Objectives Fitting spherical point clouds has a wide range of applications in the field of measurement, such as point cloud registration, instrument calibration, and laser tracker self-calibration. In the uncontaminated point cloud or the data processed by robust estimation, existing geometric and algebraic fitting methods use the identity matrix as the cofactor matrix, which ignores the correlation of point cloud coordinates. In order to consider the rigorous statistical models of the point cloud, the nonlinear Gauss-Helmert model (GHM) needs to be established for the iterative solution. The method based on GHM is verified to be more accurate than the existing methods, including geometric fitting and algebraic fitting. However, when estimating a large number of spherical point clouds, this method generates a serious computational burden.

Methods The spherical fitting method based on GHM requires large matrix inversion operations, which is the main reason for reducing the efficiency of the algorithm. Because the complexity of matrix inversion is the cube of its dimension. For example, when the number of observations doubles, the time consumed by matrix inversion will be 8 times the original operation. In order to reduce the dimension of the matrix, all point cloud coordinates are grouped in many subsets. Only the three coordinate components of the same point are correlated, so the point clouds of different groups are independent of each other. When the existing parameter estimators are fused with the unprocessed point cloud data, the center and radius parameters are updated sequentially. In the proposed method, the matrix dimension is significantly smaller than the existing spherical fitting method based on GHM, which will help reduce the running time of the algorithm.

Results and Discussions In the simulation experiment, we used five methods to fit 1000 spherical points with different coverages (Fig.2), including geometric fitting, algebraic fitting, linear least squares, the method based on GHM, and the proposed method. All the observations are divided into 20 subsets randomly in the proposed method. The results of 1000 Monte Carlo experiments show that high coverage will increase the accuracy and computational efficiency of the five schemes. The results of the algorithm based on the GH model and the proposed algorithm are consistent in the root mean square errors (RMSE) of the sphere center and radius, and both of them are better than the existing benchmarking methods. Therefore, it is necessary to consider the statistical models of the point cloud in spherical fitting. When the coverage is 50%, the average running time of the proposed method is only 0.021s, which is less than 1.633 s of the method based on the GH model (Tab.1). Our algorithm has significant superiority in efficiency compared with the method based on GHM in the different coverages. When the number of point clouds increases from 1 000 to 5 000, the running time of the proposed algorithm is much lower than that of the method based on GHM (Fig.3). Further experiment by the point cloud data of 5 target balls in the Tongji University model (Fig.4) shows that the time the proposed method consumes is only 0.98%-1.74% of the method based on the GH model (Tab.2). Different grouping strategies covering from 10 to 100 were tested for the second spherical point cloud. As the number of groups increases, the running time first decreases and then increases (Fig.5). Although more grouping can reduce the matrix dimension in the computation, it will increase the number of recursions. In general, the grouping strategy can still significantly improve the efficiency of the original algorithm by more than an order of magnitude.

Conclusions The efficiency of the algorithm is as important as its numerical performance. We derive a sequential solution based on the spherical fitting of the GH model, which reduced the matrix dimension and greatly increase the computation efficiency. The next step is to obtain a more rigorous statistical model by variance component estimation and fitting more geometric shapes.