Accelerated Image Reconstruction Using Ordered Subsets of Projection Data Iii. Selecting Subsets and Order

| We deene ordered subset processing for standard algorithms (such as Expectation Maximization, EM) for image restoration from projections. Ordered subsets methods group projection data into an ordered sequence of subsets (or blocks). An iteration of ordered subsets EM is deened as a single pass through all the subsets, in each subset using the current estimate to initialise application of EM with that data subset. This approach is similar in concept to block-Kaczmarz methods introduced by Eggermont et al 1] for iterative reconstruction. Simultaneous iterative reconstruction (SIRT) and multiplicative algebraic reconstruction (MART) techniques are well known special cases. Ordered subsets EM (OS-EM) provides a restoration imposing a natural positiv-ity condition and with close links to the EM algorithm. OS-EM is applicable in both single photon (SPECT) and positron emission tomography (PET). In simulation studies in SPECT the OS-EM algorithm provides an order-of-magnitude acceleration over EM, with restoration quality maintained. The application of Expectation Maximization (EM) algorithms in emission tomography by Shepp and Vardi 2] has led to the introduction of many related techniques. These include a number of important Bayesian (or equivalently penalized likelihood) adaptions of EM. Recent papers have emphasized the quality of the reconstruction ooered by these algorithms. See Hebert and Leahy 3] and Green 4] for some recent Bayesian developments, and Chornboy et al 5] for an evaluation of the beneets of EM in single photon emission tomography (SPECT). While quality of reconstruction is good, the application of EM is computer intensive, and convergence slow, even with standard acceleration techniques (Kaufman, 6]). We provide here an ordered subset (OS) algorithm that processes the data in subsets (blocks) within each iteration and show that this procedure accelerates convergence by a factor proportional to the number of subsets. In SPECT, the sequential processing of ordered subsets is very natural, as projection data is collected separately for each projection angle (as a camera rotates around the patient in SPECT); counts on single projections can form successive subsets. Computational beneet from such`divide and conquer' processing may be anticipated, as in applications in sorting and fast Fourier transforms. Related approaches to the solution of linear systems have been used in tomography (see Section IV). With data acquired in time order, sequential processing is also an option. General approaches to recursive estimation in processing sequences of images are discussed by Green and Titterington 7]. Titterington 8] has provided a recursive EM algorithm for …