An overview of fast convergent ordered-subsets reconstruction methods for emission tomography based on the incremental EM algorithm

Statistical reconstruction has become popular in emission computed tomography but suffers slow convergence (to the MAP or ML solution). Methods proposed to address this problem include the fast but non-convergent OSEM and the convergent RAMLA [J. Browne, A. De Pierro, IEEE Trans. Med. Imaging 15 (5) (1996) 687.] for the ML case, and the convergent BSREM [A. De Pierro, M. Yamagishi, IEEE Trans. Med. Imaging 20 (4) (2001) 280.], relaxed OS-SPS and modified BSREM [S. Ahn, J.A. Fessler, IEEE Trans. Med. Imaging 22 (5) (2003) 613.] for the MAP case. The convergent algorithms required a user-determined relaxation schedule. We proposed fast convergent OS reconstruction algorithms for both ML and MAP cases, called COSEM (Complete-data OSEM), which avoid the use of a relaxation schedule while maintaining convergence. COSEM is a form of incremental EM algorithm. Here, we provide a derivation of our COSEM algorithms and demonstrate COSEM using simulations. At early iterations, COSEM-ML is typically slower than RAMLA, and COSEM-MAP is typically slower than optimized BSREM while remaining much faster than conventional MAP-EM. We discuss how COSEM may be modified to overcome these limitations.