Degenerate in Whose Frame? An Equivariance Condition for Degeneracy Detection in LiDAR Registration

LiDAR point cloud registration is fundamental to robotic perception and navigation. In geometrically degenerate environments (e.g., corridors), registration becomes ill-conditioned: certain motion directions are weakly constrained, causing unstable solutions and degraded accuracy. Existing detect-then-mitigate methods fail to reliably detect, physically interpret, and stabilize this ill-conditioning without corrupting the optimization. We introduce DCReg (Decoupled Characterization for Ill-conditioned Registration), establishing a detect-characterize-mitigate paradigm that systematically addresses ill-conditioned registration via three innovations. First, DCReg achieves reliable ill-conditioning detection by employing Schur complement decomposition on the Hessian matrix. This decouples the 6-DoF registration into 3-DoF clean rotational and translational subspaces, eliminating coupling effects that mask degeneracy in full-Hessian analyses. Second, within these subspaces, we develop interpretable characterization techniques resolving eigen-basis ambiguities via basis alignment. This establishes stable mappings between eigenspaces and physical motion directions, providing actionable insights on which motions lack constraints and to what extent. Third, leveraging this spectral information, we design a targeted mitigation via a structured preconditioner. Guided by MAP regularization, we implement eigenvalue clamping exclusively within the preconditioner rather than modifying the original problem. This preserves the least-squares objective and minimizer, enabling efficient optimization via Preconditioned Conjugate Gradient with a single interpretable parameter. Experiments demonstrate DCReg achieves 20-50% higher long-duration localization accuracy and 5-30x speedups (up to 116x) over degeneracy-aware baselines across diverse environments. Code: https://github.com/JokerJohn/DCReg

In this paper, we study 3D LiDAR scan alignment in challenging scenarios with degeneracies, such as straight corridors or flat fields, where the alignment solution is not unique and compromises localization and mapping accuracy. Existing degeneracy detection methods that neglect the potential for reassociating data points are prone to being sensitive to noise and complex degeneracies. Therefore, we propose LiLi - a novel method that leverages Lie theory to identify the full set of degenerate transformations within the SE(3) Lie group of rigid transformations. The method employs perturbations of the optimized solution and compares the resulting optimized poses to ensure robust detection of degeneracies. By leveraging generators from the Lie algebra se(3), the method provides a systematic approach to describing the set of degenerate transformations. Quantitative evaluations on synthetic data show significant improvement over the state-of-the-art Hessian-based method, reducing alignment error by 50%, with more significant improvements for datasets featuring noise. In the real-world degenerate datasets, the proposed method integrated into LiDAR-based odometry yields superior localization performance compared to the reference solution based on the Hessian-based degeneracy detector on a 260 m long trajectory, and succeeds on a 430 m long round-trip tunnel trajectory where the reference fails.

Scan-to-map LiDAR odometry drifts unboundedly along the unobservable axes of geometrically degenerate environments like tunnels and corridors, and existing degeneracy handling requires environment-specific parameter tuning. This paper presents a parameter-free approach. We show that in voxelized GICP the Gauss--Newton (GN) Hessian masks translational degeneracy, because covariance regularization keeps the translation block artificially well-conditioned. We bypass this with a regularization-free voxel-normal localizability field and two of its statistics: a normalized fraction $f_0$ detecting directional anisotropy, and an absolute per-voxel mass $λ_0$ distinguishing information absence (tunnels) from dilution (dense open scenes). A temporal-median gate combines both to trigger Fisher-information correspondence weighting. Calibrated once by fixed rules on two short sequences and then frozen, LF-GICP achieves the lowest KITTI relative translation error ($0.865\%$) under an identical evaluation protocol against re-run baselines, outperforms them on GEODE tunnels and MulRan, leads the HeLiPR mean, and generalizes across four sensor types without re-tuning. We further demonstrate empirically that straight, uniform tunnels remain unobservable along their axis for LiDAR-only registration.