Inversion of 2D planogram data for finite-length detectors

In this paper we investigate the problem of inverting data acquired from finite-length linear detectors in the 2D case. In our previous modeling of the forward problem we dealt with explicit formulas for the elements of system and Gram matrices involved in 2D and 3D algebraic reconstruction from planograms. In this paper we continue our efforts to model flat panel detectors, from the discrete algebraic approach with huge Gram matrices arising in practical 3D PET situations to more compact and fast representation in terms of integral operators. Integral equations for a single pair and for two pairs of linear detectors taking into account the finite length of the detectors are derived. As first applications of our theoretical results, fast filtered backprojection-like algorithms based on the Hilbert and Fourier transforms are proposed. Test numerical experiments are presented.