3D image reconstruction for a Compton SPECT camera model

In this paper we propose a 3D image reconstruction algorithm for a 3D Compton camera being developed at the University of Michigan. We present a mathematical model of the transition matrix of the camera which exploits symmetries by using an adapted spatial sampling pattern in the object domain. For each projection angle, the sampling pattern is uniform over a set of equispaced nested hemispheres. By using this sampling pattern the system matrix is reduced to a product of a (approximately) block circulant matrix and a sparse interpolation matrix. This representation reduces the very high storage and computation requirement inherent to 3D reconstruction using transition matrix inversion methods. We geometrically optimize our hemispherical sampling and propose a 3D volumetric interpolation. Finally, we present a 3D image reconstruction method which uses the Gauss-Seidel algorithm to minimize a penalized least square objective.