Non-Uniqueness in Dual-Energy CT with Beam Hardening
Zachary H. Levine
Purpose: The goal is to determine whether dual-energy computed tomography (CT) leads to a unique reconstruction into two basis materials. Methods and Materials: The beam hardening equation is simplified to the single voxel case. The simplified equation is rewritten to show that the solution can be considered to be linear operations in a vector space followed by a measurement model which is the sum of the exponential of the coordinates. The case of finding the concentrations of two material from measurements of two spectra with three photon energies is shown to be the simplest non-trivial case and then is considered in more detail. As an intermediate step, a piecewise linear approximation to the universal detector function is given which yields a heuristic for how to vary the spectrum so that two solutions will result. Results: Using a material basis of water and bone, with photon energies of 30~keV, 60~keV, and 100~keV, a case with two solutions is demonstrated. Conclusions: Dual-energy reconstruction into two materials is not unique as shown by an example. Algorithms for dual-energy, dual-material reconstructions need to be aware of this potential ambiguity in the solution.