In this paper, we propose a new method to solve the minimization problem in a simultaneous reconstruction and segmentation (SRS) model for X-ray computed tomography (CT). The SRS model uses Bayes' rule and the maximum a posteriori (MAP) estimate on the hidden Markov measure field model (HMMFM). The original method [Romanov M, Dahl AB, Dong Y, Hansen PC. Simultaneous tomographic reconstruction and segmentation with class priors. Inverse Problems Sci Eng. 2016;24(8):1432–1453] includes a subproblem with logarithmic-summation (log-sum) term, which is non-separable to the classification index. This subproblem was solved by Frank–Wolfe algorithm, which is very time consuming especially when dealing with large-scale CT problems. The starting point of this paper is the commutativity of log-sum operations, where the log-sum problem could be transformed into a sum-log problem by introducing an auxiliary variable. The corresponding sum-log problem for the SRS model is separable. By applying the primal-dual algorithm, the sum-log problem turns into several easy-to-solve convex subproblems. In addition, we introduce an improved model by adding Tikhonov regularization on the SRS model, and give some convergence results for the proposed methods. Experimental results demonstrate that the proposed methods produce comparable results compared with the original SRS method with much less CPU time.

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