Purpose: Grating-based x-ray phase-contrast imaging (GPCI) is a promising technique for clinical applications as it can provide two newly emerging imaging modalities (differential phase-contrast and dark-field contrast) in addition to the conventional absorption contrast. As far, phase-stepping strategy is the most commonly used approach in GPCI to indirectly acquire differential phase-contrast and dark-field contrast. It is known that the obtained phase-stepping curves (PSCs) have the cosine property and the convolution property, leading to two types of information retrieval approaches in literature: the Fourier component analysis and the multi-order moment analysis. The purpose of this paper is to derive a new property of PSCs and apply the property to noise optimization for information retrieval.
Methods: Based on the cosine expression of the flat PSC without the sample and the well-established convolution relationship between the flat PSC and the sample PSC, we reveal an important integral property of PSCs: the inner product of PSCs and an arbitrary function contains only zero-order and first-order components in the Fourier series. Furthermore, we apply the property to the direct multi-order moment analysis and propose a set of generalized forms including an optimal one in the presence of noise.
Results: To validate the effectiveness of our analysis, we compared the simulated and real experiment results retrieved by the original direct multi-order moment analysis with the ones retrieved by our proposed noise-optimal form. A significant improvement of noise performance by our method is observed and the improvement ratio in differential phase-contrast is consistent with our theoretical calculation (39.2%).
Conclusions: In this paper, we reveal a new integral property of the acquired PSCs with and without samples in GPCI, which can be applied to information retrieval approaches like the direct multi-order moment analysis. Then we optimize these approaches to improve the noise performance, offering great potentials of dose reduction in practical applications.