Most optical sensors obtain a large number of observations in a short time. Image data obtained by image sensors have many feature points, and point cloud data obtained by laser scanners contain a large number of three-dimensional coordinates. In image matching and point cloud fitting, weighted total least squares (WTLS) consider the errors in the observation vector and coefficient matrix, displaying higher accuracy and better statistical properties than least squares (LS). However, WTLS has not been widely used in image registration and point cloud fitting because the computational efficiency is too low to satisfy real-time requirements when the adjustment model contains a large number of observations.

One of the main reasons for the decreased efficiency in WTLS is the inversion of matrices with large dimensions. The complexity of a matrix inversion operation is proportional to the cube of its dimension. For example, if the matrix dimension increases 10 times, its time complexity increases 1000 times. To reduce the calculation time, we propose a solution based on the grouping strategy, named sequential total least squares (STLS), which is inspired by the sequential solution of the Gauss-Helmert model that has a calculation time less than that of the batch solution. The proposed algorithm groups all observations into many independent parts, recursively updating parameters, as new observations are added to existing estimated results. STLS consists of an outer iteration and inner recursion. The matrix dimension involved in the operation process is much smaller than that of the existing WTLS algorithm, thereby overcoming the efficiency issues of the existing WTLS method.

Simulated and measured data were used to test the efficiency of the proposed algorithm, comparing it to several existing WTLS algorithms, including traditional WTLS, structured total least squares (TLS), and the alternative method in partial errors-in-variables models. In the following three experiments, STLS divided all observations into 20 groups on average. First, in the simulated image registration experiment involving 1000 identical points, the root mean-squared error (RMSE) of LS was larger than that of WTLS (Table 1), which illustrated the necessity of considering the errors in the source image. STLS obtained the same RMSE as the existing WTLS methods because there was no difference between their mathematical models and objective functions. The average running time of STLS was only 0.176 s, whereas the existing WTLS method required at least 10.078 s (Table 2), which corresponded to a time ratio of 1.75%. With an increase in the number of identical points in the registration, the running time of STLS was still much smaller than that of the structured TLS (Fig. 2). Second, STLS was more accurate than LS with less bias when using wall images for affine transformation (Fig. 3), and the running time was less than 2% of the existing WLTS schemes (Table 3). Third, when fitting the plane containing 3120 wall point clouds in the WHU-TLS database (Fig. 4), STLS only took 0.381 s, whereas the existing WTLS algorithm required at least 22.317 s (Table 4), which corresponded to an increase in efficiency by more than 50 times.

In this era of big data, the computational efficiency of an algorithm is as important as its numerical performance. The observation values obtained by optical sensors are very large, imposing a heavy computational burden on the existing WTLS algorithm. By grouping observations, the dimension of the matrix is reduced significantly during the operation, as proven in the three experiments demonstrating the obvious efficiency improvement provided by STLS. Further research can still be done on the proposed algorithm. In the presence of outliers, the estimation results of this algorithm are seriously distorted, necessitating an increase in robustness. Numerous observation values are obtained by optical sensors. Therefore, the next step is to extend this algorithm to other models that contain a large number of observations, such as point cloud registration.