System and Gram matrices of 3D planogram data

This paper addresses the topic of geometric voxel-based approaches to forward and inverse problems of image reconstruction from planogram data. We investigate 2D and 3D planar detector configurations for which we reduce complex geometrical calculations of Gram and system matrices to a series of simple transformations like shifts and parallel projections. We use a direct algebraic approach, or natural pixels approach in reconstruction. The approach needs precomputation and inversion of the Gram matrix with entities as volumes of intersection of two arbitrary natural pixels - strips or parallelepipeds. Numerical experiments with test small-scale data are presented.