應用對偶四元數於攝影測量方位的封閉式解算方法

Abstract

近年無人飛行載具以及近景攝影測量技術的應用需求日益增加，而攝影測量方位解算是攝影測量作業中的重要問題。在傳統的攝影測量方位解算上，通常是利用共線式、共面式或是七參數轉換公式等非線性的數學模式進行之，此類數學模型是以尤拉角所組成的，其解算方式為將觀測方程式線性化，並給予良好的方位參數初始值以進行迭代計算，而若方位參數的初始值給定不佳，或是攝影姿態變化較大，則可能造成解算系統無法收斂，或是出現奇異解的情形。本研究引進四元數與對偶四元數的概念，並以之求解攝影測量的方位參數，提出不須給定初始值以及無角度奇異解的解算模式。 四元數為描述向量旋轉的工具，其描述向量旋轉的方式為以一個旋轉軸和一個旋轉角描述之，故沒有角度奇異的問題，且具描述方式簡潔的特性。對偶四元數保有四元數描述旋轉的優勢，且其同時描述旋轉與平移。本研究以此種方式描述坐標系統間的轉換關係。 本研究將四元數方法與對偶四元數演算法應用於若干攝影測量方位參數解算議題，如單片後方交會、立體像對相對方位以及獨立模型連結等，達到不須給定初始值、無角度奇異解以及運算效率高等目標。最後將本研究所使用的方法與過去的方法進行比較，成果顯示Walker對偶四元數演算法的成果精度優於SVD對偶四元數演算法者，而以四元數為基礎的方法在姿態準確度的部分有機會與傳統迭代方法者相當。

In recent years, the application of UAV and close-range photogrammetry are increasing. However, the calculation of orientation parameters is an important issue. In the traditional photogrammetry, orientation parameters are usually calculated by using a nonlinear mathematical model such as collinear condition equation or coplanar condition equation. The method is to linearize the observation equation and give a good initial value of the orientation parameters for iterative calculation. If the initial value of the orientation parameters are given poorly, the calculation system cannot converge. And if the shooting attitude changes greatly, singular solutions would appear. This study introduces the concept of quaternion and dual quaternion to solve the orientation parameters of photogrammetry. The quaternion is a tool for describing the rotation of 3D vectors. It describes the rotation with a rotation axis and a rotation angle, so there is no problem of angular singularity. The dual quaternion can describe both rotation and translation simultaneously. This study shows the transformation between coordinate systems in dual quaternion way. The study applys quaternion method and dual quaternion method to several photogrammetry issues such as Single Photo Resection, Relative Orientation, and Independent Model etc, achieving initial values unrequirment, none of angular singularties, and efficient calculation. At last, the study compares two common-used dual quaternion methods, and makes comparisons between quaternion-based methods and traditional iterative methods. It reveals that outcome precision from Walker method is better than SVD method, and the attitude accuracy from quaternion-based methods is almost equivalent to traditional iterative methods.