Bayesian algorithms for PET image reconstruction with mean curvature and Gauss curvature diffusion regularizations

The basic mathematical problem behind PET is an inverse problem. Due to the inherent ill-posedness of this inverse problem, the reconstructed images will have noise and edge artifacts. A roughness penalty is often imposed on the solution to control noise and stabilize the solution, but the difficulty is to avoid the smoothing of edges. In this paper, we propose two new types of Bayesian one-step-late reconstruction approaches which utilize two different prior regularizations: the mean curvature (MC) diffusion function and the Gauss curvature (GC) diffusion function. As they have been studied in image processing for removing noise, these two prior regularizations encourage preserving the edge while the reconstructed images are smoothed. Moreover, the GC constraint can preserve smaller structures which cannot be preserved by MC. The simulation results show that the proposed algorithms outperform the quadratic function and total variation approaches in terms of preserving the edges during emission reconstruction.