Invertibility of the dual energy x-ray data transform

Abstract

Purpose: The Alvarez–Macovski method extracts the x-ray energy-dependent information by expanding the attenuation coefficient as a linear combination of functions of energy multiplied by basis set coefficients. Since the basis functions are known a priori, the coefficients represent all the energy-dependent information. The method then computes the line integrals of these coefficients, summarized as a vector A, from measurements with multiple x-ray spectra, summarized as a vector L. The purpose of this paper is to determine the factors that affect the invertibility of the L(A) transformation with a two function basis set and two spectral measurements, the dual energy transformation. Methods: A general invertibility theorem is applied that requires testing for zero values of the Jacobian of the transformation in its input domain. General conditions for invertibility are proved. It is shown that the generalized A vector noise variance is proportional to the generalized measurement noise variance divided by the square of the Jacobian. The relationship between the zero Jacobian values and ambiguous sets of A vector points with the same L values is determined. The effect of zero Jacobian values on an iterative algorithm that inverts L(A) is simulated. Results: The choice of a particular valid basis set does not affect invertibility. Nonoverlapping measurement spectra such as those from photon counting detectors with perfect pulse height analysis are invertible. The widely used x-ray tube spectra with different voltages are shown to be invertible. Spectra with the same maximum energy, such as those from layered detectors, approach noninvertibility with small absolute value Jacobian for large object thicknesses. The zero Jacobian values fall on curves in A vector space that, except for a simple artificial case, are close to but not exactly straight lines. With noninvertible spectra, pairs of ambiguous points are located on opposite sides of the zero Jacobian curve. The iterative algorithm has large convergence errors near zero Jacobian curves and converges to the closest ambiguous point to the initial estimate for other points. Conclusion: The invertibility of dual energy systems is determined by the presence of zero values of the Jacobian of the dual x-ray energy data transformation L(A) in the input domain.