Extended Abstract

Introduction

3D similar transformation is used in various applications such as photogrammetry, geodesy, robotics and machine vision. Calculating the parameters of this transformation using the least squares method requires determining the initial values as close as to the final values. If the initial values used are not close to the final values, especially in the case that the rotation angles related to this transformation have large values, the least squares method will either not converge or converge to a wrong solution. In this paper, a direct and new closed-form method for determining the parameters of this transformation is presented. This method is able to determine 3D similar transmission parameters by using at least three corresponding points in both model and ground coordinate systems. In general, direct and non-iterative methods are faster and have lower computational cost, and most importantly, they do not require initial values. In contrast to these advantages, these methods are sensitive to noise in observations and outliers and have less accuracy than iterative methods. Iterative methods, although they have better accuracy, on the other hand, have more computational cost and their speed is low. Most importantly, these methods require initial values and if the initial values used in these methods are not close enough to the final values of the parameters, these methods will either not converge to the correct solution or converge to a wrong solution.

Methods and Materials

The method presented in this article is based on one of the characteristics of 3D similar transformation, i.e., establishing the same 3D similar transformation relationship between the gravity centers of corresponding points. By transferring the origin of the coordinate systems of the corresponding points to the gravity center points, the 3D similar transformation parameters between these two sets of points can be calculated in a closed-form manner, with the presented method. Two datasets were used to show the effectiveness of the presented method. The first dataset was created by simulation with large rotation angles and four times scale factor and with the minimum number of required points, i.e., three points. To simulate the real state in this dataset, random errors with normal distribution were added to each set of the corresponding points. The second dataset was selected from the real data obtained from LiDAR operations.

Results and discussion

The results of the method presented in this article were compared and evaluated with the results of the least squares method and two other closed-form and direct methods, i.e., the SVD method and the dual quaternion method. The results of the method presented in this article are close to the final values of these parameters and the values obtained from other methods. Tables (6) and (8), respectively, show the difference values of 3D similar transmission parameters between the results of using direct and closed-form methods with the least squares method for simulated dataset and real LiDAR dataset.
 
The data in Tables (5) and (8) show that the presented closed-form method in this paper provides similar 3D transmission parameters for both simulated data sets and real data with a slight difference of about 0.02° for rotational parameters and with a slight difference of less than 0.2m in the displacement vector parameters and with a slight difference of less than 0.002 in the scale parameter.

Conclusions

As can be seen from the obtained results, the accuracy of the values calculated by the presented method in this article is to the extent that it can be used directly for most applications, especially in online applications. On the other hand, the lower volume of calculations of the method presented in this article, compared to the SVD and dual quaternion methods as well as the iterative least squares method, justifies the use of this method for online applications. Also, the results of this method can be used as accurate initial values for the least squares method, in Close-range and UAV photogrammetry applications, where the rotational angular parameters can have large values.