## 4 Discussion  
### Assessment of Accuracy of MS DVSP  
For evaluating the robustness of an algorithm, numerical experiments are usually used. There are two methods to generate the reference and the deformed images. The first method is to apply computer simulation to digitally generating images. This method can isolate the possible errors caused by the image acquisition. In most DVC and DVSP applications, natural microstructures are seen as the speckle patterns. It is difficult to find a mathematic function to describe and simulate the natural microstructures. The second method is to employ the original image of a specimen as the reference image, and the deformed images can be generated by using synthetic shifts with Fourier transform. In this paper we use the second method. Two volumetric images of 200 \(\times\) 200 \(\times\) 200 voxels are isolated from the reconstructed images of step 1 in the above two experiments, which are defined as the reference images, respectively. The \"deformed\" volumetric images are generated by imposing a rigid displacement ranging from 0.1 to 1.0 voxel with an increment of 0.1 voxel along the \(z\) direction. By using the MS-DVSP algorithm, pre-imposed sub-voxel translation displacements are  
Figure 11: \(\epsilon_{yz}\) strain contours of two transverse sectional images at step 7, step 9 and step 10, respectively; (**a**) step 7; (**b**) step 9 (**c**) step 10. Figure 10: Displacement and strain fields of the middle longitudinal sectional image at step 7, step 9 and step 10, respectively; (**a**–**c**) \(v\) fields; (**d**–**f**) \(w\) fields; (**g**–**i**) \(\epsilon_{yy}\) fields; (**j**–**l**) \(\epsilon_{zz}\) fields; (**m**–**o**) \(\epsilon_{yz}\) fields. calculated. The predetermined subset sizes are defined as 16 \(\times\) 16 \(\times\) 16 voxels, 32 \(\times\) 32 \(\times\) 32 voxels and 64 \(\times\) 64 \(\times\) 64 voxels, respectively, and the shift of the subset is 10 voxels. Figure 12 shows both the mean bias error and the standard deviation error as a function of the imposed displacement for different subset sizes. With a predetermined subset size, the error reaches maximum at 0.5 voxels. In this situation the information between each subset in the reference and deformed images is the most biased. Furthermore the larger is the subset, the smaller the error. But, increasing the subset size would degrade the spatial resolution. We define the predetermined subset size as having 32 \(\times\) 32 \(\times\) 32 voxels. With this subset size the maximum mean bias error of the red sandstone specimen is 0.0160 voxels and the corresponding standard deviation error is and 0.0060. For the woven composite specimen, the maximum mean bias error is 0.0062 voxels, and the corresponding standard deviation error is and 0.0089 voxels. The difference of accuracy between those two materials depends on their natural structures. The Shannon entropy values Shannon (1954) of the red sandstone and the woven composite are 2.7606 and 3.0337, respectively, which means the woven composite has more feature information seen as speckle patterns resulting in a bit higher accuracy in the measurement. In reference [39] a multi-scale coarse-fine subset calculation process was used in DVSP, which also increased the performance of DVSP. We call this algorithm M-DVSP. To compare the performance of MS-DVSP, M-DVSP, and DVSP, we calculate the displacements of these volumetric images with imposed sub-voxel translation using these three algorithms, respectively. The subset size is predetermined at 32 \(\times\) 32 \(\times\) 32 voxels, and the shift of the subset is 10 voxels. The error curves are plotted as shown in Figure 13. The performance of M-DVSP algorithm is slightly better than that of DVSP, especially in standard deviation errors after 0.5 voxels. When the imposed displacement is increased continuously, the overlapping area between the reference subset and the deformed subset is decreased, and the noise caused by non-overlapping becomes stronger.