Diffusion regularization for iterative reconstruction in emission tomography

We have recently proposed a regularized least square criterion for adaptive regularization of single photon emission computed tomography (SPECT) reconstruction with nonuniform attenuation correction. In the present study, we show that this regularization 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 positron emission tomography (PET) data (i.e., without attenuation) solved by the expectation maximization (EM) algorithm. For all these criteria, the regularization level can thus be set as the FWHM of a Gaussian filter.