# HiGoReg: A Hierarchical Grouping Strategy for Point Cloud Registration
## 3 Experiments
### Simulation Experiment  
To rigorously assess the efficacy of our method, we selected Bunny, Monkey, and Dragon point clouds from Stanford University's publicly available point cloud dataset ([[http://graphics.stanford.edu/data/3](http://graphics.stanford.edu/data/3) Dscanrep/]([http://graphics.stanford.edu/data/3](http://graphics.stanford.edu/data/3) Dscanrep/), accessed on 17 March 2025), as depicted in Figure 2. The target point cloud (in green) was realized by slight rotation and translation from the source point cloud (in red). Before solving for the transformation parameters, keypoints were first detected using the intrinsic shape signature (ISS) method [69]. This step significantly reduced the number of coordinates involved in the registration of the Bunny, Monkey, and Dragon point clouds from 35,947, 125,952, and 437,645 to a more manageable 330, 233, and 936, respectively. Subsequently, point correspondences were established and selected for parameter estimation. Following the picky ICP algorithm [70], only point pairs whose Euclidean distances were less than the mean value with 2\(\times\) median error were retained. Considering the limited number of points in the Stanford dataset and the dimensionality reduction resulting from ISS keypoint extraction, two additional synthetic point clouds were generated to simulate large-scale registration scenarios. Each synthetic point cloud contained 7000 corresponding keypoints. The source point cloud consisted of randomly distributed coordinates within the ranges \(x\in[-100,200]\) m, \(y\in[-50,200]\) m, and \(z\in[-10,50]\) m. The target point cloud was obtained by applying a rigid transformation to the source, characterized by Euler angles \(\omega=0.3491\) rad (\(20^{\circ}\)), \(\phi=-0.3491\) rad (\(20^{\circ}\)), and \(\kappa=0.1745\) rad (\(10^{\circ}\)) and a translation vector \(\left[1\quad 0.5\quad-0.2\right]^{\top}\) m. The simulated point clouds are illustrated in Figure 3.  
Given the superior performance of GH, the GM model was excluded from further analysis (see [62] for details), and the comparison was focused on the following two approaches:  
* Batch solution. * HiGoReg method. Due to the absence of instrument-specific details, all point cloud observations were equally weighted. Within the HiGoReg scheme, the point clouds from the Stanford dataset were segmented into 2, 3, 4, 5, and 6 groups, while the large-scale synthetic dataset was divided into 10, 20, 50, 100, and 200 groups, aiming for a balanced distribution of points across groups. As an illustrative example, dividing 330 points into four groups would yield an approximately uniform distribution of 83, 83, 83, and 81 points, respectively. During the registration phase, the number of iterations was conservatively set with a minimum of 5 and a maximum of 20. The hierarchical iteration process was terminated early if the difference between two successive Root Mean Squared Error (RMSE) values fell below a threshold of \(\varepsilon_{\text{rms}}=10^{-4}\) m, as defined in Equation (28). Within each iteration, to solve for  
Figure 3: Simulated large\(-\)scale point cloud dataset, where red indicates the source point cloud and green stands for the target point cloud. Figure 2: Stanford point cloud dataset, where red indicates the source point cloud and green stands for the target point cloud. (**a**) Bunny; (**b**) Monkey; (**c**) Dragon. the nonlinear rotation and translation parameters, convergence was determined based on the thresholds \(\varepsilon_{R}=10^{-5}\) rad and \(\varepsilon_{T}=10^{-5}\) m, respectively. \[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\Bigl{(}f_{pp,i}-g_{pp,i}\Bigr{)}^{2}} \tag{28}\]  
where \(pp\) is the abbreviation of point pairs. All experiments and operations were executed on Matlab R2022a, hosted on a Windows 10 operating system, leveraging the Intel Core i9-10900K central processing unit (CPU) and 32 GB of random access memory (RAM). The visualized outcomes of the registration are presented in Figure 4, in which the registered source point clouds are depicted in blue.