Opis: Iterative Optimization in Inverse Problems - Charles L. Byrne

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.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

Szczegóły: Iterative Optimization in Inverse Problems - Charles L. Byrne

Tytuł: Iterative Optimization in Inverse Problems
Autor: Charles L. Byrne
Producent: Apple
ISBN: 9781482222333
Rok produkcji: 2014
Ilość stron: 300
Oprawa: Twarda
Waga: 0.54 kg

Recenzje: Iterative Optimization in Inverse Problems - Charles L. Byrne