Accelerating the EMML algorithm and related iterative algorithms by rescaled block-iterative methods

Abstract

Analysis of convergence of the algebraic reconstruction technique (ART) shows it to be predisposed to converge to a solution faster than simultaneous methods, such as those of the Cimmino-Landweber type, the expectation maximization maximum likelihood method for the Poisson model (EMML), and the simultaneous multiplicative ART (SMART), which use all the data at each step. Although the choice of ordering of the data and of relaxation parameters are important, as Herman and Meyer have shown, they are not the full story. The analogous multiplicative ART (MART), which applies only to systems y=Px in which y>0, P= or >0 and a nonnegative solution is sought, is also sequential (or "row-action"), rather than simultaneous, but does not generally exhibit the same accelerated convergence relative to its simultaneous version, SMART. By dividing each equation by the maximum of the corresponding row of P, we find that this rescaled MART (RMART) does converge faster, when solutions exist, significantly so in cases in which the row maxima are substantially less than one. Such cases arise frequently in tomography and when the columns of P have been normalized to have sum one. Between simultaneous methods, which use all the data at each step, and sequential (or row-action) methods, which use only a single data value at each step, there are the block-iterative (or ordered subset) methods, in which a single block or subset of the data is processed at each step. The ordered subset EM (OSEM) of Hudson et al. is significantly faster than the EMML, but often fails to converge. The "rescaled block-iterative" EMML (RBI-EMML) is an accelerated block-iterative version of EMML that converges, in the consistent case, to a solution, for any choice of subsets; it reduces to OSEM when the restrictive "subset balanced" condition holds. Rescaled block-iterative versions of SMART and MART also exhibit accelerated convergence.

Csiszár’s Divergences for Non-negative Matrix Factorization: Family of New Algorithms

“…The above algorithm can be considered as an alternating minimization/projection extension of the well known SMART (Simultaneous Multiplicative Algebraic Reconstruction Technique) [9].…”

“…which are extensions (by alternating minimization) of the well known EMML or Richardson-Lucy algorithm (RLA) [9]. It should be noted that he most existing NMF algorithms perform blind source separation rather very poorly due to the non-uniqueness of solution and/or the lack of additional constraints which should be satisfied.…”

“…which is an extension of the well known ISRA (Image Space Reconstruction Algorithm) algorithm [9]. Alternative mostly used loss function that intrinsically ensures non-negativity constraints and it is related to the Poisson likelihood is a functional based on the Kullback-Leibler divergence [3,5]:…”

Attenuation correction using SPECT emission data only

“…In our experiments, we have solved § directly using nonlinear optimization methods, since the standard versions of the EM-algorithm, along with its modifications like OS-EM [18] or row action methods [7], are no longer applicable.…”

Accelerated time-of-flight (TOF) PET image reconstruction using TOF bin subsetization and TOF weighting matrix pre-computation

“…Block-iterative or row-action schemes aim to geometrically or chronologically partition the emission data into a number of subsets or blocks and to sequentially update the entire image with each block. The ordered subsets EM (OSEM) (Hudson and Larkin 1994), rescaled block-iterative EM (RBI-EM) (Byrne 1998) and rowaction maximum likelihood (RAMLA) (Browne and De Pierro 1996) algorithms are the best instances of these approaches. The OSEM algorithm, which is based on the subsetization of angular projection bins into a series of mutually exclusive (balanced) subsets, shows a considerably improved initial convergence and therefore was the first sufficiently fast algorithm paving the way for clinical adoption of iterative reconstruction methods.…”