## 4 Discussion
### Influence of Artifacts in CT Image  
During reconstruction of CT slice image, the non-uniformity of detector units and the polychromatic nature of the X-ray will give rise to ring and beam hardening artifacts. The self-heating effect of X-ray tube may result in distance variations that induce magnification changes and thus spurious dilatational strain. In addition, the image noise and contrast, the imperfect motion of the rotation stage, and the possible rigid body motion of the specimen will all influence the measurement result. Among all these effects, the spurious dilatational strain is definitely the most important to correct. More detailed information on the influence of these effects can be found in references [48; 49; 50; 51]. Here, we just discuss the influence of the image noise and the imperfect motion. To analyze the effect of artifacts of the micro X-ray CT system on the MS-DVSP algorithm, we take two consecutive scans in step 1 of the red sandstone specimen in Section 3.1 as the baseline experiment. Noise and artifacts of the system are included in the reconstructed images. For saving the computation time, we isolate two volumetric images having 200 \(\times\) 200 \(\times\) 200 voxels from the center of reconstructed images of Scan 1 and Scan 2, define them as the reference image and the deformed image, respectively, and calculate displacements by using the MS-DVSP algorithm. The predetermined subset size is 32 \(\times\) 32 \(\times\) 32 voxels, and the shift of the subset is 10 voxels. The average measured displacements  
Figure 13: Accuracy comparison among MS-DVSP, M-DVSP and DVSP algorithm: (**a**) Mean bias error of displacement of the red sandstone specimen; (**b**) Standard deviation error of displacement of the red sandstone specimen; (**c**) Mean bias error of displacement of the woven composite specimen; (**d**) Standard deviation error of displacement of the woven composite specimen. are \(u=-0.0059\) voxels, \(v=-0.0147\) voxels, and \(w=0.0175\) voxels, and the standard deviation errors are 0.0107 voxels, 0.0115 voxels and 0.0105 voxels, respectively. The measured displacement along \(y\) and \(z\) axes is greater than the standard deviation error, indicating that some motion occurred along \(y\) and \(z\) axes between these two scans due to physical perturbations in the CT system. Because there are real noise and artifacts in the volumetric images of Scans 1 and 2, the above results indicate that the uncertainty of displacement measurement in all three directions is about 0.02 voxels with the MS-DVSP algorithm in the red sandstone specimen experiment. For investigating the influence of noise and artifacts, two groups of deformed volumetric images are obtained by imposing subvoxel displacements on above two cropped volumetric images, and they are marked as Group 1, and Group 2, respectively. We define the cropped volumetric image of Scan 1 as the reference image, and the images of Group 1 and Group 2 as the deformed images, respectively. Then we calculate the displacements with the MS-DVSP algorithm. The predetermined subset size is 32 \(\times\) 32 \(\times\) 32 voxels, and the shift of the subset is 10 voxels. Figure 14 illuminates the results of the displacement measurement. Due to the influence of noises and artifacts, the error increases almost two times. The uncertainty of displacement measurement is 0.015 voxel. As a more accurate alternative to fitting just rigid body movements, a least squares fit is used to calculate the rigid body translations and rotations, assuming small angles [52],  
\[ \begin{array}{l}u_{rigid}+\theta_{z}Y-\theta_{y}Z\cong U\\ v_{rigid}-\theta_{z}X+\theta_{x}Z\cong V\\ w_{rigid}+\theta_{y}X-\theta_{x}Y\cong W\end{array}  \tag{2}\]  
where \(\theta_{x}\), \(\theta_{y}\), and \(\theta_{z}\) are the rotations about the \(x\), \(y\), and \(z\) axes, respectively, and \(X\), \(Y\), \(Z\) and \(U\), \(V\), \(W\) are the vectors of the \(x\), \(y\), \(z\) coordinates and \(u\), \(v\), \(w\) displacements, respectively, for all of the correlation points.