Fundamental relationship between the noise properties of grating-based differential phase contrast CT and absorption CT: Theoretical framework using a cascaded system model and experimental validation

Purpose: Using a grating interferometer, a conventional x-ray cone beam computed tomography (CT) data acquisition system can be used to simultaneously generate both conventional absorption CT (ACT) and differential phase contrast CT (DPC-CT) images from a single data acquisition. Since the two CT images were extracted from the same set of x-ray projections, it is expected that intrinsic relationships exist between the noise properties of the two contrast mechanisms. The purpose of this paper is to investigate these relationships. Methods: First, a theoretical framework was developed using a cascaded system model analysis to investigate the relationship between the noise power spectra (NPS) of DPC-CT and ACT. Based on the derived analytical expressions of the NPS, the relationship between the spatial-frequency-dependent noise equivalent quanta (NEQ) of DPC-CT and ACT was derived. From these fundamental relationships, the NPS and NEQ of the DPC-CT system can be derived from the corresponding ACT system or vice versa. To validate these theoretical relationships, a benchtop cone beam DPC-CT/ACT system was used to experimentally measure the modulation transfer function (MTF) and NPS of both DPC-CT and ACT. The measured three-dimensional (3D) MTF and NPS were then combined to generate the corresponding 3D NEQ. Results: Two fundamental relationships have been theoretically derived and experimentally validated for the NPS and NEQ of DPC-CT and ACT: (1) the 3D NPS of DPC-CT is quantitatively related to the corresponding 3D NPS of ACT by an inplane-only spatial-frequency-dependent factor 1/f ², the ratio of window functions applied to DPC-CT and ACT, and a numerical factor C_g determined by the geometry and efficiency of the grating interferometer. Note that the frequency-dependent factor is independent of the frequency component f_z perpendicular to the axial plane. (2) The 3D NEQ of DPC-CT is related to the corresponding 3D NEQ of ACT by an f ² scaling factor and numerical factors that depend on both the attenuation and refraction properties of the image object, as well as C_g and the MTF of the grating interferometer. Conclusions: The performance of a DPC-CT system is intrinsically related to the corresponding ACT system. As long as the NPS and NEQ of an ACT system is known, the corresponding NPS and NEQ of the DPC-CT system can be readily estimated using additional characteristics of the grating interferometer.