet Transform 162.1 Introduction 162.2 The Continuous Wavelet Transform 162.3 Examples of Wavelet Functions 182.4 Continuous Wavelet Transform Algorithm 212.5 The Discrete Wavelet Transform 222.6 Nondyadic Resolution Factor 282.7 The Lifting Scheme 312.8 Wavelet Packets 342.9 Guided Numerical Experiments 382.10 Summary 443 Redundant Wavelet Transform 453.1 Introduction 453.2 The Undecimated Wavelet Transform 463.3 Partially Decimated Wavelet Transform 493.4 The Dual-Tree Complex Wavelet Transform 513.5 Isotropic Undecimated Wavelet Transform: Starlet Transform 533.6 Nonorthogonal Filter Bank Design 583.7 Pyramidal Wavelet Transform 643.8 Guided Numerical Experiments 693.9 Summary 744 Nonlinear Multiscale Transforms 754.1 Introduction 754.2 Decimated Nonlinear Transform 754.3 Multiscale Transform and Mathematical Morphology 774.4 Multiresolution Based on the Median Transform 814.5 Guided Numerical Experiments 864.6 Summary 885 The Ridgelet and Curvelet Transforms 895.1 Introduction 895.2 Background and Example 895.3 Ridgelets 915.4 Curvelets 1005.5 Curvelets and Contrast Enhancement 1105.6 Guided Numerical Experiments 1125.7 Summary 1186 Sparsity and Noise Removal 1196.1 Introduction 1196.2 Term-By-Term Nonlinear Denoising 1206.3 Block Nonlinear Denoising 1276.4 Beyond Additive Gaussian Noise 1326.5 Poisson Noise and the Haar Transform 1346.6 Poisson Noise with Low Counts 1366.7 Guided Numerical Experiments 1436.8 Summary 1457 Linear Inverse Problems 1497.1 Introduction 1497.2 Sparsity-Regularized Linear Inverse Problems 1517.3 Monotone Operator Splitting Framework 1527.4 Selected Problems and Algorithms 1607.5 Sparsity Penalty with Analysis Prior 1707.6 Other Sparsity-Regularized Inverse Problems 1727.7 General Discussion: Sparsity, Inverse Problems, and Iterative Thresholding 1747.8 Guided Numerical Experiments 1767.9 Summary 1788 Morphological Diversity 1808.1 Introduction 1808.2 Dictionary and Fast Transformation 1838.3 Combined Denoising 1838.4 Combined Deconvolution 1888.5 Morphological Component Analysis 1908.6 Texture-Cartoon Separation 1988.7 Inpainting 2048.8 Guided Numerical Experiments 2108.9 Summary 2169 Sparse Blind Source Separation 2189.1 Introduction 2189.2 Independent Component Analysis 2209.3 Sparsity and Multichannel Data 2249.4 Morphological Diversity and Blind Source Separation 2269.5 Illustrative Experiments 2379.6 Guided Numerical Experiments 2429.7 Summary 24410 Multiscale Geometric Analysis on the Sphere 24510.1 Introduction 24510.2 Data on the Sphere 24610.3 Orthogonal Haar Wavelets on the Sphere 24810.4 Continuous Wavelets on the Sphere 24910.5 Redundant Wavelet Transform on the Sphere with Exact Reconstruction 25310.6 Curvelet Transform on the Sphere 26110.7 Restoration and Decomposition on the Sphere 26610.8 Applications 26910.9 Guided Numerical Experiments 27210.10 Summary 27611 Compressed Sensing 27711.1 Introduction 27711.2 Incoherence and Sparsity 27811.3 The Sensing Protocol 27811.4 Stable Compressed Sensing 28011.5 Designing Good Matrices: Random Sensing 28211.6 Sensing with Redundant Dictionaries 28311.7 Compressed Sensing in Space Science 28311.8 Guided Numerical Experiments 28511.9 Summary 286References 289List of Algorithms 311Index 313
