## 2 Materials and Methods  
### The Sparse-View LDCT Problem  
During a CT scan, the X-ray source rotates in a circle surrounding the body. The X-ray beam sent from the X-ray tube cuts across the body and is collected by the detector. The sinogram is obtained by capturing the attenuation values of X-rays as they pass through the scanned object during the CT scan. CT images can be recovered from full-view sinograms by CT reconstruction.  
Assume that \(\mathbf{Y}\in\mathbb{R}^{D\times V}\) are the full-dose projection data (sinogram) acquired by a CT scan, where \(D\) is the the number of detectors and \(V\) is the number of projection views. \(\mathbf{X}\in R^{M\times M}\) is an underlying CT image of size \(M\times M\). CT imaging can be represented as a linear mapping from the image domain to the sinogram domain:  
\[\mathbf{y}=\mathbf{A}\mathbf{x} \tag{1}\]  
where \(\mathbf{A}\) is the system-specific matrix of the imaging protocol used by the CT scanner, and \(\mathbf{x}\) and \(\mathbf{y}\) are column vectors reshaped by \(\mathbf{X}\) and \(\mathbf{Y}\), respectively.  
CT reconstruction can be viewed as the inverse problem of Equation (6), i.e., estimating the image \(\mathbf{x}\) from projection data \(\mathbf{y}\) given the imaging system matrix \(\mathbf{A}\). For full-dose CT, the system matrix \(\mathbf{A}\) is usually a tall matrix with column full rank (i.e., \(D\times V>M^{2}\)), and the image \(\mathbf{x}\) can be theoretically solved by \(\mathbf{x}=\mathbf{A}^{+}\mathbf{y}\), where \(\mathbf{A}^{+}\) denotes the pseudo-inverse of the matrix \(\mathbf{A}\). However, in practice, \(\mathbf{A}^{+}\) is difficult to compute analytically because the dimensionality of \(\mathbf{A}\) is too high. The classic FBP algorithm can be written as a linear mapping from the sinogram domain to the image domain,  
\[\hat{\mathbf{x}}=\mathbf{A}^{*}\mathbf{y} \tag{2}\]  
where \(\mathbf{A}^{*}\) can be viewed as an approximation to \(\mathbf{A}^{+}\). Iterative algorithms are also commonly used for full-dose CT, and are of non-closed form and gradually improve the intermediate image with deterministic and/or statistic knowledge [28].  
Sparse-view LDCT is achieved by reducing the number of projected views in the sine diagram, i.e., increasing the angular separation between adjacent projection views. Let the full-dose sinogram \(\mathbf{Y}\) consist of \(V\) projection views with a sampling step of 1, denoted as \(\mathbf{Y}=\{\mathbf{y}_{1:V}\}\). The sparse-view sinogram at \(1/S\) radiation dose can be expressed as \(\mathbf{Y}_{S}=\{\mathbf{y}_{1:S:V}\}\subset\mathbf{Y}\), a subset of the columns of \(\mathbf{Y}\), where the sampling step \(S\in\mathbb{N}\) and is divisible by \(V\). Recovering a full-dose sinogram from a sparse-view sinogram is an ill-posed problem that requires some prior knowledge about sinograms. For this purpose, a full convolutional network is designed to learn this interpolated nonlinear mapping  
\[f:\mathbf{Y}_{S}\rightarrow\mathbf{Y}_{S} \tag{3}\]  
and to predict the missing projection views \(\mathbf{Y}_{S}\), which are the complement of subset \(\mathbf{Y}_{S}\) in \(\mathbf{Y}\), i.e., \(\mathbf{Y}=\mathbf{Y}_{S}\cup\mathbf{Y}_{\bar{S}}\).