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

TL;DR

This paper proves that in a two-spectrum x-ray system with photon counting detectors, pileup does not compromise the invertibility of the measurement-to-parameter transformation, ensuring reliable spectral analysis despite detector effects.

Contribution

It provides a mathematical proof that pileup does not affect invertibility in a two-spectrum x-ray system, extending understanding of spectral data reconstruction under realistic detector conditions.

Findings

Pileup does not affect invertibility in the two-spectrum case.

Invertibility is maintained with pileup, though noise may increase.

The reduced Jacobian can lead to higher noise levels.

Abstract

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.