You are previewing Sparse Image and Signal Processing.
O'Reilly logo
Sparse Image and Signal Processing

Book Description

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 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 are available for download at the associated web site.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Contents
  5. Acronyms
  6. Notation
  7. Preface
  8. 1 Introduction to the World of Sparsity
    1. 1.1 Sparse Representation
    2. 1.2 From Fourier to Wavelets
    3. 1.3 From Wavelets to Overcomplete Representations
    4. 1.4 Novel Applications of the Wavelet and Curvelet Transforms
    5. 1.5 Summary
  9. 2 The Wavelet Transform
    1. 2.1 Introduction
    2. 2.2 The Continuous Wavelet Transform
    3. 2.3 Examples of Wavelet Functions
    4. 2.4 Continuous Wavelet Transform Algorithm
    5. 2.5 The Discrete Wavelet Transform
    6. 2.6 Nondyadic Resolution Factor
    7. 2.7 The Lifting Scheme
    8. 2.8 Wavelet Packets
    9. 2.9 Guided Numerical Experiments
    10. 2.10 Summary
  10. 3 Redundant Wavelet Transform
    1. 3.1 Introduction
    2. 3.2 The Undecimated Wavelet Transform
    3. 3.3 Partially Decimated Wavelet Transform
    4. 3.4 The Dual-Tree Complex Wavelet Transform
    5. 3.5 Isotropic Undecimated Wavelet Transform: Starlet Transform
    6. 3.6 Nonorthogonal Filter Bank Design
    7. 3.7 Pyramidal Wavelet Transform
    8. 3.8 Guided Numerical Experiments
    9. 3.9 Summary
  11. 4 Nonlinear Multiscale Transforms
    1. 4.1 Introduction
    2. 4.2 Decimated Nonlinear Transform
    3. 4.3 Multiscale Transform and Mathematical Morphology
    4. 4.4 Multiresolution Based on the Median Transform
    5. 4.5 Guided Numerical Experiments
    6. 4.6 Summary
  12. 5 The Ridgelet and Curvelet Transforms
    1. 5.1 Introduction
    2. 5.2 Background and Example
    3. 5.3 Ridgelets
    4. 5.4 Curvelets
    5. 5.5 Curvelets and Contrast Enhancement
    6. 5.6 Guided Numerical Experiments
    7. 5.7 Summary
  13. 6 Sparsity and Noise Removal
    1. 6.1 Introduction
    2. 6.2 Term-By-Term Nonlinear Denoising
    3. 6.3 Block Nonlinear Denoising
    4. 6.4 Beyond Additive Gaussian Noise
    5. 6.5 Poisson Noise and the Haar Transform
    6. 6.6 Poisson Noise with Low Counts
    7. 6.7 Guided Numerical Experiments
    8. 6.8 Summary
  14. 7 Linear Inverse Problems
    1. 7.1 Introduction
    2. 7.2 Sparsity-Regularized Linear Inverse Problems
    3. 7.3 Monotone Operator Splitting Framework
    4. 7.4 Selected Problems and Algorithms
    5. 7.5 Sparsity Penalty with Analysis Prior
    6. 7.6 Other Sparsity-Regularized Inverse Problems
    7. 7.7 General Discussion: Sparsity, Inverse Problems, and Iterative Thresholding
    8. 7.8 Guided Numerical Experiments
    9. 7.9 Summary
  15. 8 Morphological Diversity
    1. 8.1 Introduction
    2. 8.2 Dictionary and Fast Transformation
    3. 8.3 Combined Denoising
    4. 8.4 Combined Deconvolution
    5. 8.5 Morphological Component Analysis
    6. 8.6 Texture-Cartoon Separation
    7. 8.7 Inpainting
    8. 8.8 Guided Numerical Experiments
    9. 8.9 Summary
  16. 9 Sparse Blind Source Separation
    1. 9.1 Introduction
    2. 9.2 Independent Component Analysis
    3. 9.3 Sparsity and Multichannel Data
    4. 9.4 Morphological Diversity and Blind Source Separation
    5. 9.5 Illustrative Experiments
    6. 9.6 Guided Numerical Experiments
    7. 9.7 Summary
  17. 10 Multiscale Geometric Analysis on the Sphere
    1. 10.1 Introduction
    2. 10.2 Data on the Sphere
    3. 10.3 Orthogonal Haar Wavelets on the Sphere
    4. 10.4 Continuous Wavelets on the Sphere
    5. 10.5 Redundant Wavelet Transform on the Sphere with Exact Reconstruction
    6. 10.6 Curvelet Transform on the Sphere
    7. 10.7 Restoration and Decomposition on the Sphere
    8. 10.8 Applications
    9. 10.9 Guided Numerical Experiments
    10. 10.10 Summary
  18. 11 Compressed Sensing
    1. 11.1 Introduction
    2. 11.2 Incoherence and Sparsity
    3. 11.3 The Sensing Protocol
    4. 11.4 Stable Compressed Sensing
    5. 11.5 Designing Good Matrices: Random Sensing
    6. 11.6 Sensing with Redundant Dictionaries
    7. 11.7 Compressed Sensing in Space Science
    8. 11.8 Guided Numerical Experiments
    9. 11.9 Summary
  19. References
  20. List of Algorithms
  21. Index
  22. Color Plates