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Total Variation Regularization in PET Reconstruction

Positron Emission Tomography reconstruction is ill posed. The result obtained with iterative maximum likelihood estimation has maximum probability, but is often unrealistic and has noisy behavior. The introduction of additional knowledge in the solution process is called regularization. Common regularization methods enforce continuity, smoothness or finite band-limits, but these are inappropriate in PET since typical solutions may have sharp features. Total variation regularization does not impose smoothness requirements and preserves edges and boundary surfaces, thus we incorporate this scheme in the reconstruction process. However, total variation involves the absolute value of the derivative, and the absolute value function cannot be differentiated at zero. We investigate different options to handle this problem, including the addition of a small constant to avoid singularity and the primal-dual algorithm, and also address the efficient evaluation of the total variation on the massively parallel GPU architecture.