Title: Conditions for the invertibility of dual energy data,
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.