Differential phase contrast imaging using the phase retrieval of a Hilbert transform and noise filtering by low rank method

In this paper, a phase retrieval method using the Hilbert transform (HT) and low-rank method is proposed to obtain differential phase contrast (DPC) imaging. The method has the following advantages: 1. A single grating system can be implemented without the mechanical movement of the grating. 2. The complex computation of the phase retrieval method by the fast Fourier transform (FFT) method can be avoided. 3. The noise rejection by the low-rank can be handled due to fringes from the various energy bins. Specifically, the low-rank method is the singular value decomposition (SVD) by the rank-one property. The phase retrieval of the HT method and noise filtering by the low-rank method have been performed to validate the proposed method. The proposed method provided the clear boundary division between the sample area and the air area. The boundary division of the high energy image between the sample area and the air area was improved with the low rank method (resulting in a sharp division). Moreover, the profile of the DPC image obtained by HT had symmetrical form similar to the theoretical profile.