Ultra-fast ToF-PET Reconstruction with an Optimized Preconditioned Gradient Descent

The most common algorithm for clinical PET image reconstruction is MLEM, in which higher frequencies are the slowest to converge. This is exploited to control noise by early stopping. If noise is to be controlled using a more accurate method, e.g.: regularization, convergent reconstructions require thousands of iterations through the whole datasets. In this work, we revise an old, preconditioned conjugate gradient strategy. An ideal preconditioner would involve the backprojection of the inverse of the estimated sinogram, which is unknown before the reconstruction. The measured sinogram cannot be used as an approximation, due to the presence of zeros, especially when using Time Of Flight. We overcome this limit by introducing approximations, specifically tailored to this problem. The method was validated using different digital phantoms to study the algorithm behaviour with different contrasts, object sizes and different amounts of random and scattered coincidences In all the simulated conditions the proposed algorithm reached almost full convergence using less than 10 cycles over the whole datasets, with most of the recovery happening already in the first 2 iterations. This is achieved without the help of accelerating strategies like ordered subsets. The proposed algorithm has the potential to allow accurate, convergent, PET image reconstruction using a very limited number of operations.