空间相似变换的直接解法及比较研究

Comparative study on closed-form solutions for spatial similar transformation

求解空间相似变换问题是指在多余控制条件下确定尺度、平移和旋转参数,测绘领域的传统解法是给定待求参数的初值后按最小二乘法迭代求解,其缺点是需要预先给定参数初值以保证正确收敛。而直接解是以旋转矩阵正交为约束条件在最小二乘准则下通过奇异值分解得出的,无需参数初值和迭代计算,可以解决大旋转角度的估计问题。本文给出了基于正交Procrustes分析的直接解法,基于正交规范分解的直接解法和基于单位元数的直接解法的解算原理和过程,对计算模型、解算精度、计算速度和抗粗差能力与传统方法进行了模拟比较分析,旨在推广直接解法在测绘领域的应用。

Spatial similar transformation problem is to estimate one scale factor,three translations and three independent rotation parameters with sufficient control points,traditional approach is the least squares adjustment for over-determined equations,which needs initial values to assure convergence.Whereas the direct solutions are decomposition of a certain matrix to obtain optimal estimates from the cost function directly.Three direct methods named Orthogonal Procrustes Analysis,Orthonormal Matrices Decomposition and unit quaternions are presented in this paper,and comparative studies on computational accuracy and speed are made of the mentioned approaches.The paper is to promote the applications of the direct solutions of spatial similar transformation in surveying and mapping.