ARCS: Accurate Rotation and Correspondence Search

TL;DR

ARCS introduces a fast, robust algorithm for simultaneous rotation and correspondence search in large 3D point sets, outperforming existing methods in speed and accuracy.

Contribution

The paper presents ARCS, a novel solver for the generalized Wahba problem that is efficient, scalable, and robust to noise, with an extension ARCS+ for joint rotation and correspondence estimation.

Findings

ARCS solves large-scale problems in about 0.1 seconds.

ARCS+ achieves state-of-the-art accuracy on datasets with over 10^6 points.

The methods outperform existing approaches by a factor of 10^4 in speed.

Abstract

This paper is about the old Wahba problem in its more general form, which we call "simultaneous rotation and correspondence search". In this generalization we need to find a rotation that best aligns two partially overlapping $3$D point sets, of sizes $m$ and $n$ respectively with $m\geq n$. We first propose a solver, $\texttt{ARCS}$, that i) assumes noiseless point sets in general position, ii) requires only $2$ inliers, iii) uses $O(m\log m)$ time and $O(m)$ space, and iv) can successfully solve the problem even with, e.g., $m,n\approx 10^6$ in about $0.1$ seconds. We next robustify $\texttt{ARCS}$ to noise, for which we approximately solve consensus maximization problems using ideas from robust subspace learning and interval stabbing. Thirdly, we refine the approximately found consensus set by a Riemannian subgradient descent approach over the space of unit quaternions, which we show converges globally to an $\varepsilon$-stationary point in $O(\varepsilon^{-4})$ iterations, or locally to the ground-truth at a linear rate in the absence of noise. We combine these algorithms into $\texttt{ARCS+}$, to simultaneously search for rotations and correspondences. Experiments show that $\texttt{ARCS+}$ achieves state-of-the-art performance on large-scale datasets with more than $10^6$ points with a $10^4$ time-speedup over alternative methods. \url{https://github.com/liangzu/ARCS}