Convergence and resolution recovery of block-iterative EM algorithms modeling 3D detector response in SPECT

The authors evaluate fast reconstruction algorithms including ordered subsets EM (OS-EM) and Rescaled Block Iterative EM (RBI-EM) in fully 3D SPECT applications on the basis of their convergence and resolution recovery properties as iterations proceed. Using a 3D computer-simulated phantom consisting of 3D Gaussian objects, the authors simulated projection data that includes only the effects of sampling and detector response of a parallel-hole collimator. Reconstructions were performed using each of the 3 algorithms (ML-EM, OS-EM, and RBI-EM) modeling the 3D detector response in the projection function. Resolution recovery was evaluated by fitting Gaussians to each of the 4 objects in the iterated image estimates at selected intervals. Results show that OS-EM and RBI-EM behave identically in this case; their resolution recovery results are virtually indistinguishable. Their resolution behavior appears to be very similar to that of ML-EM, but accelerated by a factor of 20. For all 3 algorithms, smaller objects take more iterations to converge. Next, the authors consider the effect noise has on convergence. For both noise-free and noisy data, they evaluate the log likelihood function at each subiteration of OS-EM and RBI-EM, and at each iteration of ML-EM. With noisy data, both OS-EM and RBI-EM give results for which the log-likelihood function oscillates. Especially for 180-degree acquisitions, RBI-EM oscillates less than OS-EM. Both OS-EM and RBI-EM appear to converge to solutions, but not to the ML solution. It is concluded that both OS-EM and RBI-EM can be effective algorithms for fully 3D SPECT reconstruction. Both recover resolution similarly to ML-EM, only more quickly.