Preconditioned Landweber iteration for nonlinear least-squares conical surface fitting

Shangzuo Xie; Gangrong Qu; Liyi Feng; Shangzuo Xie; Gangrong Qu; Liyi Feng

doi:10.3934/era.2025290

Electronic Research Archive

2025, Volume 33, Issue 11: 6558-6576. doi: 10.3934/era.2025290

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Preconditioned Landweber iteration for nonlinear least-squares conical surface fitting

School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China

Received: 20 August 2025 Revised: 09 October 2025 Accepted: 28 October 2025 Published: 04 November 2025

To address the slow convergence and sensitivity to initial values of the traditional Levenberg-Marquardt (LM) algorithm in 3D point cloud conical surface fitting-issues caused by ill-conditioned Hessian matrices—this paper developed a preconditioned Landweber iterative framework for nonlinear least-squares optimization. A residual model based on point-to-cone geometric distances was constructed, the analytical Jacobian was derived, and a spectral preconditioning strategy was introduced. Theoretical analysis showed that the preconditioner effectively reduced the condition number of the normal matrix, thereby improving numerical stability and accelerating practical convergence. A relaxation-factor rule was further derived to balance stability and efficiency. Experiments on synthetic and industrial datasets demonstrated that, compared with the Levenberg-Marquard(LM) algorithm, the proposed method reduced iteration counts by 51.9%, decreased root mean square error (RMSE) by 98%, and shortened computation time by 73%. It also maintained robust performance under high-noise and non-uniform sampling conditions, providing an efficient and reliable tool for accurate conical surface fitting in industrial inspection and reverse engineering.
- preconditioned Landweber iteration,
- nonlinear least-squares optimization,
- conical surface fitting,
- spectral preconditioning,
- convergence analysis
Citation: Shangzuo Xie, Gangrong Qu, Liyi Feng. Preconditioned Landweber iteration for nonlinear least-squares conical surface fitting[J]. Electronic Research Archive, 2025, 33(11): 6558-6576. doi: 10.3934/era.2025290

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Abstract

To address the slow convergence and sensitivity to initial values of the traditional Levenberg-Marquardt (LM) algorithm in 3D point cloud conical surface fitting-issues caused by ill-conditioned Hessian matrices—this paper developed a preconditioned Landweber iterative framework for nonlinear least-squares optimization. A residual model based on point-to-cone geometric distances was constructed, the analytical Jacobian was derived, and a spectral preconditioning strategy was introduced. Theoretical analysis showed that the preconditioner effectively reduced the condition number of the normal matrix, thereby improving numerical stability and accelerating practical convergence. A relaxation-factor rule was further derived to balance stability and efficiency. Experiments on synthetic and industrial datasets demonstrated that, compared with the Levenberg-Marquard(LM) algorithm, the proposed method reduced iteration counts by 51.9%, decreased root mean square error (RMSE) by 98%, and shortened computation time by 73%. It also maintained robust performance under high-noise and non-uniform sampling conditions, providing an efficient and reliable tool for accurate conical surface fitting in industrial inspection and reverse engineering.

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