基于Moore-Penrose广义逆及立体矩阵的可分离非线性最小二乘解算方法

针对测绘领域中函数模型为非线性函数的线性组合的特殊结构，本文提出了基于Moore-Penrose广义逆和立体矩阵的可分离非线性最小二乘解算方法。该方法首先利用变量投影算法消除可分离非线性模型中的线性参数，将包含两类参数的原非线性优化问题转化为仅含有非线性参数的最小二乘问题。然后，基于Moore-Penrose广义逆矩阵的微分和立体矩阵理论计算最小二乘目标函数的一阶导数，进而采用非线性优化的LM方法求解非线性参数的最优估值。最后，根据最小二乘方法求解线性参数的最优估值。通过指数函数模型拟合和机载LiDAR全波形参数求解试验与传统参数不分离优化方法进行对比，结果表明，基于Moore-Penrose广义逆和立体矩阵的可分离非线性最小二乘解算方法对待求参数初值依赖性低，同时避免了迭代过程中线性参数导致的病态问题，算法稳定性好，为测绘领域中可分离非线性最小二乘问题的解算提供了一种思路，也拓展了可分离非线性最小二乘方法的应用。

A separable nonlinear least squares solution method based on Moore-Penrose generalized inverse and solid matrix

A separable nonlinear least squares algorithm based on Moore-Penrose generalized inverse and solid matrix is proposed to solve the special structure of linear combination of nonlinear functions in the field of surveying and mapping. Firstly, the variable projection algorithm is used to eliminate the linear parameters in the separable nonlinear model, and the original nonlinear optimization problem with two kinds of parameters is transformed into the least squares problem with only nonlinear parameters. Then, the first-order partial derivative of the least squares objective function is calculated based on the theory of differentiation of Moore-Penrose inverse matrix and solid matrix. Then the LM method of nonlinear optimization is used to solve the optimal estimation of nonlinear parameters. Finally, the optimal solution of linear parameters is obtained by linear least square method. The exponential model fitting experiment and airborne LiDAR full-waveform parameter solving experiment are used to compare the proposed method with the traditional optimization method without separation of parameters. The results show that the separable nonlinear least squares solution method based on the Moore-Penrose generalized inverse and the solid matrix is less dependent on the initial value of the parameter, avoids the ill-conditioned problem caused by the linear parameter in the iterative process, and the algorithm is robust. It provides a new idea for solving the separable nonlinear least squares problem in the field of surveying and mapping, and also expands the application of separable nonlinear least squares method.