Simultaneous Coordinate Maximization Algorithm for Maximum A Posteriori Compton Camera Imaging With Markov Random Field Prior

It is widely acknowledged that maximum-a-posteriori (MAP) estimation, when combined with a Markov random field (MRF) prior, is an effective tool for Compton camera imaging from Poisson data. While MAP estimation involves solving an optimization problem, the primary challenge arises from the correlation inherent in the MRF prior. Unlike most existing expectation maximization (EM)-like algorithms that address this challenge indirectly, we propose a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">simultaneous</i> coordinate maximization (SCM) algorithm to directly handle convex MRF priors. Basically, the proposed algorithm breaks the correlation within MRF in the same way as sequential coordinate ascent algorithms; however, it allows updating all coordinates simultaneously at each iteration, rather than one coordinate or one block of coordinates sequentially. It is thus applicable to large-scale optimization problems, and hence especially suitable for high-dimensional Compton image reconstruction in real-time. We prove the convergence of the SCM algorithm and analyze its convergence rate and complexity using both analytical and numerical methods. In light of the SCM algorithm, we develop a closed-form algorithm called MAP-SCM-EM for Compton camera imaging under the assumption of the EM surrogate of the Poisson log-likelihood function and the zero-mean Gaussian MRF prior. Numerous comparative studies with more classical reconstruction algorithms using real-world data, conducted with hand-held CeBr3 Temporal Compton cameras developed by Damavan company, have confirmed that our algorithm offers a good compromise between speed and accuracy of reconstruction.