# Limited-Angle Computer Tomography with Truncated Projection Artifacts Removal
## 2 Materials and Methods
### TPA Reduction Method  
According to the previous description, the cause of TPA is that the FOV does not include the entire object. If the reconstructed FOV can be expanded to cover the whole target, using SART directly can reconstruct the images without TPA. To expand the FOV, not only expanding the scope of reconstruction was needed but completing the truncated parts of the original data was also required. Firstly, SART reconstruction was performed on the original projection data. Then, the re-projection was virtually performed according to the position of the detector panel and the X-ray source. During this process, the modeling of the detector panel was expanded so that it could receive all the X-rays as shown in Figure 5. Finally, since each FOV can contain the entire imaged object, SART projection data were used to complete the original data to obtain the completed data. In the following truncation data completion algorithm description diagram (as listed in Algorithm 1), we traversed each row of the original projection data to complete the pixel value of the truncated area. For each row of the projection data, \(p\) is the column index of the leftmost non-zero number; \(q\) represents the column index of the rightmost non-zero number. \(y_{m,n}^{\prime}\) is value of the \(n\)th column, \(m\)th row in completed data; \(y_{m,n}^{\prime}\) represents the value of same voxel in projection data, and \(y_{m,n}\) is the value of the voxel in original data. \(N\) represents the total columns of original data. For each PV, the truncation data were completed row by row. Figure 4: (a) Breast slice for the first update in the first SART iteration. (b,c) Breast slice for the second, the third update in the first SART iteration. **Step 1. ** For each row of projection data, center-align original data with completed data as shown in step 1 of Figure 6. The values of the middle part of the completed data are defined as  
\[\left[\;y^{\prime}_{m,\frac{N}{2}}\;,\;y^{\prime}_{m,\frac{N}{2}+1},\ldots,y^{ \prime}_{m,\frac{N}{2}-1}\right]=\left[y_{m,1},y_{m,2},\ldots,y_{m,N}\right] \tag{3}\]  
**Step 2. ** Find the positions of \(\mathrm{p}\) and \(q\) at the \(m\)th row. The part from \(p\) to the left truncation boundary of the original data and the part from \(q\) to the right truncation boundary are the ranges that need to be completed. The positions and ranges are shown in step 2 in Figure 6. The projection value of the \(n\)th column, \(m\)th row is defined as:  
\[y^{\prime\prime}_{m,n}=\sum_{r=1}^{R}a_{r,L}*x^{k+1,c}_{r} \tag{4}\]  
**Step 3. ** Take the gradient from the left end to the right end of the projection data and use the gradient information to complete truncation data based on the original data. When completing the left side of the data, the calculation method is defined as:  
\[\left[\;y^{\prime}_{m,1}\;,\;y^{\prime}_{m,2},\ldots,y^{\prime}_{m,\frac{N}{2 }-1}\right]=\left[f_{1}(1),f_{1}(2),\ldots,f_{1}\left(\frac{N}{2}-1\right)\right] \tag{5}\]  
\[f_{1}(n)= \begin{cases}0&1\leq n<p\\ y_{m,1}+\
abla y^{\prime}_{m,\frac{N}{2}-1}+\
abla y^{\prime}_{m,\frac{N}{2}-2 }+\ldots+\
abla y^{\prime\prime}_{m,n}&p\leq n\leq\frac{N}{2}-1\end{cases}  \tag{6}\]  
**Step 4. ** To complete the right side of the data, the method is defined as:  
\[\left[y^{\prime}_{m,\frac{N}{2}}\;,\;y^{\prime}_{m,\frac{NN}{2}+1},\ldots,y^{ \prime}_{m,2N}\right]=\left[f_{2}\left(\frac{3N}{2}\right),f_{2}\left(\frac{3 N}{2}+1\right),\ldots,f_{2}(2N)\right] \tag{7}\]  
\[f_{2}(n)= \begin{cases}y_{m,N}-\
abla y^{\prime\prime}_{m,N}-\
abla y^{\prime \prime}_{m,N+1}-\ldots-\
abla y^{\prime\prime}_{m,n}&\frac{3N}{2}\leq n\leq q \\ 0&q<n\leq 2N\end{cases}  \tag{8}\]  
The gradient reconstruction process is shown in step 3-4 in Figure 6, and the gradient \(\
abla y^{\prime\prime}_{m,n}\) in projection data is  
\[\
abla y^{\prime\prime}_{m,n}=y^{\prime\prime}_{m,n}-y^{\prime\prime}_{m,n+1} \tag{9}\]  
**Algorithm 1**Complete truncation data.