Invertibility of spectral x-ray data with pileup--two dimension-two spectrum case

In the Alvarez-Macovski method, the line integrals of the x-ray basis set coefficients are computed from measurements with multiple spectra. An important question is whether the transformation from measurements to line integrals is invertible. This paper presents a proof that for a system with two spectra and a photon counting detector, pileup does not affect the invertibility of the system. If the system is invertible with no pileup, it will remain invertible with pileup although the reduced Jacobian may lead to increased noise.