A Hybrid MLEM/Split-Bregman approach for constrained, robust estimation of the attenuation map in PET

Joint estimation of activity and attenuation in PET is promising but has not achieved widespread use due to its lack of robustness. In this work, we propose the use of a different minimizer, the Split-Bregman (SB) algorithm, instead of the common expectation maximization algorithm. The SB is widely used in compressive sensing for its ability to easily integrate a large number of L1 regularization terms. The L1 norm has been shown to be superior to the L2 norm in recovering signals that are sparse in a proper basis. In this work we review this minimizer and then perform two simulations of representative settings to analyse its performances to reconstruct the transmission part of the joint emission/transmission problem. We first reconstruct the transmission image of a digital phantom supposing that the emission image has been correctly reconstructed and we compare the results with the standard MLTR algorithm, implementing the total variation regularization in both algorithms. Both show comparable levels of noise but the SB minimization has better edge preserving proprieties due to the use of the L1 norm. In a second simulation we analyse a setup similar to a condition that could be encountered in a PET/MR study: some prior information is available about the attenuation map, but it is not correct everywhere; the prior information does not feature the MR coils and they lie outside of the region where the attenuation can be univocally recovered from PET coincidences. We show promising results which demonstrate that imposing the total variation together with the prior attenuation information is successful in recovering the attenuation map everywhere. This prototype algorithm which we present, can be easily modified to insert as many L1 constraints as desired, optimized for specific tasks.