PHD intensity filtering is one step of a MAP estimation algorithm for positron emission tomography

The well-known Shepp-Vardi algorithm (1982) for positron emission tomography (PET) is used to estimate the intensity function of the emissions of short-lived radioisotopes absorbed by the brain or other tissues. In the PET application, radioisotope emissions are modeled as a nonhomogeneous Poisson point process. The Shepp-Vardi algorithm produces the maximum a posteriori (MAP) estimate of the intensity function of this process. The intensity function provides an image of the radioisotope density in the brain or other tissues. The PHD intensity filter is a multiple target tracking filter. Its information update is shown to be the first step of the Shepp-Vardi algorithm. The correspondence with PET reveals that the filter is estimating the intensity function of a nonhomogeneous Poisson point process that approximates the Bayes posterior multi-target point process. The iterated PHD intensity filter uses the Shepp-Vardi algorithm to the compute the MAP optimal approximation. PET estimation has a fundamental noise instability that is overcome in practice by various regularization procedures. Grenander's method of sieves is a regularization procedure that is compatible with the PHD intensity filter.