A common problem present in Positron Emission Tomography (PET) images is the low spatial resolution due to physical factors. Traditionally, these factors are not considered within the image reconstruction and therefore cannot be corrected. Here we propose a new approach to increase the resolution of our PET images by advanced mathematical modelling. We use a variational approach which involves a minimization problem that will be faced using some well-studied optimization algorithms:  the Proximal Forward-backward splitting and the Linearized Bregman iteration algorithms. 

Furthermore, we propose to use directional total variation regularization (dTV) to incorporate a-priori knowledge of the radiotracer distribution which is readily available from modern PET-MR scanners. Additionally, in these considered applications, the resolution model (point spread function) is not well-identified, so we will take a blind approach and estimate it during the reconstruction. This approach has been successfully tested in non-medical applications and now we extend some of those ideas for PET imaging.