## 4 Results and Discussions  
Table 2 shows the RMSE of check plane residuals for the registration with Dataset A based on the proposed method with the 0.125 m\({}^{3}\) die. The RMSEs after the registration should be as close as possible to that before the registration, indicating the registrations bring in the least additional errors. From Table 2, it can be seen that the proposed method with the dimension calibration embedded can deliver comparable results to the traditional method based on the spherical targets. Since the dice are made of foam, the dimension may  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Datasets** & **Point Cloud** & **NO. of Points** & \begin{tabular}{c} **Scanner-to-Die** \\ **Distance (m)** \\ \end{tabular}  & \begin{tabular}{c} **Scan the Plane** \\ **of the Dice** \\ \end{tabular}  \\ \hline A & \begin{tabular}{c} 1 (master) \\ 2 (slave) \\ \end{tabular}  & \begin{tabular}{c} 91,628 \\ 76,783 \\ \end{tabular}  & \ \begin{tabular}{c} 6.1 \\ 8.1 \\ \end{tabular}  &
\begin{tabular}{c} \text{Pips1}\backslash\text  
deviate considerably from the nominal values provided by the manufacturer. As a result, dimension calibration becomes important to maintain registration accuracy. The calibration does not need to be carried out for every registration. The calibration is only required when the die's facets for the scanning are altered, which may occur following handling during a long period of scanning or a sudden change in environmental conditions such as an abrupt temperature change. The RMSEs of the check planes for Datasets B and C are shown in Tables 3 and 4, respectively. These registrations were performed using the same 0.125 m\({}^{3}\) die. Similar to Dataset A, the die-based registrations with Datasets B and C are commensurate with the conventional sphere-based method, whereas the largest RMSE discrepancy is only 0.5 mm. In addition, it can be seen from Tables 2-4 that the method can be flexibly carried out for a scanner-to-die distance within several meters (e.g., three meters) and up to about twelve meters. The proposed die-based method only requires the setup of a single low-cost die, which is very convenient compared to the installation of six higher-cost and evenly distributed spheres. Setting six spherical targets requires empirical knowledge as the collinearity of the distributed spheres and uneven distribution of the spheres may weaken the network geometry. In contrast, the proposed die-based method will never suffer from this problem as the reference points for registration are always lying on axes that are mutually orthogonal. Overall, the proposed method can be used as an alternative to the sphere-based method, with higher flexibility and lower instrument cost. When a smaller die (volume of 0.027 m\({}^{3}\)) is used, the proposed dice-based method can still achieve comparable accuracy (within RMSE of 0.5 mm compared to the results obtained from the sphere-based method), as can be seen from the Dataset D results tabulated in Table 5 Since the volume of the die is reduced by 40%, the effective range for the registration is reduced to approximately 6 m. This range can still satisfy the requirement of many close-range metrological applications [32]. The 0.027 m\({}^{3}\) die can be even more flexibly placed in the scene since it occupies less space.