A general theoretical formalism for X-ray phase contrast imaging

Abstract

The in-line phase-contrast imaging has great potential for clinical imaging applications. This work presents a general theoretical formalism for the in-line phase-contrast imaging. The theoretical formalism developed in this work is derived by taking a new strategy to calculate the Fourier transform of image intensity directly. Different from the transport of intensity equation (TIE) formalism for phase-contrast imaging in literature [6], this general formalism covers both the near field regime and the holography regime of phase-contrast imaging. The image intensity formulas have been derived in both the image space and frequency space. Especially our results show that the Fresnel diffraction image intensity is a sum of convolutions of the cosine- and sine-Fresnel filters with the object attenuation $A_0^2(x)$ and attenuated phase $A_0^2(x)φ(x)$ , respectively. The Pogany-Gao-Wilkins (PGW) formalism is recovered as a special case of our general formalism. In addition, in the low-resolution approximation, the general formula is reduced a spherical wave-generalization of the TIE-based formula for phase-contrast imaging. This spherical wave-generalization will be useful for phase-contrast imaging with a micro-focus x-ray tube. The transition of the formalism from 1-D to 2-D cases has been provided as well.