Accelerated penalized weighted least-squares and maximum likelihood algorithms for reconstructing transmission images from PET transmission data

We present penalized weighted least-squares (PWLS) and penalized maximum-likelihood (PML) methods for reconstructing transmission images from positron emission tomography transmission data. First, we view the problem of minimizing the weighted least-squares (WLS) and maximum likelihood objective functions as a sequence of nonnegative least-squares minimization problems. This viewpoint follows from using certain quadratic functions as surrogate functions for the WLS and maximum likelihood objective functions. Second, we construct surrogate functions for a class of penalty functions that yield closed form expressions for the iterates of the PWLS and PML algorithms. Due to the slow convergence of the PWLS and PML algorithms, accelerated versions of them are developed that are theoretically guaranteed to monotonically decrease their respective objective functions. In experiments using real phantom data, the PML images produced the most accurate attenuation correction factors. On the other hand, the PWLS images produced images with the highest levels of contrast for low-count data.