臺灣博碩士論文加值系統

在攝影測量中通常以尤拉角表示外方位參數中的姿態參數，以共線式為物理模式，物點、相機中心、像點在同一直線上。解外方位參數時，需要物空間坐標、像點坐標，將共線式線性化再以最小二乘法迭代解算外方位參數，此方法需要外方位參數的初值，初值不佳容易造成求解失敗。本研究將以四元數及對偶四元數取代以尤拉角表示的旋轉矩陣及平移向量，以解決共線式線性化遇到的問題。
四元數是複數空間的概念，可以直觀的表示三維空間中的向量旋轉，以四元數表示法取代共線式中尤拉角表示的旋轉矩陣在測量領域已有相關的研究。對偶四元數則是以四元數的基礎提出的概念，可同時表達空間中的轉動及移動，本研究將以此性質表示攝影測量中不同坐標系之間的轉換關係。
對偶四元數可以表示三維空間中的旋轉及平移關係，若要解決二維與三維之間關係，結合電腦視覺方法，已知相機內參數，將二維影像坐標轉換到三維相機坐標系坐標，再與對應的物空間坐標，依照對偶四元數解算模式，有三組對應點即可以求得一個表示物空間坐標系及相機空間坐標系關係的對偶四元數，由此轉換所得角度及平移量，因此此解算模式可以作為攝影測量求解外方位參數的模式。

In photogrammetry, the Euler angular are commonly used to describe the attitude parameters for exterior orientation. The basic formula is the collinearity condition equation, which connect the perspective center of camera, the photo point, and the ground point along a straight line. With the given coordinates of ground control points and the measured coordinates of the photo point, the exterior orientation parameters can be determined based on the linearization of the collinearity equation using least squared adjustment. Because of the linearization, the prior estimation of the initial values of the unknown parameters are highly required. In this study, the quaternion and the dual quaternions are used to replace the rotation matrix by Euler anglers and translations to avoid the problem caused by the linearization.
Dual quaternion has the ability to represent the rotation and translation simultaneously in three dimensional vector space. In this study, the 2D image coordinates are transformed to 3D camera coordinate system by computer vision techniques, then the 3D object coordinates can by determined by dual quaternion algorithm. Only three control points are needed to calculate the rotation and translation between the camera and object coordinates and then the camera orientation can be determined.

摘要 i
Abstract ii
第1章 前言 1
1.1 研究動機與目的 1
1.2 文獻回顧 2
1.3 研究方法與流程 4
1.4 論文架構 4
第2章 四元數及對偶四元數 6
2.1 四元數 6
2.2 對偶數及對偶向量 12
2.3 對偶四元數 14
第3章 研究流程及方法 19
3.1 旋轉矩陣與四元數關係 20
3.2 對偶四元數解算模式 23
3.3 電腦視覺方法 28
3.4 對偶四元數應用於攝影測量 30
第4章 實驗與成果分析 32
4.1 實驗一、以對偶四元數表示旋轉角測試 33
4.2 實驗二、以對偶四元數模式取代七參數轉換 36
4.3 實驗三、控制點分佈實驗 38
4.4 實驗四、控制點數量實驗 42
4.5 實驗五、外方位參數實驗 45
4.6 實驗六、相對方位參數實驗 48
4.7 小結 51
第5章 結論與未來展望 52
參考文獻 53

Bayoud, F. A. (2006). Development of a robotic mobile mapping system by vision-aided inertial navigation: A geomatics approach: Institut für Geodäsie und Photogrammetrie.
Daniilidis, K. (1999). Hand-eye calibration using dual quaternions. The International Journal of Robotics Research, 18(3), 286-298.
Funda, J., Taylor, R. H., & Paul, R. P. (1990). On homogeneous transforms, quaternions, and computational efficiency. IEEE Transactions on Robotics and Automation, 6(3), 382-388.
Harauz, G. (1990). Representation of rotations by unit quaternions. Ultramicroscopy, 33(3), 209-213.
Horn, B. K. (1987). Closed-form solution of absolute orientation using unit quaternions. JOSA A, 4(4), 629-642.
Horn, B. K., Hilden, H. M., & Negahdaripour, S. (1988). Closed-form solution of absolute orientation using orthonormal matrices. JOSA A, 5(7), 1127-1135.
Kenwright, B. (2012). A beginners guide to dual-quaternions: what they are, how they work, and how to use them for 3D character hierarchies.
Kuipers, J. B. (1999). Quaternions and rotation sequences : a primer with applications to orbits, aerospace, and virtual reality. Princeton: Princeton University Press.
Szeliski, R. (2010). Computer vision: algorithms and applications: Springer Science & Business Media.
Walker, M. W., Shao, L., & Volz, R. A. (1991). Estimating 3-D location parameters using dual number quaternions. CVGIP: image understanding, 54(3), 358-367.
Wolf, P. R., & Dewitt, B. A. (2000). Elements of Photogrammetry: with applications in GIS (Vol. 3): McGraw-Hill New York.
江剛武、姜挺、王勇、龔輝. (2007). 基于單位四元數的無初值依賴空間后方交會. 測繪學報, 36(2), 169-175.
黃博彥. (2007). 應用雙四元數同時估測轉動與移動. 台灣大學電機工程研究所.