Spatial Modulation for L1 penalty terms in emission tomography

In emission tomography the Fisher information matrix is highly non-uniform along the diagonal due to the Poisson nature of photon counting. For this reason, quadratic penalties result in non-uniform spatial resolution, with the poorest resolution in regions with most counts. A strategy had been proposed to spatially modulate the penalty strength to achieve uniform resolution. In recent years, however, regularizations based on the L1 norm have generated much interest for their sparsity inducing proprieties. Nonetheless, we observed that, in emission tomography, if the penalty is not spatially modulated it suffers from the same problem observed in the L2 case. We propose here a spatial modulation factor for L1 penalties. We test this approach in two independent tasks. The first one involves the reconstruction of 2D PET digital phantoms in a sparse wavelet basis. The second task is the reconstruction of the attenuation map from emission data in 3D human studies with two L1 terms. In both tasks the proposed strategy greatly improves image quality and produces images that appear to be of uniform spatial resolution.