R-LM optimization algorithm for parametric fitting of aspheric surfaces

Abstract

The high-accuracy parametric fitting of aspheric optical elements is a fundamental challenge in precision optical manufacturing and metrology. Conventional fitting methods often suffer from limited robustness in the presence of outlier contamination and insufficient accuracy in regions characterized by complex surface shapes. To overcome these limitations, we propose an improved hybrid fitting framework that integrates robust random sample consensus (RANSAC) initialization with the Levenberg–Marquardt (LM) optimization algorithm, referred to as the R-LM method. The proposed algorithm introduces two principal enhancements. First, in the RANSAC initialization phase, an improved inlier discrimination criterion that combines aspheric geometric features and machining-oriented physical constraints is introduced. The preliminary inlier set obtained via iterative sampling is further refined through a secondary screening mechanism, thereby enhancing the reliability of the initial model and strengthening the resistance to outliers. Second, in the LM nonlinear optimization phase, an adaptive damping strategy guided by surface-region partitioning is developed. The damping factor is dynamically regulated by evaluating the local condition number of different surface regions, which effectively balances the convergence rate and numerical stability during optimization. The performance of the proposed algorithm was validated using a systematic simulation. The proposed method outperforms comparable algorithms, yielding a more uniform residual distribution and better representing global and local surface-form features. Overall, the proposed method improves the stability and accuracy of aspheric-profile parameter fitting under the tested conditions.