The trigonometric orthogonality of phase‐stepping curves in grating‐based x‐ray phase‐contrast imaging: Integral property and its implications for noise optimization

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.