166 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 2, APRIL 1997 Grouped-Coordinate Ascent Algorithms for Penalized-Likelihood Transmission Image Reconstruction Jeffrey A. Fessler,* Member, IEEE, Edward P. Ficaro, Member, IEEE, Neal H. Clinthorne, Member, IEEE, and Kenneth Lange Abstract— This paper presents a new class of algorithms for penalized-likelihood reconstruction of attenuation maps from low-count transmission scans. We derive the algorithms by apply- ing to the transmission log-likelihood a version of the convexity technique developed by De Pierro for emission tomography. The new class includes the single-coordinate ascent (SCA) algorithm and Lange’s convex algorithm for transmission tomography as special cases. The new grouped-coordinate ascent (GCA) al- gorithms in the class overcome several limitations associated with previous algorithms. 1) Fewer exponentiations are required than in the transmission maximum likelihood-expectation maxi- mization (ML-EM) algorithm or in the SCA algorithm. 2) The algorithms intrinsically accommodate nonnegativity constraints, unlike many gradient-based methods. 3) The algorithms are easily parallelizable, unlike the SCA algorithm and perhaps line-search algorithms. We show that the GCA algorithms converge faster than the SCA algorithm, even on conventional workstations. An example from a low-count positron emission tomography (PET) transmission scan illustrates the method. Index Terms— Biomedical nuclear imaging, Gauss–Seidel method, iterative methods, maximum likelihood estimation, nuclear tomography, positron emission tomography, single photon emission computed tomography. I. INTRODUCTION S TATISTICAL methods for reconstructing attenuation im- ages from transmission scans have increased in impor- tance recently for several reasons, including the necessity of reconstructing two-dimensional (2-D) attenuation maps for reprojection to form three-dimensional (3-D) attenuation correction factors in septaless positron emission tomography (PET) [1], [2], the widening availability of single photon emission computed tomography (SPECT) systems equipped with transmission sources [3], and the potential for reducing transmission noise in whole body PET images and in other pro- tocols requiring short transmission scans [4]. The nonstatistical filtered backprojection (FBP) method and the data-weighted least-squares method [5] for transmission image reconstruction Manuscript received April 12, 1996; revised October 9, 1996. This work was supported in part by the National Institutes of Health under Grants CA- 60711 and CA-54362. The Associate Editor responsible for coordinating the review of this paper and recommending its publication was R. Leahy. Asterisk indicates corresponding author. *J. A. Fessler is with the University of Michigan, 4240 EECS Bldg., Ann Arbor, MI 48109-2122 USA (email: fessler@umich.edu). E. P. Ficaro, N. H. Clinthorne, and K. Lange are with the University of Michigan, Ann Arbor, MI 48109-2122 USA. Publisher Item Identifier S 0278-0062(97)02405-1. lead to systematic biases for low-count scans [6]–[8]. These biases are due to the nonlinearity of the logarithm applied to the transmission data. To eliminate these biases, one can use statistical methods based on the Poisson measurement statistics, which use the raw measurements rather than its logarithms [9]–[11], [6]. Statistical methods also offer reduced variance relative to FBP [6], [8], [12]. Several reconstruction algorithms based on the Poisson statistical model for transmission scans [13] have appeared recently [6], [10], [11], [14]–[20], all of which converge faster than the original transmission maximum-likelihood expectation-maximization (ML-EM) algorithm [9]. Never- theless, each of these methods is still less than ideal due to one or more of the following reasons. • The EM algorithms [9], [18] and single-coordinate ascent (SCA) algorithms [6], [10], [11] require at least one exponentiation per nonzero element of the system matrix per iteration, which is a large computational expense. • Enforcing nonnegativity in gradient-based algorithms [19]–[22] is possible but somewhat awkward. • Many algorithms are poorly suited to parallel processors such as the i860 arrays that are common at septaless PET sites. This is true of SCA methods and of algorithms that use line searches, since a line-search step may not parallelize easily. This paper describes a new class of algorithms for recon- structing attenuation maps from low-count transmission scans. These algorithms are parallelizable, easily accommodate non- negativity constraints and nonquadratic convex penalties, and require a moderate number of exponentiations. The derivation of these transmission algorithms exploits two ideas underlying recent developments in algorithms for emission tomography: updating the parameters in groups [23], [24], and the convexity technique of De Pierro [25], [26]. Integrating these two ideas leads to a new class of algorithms [27] that converge quickly and with less computation than previous statistical methods for transmission tomography. This work can be considered a generalization of previous methods for tomographic image reconstruction based on se- quential updates [5], [10], [11], [23], [24], [28], [29]. The fast convergence of sequential updates for tomographic problems was analyzed by Fourier methods and shown empirically to converge faster than simultaneous updates in [5]. Tomographic 0278–0062/97$10.00  1997 IEEE