# A Rigid Motion Artifact Reduction Method for CT Based on Blind Deconvolution
## 3 Results  
### Numerical Phantom Experiments  
For the numerical simulations, the modified three-dimensional version of the Shepp-Logan phantom was used to simulate the CBCT system. The additional tiny circles in the 128 th slice of modified 3D Shepp-Logan head phantom are used to simulate the small tissue, as shown in Figure 2a. The size of the phantom was \(256\times 256\times 256\) voxels and the display window was [0 HU, 1000 HU]. The scanning time was set to \(t_{m}=1\). The interval angle was \(1^{\circ}\), and the number of projections was 360. The distance from \(X\)-ray source to detector was 100 cm and source to object was 50 cm. Poisson noise was added to the raw simulated data before the reconstruction (assuming that 10,000 photons were detected on each detector element in the blank scan). The projection data were obtained by applying the motion to a phantom during the scan. Figure 3 shows the motion segment that was used in the simulations. The slight motion of up \(4^{\circ}\) and 7 mm for rotations and translations obtained by [PERSON] et al. [7] (a motion segment in Figure 15 of [7]) are presented in Figure 3a. Figure 3b shows the large motion exceeding \(30^{\circ}\) and 30 mm, which is much larger than that typically expected in patients [7]. In simulations, a parameter \(\sigma=1\) and the smoothness parameter \(\varepsilon=10^{-6}\) were used for computations and the iterations were stopped when the difference between two images is less than a chosen threshold \(r=10^{-4}\). For the OSEM reconstruction, four iterations with 60 subsets were applied. The deconvolution were stopped at the third iteration for the BD-iteration. Figure 4 shows the 128 th slice of reconstructed images obtained by four methods. In this paper, \"OSEM-BD\" refers the method based on OSEM reconstruction algorithm with motion correction by BD-iteration, and \"RL-OSEM\" represents the method based on the combination of RL deconvolution and OSEM reconstruction, that is the proposed method without SATV regularization. It can be seen that the motion-corrected image obtained by the proposed method is very close to the real phantom. Since the motion information is constantly superimposed in the reconstruction process, the OSEM-BD method cannot completely eliminate the motion artifacts. In addition, the RL-OSEM method cannot balance edge preservation and noise amplification very well because it has no regularization constraints, so there are some motion artifacts and intensity oscillations in the reconstructed image. The proposed  
Figure 2: Head phantoms for simulation: (**a**) The 128 th slice of modified 3D Shepp–Logan head phantom; (**b**) a selected slice of head phantom. method can overcome the problems of RL-OSEM method. Thus, the reconstructed image obtained by the proposed method is the clearest and most of the motion artifacts are eliminated. The effects of motion artifact reduction were evaluated by visual assessment and with quantitative analysis. The root-mean-square error (RMSE), Pearson correlation coefficient (CC), and mean structural similarity (MSSIM) were chose as the metrics. RMSE was calculated in HU, while CC and MSSIM were dimensionless [7]. Table 1 provides quantitative comparisons of all reconstructed images among four methods. The mean and standard deviation of each metric were calculated over 256 reconstructed slices. The means (RMSE, CC, and MSSIM) indicated the overall quality of the reconstructed images. Standard deviations were taken as indices of the slice-to-slice variation in the calculated metrics. As expected, images reconstructed by using the proposed algorithm provide much higher CC and MSSIM mean values and much lower RMSE mean values. All the standard deviations of the proposed method are much lower than the other three methods, which demonstrate the proposed method has better stability. The simulation results of slight motion show that the proposed method has better motion artifact reduction effect than OSEM-BD and RL-OSEM method.