# Limited-Angle Computer Tomography with Truncated Projection Artifacts Removal
## 2 Materials and Methods
### Truncated Projection Artifacts  
In this study, we used SART to reconstruct DBT images [11; 12; 13; 14; 15; 16]. The DBT system has 21 projection angles, which means that each voxel was updated 21 times in one SART iteration. The reconstructed images were updated once for each PV. Each update traversed all detector values and updated the corresponding voxel values. For the \((k+1)\)th iteration during the processing of the PVs, the update of the linear attenuation coefficient of the voxel is defined in Equation (1) [15; 16],  
\[x_{r}^{k+1,c}=x_{r}^{k+1,c-1}+\lambda*\frac{\sum\limits_{t=1}^{M_{c}}a_{r,t,c} \left(\frac{\frac{v_{r,c}-\sum\limits_{t=1}^{R}a_{r,t,c}*x_{r}^{k+1,c-1}}{\sum \limits_{t=1}^{M_{c}}a_{r,t,c}}}{\sum\limits_{t=1}^{M_{c}}a_{r,t,c}}\right)}{ \sum\limits_{t=1}^{M_{c}}a_{r,t,c}} \tag{1}\]  
where \(t\) is the index of detector in the detector panel; \(c\) represents the index of PV; the radiation path between the center point of a detector and the X-ray source is regarded as a beam of X-ray; \(r\) is the index of voxels in the reconstructed volume; \(R\) is the total number of voxels that a beam of X-ray passes through; \(a_{r,t,c}\) represents the weight of the \(r\)th voxel when traversing to the value of the \(t\)th detector in \(c\)th PV, which equals to the length of the beam of X-ray intersected by the voxel; \(v_{t,c}\) is the data detected by the \(t\)th detector in \(c\)th PV; \(M_{c}\) is the total amount of times that the voxel \(x_{r}^{k+1,c}\) updated in this iteration, and \(\lambda\) is a relaxation coefficient.  
Since the X-rays are irradiated by an approximate point source and the area of the detector panel is limited, parts of the X-rays passing through the object cannot be received by the detector, which leads to truncated projection data. When the images were reconstructed using truncated projection data, the truncated parts of the object were not reconstructed. In the reconstruction of the next angle, the voxel values of the truncated parts that were not updated last time will become dim. These are the causes of the TPA at the boundaries of each FOV. The generation process of TPA in the first round of SART is shown in Figure 4.  
Figure 3: Projection data generation.  
After the projection data were used for image reconstruction, by observing Figure 4a-c, it can be seen that there were obvious truncations on the left side. X-rays passing through the left side of the object were not received by the detector panel, which lead to the voxel values on the left side of the image not being updated in this iteration. Thus, the pixel values on the left side of Figure 4a are all zero. During the subsequent iteration process, since some regions of the reconstruction target were assigned with non-zero values, according to Equation (1); it can be found that the difference between the actual value detected by the detector and the calculated value becomes smaller. The difference can be calculated as:  
\[d=v_{t,c}-\sum_{r=1}^{R}a_{r,t,c}*x_{r}^{k+1,c} \tag{2}\]  
Thus, the updated voxel values will become smaller at this PV, resulting in darker stripes on the left side of the image. Therefore, in Figure 4b,c, the artifacts of gradually darkening voxel values appear in the red box in the images.