Inversion of 2D planogram data for finite-length detectors

In this work we investigate the problem of inverting data acquired from finite-length linear detectors in the 2D case. Image reconstruction algorithms using planogram data are usually derived under the initial assumption that the detectors are infinitely long and complete data can be acquired. This work deals with an analytical approach taking into account the finite length of the detectors explicitly and does not require operations on huge matrices. We derive integral equations for a single pair and for two pairs of linear detectors. This approach extends our previous work that dealt with explicit formulas for the elements of system and Gram matrices involved in 3D algebraic reconstruction from planograms, to a mathematical model in terms of continuous variables and a Fredholm-Volterra integral operator. As a result, fast filtered backprojection-like (FBP) algorithms based on the Hilbert and Fourier transforms (HFBP) are proposed