  In this paper, we propose a fast method for simultaneous reconstruction and
segmentation (SRS) in X-ray computed tomography (CT). Our work is based on the
SRS model where Bayes' rule and the maximum a posteriori (MAP) are used on
hidden Markov measure field model (HMMFM). The original method leads to a
logarithmic-summation (log-sum) term, which is non-separable to the
classification index. The minimization problem in the model was solved by using
constrained gradient descend method, 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. After applying alternating minimization method, this problem turns
into several easy-to-solve convex sub-problems. In the paper, we also study an
improved model by adding Tikhonov regularization, and give some convergence
results. Experimental results demonstrate that the proposed algorithms could
produce comparable results with the original SRS method with much less CPU
time.
