Diffusion regularization for iterative reconstruction in emission tomography

Recently, we have proposed a regularized least square criterion for adaptive regularization for reconstructing noisy SPECT data with non-uniform attenuation correction. In the present study, we show that the regularization penalty that was used is closely related to a diffusion scheme used for Gaussian filtering. For a given value of the regularization parameter, the amount of smoothing is independent from the patient attenuation map, and it is mathematically related to the full width at half maximum (FWHM) of a Gaussian filter. A second regularized least square criterion is then derived for which regularization also behaves as a diffusion scheme. The new penalty is then shown to be also applicable to the weighted least square criterion, and to the Poisson maximum likelihood criterion for PET data (i.e. without attenuation) solved by the EM algorithm. For all these criteria, the regularization strength can thus be set as the FWHM of a Gaussian filter.