Algorithms for the Solution of the Nonlinear Least-Squares Problem

1978 ◽  
Vol 15 (5) ◽  
pp. 977-992 ◽  
Author(s):  
Philip E. Gill ◽  
Walter Murray
Keyword(s):  
Least Squares ◽  
Nonlinear Least Squares ◽  
Least Squares Problem ◽  
Nonlinear Least Squares Problem

scholarly journals A Method for Solving LiDAR Waveform Decomposition Parameters Based on a Variable Projection Algorithm

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Ke Wang ◽  
Guolin Liu ◽  
Qiuxiang Tao ◽  
Luyao Wang ◽  
Yang Chen
Keyword(s):  
Parameter Estimation ◽  
Least Squares ◽  
Gaussian Function ◽  
Nonlinear Least Squares ◽  
Projection Algorithm ◽  
Nonlinear Parameters ◽  
Least Squares Problem ◽  
Variable Projection ◽  
Nonlinear Least Squares Problem ◽  
Lidar Waveform

Light detection and ranging (LiDAR) is commonly used to create high-resolution maps; however, the efficiency and convergence of parameter estimation are difficult. To address this issue, we evaluated the structural characteristics of received LiDAR signals by decomposing them into Gaussian functions and applied the variable projection algorithm of the separable nonlinear least-squares problem to the process of waveform fitting. First, using a variable projection algorithm, we separated the linear (amplitude) and nonlinear (center position and width) parameters in the Gaussian function model; the linear parameters are expressed with nonlinear parameters by the function. Thereafter, the optimal estimation of the characteristic parameters of the Gaussian function components was transformed into a least-squares problem only comprising nonlinear parameters. Finally, the Levenberg–Marquardt algorithm was used to solve these nonlinear parameters, whereas the linear parameters were calculated simultaneously in each iteration, and the estimation results satisfying the nonlinear least-square criterion were obtained. Five groups of waveform decomposition simulation data and ICESat/GLAS satellite LiDAR waveform data were used for the parameter estimation experiments. During the experiments, for the same accuracy, the separable nonlinear least-squares optimization method required fewer iterations and lesser calculation time than the traditional method of not separating parameters; the maximum number of iterations was reached before the traditional method converged to the optimal estimate. The method of separating variables only required 14 iterations to obtain the optimal estimate, reducing the computational time from 1128 s to 130 s. Therefore, the application of the separable nonlinear least-squares problem can improve the calculation efficiency and convergence speed of the parameter solution process. It can also provide a new method for parameter estimation in the Gaussian model for LiDAR waveform decomposition.

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