Convergence results for scaled gradient algorithms in positron emission tomography

Abstract

Positron emission tomography is a well-known technique aiming at reconstructing the emission density of a compound tagged by an artificial isotope inside the body in order to study a given physiological process. The maximum likelihood (ML) approach to PET has been proven to be adequate for modelling this problem and iterative methods provide the option to compute the solutions. In the 1980s the EM (expectation maximization) algorithm was the accepted choice, which was substituted by the faster OS-EM (ordered subsets EM) and its convergent version, RAMLA (row action ML algorithm). In this paper, we present an improved and extended convergence proof for RAMLA, which includes another recently proposed algorithm.

Bounded perturbation resilience of projected scaled gradient methods

“…with {τ k } ∞ k=0 as relaxation parameters [18, Section 5.1] and {D KL (x k )} ∞ k=0 as diagonal scaling matrices. The diagonal scaling matrices for the generalized EM-type methods are typically of the form, see, e.g., [29],…”

“…k=0 be any sequence generated by the PSG method with bounded outer perturbations of (29). Let η 1 be given by (55) and {δ k } ∞ k=0 be given by (49).…”

“…k=0 is any sequence generated by the PSG method with bounded outer perturbations of (29), then there exists a constant η 3 > 0 and an index K 3 > 0 such that for all k > K 3…”

Bounded Perturbation Resilience and Superiorization of Proximal Scaled Gradient Algorithm with Multi-Parameters

“…The scaled method was proposed by Strand [20] for increasing the rate of convergence of some algorithm. In a finite dimensional space, the selection of scaling matrices depends on the particular problem [21,22]. Jin, Censor and Jiang [13] introduced the following projected scaled gradient (PSG) algorithm:…”

String-averaging incremental subgradient methods for constrained convex optimization problems

“…, m}, are convex functions. Problems that can be modeled as in (1.1) appear frequently in practice and have been studied often in the literature: for instance, we can mention tomographic image reconstruction problems (BROWNE;PIERRO, 1996;PIERRO, 2005;PIERRO, 2009;OLIVEIRA;COSTA, 2016), assignment problems BERTSEKAS, 2001b), least squares problems (BERTSEKAS, 1997), distributed optimization problems in wireless sensor networks (BLATT; HERO; GAUCHMAN, 2007) and neural network training (BERTSEKAS;TSITSIKLIS, 1996).…”