Preconditioned Landweber iteration for nonlinear least-squares conical surface fitting

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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.

Keywords

preconditioned Landweber iteration / nonlinear least-squares optimization / conical surface fitting / spectral preconditioning / convergence analysis

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