Variational image reconstruction methods for tomography - PET and transmission CT with mixed Poisson-Gaussian noise

This thesis focuses on total variation based variational image reconstruction models that arise in linear and non-linear inverse problems with measurements corrupted by mixed Poisson and Gaussian noise. An inverse problem can be described as identifying model parameters given the observed measurements. More specifically in this work, we consider inverse problems arises in positron emission tomography and transmission computed tomography. For these problems in the literature, a single noise assumption on the measurements is usually taken. Here, in both tomographic scenarios, we consider measurements corrupted not only by Poisson but also by Gaussian noise. In the first part of this thesis, we introduce the basic mathematical theory that we will use throughout this work. The second part aims to give a comprehensive and detailed description of the mathematical framework in tomography. In this part, we get more into the mathematical details about the X-ray and divergent X-ray transforms and we prove important results that will be used in the analysis of the variational models. The last part contains the main contributions of this thesis. More specifically, a complete reconstruction framework is described for PET and transmission micro-CT measurements generated by a monoenergetic X-ray source. The measurements are corrupted by Poisson and Gaussian noise. We start by giving a statistical motivation behind the noise model and then by using Bayesian modeling we derive our corresponding reconstruction models. Next, a detailed theoretical analysis for these models is presented where the main results are proven such as well-posedness and stability. Next, we proceed by deriving the corresponding algorithms. A theoretical convergence analysis is carried out for the corresponding algorithms as well as numerical results are presented to demonstrate their efficiency. Finally, we propose a Bregman refinement technique to overcome the contrast loss that the total variation regularization introduces to the reconstructed images. Similarly, for the updated models we derive the corresponding algorithms and numerical results are presented to demonstrate their efficiency