Monotonic Algorithms for Transmission Tomography

We present a framework for designing fast and monotonic algorithms for transmission tomography penalized-likelihood image reconstruction. The new algorithms are based on paraboloidal surrogate functions for the log likelihood. Due to the form of the log-likelihood function it is possible to find low curvature surrogate functions that guarantee monotonicity. Unlike previous methods, the proposed surrogate functions lead to monotonic algorithms even for the nonconvex log likelihood that arises due to background events, such as scatter and random coincidences. The gradient and the curvature of the likelihood terms are evaluated only once per iteration. Since the problem is simplified at each iteration, the CPU time is less than that of current algorithms which directly minimize the objective, yet the convergence rate is comparable. The simplicity, monotonicity, and speed of the new algorithms are quite attractive. The convergence rates of the algorithms are demonstrated using real and simulated PET transmission scans.