## 2 Materials and Methods
### Neural Network Architecture
#### 2.2.2 Phase Contrast Radon Inversion Layer  
The PCCT reconstruction is required since it can achieve mapping from the projection sinogram domain to the image domain. However, conventional reconstruction algorithms do not allow for the backpropagation of gradients, resulting in that only parameters of the enhanced network in the image domain are updated. The Radon inversion layer (RIL) proposed by [PERSON] [40], acting as an efficient and differentiable variant of FBP, allows for the backpropagation of gradients. It is adopted in the absorption-based CT and obtains excellent performance on reducing metal artifacts. Based on RIL, the PCRIL is derived in this work, which consists of the phase contrast filter, the back-projection derivation and the gradients of the backpropagation. The fan-beam back-projection is required since the grating-based PCCT allows for the use of the laboratory source.  
Hilbert Transform Filter: The phase contrast filter is performed with the Hilbert transform in the PCCT reconstruction. It can provide a phase shift of 90\({}^{\circ}\) without affecting the amplitude. Therefore, the Hilbert transform is equivalent to the quadrature phase shift of the signal, making them quadrature pairs [41]. As presented in Equations (6) and (7), the sinogram is filtered with the Hilbert transform filter, where \(H\) represents the filter, \(\omega\) represents the frequency, \(x\) represents the initial sinogram, \(X\) represents the filtered sinogram, \(F\) and \(F^{-1}\) represent the discrete Fourier transform (DFT) and inverse discrete Fourier transform (iDFT), respectively.  
\[H(\omega)=-i\cdot sgn(\omega)=\begin{cases}-i,&\omega\geq 0\\ i,&\omega<0\end{cases} \tag{6}\]  
\[X=F^{-1}[-i\cdot sgn(\omega)\cdot F(x)] \tag{7}\]  
Back-projection Module: Back-projection is when the value of each pixel in the reconstructed image is regarded as the sum of all projections passing through it. Equations (8) and (9) present the back-projection process, where \(Y\) represents the reconstructed image with a size of \(row\times col\), \(\theta\) represents the rotation angle, \(D\) and \(D_{0}\), respectively, represent the distance between the source and detector and that between the source and object, \(offset\) represents the offset between the rotation center and the detector center, \(i\) represents the sinogram index, and \(\left\lceil\right\rceil\) and \(\left\lfloor\right\rfloor\) represent the round up and round down operators. Moreover, the computation can be highly parallel since the back-projection at each view is independent.  
\[Y =\int_{0}^{2\pi}X(\theta,\frac{D_{0}*(row\cdot\cos\theta+col\cdot \sin\theta)}{D-row\cdot\sin\theta+col\cdot\cos\theta}+offset)d\theta \tag{8}\] \[\approx\Delta\theta\sum_{i}X(\theta_{i},\frac{D_{0}*(row\cdot \cos\theta+col\cdot\sin\theta)}{D-row\cdot\sin\theta+col\cdot\cos\theta}+offset)\] \[\approx\Delta\theta\sum_{i}(\left\lceil t_{i}\right\rceil-t_{i} )X(\theta_{i}\left\lfloor t_{i}\right\rfloor)+(t_{i}-\left\lfloor t_{i} \right\rfloor)X(\theta_{i}\left\lceil t_{i}\right\rceil)\] \[t_{i}=\frac{D_{0}*(row\cdot\cos\theta+col\cdot\sin\theta)}{D-row \cdot\sin\theta+col\cdot\cos\theta}+offset \tag{9}\]Backpropagation gradients: While backpropagating, the gradients from the image domain to the projection sinogram domain are presented as Equation (10), where the symbols have the same representation as those of Equation (8).  
\[\frac{\partial Y}{\partial X}=\begin{cases}\Delta\theta(\left[t_{i}\right]-t_{i} ),&t=\left[t_{i}\right]\\ \Delta\theta(t_{i}-\left[t_{i}\right]),&t=\left[t_{i}\right]\\ 0,\quad\text{{otherwise}}\end{cases} \tag{10}\]