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

2026-08-16

AI Summary

This study addresses the inconsistency of LiDAR registration degeneracy detection labels caused by coordinate frame variations. We propose a generalized eigenvalue criterion based on an equivariant point displacement metric to resolve this issue. By employing adjoint reparameterization and generalized eigenvalue decomposition, we establish a novel frame-independent and scale-invariant paradigm for degeneracy determination that fundamentally eliminates reliance on specific reference frames. Experimental results demonstrate that the proposed criterion enables robust cross-sequence threshold transferability. Furthermore, our analysis reveals that correction magnitudes in 44.5%–69.5% of frame pairs are significantly affected by coordinate system choices, thereby validating the effectiveness and robustness of our approach in overcoming traditional frame-dependent limitations.

Abstract

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.

Problem

• LiDAR registration
• degeneracy detection
• frame dependence
• equivariance
• coordinate invariance

Innovation

• Equivariant Degeneracy Detection
• Frame Independence
• Generalized Eigenvalue Criterion
• Point-Displacement Metric
• LiDAR Registration