Fast Globally Convergent Reconstruction in Emission Tomography Using COSEM, an Incremental EM Algorithm

We present globally convergent incremental EM algorithms for reconstruction in emission tomography, COSEMML for maximum likelihood and COSEM-MAP for maximum a posteriori reconstruction. The COSEM (Complete data Ordered Subsets Expectation Maximization) algorithms use ordered subsets (OS) for fast convergence, but unlike other globally convergent OS-based ML and MAP algorithms such as RAMLA (Browne and De Pierro, 1996), BSREM (De Pierro and Yamagishi, 2001) and modified BSREM and relaxed OS-SPS (Ahn and Fessler, 2003), COSEM does not require a user-specified object-dependent relaxation schedule. For the ML case, the COSEM-ML algorithm was independently derived previously (Gunawardana, 2001), but our theoretical approach differs. We present convergence proofs for COSEM-ML and COSEM-MAP and we also demonstrate COSEM in SPECT simulations. The monotonicity of COSEM remains an open question. At early iterations, COSEM-ML is typically slower than RAMLA and COSEM-MAP is typically slower than optimized BSREM. For COSEM, the usual speed increase with subset number is slower than that typically observed for OS-type algorithms. We discuss how COSEM may be modified to overcome these limitations.