Analytic spectra for spectral x-ray imaging

Overview & Synopsis

Purpose: Analytic x-ray energy spectra are found with closed form mathematical formulas for the transmittance and the elements of the Jacobian matrix. The purpose is to provide formulas that can be used to develop and to test results on spectral x-ray imaging[#Alvarez1976]. Example applications are the invertibility of the dual energy transformation and correction for beam hardening artifacts. Methods: Analytic expressions are derived and then verified with numerical simulations. The formulas are derived first by assuming the Compton scattering basis set coefficient is equal to zero. Then a linear approximation to the basis function is studied. The formulas are verified by comparing them to numerical simulations. Results: Formulas are derived and verified for the transmittance and the Jacobian matrix elements with constant, linear, and parabola spectra over a fixed x-ray energy range and also a spectrum derived by Rigaud [#rigaud2017analytical] The Jacobian matrices are used to test for invertibility. The formulas are used to derive inverses for beam hardening correction for an object with zero Compton coefficient and Rigaud's derivation for a fixed composition object is verified. The use of a linear approximation to the Compton scattering basis function is studied. Conclusion: The analytic formulas are accurate and useful for studying spectral x-ray imaging.