System and gram matrices of 3-D 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 2- and 3-D (two-, three-dimensional) 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 entries as volumes of intersection of two arbitrary natural pixels-strips or parallelepipeds. Numerical experiments with small-scale test data are presented.