摄影测量数学分析问题的计算机代数精密解法研究

There are many fussy mathematical analysis processes in photogrammetry, such as the power series expansions, spatial coordinate transformation and the solution of nonlinear equations relating to trigonometric function and quaternion. The derivation of these problems by hand not only consumes much time and energy but also makes mistake easily, and sometimes couldn't be realized at all because of the impossible complexity. Traditional algorithms mainly have following problems: (1) The expressions are complex and lengthy, which makes the computation process very complicated and time-consuming. (2) Some approximate disposal is adopted, which influences the computation accuracy. Based on modern computer algebra system and its powerful ability of mathematical analysis and symbolic operation, the precise solution of mathematical problems by computer algebra in basic theory of photogrammetry orientation, aerial triangulation and space photogrammetry are systematically studied in this research project. The new algorithms and mathematical models in symbolic form are established with the help of rigorous and analytical mathematical analysis, and they have more concise form, stricter theory basis and higher accuracy compared to traditional ones. The breakthrough and innovation of some mathematical analysis problems in the special field of photogrammetry can be realized, which will further renovate and enrich the photogrammetry theoretical system. The research results could be widely applied in such fields as remote sensing, surveying and mapping, geographic information system, and image processing, etc.

摄影测量涉及大量的与三角函数、四元数有关的级数展开、空间坐标变换、非线性方程解算等一系列繁琐的数学分析过程，人工推导不但费时费力，而且容易出错，有时由于难以忍受的复杂性等各种原因，甚至根本无法实现。传统算法主要存在以下问题：（1）表达式复杂冗长，计算过程繁琐，计算费时；（2）包含一定的近似处理，影响了计算精度。本项目以摄影测量中的各种数学分析过程为研究对象，借助现代计算机代数系统强大的数学分析功能和符号运算能力，系统研究摄影测量基本定位理论、空中三角测量、航天摄影测量数学分析问题的计算机代数精密解法，通过具有严格解析意义的数学分析，推导和建立理论上更为严密、形式上更为简单、精度上更为精确、符号化的摄影测量新算法和数学模型，实现摄影测量特定数学分析问题的突破和创新，进一步革新和丰富摄影测量的理论体系。研究成果可广泛应用于与摄影测量学相关的遥感、测绘、地理信息系统、图像处理等领域。

本项目主要针对摄影测量数学分析问题的计算机代数精密解法展开深入和系统的研究，通过理论推导、仿真分析和试验验证等方法，建立了摄影测量相关问题的计算机代数精密算法，并验证了相关算法的有效性和稳定性。借助罗德里格斯矩阵对旋转矩阵进行描述，利用计算机代数系统推导出基于共线方程线性化的空间后方交会算法。借助“地面点到摄站和像点决定的空间直线距离为0”这一共线描述，推导出适用于大旋转角的单像空间后方交会算法。引入稳健估计理论和借助选权迭代法，提出了基于总体最小二乘的多像空间前方交会模型。以空间距离、像点坐标作为观测量，推导出适用于解算任意焦距立体像对的相对定向元素公式。根据旋转矩阵的正交特性，利用罗德里格斯矩阵推导出更适用的无需初始值，无需线性化，无需迭代的7参数坐标变换公式。根据空间距离描述共线条件，利用计算机代数系统，光束法平差空间距离表示形式。按照研究计划，深入开展了摄影测量中单像空间后方交会、多像空间前方交会、相对定向与绝对定向、光束法平差模型的计算机代数分析等研究，取得了一些理论创新成果。