Fast, Robust, Permutation-and-Sign Invariant SO(3) Pattern Alignment

TL;DR

<3-5 sentence high-level summary> This work introduces a correspondence-free approach to aligning two sets of rotations on $ ext{SO}(3)$ by decomposing each rotation into Transformed Basis Vectors (TBVs) and aligning the resulting $S^2$ point patterns per axis. It then fuses axis-wise estimates into a single rotation, using a Permutation-and-Sign Invariant (PASI) wrapper to resolve axis relabeling and sign flips, while preserving linear-time complexity. The method leverages fast spherical matchers (SPMC, FRS, and a hybrid) and demonstrates strong performance on axis-consistent and axis-ambiguous scenarios, with substantial speedups over traditional cross-correlation methods and robustness to very high outlier ratios. The proposed PASI framework is practical for initialization in RWHE/RHE pipelines, IMU calibration, and SLAM setups, and opens pathways for certifiable extensions and SE(3) integration."

Abstract

We address the correspondence-free alignment of two rotation sets on \(SO(3)\), a core task in calibration and registration that is often impeded by missing time alignment, outliers, and unknown axis conventions. Our key idea is to decompose each rotation into its \emph{Transformed Basis Vectors} (TBVs)-three unit vectors on \(S^2\)-and align the resulting spherical point sets per axis using fast, robust matchers (SPMC, FRS, and a hybrid). To handle axis relabels and sign flips, we introduce a \emph{Permutation-and-Sign Invariant} (PASI) wrapper that enumerates the 24 proper signed permutations, scores them via summed correlations, and fuses the per-axis estimates into a single rotation by projection/Karcher mean. The overall complexity remains linear in the number of rotations (\(\mathcal{O}(n)\)), contrasting with \(\mathcal{O}(N_r^3\log N_r)\) for spherical/\(SO(3)\) correlation. Experiments on EuRoC Machine Hall simulations (axis-consistent) and the ETH Hand-Eye benchmark (\texttt{robot\_arm\_real}) (axis-ambiguous) show that our methods are accurate, 6-60x faster than traditional methods, and robust under extreme outlier ratios (up to 90\%), all without correspondence search.

Figures (9)

• Figure 1: The figure shows how to align two sets of ($SO_3$) distribution using SO3_SPMC. The major steps of the SO3_SPMC alignment algorithm in axis consistent setting are depicted through subfigures (a-h). Sub-figures show the progression from one step to another. (a) Two sets of orientations ($SO_3$) are presented in two separate blocks. (b) S2-distributions of two sets and corresponding mean directions are presented. The darker colors represent the $S2^{A}_{x}, S2^{A}_{y}, S2^{A}_{z}$, while the lighter colors correspond to $S2^{B}_{x}, S2^{B}_{y}, S2^{B}_{z}$. (c) In the three-sphere visualization, pairs of $(S2^{AN}_{x},S2^{BN}_{x})$, $(S2^{AN}_{y},S2^{BN}_{y})$, and $(S2^{AN}_{z},S2^{BN}_{z})$ are depicted. (d) The projected S2-distributions are visualized using a 2D histogram. The color legends in the histogram maintain the same notation as mentioned earlier, where darker colors represent the relevant 2D histogram of set $\mathcal{A}$, while lighter colors correspond to set $\mathcal{B}$. (e) After performing the 1D cross-correlation, the respective 2D histograms are aligned. (f) The 2D histogram data, represented in geographic coordinates, is converted to 3D Cartesian coordinates. (g) All S2-distributions are rotated by corresponding inverse of the $R^{A}_{xN}, R^{A}_{zN}, R^{A}_{zN}$. (h) Block diagram illustrates the process of extracting an set of $SO_3$ from the aligned $S2_x$, $S2_y$, and $S2_z$-distributions of set $\mathcal{B}$.
• Figure 2: $S^2$ representation of a normal vector and $S^2$ representation of the sets of Transformed Basis Vectors (TBVs). (a) In $S^2$ representation, we view an arbitrary normal vector $\overrightarrow{OP}$ as a point $P$ on the surface of the unit sphere; $P$ is the S2-point of $\overrightarrow{OP}$. Black arrows indicate the reference-frame directions (equivalently, the TBVs of the identity rotation). The red ($x$), green ($y$), and blue ($z$) dots are the S2-points of the identity rotation. (b) The $S2_x$, $S2_y$, $S2_z$ distributions are colored red, green, and blue, respectively; their mean directions are shown as arrows.
• Figure 3: RWHE on $\mathrm{SO}(3)$ via TBVs and PASI.(a) Transformed Basis Vectors (TBVs; $S^2$ points) of the end-effector ($\mathcal{A}$) and camera ($\mathcal{B}$) orientation trajectories from the ETH Hand-Eye Calibration dataset (robot_arm_real split) furrer2017evaluation. The sets exhibit an axis-ambiguous relation: the best signed-permutation is $L^\star=\bigl[0-10100001\bigr]$, i.e., $Ax\!\to\!-By$, $Ay\!\to\!+Bx$, $Az\!\to\!+Bz$. (b) Our PASI variant enumerates proper signed permutations $L=PS$ ($\det(L)=+1$), selects $L^\star$ by maximizing the summed per-axis spherical correlation, then estimates a continuous rotation $\bar{R}$ with a spherical matcher (SPMC/FRS/hybrid), yielding the two-sided alignment $\mathcal{A}\approx L^\star\,\mathcal{B}\,\bar{R}^\top$.
• Figure 4: (a) TBV $S^2$ distributions for the target set $A_5$ (EuRoC Machine Hall) and the seven source variants $B_1$-$B_7$ after applying rotation $R_4$. (b) Representative alignment produced by SO3_SPMC_FRS.
• Figure 5: Axis-Consistent Scenario: Quantitative results on simulated $SO(3)$ data derived from the EuRoC Machine Hall sequences. For each target set, the seven source configurations $B1\!\rightarrow\!B7$ (increasing outlier ratio) are shown as box plots; the median of each box is annotated under the plot.