DCReg: Decoupled Characterization for Efficient Degenerate LiDAR
  Registration
Authors: name: Xiangcheng Hu, name: Xieyuanli Chen, name: Mingkai Jia, name: Jin Wu, name: Ping Tan, name: Steven L. Waslander

LiDAR point cloud registration is fundamental to robotic perception and
navigation. However, in geometrically degenerate or narrow environments,
registration problems become ill-conditioned, leading to unstable solutions and
degraded accuracy. While existing approaches attempt to handle these issues,
they fail to address the core challenge: accurately detection, interpret, and
resolve this ill-conditioning, leading to missed detections or corrupted
solutions. In this study, we introduce DCReg, a principled framework that
systematically addresses the ill-conditioned registration problems through
three integrated innovations. First, DCReg achieves reliable ill-conditioning
detection by employing a Schur complement decomposition to the hessian matrix.
This technique decouples the registration problem into clean rotational and
translational subspaces, eliminating coupling effects that mask degeneracy
patterns in conventional analyses. Second, within these cleanly subspaces, we
develop quantitative characterization techniques that establish explicit
mappings between mathematical eigenspaces and physical motion directions,
providing actionable insights about which specific motions lack constraints.
Finally, leveraging this clean subspace, we design a targeted mitigation
strategy: a novel preconditioner that selectively stabilizes only the
identified ill-conditioned directions while preserving all well-constrained
information in observable space. This enables efficient and robust optimization
via the Preconditioned Conjugate Gradient method with a single physical
interpretable parameter. Extensive experiments demonstrate DCReg achieves at
least 20% - 50% improvement in localization accuracy and 5-100 times speedup
over state-of-the-art methods across diverse environments. Our implementation
will be available at https://github.com/JokerJohn/DCReg.