# HiGoReg: A Hierarchical Grouping Strategy for Point Cloud Registration
## 1 Introduction
This combined framework streamlines computation, enabling fast and robust PCR even in challenging conditions. While many efforts focus on algorithmic enhancements, hardware-based acceleration techniques have likewise contributed significantly to runtime performance. [PERSON] et al. [20] demonstrated the effectiveness of graphics processing unit (GPU) parallel processing to accelerate nearest neighbor searches, a key bottleneck in many registration pipelines, enabling substantial runtime reductions, especially for large-scale point clouds. Once a set of (relatively) accurate point correspondences has been established--either through handcrafted features, learning-based matching, or geometric constraints--the next critical step is to estimate the transformation parameters that best align the source and target point clouds. This parameter estimation process lies at the core of most PCR pipelines and directly affects the final alignment accuracy. Conventionally, PCR is formulated as an optimization problem. To achieve more accurate alignment, [PERSON] and [PERSON] [21] introduced the Iterative Closest Point (ICP) algorithm, which minimizes the sum of squared Euclidean distances between corresponding closest points. The rigid transformation parameters--rotation and translation--are typically estimated using the Singular Value Decomposition (SVD) method [22], which offers a closed-form solution with favorable numerical stability and computational efficiency. Despite its simplicity and widespread use, ICP is sensitive to initial alignment conditions and the efficiency of the nearest neighbor search process [23]. To address these limitations,numerous improvements have been proposed to enhance its robustness, accuracy, and convergence characteristics [24; 25; 26]. In parallel with optimization-based approaches, more rigorous mathematical model-based techniques, such as Least Squares Matching (LSM) based on the Gauss-Markov (GM) model, can further optimize parameter estimation [27; 28; 29]. Originating from the problem of image patch-matching in photogrammetry, [PERSON] et al. [30] expanded LSM to handle 3D point cloud data, called LS3D. [PERSON] [31] highlighted the ability of LS3D to handle the surface matching of diverse qualities and resolutions by contending with surface modeling errors to ensure model flexibility and accuracy. The mathematical model of LS3D only allows for the random errors in the target point cloud but omits the error factors in the source point cloud. This one-sided error sensitivity approach is incapable of comprehensively capturing and dealing with all possible potential error sources affecting the adjustment accuracy. The Errors-In-Variables (EIV) model can account for random errors in both source and target point clouds [32] and is typically solved via the Total Least Squares (TLS) method. While offering improved parameter accuracy, its applications are always limited by repeated variables or fixed elements, which can lead to a singular variance-covariance matrix [9]. With the advancement of research, the nonlinear Gauss-Helmert (GH) model emerged as a prevailing framework that uniformly considers all errors in the observation space, regardless of whether they stem from the observations or the design matrices [33]. [PERSON] et al. [34; 35] proposed a point cloud fitting approach grounded in the Gauss-Helmert model, yielding highly accurate parameter estimates. [PERSON] et al. [36] applied the recursive GH model (RGH) to the system calibration of scanners. In the context of PCR, [PERSON] and [PERSON] [37] introduced the GH-LS3D method, which demonstrated superior estimation accuracy compared to the traditional ICP algorithm. [PERSON] et al. [9] adopted quaternions to represent rotational parameters and utilized sensitivity analysis to construct test statistics, enabling the effective identification of outliers and thus improving registration accuracy.