System modeling and spatial sampling techniques for simpli cation of transition matrix in 3D Electronically Collimated SPECT

transition matrix in 3D Electronically Collimated SPECT Anne C. Sauve1, Alfred O. Hero1, W. Leslie Rogers2 and Neal H. Clinthorne2 January 15, 1997 Abstract In this paper we will present numerical studies of the performance of a 3D Compton camera being developed at the University of Michigan. We present a physical model of the camera which exploits symmetries and an adapted spatial sampling pattern in the object domain. This model increases the sparsity of the transition matrix to reduce the very high storage and computation requirements. This model allows the decomposition of the transition matrix into several small blocks that are easy to store. Finally we discuss a real time algorithm which calculates entries of the transition matrix based on a Von Mises model for the conditional scatter angle distribution given the Compton energy measurement as well as a vector reformulation of the computation of the probabilities. 1 3D Compton scatter SPECT camera Application of the Compton scatter aperture to imaging in nuclear medicine was rst proposed in 1974 by Todd and Everett. This camera uses an innovative electronical conebeam collimator based on the Compton scattering e ect. Its requires a 3D image reconstruction. Singh [1] proposed in 1983 a linear image reconstruction for the Compton camera. This reconstruction is computationnally good but uses an inaccurate model of the system since it neglects the Poisson nature of the measurements. Leahy [2] implemented an MLE reconstruction from the transition matrix that takes into account the Poisson noise for a prototype system. This algorithm is computationnally demanding since a 3D image of moderate size (1283 pixels) requires already a very big matrix T (resp 1286). Background The aperture consists of a position sensitive solid state detector (det1) with a high energy resolution. This detector is paired with a second position sensitive detector, det2, which can be a scintillation camera with lower energy resolution. 1A. Sauve (corresponding author : [email protected]) and A. Hero are with the Dept. of Electrical Engineering and Computer Science, The University of Michigan, Ann Arbor, MI 48109-2122. 2L. Rogers and N. Clinthorne are with the Dept. of Nuclear Medicine, The University of Michigan, Ann Arbor, MI 48109-0552. Figure 1: Illustration of the Compton scatter collimator The rays from the point source, X, that reach det1 are Compton scattered by the solid state detector, det1 (Fig. 2). Those scattered photons are then detected by the second detector in coincidence with the events in det1. The energy deposited in det1 increases as a function of the scattering angle according to Compton scattering statistics.