A L1 minimization strategy for robust joint activity and attenuation estimation in positron emission tomography

Positron Emission Tomography image reconstruction needs a map of photon attenuation probability to provide the correct solution. This map is generally provided by an independent imaging modality. However, it might suffer for artifacts due to patient motion in sequential systems or from intrinsic limitation of the second modality (e.g.: bones that cannot be identified in MR images). It has been shown that such map can be estimated from the PET data themselves, but the solution to this problem has much worse conditioning than the tomographic problem. In this work we propose a new algorithm based on the use of multiple L1 regularization terms in the attenuation sub-problem, to incorporate prior knowledge. We also chose optimal maximizers for both sub-problems: preconditioned gradient descent for the emission one and split-Bregman for the attenuation one. The algorithm was then tested using digital phantom simulations. The proposed algorithm proved to provide accurate quantification over a large range of strength of the regularization terms. The algorithm is also able to reconstruct objects outside of the region where the problem is uniquely determined and it is able to fix the undetermined global scaling factor of joint attenuation and emission estimation. Thanks to the maximizers chosen, the algorithm is computationally less expensive than the current standard.