Invertibility of multi-energy X-ray transform.

Area of Science:

• Medical Imaging
• Applied Mathematics
• Physics

Background:

• Multi-energy (ME) X-ray imaging offers enhanced material differentiation compared to conventional methods.
• The Alvarez-Macovski (AM) method represents energy-dependent X-ray attenuation profiles using coefficients.
• Understanding the invertibility of the ME X-ray transform is critical for accurate image reconstruction.

Purpose of the Study:

• To derive a sufficient condition for the invertibility of the multi-energy (ME) X-ray transform.
• To establish the equivalence between global and local invertibility for ME X-ray transforms.
• To analyze the factors influencing the invertibility of the ME X-ray transform.

Main Methods:

• Applied a general invertibility theorem to analyze the Jacobian of the ME X-ray transform mapping.
• Simplified the Jacobian integrand into three factors: total attenuation, basis functions, and energy-weighting functions.
• Utilized the Cramér-Rao lower bound (CRLB) for noise analysis and developed a maximum-likelihood (ML) estimator.

Main Results:

• The basis function factor is consistently negative for standard basis functions (photoelectric/Compton/Rayleigh) without K-edge materials.
• The energy-weighting factor's sign depends on source spectra and detector response; it remains constant for four specific detector types.
• Global invertibility of the ME X-ray transform is demonstrated for non-K-edge materials with these four detector types.
• The proposed ML estimator is unbiased and efficient, suitable for diverse imaging scenarios.

Conclusions:

• A framework for studying the invertibility of arbitrary ME X-ray transforms has been established.
• Global invertibility is proven for ME X-ray imaging systems employing four specific detector types for non-K-edge materials.
• The developed framework is adaptable for analyzing various ME X-ray imaging systems, including those with K-edge materials.