Low counts reconstruction remains a challenge for Positron Emission Tomography (PET) even with the recent progresses in time-of-flight (TOF) resolution. In that setting, the bias between the acquired histogram, composed of low values or zeros, and the expected histogram, obtained from the forward projector, is propagated to the image, resulting in a biased reconstruction. This could be exacerbated with finer resolution of the TOF information, which further sparsify the acquired histogram. We propose a new approach to circumvent this limitation of the classical reconstruction model. It consists of extending the parametrization of the reconstruction scheme to also explicitly include the projection domain. This parametrization has greater degrees of freedom than the log-likelihood model, which can not be harnessed in classical circumstances. We hypothesize that with ultra-fast TOF this new approach would not only be viable for low counts reconstruction but also more adequate than the classical reconstruction model. An implementation of this approach is compared to the log-likelihood model by using two-dimensional simulations of a hot spots phantom. The proposed model achieves similar contrast recovery coefficients as MLEM except for the smallest structures where the low counts nature of the simulations makes it difficult to draw conclusions. Also, this new model seems to converge toward a less noisy solution than the MLEM. These results suggest that this new approach has potential for low counts reconstruction with ultra-fast TOF.
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