3D image reconstruction for a Compton SPECT camera model

Proposes a 3D image reconstruction algorithm for a 3D Compton camera being developed at the University of Michigan. The authors 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. The authors geometrically optimize their hemispherical sampling and propose a 3D volumetric interpolation. Finally, the authors present a 3D image reconstruction method which uses the Gauss-Seidel algorithm to minimize a penalized least square objective.