Invertibility of Multi-Energy X-ray Transform

Research output: Contribution to journalArticlepeer-review


Purpose: The goal is to provide a sufficient condition on the invertibility of a multienergy (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 ≥ 2. Methods: We apply a general invertibility theorem which tests whether the Jacobian of the mapping J(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(A) 6= 0 for all A is that the integrand of J(A) is ≥ 0 (or ≤ 0) 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 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 within the energy range normally used for imaging (0—200 keV) 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 detector response functions. For four special types of X-ray detectors, the sign of this factor stays the same over the integration volume. 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. 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.

Original languageEnglish (US)
JournalUnknown Journal
StatePublished - Jul 8 2020


  • Invertibility
  • Multi-energy X-ray imaging
  • Spectral X-ray imaging
  • X-ray

ASJC Scopus subject areas

  • General

Fingerprint Dive into the research topics of 'Invertibility of Multi-Energy X-ray Transform'. Together they form a unique fingerprint.

Cite this