## 1 Introduction  
Attenuation and refraction occur when X-rays penetrate objects, which correspond to the absorption and phase contrast. Conventional absorption-based X-ray computed tomography (CT) is widely used in clinical diagnosis [1; 2; 3; 4] and industrial testing [5; 6; 7; 8]. It plays a crucial role in imaging strong absorption objects, while it performs poorly when encountering weak absorption objects such as soft tissue, rare Earth materials and composite materials. Phase contrast computed tomography (PCCT) provides better image contrast for weak absorption objects than absorption-based CT [9; 10; 11; 12; 13; 14]. Several PCCT techniques have been developed in the past years, and the results have indicated that PCCT can greatly improve image quality for weak absorption objects [15; 16; 17; 18]. Grating-based PCCT is the most sensitive and universal approach since a coherent X-ray tube is not required during imaging. It is based on the Talbot effect. However, limited by the phase stepping principle, grating-based PCCT usually requires several samplings at each view to extract the contrast signals, which results in a high radiation dose. Sparse-view sampling is usually performed to reduce imaging radiation [19; 20], while it introduces artifacts and noise in reconstruction. In recent years, deep learning (DL) has been popular in image processing [21; 22; 23; 24; 25; 26]. DL has also been applied to the field of CT [27; 28; 29], which generally is grouped into two categories. The first category can be classified as enhancement in the projection sinogram domain. By using the residual network (ResNet) for better convergence and patch-wise training to reduce memory, [PERSON] et al. proposed a DL framework to in-paint the missing data in the sparse-view projection sinogram [30]. It significantly outperformed conventional linear interpolation algorithms. Moreover, their subsequent work that utilized UNet [31] and residual learning [32] outperformed the existing interpolation methods and IR approaches [33]. Different from using UNet to correct sparse-view sinograms, [PERSON] et al. proposed a deep learning filtered back-projection (DLFBP) framework to use differential forward projection of the image reconstructed with incomplete data as input and a dense connection net to output a complete sinogram [34]. The results showed that this framework can generate high-quality reconstructed images with given incomplete data. However, these approaches may introduce unpredictable artifacts, since the reconstruction process is extremely susceptible to the inherent consistency of the sinogram. The second category can be classified as enhancement in the image domain. [PERSON] et al. developed a deep convolutional neural network (CNN) to map low-dose CT reconstructed images to their corresponding normal-dose images in a patch-by-patch fashion [35]. The results demonstrated the great potential of the proposed method for artifact reduction. By using a directional wavelet transform to extract the directional component of artifacts and to exploit the intra- and inter-band correlations, [PERSON] et al. proposed a DL method that utilized the wavelet transform coefficients of low-dose images [36]. It could effectively suppress CT-specific noise. [PERSON] et al. used Dense Net and deconvolution to remove streaking artifacts from sparse-view CT images [37]. The results showed that it can effectively suppress artifacts in reconstructed images. These approaches offer a significant advantage in reducing artifacts and noise in reconstruction, while they may oversmooth the images. Several methods working in dual domains (i.e., the projection sinogram domain and image domain) have been developed [38; 39]. They are grouped into two categories, and each of them has its own limitations: (i) using fully connected layers to connect dual domains, which incurs a huge computational overhead; (ii) training networks in dual domains separately, which superimposes the degradation of dual domains.