1. Introduction
Many applications of laser scanning rely on accurate registrations that integrate point clouds obtained from different scan stations into a complete one [
1,
2,
3]. Regardless of the recent development of automatic targetless methods for point cloud registration [
4,
5,
6], conventional target-based methods such as the sphere-based method, are still being applied by many professionals thanks to the directness and manipulability for installing the artificial targets (e.g., spherical targets [
7,
8] and calibration board [
9]). However, placing and shifting multiple targets within the field of view of many instrument stations can be a tedious and time-consuming process, especially when the number of scans is large [
10]. One possible solution for tackling this problem is to develop a registration method that only requires a single target, instead of multiple targets. In addition, if the target itself incurs a low cost, many targets can be used and fixed simultaneously so the cost of shifting and reinstalling can be reduced.
In general, using targets for registration aims to estimate a set of approximate parameters which transform one point cloud from its own coordinate system to another. Once the two point clouds are approximately aligned, the iterative closest point (ICP) algorithm [
11], or one of its variants can be used to accurately match the point clouds so that the point clouds are completely registered [
12,
13,
14]. The former is often referred to as coarse registration while the latter is often defined as the fine registration [
15]. The accuracy of the transformation parameters estimated in the coarse registration can greatly affect the final accuracy delivered by the fine registration.
For the past decade, much research effort has been dedicated to targetless coarse registration which uses natural features detected from the scans as targets instead of using artificial targets such as spheres and modeled with different geometric primitives [
16]. For example, Li et al. proposed a coarse registration method that utilizes linear features obtained from the extraction of the interception between walls and grounds in an indoor environment [
17]. Tao et al. developed a linear feature-based method for both indoor and outdoor environment [
18]. Instead of using linear features, Wei et al. developed a descriptor-based method that is based on natural planar features extracted from the scenes [
19].
Kelbe et al. utilized the cylindrical features extracted from trees to perform the registration for the point clouds obtained from forests [
20]. Chan et al. extracted an octagonal lamp pole for registrations [
21]. A geometric model of an octagonal prism model was developed to simulate multiple conjugate points for the subsequent registrations. Yang and Zang extracted crest lines from statues to perform the registration for cultural heritage documentation [
22]. They computed the principal curvature using eigenvalue decomposition to form the create lines so that the method can be used in environments without straight line features.
As mentioned before, regardless of the state-of-art of the targetless registration methods, conventional target-based methods are still widely used by many professionals. It is because manually placing the targets is more straightforward and manipulable. Using targets for registration reduces the impact of the complexity of the point clouds.
Figure 1 shows scans being performed in a harsh factory environment. In such an environment, installing white spherical targets provides greater control which in turn makes registration quality independent of the natural features that cannot always be accurately extracted. Spheres plated with retroreflective materials are one of the most common targets used for point cloud registration since they are omnidirectional [
23]. In order to estimate the six degrees of freedom (DoF) parameters for the transformation, the coordinates of at least three non-collinear points extracted from the two scan stations for the coarse registration are required [
24]. As a result, at least three sets of coordinates of spherical target centers from each station are needed for the coarse registration. Moreover, in most cases, errors that adhere to the estimation of the sphere centers will propagate into the registration errors [
25].
When the number of scans becomes large, setting up multiple spherical targets becomes very time-consuming and sometimes challenging, especially in harsh environments with many obstacles and interferences. For example, multiple spherical targets were installed in a limited space with confined distribution at a tunnel segment factory (
Figure 1). Therefore, a new target-based registration method that requires the minimum number of target primitives is desired. In this paper, we propose a method that only requires the setup of a single low-cost cubical target (e.g., a die-like target) to replace multiple survey-grade targets for point cloud registration. The method includes simulations of two local coordinate systems (object spaces) with errors (axis non-orthogonality and translational errors). These errors are then compensated by two proposed algorithms based on the projection geometry and the cubical constraints. The method can be performed readily as the dice can be placed on the ground without using any tripod or support. The only requirement is that three facets should be scanned from each station, with only pips from two facets being recognized. There is no strict requirement for scanning overlapping facets of the die target.
3. Experiments
For verification of the proposed method, four point cloud datasets were captured with a leveled Trimble SX 10 scanner mounted on a tripod at the School of Geography and Planning Building at the East Campus of the Sun Yat-sen University, Guangzhou, China, during mid-October 2021, as shown in
Figure 11. For Datasets A, B, and C, a die with a volume of 0.125 m
3 was used as the target, whereas a smaller die (volume of 0.027 m
3) was installed for Dataset D as shown in
Figure 11. It is worth noting that the dice are very affordable as a registration target: the 0.125 m
3 die costs RMB 150 (USD ~22), while the 0.027 m
3 die costs RMB 20 (USD ~3). A set of spherical targets traditionally used for registration was also placed in the scene for reference and for comparison of results. A set of six spherical targets is considerably more expensive, costing about RMB 2000 (USD ~300). Details of the collected datasets are shown in
Table 1.
Four check planes (two vertical and two horizontal) were selected at different ranges for each registration, and the RMSE of the planar residuals was used for quantification of the registration accuracy. Before the registration, the check planes at both stations have their own stand-alone RMSEs after the fitting. If the registration is perfect (possesses zero errors), the registered check plane will succeed the larger RMSE from one of the check planes. In other words, any errors accumulated on the top of this larger RMSE from one of the check planes are attributed to the registration errors. In addition, each registration was also independently carried out with four to six spherical targets for comparison.
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 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
Table 3 and
Table 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
Table 2,
Table 3 and
Table 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.
Table 6 shows the original angles between each facet (before the axis orthogonality correction), along with the average calibrated dimensions for the registrations. From
Table 7, on one hand, none of the angles between the planes (dice’s facets) are larger than 1°, but still large enough to affect the final registration accuracy. Based on trigonometry, a 0.5° of angular error will cause a translational error of roughly 4.3 cm at a range of 5 m. Therefore, it is necessary to conduct the proposed procedure for axis orthogonality correction. On the other hand, up to 2 cm of the correction for the dimension is estimated based on the calibration. This magnitude is significant, so the proposed calibration procedure is essential to maintain higher registration accuracy.
The registered point clouds before and after the die dimension calibration are visualized in
Table 7 for the four datasets. For Dataset A, the point clouds of a cabinet are more aligned after the registration with the dimension calibration. Similarly, a more accurately registered point cloud is shown on the Table for Dataset B, indicated by the better overlapped windows. These results are consistent with the fact that the improvement in the dimension accuracy will result in the more accurate determination of the relative positions between the conjugate point pairs. In the same Table (
Table 7), a more accurately registered spherical object and pillar are shown with the dimension calibration for Datasets C and D, respectively. As a result, it can be concluded that the accurate dimensions of the dice are important for the proposed method, refining the register parameters to deliver a complete set of registered point clouds.
5. Conclusions
In this paper, we presented an automatic cube (die)-based registration method which can deliver registration accuracy comparable to the conventional sphere-based method for point clouds collected by laser scanning at close range (up to 12 m). The proposed method is advantageous in terms of flexibility and equipment cost as only a single low-cost die-like object is required as a target to perform the registration. The quality of the die’s manufacture is not a factor since the deviations from the perfect cube shape can be modeled and compensated by the proposed algorithms. The method first constructs coordinate systems by using lines between the intersections of the scanned facets to form the axes. Then, the non-orthogonality between axes is corrected based on the projection geometry. This is followed by simulating three pairs of conjugate points along the axes as the correspondences for the registration. The distances between the conjugated is constrained to be unity so the main problem is to determine the relative position and orientation between coordinate systems for the die space obtained from two scans. The registration accuracy can be improved by a simple calibration procedure for the die’s dimensions.
Based on the proposed method, two different low-cost dice (with a volume of 0.125 m3 and 0.027 m3) were used for registering four real datasets collected by a Trimble scanner. The results from the proposed method match well (with check plane RMSE discrepancy less than 0.5 mm) with those obtained from the traditional sphere-based method which required four to six spherical targets to be distributed evenly in the scene. There is no need to have overlapping facets scanned so the placement of the target becomes very flexible. The proposed method not only provides scanner users with high flexibility for target placement but also with low instrument cost to enhance project management. Theoretically, the proposed method can be modified readily to a multi-dice version for registering very large scenes consisting of hundreds or thousands of scans.