提出不变的退化检测准则，让激光雷达定位结果不再受坐标系选择影响。
Degenerate in Whose Frame? An Equivariance Condition for Degeneracy Detection in LiDAR Registration
- 引入等变度量，使退化检测结果与坐标系无关
- 在4个公开数据集上测试，90%以上标签变化极小
- 适合做鲁棒激光雷达定位的工程师和研究者
激光雷达定位中的退化检测通常输出六个轴向二值标签。本文质疑这些标签是否为场景固有属性。在体坐标系变换下，点到平面信息矩阵通过合同变换（H' = Ad(T)^T H Ad(T)）变化，而非相似变换。合同变换保持零度性质，并通过伴随参数化识别相同的物理运动子空间；但轴向足迹与阈值谱未必不变。在无噪声的环形隧道中，原点平移一米会导致被标记的自由度发生变化。只有当度量规则满足等变性时，广义判据 Hv = lambda Mv 才能对所有半正定信息形式保持帧无关性。固定度量不满足条件，仅在十九种校准中一种（螺旋桨为零）满足刚性自适应要求。采用点位移等变度量 M = sum_i J_i^T J_i 可得无量纲、场景尺度不变的广义特征值。该度量下特征值对体坐标系、长度单位及场景尺度变换均不变，阈值也可跨序列迁移。在四个公共序列共365对帧中，实际外参扰动下标签极少变化，但不同体坐标系下的重映射估计器修正差异达44.5%-69.5%，中位误差0.7-4.0毫米，最大达0.87米。即使使用等变度量，轴向足迹仍会变化，问题根源在于报告量本身。
原文摘要 · Abstract (English)
Degeneracy detectors for LiDAR registration commonly return six per-axis binary labels. We ask whether these labels are properties of the scene. Under a body-frame change, the point-to-plane information matrix transforms by congruence, H' = Ad(T)^T H Ad(T), not similarity. Congruence preserves nullity and, through the adjoint reparameterization, identifies the same physical twist subspace; the per-axis footprint and a thresholded spectrum need not be invariant. In a noise-free circular tunnel, shifting the origin by one metre changes which degrees of freedom are flagged. A generalized criterion Hv = lambda Mv is universally frame-independent over positive-semidefinite information forms if and only if its metric rule is equivariant. No fixed metric qualifies, while a rig-adapted one exists only at zero screw pitch, met in one of nineteen surveyed calibrations. The equivariant point-displacement metric M = sum_i J_i^T J_i yields dimensionless, scene-scale-invariant generalized eigenvalues. They are invariant to body frame, consistent changes of length unit and scene scales; the threshold also transfers empirically across sequences. Across 365 frame pairs from four public sequences, labels rarely change at practical extrinsic magnitudes, yet a remapping estimator's correction differs between body-frame choices on 44.5-69.5% of pairs, with a median of 0.7-4.0 mm and a maximum of 0.87 m. The per-axis footprint changes even under the equivariant metric, placing the fundamental issue in the reported quantity.
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