/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Yair Censor and Stavros A. Zenios, Oxford University Press, New York, 1997, 539 pp., ISBN 0-19-510062-X, $85.00

Parallel Optimization: Theory, Algorithms and Applications

: Significant progress has been made with solution of location problems and in preprocessing and decomposition for discrete optimization. There has also been research on the application of techniques from combinational optimization to nonlinear problems.

http://www.dtic.mil/dtic/tr/fulltext/u2/a273552.pdf

Discrete Applied Mathematics

In this paper, we present an iterative reconstruction algorithm for discrete tomography, called discrete algebraic reconstruction technique (DART). DART can be applied if the scanned object is known to consist of only a few different compositions, each corresponding to a constant gray value in the reconstruction. Prior knowledge of the gray values for each of the compositions is exploited to steer the current reconstruction towards a reconstruction that contains only these gray values. Based on experiments with both simulated CT data and experimental μCT data, it is shown that DART is capable of computing more accurate reconstructions from a small number of projection images, or from a small angular range, than alternative methods. It is also shown that DART can deal effectively with noisy projection data and that the algorithm is robust with respect to errors in the estimation of the gray values.

DART: A Practical Reconstruction Algorithm for Discrete Tomography

The year 2015 marks the 30th birthday of DC (Difference of Convex functions) programming and DCA (DC Algorithms) which constitute the backbone of nonconvex programming and global optimization. In this article we offer a short survey on thirty years of developments of these theoretical and algorithmic tools. The survey is comprised of three parts. In the first part we present a brief history of the field, while in the second we summarize the state-of-the-art results and recent advances. We focus on main theoretical results and DCA solvers for important classes of difficult nonconvex optimization problems, and then give an overview of real-world applications whose solution methods are based on DCA. The third part is devoted to new trends and important open issues, as well as suggestions for future developments.

DC programming and DCA: thirty years of developments

We present a novel approach to the tomographic reconstruction of binary objects from few projection directions within a limited range of angles. A quadratic objective functional over binary variables comprising the squared projection error and a prior penalizing non-homogeneous regions, is supplemented with a concave functional enforcing binary solutions. Application of a primal-dual subgradient algorithm to a suitable decomposition of the objective functional into the difference of two convex functions leads to an algorithm which provably converges with parallel updates to binary solutions. Numerical results demonstrate robustness against local minima and excellent reconstruction performance using five projections within a range of 90^@?. Our approach is applicable to quite general objective functions over binary variables with constraints and thus applicable to a wide range of problems within and beyond the field of discrete tomography.

Discrete tomography by convex-concave regularization and D.C. programming

Discrete tomography concerns the reconstruction of functions with a finite number of values from few projections For a number of important real-world problems, this tomography problem involves thousands of variables Applicability and performance of discrete tomography therefore largely depend on the criteria used for reconstruction and the optimization algorithm applied From this viewpoint, we evaluate two major optimization strategies, simulated annealing and convex-concave regularization, for the case of binary-valued functions using various data sets Extensive numerical experiments show that despite being quite different from the viewpoint of optimization, both strategies show similar reconstruction performance as well as robustness to noise.

/pdf/a-benchmark-evaluation-of-large-scale-optimization-1x0jbjkcs9.pdf

A benchmark evaluation of large-scale optimization approaches to binary tomography

A Linear Programming Relaxation for Binary Tomography with Smoothness Priors

In this paper we improve the behavior of a reconstruction algorithm for binary tomography in the presence of noise. This algorithm which has recently been published is derived from a primal-dual subgradient method leading to a sequence of linear programs. The objective function contains a smoothness prior that favors spatially homogeneous solutions and a concave functional gradually enforcing binary solutions. We complement the objective function with a term to cope with noisy projections and evaluate its performance.

/pdf/binary-tomography-by-iterating-linear-programs-from-noisy-8446s6xnpd.pdf

Stefan Weber

Papers

Discrete tomography by convex-concave regularization and D.C. programming

A benchmark evaluation of large-scale optimization approaches to binary tomography

A Linear Programming Relaxation for Binary Tomography with Smoothness Priors

Binary tomography by iterating linear programs from noisy projections

Prior Learning and Convex-Concave Regularization of Binary Tomography