Conditions for the invertibility of dual energy data

TL;DR

This paper derives analytical conditions for the invertibility of the transformation in dual energy X-ray data, identifying when systems are non-invertible and demonstrating the impact on inverse algorithms.

Contribution

It provides explicit formulas for invertibility conditions and analyzes specific spectra, advancing understanding of dual energy data reconstruction.

Findings

Non-invertible systems have near-zero Jacobian determinants on a straight line.

Formulas determine where the line crosses axes and the Jacobian at endpoints.

Iterative inverse algorithms perform poorly with non-invertible spectra.

Abstract

The Alvarez-Macovski method [Alvarez, R. E and Macovski, A., "Energy-selective reconstructions in X-ray computerized tomography", Phys. Med. Biol. (1976), 733--44] requires the inversion of the transformation from the line integrals of the basis set coefficients to measurements with multiple x-ray spectra. Analytical formulas for invertibility of the transformation from two measurements to two line integrals are derived. It is found that non-invertible systems have near zero Jacobian determinants on a nearly straight line in the line integrals plane. Formulas are derived for the points where the line crosses the axes, thus determining the line. Additional formulas are derived for the values of the terms of the Jacobian determinant at the endpoints of the line of non-invertibility. The formulas are applied to a set of spectra including one suggested by Levine that is not invertible as well as similar spectra that are invertible and voltage switched x-ray tube spectra that are also invertible. An iterative inverse transformation algorithm exhibits large errors with non-invertible spectra.