Fast EM-like methods for maximum "a posteriori" estimates in emission tomography

Abstract

The maximum-likelihood (ML) approach in emission tomography provides images with superior noise characteristics compared to conventional filtered backprojection (FBP) algorithms. The expectation-maximization (EM) algorithm is an iterative algorithm for maximizing the Poisson likelihood in emission computed tomography that became very popular for solving the ML problem because of its attractive theoretical and practical properties. Recently, (Browne and DePierro, 1996 and Hudson and Larkin, 1994) block sequential versions of the EM algorithm that take advantage of the scanner's geometry have been proposed in order to accelerate its convergence. In Hudson and Larkin, 1994, the ordered subsets EM (OS-EM) method was applied to the ML problem and a modification (OS-GP) to the maximum a posteriori (MAP) regularized approach without showing convergence. In Browne and DePierro, 1996, we presented a relaxed version of OS-EM (RAMLA) that converges to an ML solution. In this paper, we present an extension of RAMLA for MAP reconstruction. We show that, if the sequence generated by this method converges, then it must converge to the true MAP solution. Experimental evidence of this convergence is also shown. To illustrate this behavior we apply the algorithm to positron emission tomography simulated data comparing its performance to OS-GP.

Superiorization of incremental optimization algorithms for statistical tomographic image reconstruction

“…Algorithm (6) is correspondingly named the superiorized version of Algorithm (4). Notice that if we definẽ…”

“…, x n are the components of x. Each of these operators is no more than a block-ramla (bramla) iteration over a specific data subset [2,4]. A bramla iteration is a generalization of a ramla iteration whereby blocks of data can be used at each subiteration.…”

Reconstruction of coronary arteries from X-ray angiography: A review

“…These algorithms are used without any modifications, however we mention these algorithms in this section for completeness. Alternatives for energy minimization include gradient descent (Hansis et al, 2009), stochastic gradient descent (Rohkohl et al, 2009a(Rohkohl et al, , 2010a, L-BFGS-B (Rohkohl et al, 2009b(Rohkohl et al, , 2010b, separable paraboloidal surrogates (SPS) (Erdogan and Fessler, 1999;Hu et al, 2010Hu et al, , 2012, and block sequential regularized expectation maximization (BSREM) (de Pierro and Beleza Yamagishi, 2001;Zhou et al, 2008).…”

Accelerating ordered-subsets image reconstruction for x-ray CT using double surrogates

“…9,10 Despite their success in speeding up the initial convergence, ordinary OS algorithms are not convergent in general but rather approach a suboptimal limit-cycle without relaxation, i.e., when α n = α. To address this issue, several families of convergent OS type algorithms have been proposed, [11][12][13][14] although those modifications tend to slow down convergence. In this paper, we focus on the initial convergence characteristics of OS algorithms rather than their final convergence properties.…”