Yair Censor

Yair Censor, Alexander J. Zaslavski

We consider the convex feasibility problem (CFP) in Hilbert space and concentrate on the study of string-averaging projection (SAP) methods for the CFP, analyzing their convergence and their perturbation resilience. In the past, SAP methods were formulated with a single predetermined set of strings and a single predetermined set of weights. Here we extend the scope of the family of SAP methods to allow iteration-index-dependent variable strings and weights and term such methods dynamic string-averaging projection (DSAP) methods. The bounded perturbation resilience of DSAP methods is relevant and important for their possible use in the framework of the recently developed superiorization heuristic methodology for constrained minimization problems.

Dan Butnariu, Yair Censor, Pini Gurfil et al.

We study some methods of subgradient projections for solving a convex feasibility problem with general (not necessarily hyperplanes or half-spaces) convex sets in the inconsistent case and propose a strategy that controls the relaxation parameters in a specific self-adapting manner. This strategy leaves enough user-flexibility but gives a mathematical guarantee for the algorithm's behavior in the inconsistent case. We present numerical results of computational experiments that illustrate the computational advantage of the new method.

Ron Aharoni, Yair Censor, Zilin Jiang

Given two disjoint convex polyhedra, we look for a best approximation pair relative to them, i.e., a pair of points, one in each polyhedron, attaining the minimum distance between the sets. Cheney and Goldstein showed that alternating projections onto the two sets, starting from an arbitrary point, generate a sequence whose two interlaced subsequences converge to a best approximation pair. We propose a process based on projections onto the half-spaces defining the two polyhedra, which are more negotiable than projections on the polyhedra themselves. A central component in the proposed process is the Halpern--Lions--Wittmann--Bauschke algorithm for approaching the projection of a given point onto a convex set.

Kay Barshad, Yair Censor

We study the strong convergence and bounded perturbation resilience of iterative algorithms based on the Generalized Modular String-Averaging (GMSA) procedure for infinite sequences of input operators under a general admissible control. These methods address a variety of feasibility-seeking problems in real Hilbert spaces, including the common fixed point problem and the convex feasibility problem. In addition to the general case, involving certain strongly quasi-nonexpansive input operators, we consider a specific subclass of their corresponding relaxed firmly nonexpansive operators. This subclass proves useful for establishing bounded perturbation resilience. We further demonstrate the applicability of our strong convergence results, within the GMSA framework, to the Superiorization Methodology and to Dynamic String-Averaging, analyzing the behavior of a superiorized version of our main algorithm. The novelty and significance of this work is that it not only includes a variety of earlier algorithms as special cases but, more importantly, it allows the use of modular options of string-averaging that give rise to new, hitherto unavailable, algorithmic schemes with emphasis on infinitely many input operators. The strong convergence guarantees and the applications for superiorization and dynamic string-averaging are also important facets.

Blake Schultze, Micah Witt, Yair Censor et al.

Proton computed tomography (pCT) is a novel imaging modality developed for patients receiving proton radiation therapy. The purpose of this work was to investigate hull-detection algorithms used for preconditioning of the large and sparse linear system of equations that needs to be solved for pCT image reconstruction. The hull-detection algorithms investigated here included silhouette/space carving (SC), modified silhouette/space carving (MSC), and space modeling (SM). Each was compared to the cone-beam version of filtered backprojection (FBP) used for hull-detection. Data for testing these algorithms included simulated data sets of a digital head phantom and an experimental data set of a pediatric head phantom obtained with a pCT scanner prototype at Loma Linda University Medical Center. SC was the fastest algorithm, exceeding the speed of FBP by more than 100 times. FBP was most sensitive to the presence of noise. Ongoing work will focus on optimizing threshold parameters in order to define a fast and efficient method for hull-detection in pCT image reconstruction.