Compressive imaging : structure, sampling, learning / Ben Adcock, Anders C. Hansen, Vegard Antun.

Format:

Book

Author/Creator:

Adcock, Ben, author.

Hansen, Anders C., author.

Antun, Vegard, author.

Language:

English

Subjects (All):

Image compression.

Digital images--Deconvolution.

Digital images.

Physical Description:

1 online resource (xv, 601 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Cambridge : Cambridge University Press, 2021.

Summary:

Accurate, robust and fast image reconstruction is a critical task in many scientific, industrial and medical applications. Over the last decade, image reconstruction has been revolutionized by the rise of compressive imaging. It has fundamentally changed the way modern image reconstruction is performed. This in-depth treatment of the subject commences with a practical introduction to compressive imaging, supplemented with examples and downloadable code, intended for readers without extensive background in the subject. Next, it introduces core topics in compressive imaging - including compressed sensing, wavelets and optimization - in a concise yet rigorous way, before providing a detailed treatment of the mathematics of compressive imaging. The final part is devoted to recent trends in compressive imaging: deep learning and neural networks. With an eye to the next decade of imaging research, and using both empirical and mathematical insights, it examines the potential benefits and the pitfalls of these latest approaches.

Contents:

Cover

Half-title

Title page

Copyright information

Dedication

Contents

Preface

1 Introduction

1.1 Imaging and Inverse Problems

1.2 What is Compressive Imaging?

1.3 Terminology

1.4 Imaging Modalities

1.5 Conventional Compressed Sensing

1.6 Imaging with Compressed Sensing

1.7 Neural Networks and Deep Learning for Compressive Imaging

1.8 Overview and Highlights

1.9 Disclaimers

1.10 Reading this Book

Part I The Essentials of Compressive Imaging

Summary of Part I

2 Images, Transforms and Sampling

2.1 Images

2.2 Sampling with the Fourier Transform

2.3 Sampling with the Radon Transform

2.4 Binary Sampling with the Walsh Transform

3 A Short Guide to Compressive Imaging

3.1 The Six Stages of the Compressive Imaging Pipeline

3.2 First Example: MRI

3.3 Second Example: X-ray CT

3.4 Third Example: Optical Imaging

3.5 Recovery of Image Sequences

4 Techniques for Enhancing Performance

4.1 Sampling Strategies

4.2 How to Design a Sampling Strategy

4.3 Resolution Enhancement

4.4 Discretization, Model Mismatch and the Inverse Crime

4.5 Sparsifying Transforms

4.6 Iterative Reweighting

4.7 Learning: Incorporating Training Data

4.8 An All-Round Model-Based Compressive Imaging Strategy

Part II Compressed Sensing, Optimization and Wavelets

Summary of Part II

5 An Introduction to Conventional Compressed Sensing

5.1 Projections

5.2 Sparsity and Compressibility

5.3 Measurements and Measurement Matrices

5.4 [ell[sup(1)]]-Minimization

5.5 Recovery Guarantees in Compressed Sensing

5.6 Techniques for Uniform Recovery

5.7 Techniques for Nonuniform Recovery

5.8 Oracle Estimators

6 The LASSO and its Cousins

6.1 Definitions

6.2 Summary

6.3 Uniform Recovery Guarantees

6.4 Nonuniform Recovery Guarantees.

7 Optimization for Compressed Sensing

7.1 Minimizers, Not Minimum Values

7.2 Flavours of Optimization for Compressive Imaging

7.3 Gradient Descent

7.4 Forward-Backward Splitting

7.5 The Primal-Dual Iteration

7.6 Nesterov's Method and NESTA for QCBP

8 Analysis of Optimization Algorithms

8.1 The rNSP and Inexactness

8.2 Compressed Sensing Analysis of the Primal-Dual Iteration

8.3 Compressed Sensing Analysis of NESTA

8.4 Computability and Complexity of Finding Minimizers

8.5 Impossibility of Computing Minimizers from Inexact Input

8.6 Complexity Theory for Compressed Sensing Problems

9 Wavelets

9.1 Introduction

9.2 Multiresolution Analysis

9.3 Wavelet Construction from an MRA

9.4 Wavelet Design

9.5 Compactly Supported Wavelets

9.6 Daubechies Wavelets

9.7 Wavelets on Intervals

9.8 Higher Dimensions

9.9 Summary and Orderings

9.10 Discrete Wavelet Computations

9.11 Beyond Wavelets

10 A Taste of Wavelet Approximation Theory

10.1 Linear and Nonlinear Approximation

10.2 Linear and Nonlinear Wavelet Approximation Rates

10.3 Discussion

10.4 Proofs of the Approximation Results

10.5 Higher Dimensions

Part III Compressed Sensing with Local Structure

Summary of Part III

11 From Global to Local

11.1 The Fourier-Wavelets Problem

11.2 Structured Sparsity and the Flip Test

11.3 Local Sparsity in Levels

11.4 Sampling Operators

11.5 Notions of Coherence and Recovery Guarantees

11.6 The One-Dimensional Discrete Fourier-Haar Wavelet Problem

12 Local Structure and Nonuniform Recovery

12.1 Weighted [ell[sup(1)]]-Minimization

12.2 Local Recovery Guarantee

12.3 The Sparse Model

12.4 The Sparse in Levels Model

12.5 The Fourier-Haar Wavelet Problem

12.6 Comparison with Oracle Estimators

12.7 Coherences of the Discrete Fourier-Haar Matrix.

12.8 Proof of Theorem 12.4

12.9 Improvements for Random Signal Models

13 Local Structure and Uniform Recovery

13.1 Decoders and Recovery Guarantees

13.2 Restricted Isometry and Robust Null Space Properties

13.3 Stable and Accurate Recovery via the rNSPL and G-RIPL

13.4 Measurement Conditions for Uniform Recovery

13.5 Proof of Theorem 13.12

14 Infinite-Dimensional Compressed Sensing

14.1 Motivations

14.2 When Vectors Become Functions

14.3 Bandwidth

14.4 Generalized Sampling

14.5 Compressed Sensing in Infinite Dimensions

14.6 Recovery Guarantees for Weighted QCBP

14.7 Feasibility and the SR-LASSO

Part IV Compressed Sensing for Imaging

Summary of Part IV

15 Sampling Strategies for Compressive Imaging

15.1 Overview

15.2 The DS, DIS and DAS Schemes

15.3 Summary of the Measurement Conditions

15.4 When and Why Do Certain Patterns Work Well?

16 Recovery Guarantees for Wavelet-Based Compressive Imaging

16.1 Decoders and Recovery Guarantees

16.2 What does this Mean? Near-Optimal Wavelet Approximation

16.3 Main Recovery Guarantees

16.4 Proofs of the Recovery Guarantees

17 Total Variation Minimization

17.1 Definitions

17.2 Two Curious Experiments

17.3 Recovery Guarantees for Fourier Sampling

17.4 Proofs

17.5 Recovery Guarantees for Haar-Incoherent Measurements

17.6 Structure-Dependent Sampling

Part V From Compressed Sensing to Deep Learning

Summary of Part V

18 Neural Networks and Deep Learning

18.1 Supervised Machine Learning

18.2 Neural Networks

18.3 The Universal Approximation Theorem

18.4 Architecture Design and Extensions

18.5 Training a Neural Network

18.6 The Success of Deep Learning

18.7 Instabilities in Deep Learning

18.8 Why do Instabilities Occur?

19 Deep Learning for Compressive Imaging.

19.1 Why Go Beyond Compressed Sensing

19.2 Deep Learning for Inverse Problems

19.3 Experimental Setup

19.4 Testing for Instabilities in Neural Networks for Compressive Imaging

20 Accuracy and Stability of Deep Learning for Compressive Imaging

20.1 Optimal Maps

20.2 Examining Instabilities in Deep Learning for Compressive Imaging

20.3 The Stability versus Performance Tradeoff

20.4 The Tradeoff for Compressed Sensing and Deep Learning

20.5 Can Instabilities be Remedied?

21 Stable and Accurate Neural Networks for Compressive Imaging

21.1 Algorithms for Computing Neural Networks

21.2 Unravelled ISTA

21.3 Unravelling the Primal-Dual Iteration

21.4 Stable and Accurate Neural Networks for Compressed Sensing

21.5 Stable and Accurate Neural Networks for Compressive Imaging

Epilogue

Appendices

Summary of the Appendices

Appendix A Linear Algebra

A.1 Norms

A.2 Orthogonality and Orthonormal Bases

A.3 Matrices

A.4 Matrix Norms

A.5 Further Properties of the Matrix [ell[sup(2)]]-Norm

A.6 The Singular Value Decomposition

A.7 Least Squares and the Pseudoinverse

A.8 Circulant Matrices and Convolution

A.9 Kronecker products of matrices

Appendix B Functional Analysis

B.1 Metric Spaces

B.2 Banach and Hilbert Spaces

B.3 The [ell[sup(p)]] and L[sup(p)] Spaces

B.4 Operators between Normed Spaces

B.5 Orthogonality in Hilbert Spaces

B.6 Orthonormal Bases of Hilbert Spaces

Appendix C Probability

C.1 Definition and Basic Properties

C.2 Random Variables

C.3 Joint Distributions and Independence

C.4 Expectation and Variance

Appendix D Convex Analysis and Convex Optimization

D.1 Convex Sets and Convex Functions

D.2 Subdifferentials

D.3 Convex Optimization

D.4 The Convex Conjugate

D.5 The Proximal Map

Appendix E Fourier Transforms and Series.

E.1 The Fourier Transform

E.2 The Fourier Basis

E.3 The Discrete Fourier Transform

Appendix F Properties of Walsh Functions and the Walsh Transform

F.1 Sign Changes

F.2 Binary Addition, Translation and Scaling

F.3 Orthonormality and Relation to Haar Wavelets

F.4 The Paley and Sequency Orderings

F.5 Properties of the Hadamard Matrix

F.6 The Fast Walsh-Hadamard Transform

Notation

Abbreviations

References

Index.

Notes:

Title from publisher's bibliographic system (viewed on 16 Jul 2021).

ISBN:

1-108-38391-2

1-108-37744-0

OCLC:

1266907549