نوع مقاله : مقاله پژوهشی
نویسنده
استادیارگروه نقشه برداری، دانشکده عمران، دانشگاه صنعتی نوشیروانی بابل
چکیده
تبدیل متشابه سه بعدی در کاربردهای مختلفی مانند فتوگرامتری، ژئودزی، رباتیک و بینایی‌ماشین مورد استفاده قرار می‌گیرد. محاسبه پارامتر‌های این تبدیل با استفاده از روش سرشکنی کمترین مربعات مستلزم تعیین مقادیر اولیه نزدیک به مقادیر نهایی می‌باشد. در صورتی که مقادیر اولیه مورد استفاده به مقادیر نهایی نزدیک نباشد و به خصوص در حالتی که زوایای دوران مربوط به این تبدیل دارای مقادیر بزرگی باشند، روش کمترین مربعات یا همگرا نخواهد شد و یا به یک جواب اشتباه همگرا می‌شود. در این مقاله، یک روش مستقیم و بسته‌ی جدید برای تعیین پارامتر‌های این تبدیل با استفاده از حداقل سه زوج نقطه متناظر در دو سیستم مختصات ارائه شده است. نتایج حاصل از این روش به مقادیر نهایی این پارامترها نزدیک بوده و در کاربردهای کم دقت می‌تواند به صورت مستقیم مورد استفاده قرار گیرد. در کاربردهای دقیق نیز می‌توان از نتایج این روش به عنوان مقادیر اولیه برای محاسبات کمترین مربعات استفاده کرد. نتایج روش ارائه شده با نتایج روش کمترین مربعات و دو روش بسته و مستقیم دیگر یعنی روش SVD و روش کواترنیون دوگانه مقایسه و مورد ارزیابی قرار گرفت. در ارزیابی روش ارائه شده از دو سری داده شبیه‌سازی شده و داده واقعی استفاده شد. اختلاف نتایج بدست آمده از این روش با نتایج به دست آمده از روش کمترین مربعات دارای خطایی در حدود 0.02° در مورد پارامتر‌های دورانی و خطایی در حدود 0.2m در مورد پارامتر‌های بردار جابجایی و خطایی کمتر از 0.002 در مورد پارامتر مقیاس هستند. این ارزیابی، کارآیی خوب این روش را نشان می دهد.
کلیدواژه‌ها
موضوعات
عنوان مقاله [English]
A new closed-form solution in direct determination of 3D similarity transformation parameters in photogrammetric absolute orientation computations
نویسنده [English]
- Alireza Afary
Assistant professor, Dept. of surveying, civil engineering Faculty, Babol Noshirvani University of Technology
چکیده [English]
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.
کلیدواژه‌ها [English]
- 3D similar transformation
- Direct and closed-form methods
- Least square
- SVD
- Quaternion
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