OCE: an optimization convergence metric for LiDAR SLAM degeneracy detection

LiDAR-based Simultaneous Localization and Mapping (SLAM) systems suffer from environment-induced degeneracy in geometrically self-similar scenes such as tunnels and cylindrical structures, where the point cloud registration problem becomes ill-conditioned. Existing degeneracy detection methods predominantly rely on eigenvalue analysis of the Hessian matrix, which assesses geometric conditioning at a fixed linearization point. However, in semi-degenerate environments containing partial but sufficient geometric constraints, these methods tend to produce false positive detections because measurement noise inflates eigenvalues and obscures the contribution of minor structural features. In this paper, we propose the Optimization Convergence Efficiency (OCE) metric, which evaluates the relative contraction of the directional optimizer gradient across Levenberg-Marquardt evaluations and directly characterizes the convergence response of LiDAR registration along the monitored state direction. The proposed method is evaluated through controlled simulation with ground-truth-based quantitative assessment and real-world UAV experiments in a wind-turbine tower and a long tunnel. In the nominal simulation, OCE achieves 98.21% recall with a 2.33% false-positive rate, compared with 100% recall and a 49.42% false-positive rate for the conventional eigenvalue-ratio detector. Controlled measurement-noise experiments further demonstrate that OCE maintains substantially lower false-alarm rates as spectral discrimination becomes less reliable. These results show that OCE can retain usable LiDAR pose estimates in semi-degenerate environments while reliably identifying actual registration failure.