Invertibility of Multi-Energy X-ray Transform

Purpose: The goal is to provide a sufficient condition on the invertibility of a multi-energy (ME) X-ray transform. The energy-dependent X-ray attenuation profiles can be represented by a set of coefficients using the Alvarez-Macovski (AM) method. An ME X-ray transform is a mapping from $N$ AM coefficients to $N$ noise-free energy-weighted measurements, where $N\geq2$. Methods: We apply a general invertibility theorem which tests whether the Jacobian of the mapping $J(\mathbf A)$ has zero values over the support of the mapping. The Jacobian of an arbitrary ME X-ray transform is an integration over all spectral measurements. A sufficient condition of $J(\mathbf A)\neq0$ for all $\mathbf A$ is that the integrand of $J(\mathbf A)$ is $\geq0$ (or $\leq0$) everywhere. Note that the trivial case of the integrand equals to zero everywhere is ignored. With symmetry, we simplified the integrand of the Jacobian into three factors that are determined by the total attenuation, the basis functions, and the energy-weighting functions, respectively. The factor related to total attenuation is always positive, hence the invertibility of the X-ray transform can be determined by testing the signs of the other two factors. Furthermore, we use the Cramer-Rao lower bound (CRLB) to characterize the noise-induced estimation uncertainty and provide a maximum-likelihood (ML) estimator. Conclusions: We have provided a framework to study the invertibility of an arbitrary ME X-ray transform and proved the global invertibility for four types of systems.