Iterative Optimization in Inverse Problems: 1st Edition (Hardback) book cover

Iterative Optimization in Inverse Problems

1st Edition

By Charles L. Byrne

Chapman and Hall/CRC

300 pages | 6 B/W Illus.

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pub: 2014-02-12
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Description

Iterative Optimization in Inverse Problems brings together a number of important iterative algorithms for medical imaging, optimization, and statistical estimation. It incorporates recent work that has not appeared in other books and draws on the author’s considerable research in the field, including his recently developed class of SUMMA algorithms. Related to sequential unconstrained minimization methods, the SUMMA class includes a wide range of iterative algorithms well known to researchers in various areas, such as statistics and image processing.

Organizing the topics from general to more specific, the book first gives an overview of sequential optimization, the subclasses of auxiliary-function methods, and the SUMMA algorithms. The next three chapters present particular examples in more detail, including barrier- and penalty-function methods, proximal minimization, and forward-backward splitting. The author also focuses on fixed-point algorithms for operators on Euclidean space and then extends the discussion to include distance measures other than the usual Euclidean distance. In the final chapters, specific problems illustrate the use of iterative methods previously discussed. Most chapters contain exercises that introduce new ideas and make the book suitable for self-study.

Unifying a variety of seemingly disparate algorithms, the book shows how to derive new properties of algorithms by comparing known properties of other algorithms. This unifying approach also helps researchers—from statisticians working on parameter estimation to image scientists processing scanning data to mathematicians involved in theoretical and applied optimization—discover useful related algorithms in areas outside of their expertise.

Table of Contents

Background

Overview

An Urns Model for Remote Sensing

Hidden Markov Models

Measuring the Fourier Transform

Transmission Tomography

Emission Tomography

A Unifying Framework

Sequential Optimization

Overview

Examples of SUM

Auxiliary-Function Methods

The SUMMA Class of AF Methods

Barrier-Function and Penalty-Function Methods

Barrier Functions

Examples of Barrier Functions

Penalty Functions

Examples of Penalty Functions

Basic Facts

Proximal Minimization

The Basic Problem

Proximal Minimization Algorithms

Some Obstacles

All PMA Are SUMMA

Convergence of the PMA

The Non-Differentiable Case

The IPA

Projected Gradient Descent

Relaxed Gradient Descent

Regularized Gradient Descent

The Projected Landweber Algorithm

The Simultaneous MART

A Convergence Theorem

Another Job for the PMA

The Goldstein-Osher Algorithm

A Question

The Forward-Backward Splitting Algorithm

Moreau’s Proximity Operators

The FBS Algorithm

Convergence of the FBS Algorithm

Some Examples

Minimizing f2 over a Linear Manifold

Feasible-Point Algorithms

Operators

Overview

Operators

Contraction Operators

Convex Sets in RJ

Orthogonal Projection Operators

Firmly Nonexpansive Gradients

Exercises

Averaged and Paracontractive Operators

Averaged Operators

Gradient Operators

Two Useful Identities

The Krasnosel’skii-Mann-Opial Theorem

Affine Linear Operators

Paracontractive Operators

Exercises

Convex Feasibility and Related Problems

Convex Constraint Sets

Using Orthogonal Projections

The ART

Regularization

Avoiding the Limit Cycle

Exercises

Eigenvalue Bounds

Introduction and Notation

Overview

Cimmino’s Algorithm

The Landweber Algorithms

Some Upper Bounds for L

Simultaneous Iterative Algorithms

Block-Iterative Algorithms

Exercises

Jacobi and Gauss-Seidel Methods

The Jacobi and Gauss-Seidel Methods: An Example

Splitting Methods

Some Examples of Splitting Methods

Jacobi’s Algorithm and JOR

The Gauss-Seidel Algorithm and SOR

Summary

The SMART and EMML Algorithms

The SMART Iteration

The EMML Iteration

The EMML and the SMART as AM

The SMART as SUMMA

The SMART as PMA

Using KL Projections

The MART and EMART Algorithms

Extensions of MART and EMART

Convergence of the SMART and EMML

Regularization

Modifying the KL Distance

The ABMART Algorithm

The ABEMML Algorithm

Alternating Minimization

Alternating Minimization

Exercises

The EM Algorithm

Overview

A Non-Stochastic Formulation of EM

The Stochastic EM Algorithm

The Discrete Case

Missing Data

The Continuous Case

EM and the KL Distance

Finite Mixture Problems

Geometric Programming and the MART

Overview

An Example of a GP Problem

The Generalized AGM Inequality

Posynomials and the GP Problem

The Dual GP Problem

Solving the GP Problem

Solving the DGP Problem

Constrained Geometric Programming

Exercises

Variational Inequality Problems and Algorithms

Monotone Functions

The Split-Feasibility Problem

The Variational Inequality Problem

Korpelevich’s Method for the VIP

On Some Algorithms of Noor

Split Variational Inequality Problems

Saddle Points

Exercises

Set-Valued Functions in Optimization

Overview

Notation and Definitions

Basic Facts

Monotone Set-Valued Functions

Resolvents

Split Monotone Variational Inclusion

Solving the SMVIP

Special Cases of the SMVIP

The Split Common Null-Point Problem

Exercises

Fenchel Duality

The Legendre-Fenchel Transformation

Fenchel’s Duality Theorem

An Application to Game Theory

Exercises

Compressed Sensing

Compressed Sensing

Sparse Solutions

Minimum One-Norm Solutions

Why Sparseness?

Compressed Sampling

Appendix: Bregman-Legendre Functions

Essential Smoothness and Essential Strict Convexity

Bregman Projections onto Closed Convex Sets

Bregman-Legendre Functions

Bibliography

Index

About the Originator

About the Series

Chapman & Hall/CRC Monographs and Research Notes in Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
BUS049000
BUSINESS & ECONOMICS / Operations Research
MAT003000
MATHEMATICS / Applied
TEC029000
TECHNOLOGY & ENGINEERING / Operations Research