PHD Intensity Filtering Is One Step of a MAP Estimation Algorithm for Positron Emission Tomography Roy L. Streit Metron 1818 Library Street, Suite 600 Reston, VA 20190 USA streit@metsci.com ; r.streit@ieee.org Abstract – 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 ra- dioisotopes 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) esti- mate 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 fil- ter. 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 iter- ated PHD intensity filter uses the Shepp-Vardi algorithm to the compute the MAP optimal approximation. PET esti- mation has a fundamental noise instability that is overcome in practice by various regularization procedures. Grenan- der’s method of sieves is a regularization procedure that is compatible with the PHD intensity filter. Keywords: Shepp-Vardi algorithm, positron emission tomography, PET, PET imaging, PHD filter, intensity filter, multisensor intensity filter, Poisson point pro- cesses, nonhomogeneous Poisson point processes. 1 Introduction Positron emission tomography (PET) is an established and accepted method for imaging the brain and other tissues. One of the reasons for its widespread ac- ceptance is that maximum a posteriori estimators are available for optimally reconstructing tissue density or, rather more accurately, the intensity of positron- electron annihilation events that are indicative of ra- dioisotope density. The spatial map of the annihilation intensity is thus an image that shows the spatial density of the radioisotope. Image reconstruction algorithms solve a difficult stochastic inverse problem. The well known, and perhaps most influential, of these is the Shepp-Vardi algorithm [1]. This algorithm is derived via the Expectation-Maximization (EM) method, which was then not widely known (1982). The purpose of this paper is to show that the prob- ability hypothesis density (PHD) filter for multitarget tracking and the Shepp-Vardi algorithm for PET are very closely related. This surprising connection is es- tablished by proving that the first step of the Shepp- Vardi algorithm is essentially identical to the informa- tion update of the PHD intensity filter [2, 3]. The proof is straightforward. Section 2 discusses the importance of the connection between the PHD intensity filters and PET imaging. Section 3 reviews the PET problem with an emphasis on the problem itself without any reference to tracking. The derivation of the Shepp-Vardi algorithm closely follows that of [6, Chap. 3]. This section establishes a baseline for the subsequent tracking discussion. The single-sensor PHD intensity filter is derived in Section 4. Given the connections to PET and the PPP (Poisson point process), the filter derivation is short, mathematically rigorous, and easy to understand. The PET connection clarifies the interpretation of the target model as a PPP and, as discussed in Section 5, it helps clarify the differences between intensity and PHD fil- ters in the multisensor case. 2 Importance of PET Connection There are several important consequences that stem from the connection between PET imaging and PHD Presented at ISIF Fusion Conference, July 6-9, 2009, Seattle, WA.