  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.
