Invertibility of multi-energy X-ray transform

Abstract

Purpose: The goal is to provide a sufficient condition for 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 $N\ge 2$ .

Methods: We apply a general invertibility theorem to prove the equivalence of global and local invertibility for an ME X-ray transform. We explore the global invertibility through testing whether the Jacobian of the mapping $J\left(A\right)$ 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 for $J\left(A\right)\ne 0$ for all $A$ is that the integrand of $J\left(A\right)$ is $\ge 0$ (or $\le 0$ ) everywhere. Note that the trivial case of the integrand equals 0 everywhere is ignored. Using symmetry, we simplified the integrand of the Jacobian to three factors that are determined by the total attenuation, the basis functions, and the energy-weighting functions, respectively. The factor related to the 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 Cramér-Rao lower bound (CRLB) to characterize the noise-induced estimation uncertainty and provide a maximum-likelihood (ML) estimator.

Results: The factor related to the basis functions is always negative when the photoelectric/Compton/Rayleigh basis functions are used and K-edge materials are not considered. The sign of the energy-weighting factor depends on the system source spectra and the detector response functions. For four special types of X-ray detectors, the sign of this factor stays the same over the integration range. Therefore, when these four types of detectors are used for imaging non-K-edge materials, the ME X-ray transform is globally invertible. The same framework can be used to study an arbitrary ME X-ray imaging system, for example, when K-edge materials are present. Furthermore, the ML estimator we presented is an unbiased, efficient estimator and can be used for a wide range of scenes.

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.

Keywords: X-ray; invertibility; multi-energy X-ray imaging; spectral X-ray imaging.