Mathematical aspects of 2D PET using dual curvilinear detectors

In this work we investigate the problem of inverting data acquired from finite-length curvilinear 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 curvilinear 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 of curvilinear detectors taking into account the finite length of the detectors are derived. As first application of our theoretical results, fast filtered backprojection-like algorithm based on the Hilbert transform is proposed. Test numerical experiments are presented.