Total Variation Regularisation in Measurement and Image space for PET reconstruction

The aim of this paper is to test and analyse a novel technique for image reconstruction in positron emission tomography, which is based on (total variation) regularisation on both the image space and the projection space. We formulate our variational problem considering both total variation penalty terms on the image and on an idealised sinogram to be reconstructed from a given Poisson distributed noisy sinogram. We prove existence, uniqueness and stability results for the proposed model and provide some analytical insight into the structures favoured by joint regularisation. For the numerical solution of the corresponding discretised problem we employ the split Bregman algorithm and extensively test the approach in comparison to standard total variation regularisation on the image. The numerical results show that an additional penalty on the sinogram performs better on reconstructing images with thin structures.