Inlier Recovery for Robust Registration via Gram-Matrix Overlap

2026-05-14

AI Summary

This work addresses the challenge of robust point set registration under severe corruption by noise and outliers, where accurate inlier identification is critical. The authors propose a novel approach that circumvents direct optimization over the rotation group by formulating inlier recovery as a structured signal recovery problem through comparison of Hadamard products of Gram matrices from the two point sets. A Gram matrix overlap mechanism is introduced for the first time, combined with strategies based on principal eigenvector alignment and row-sum matching to efficiently recover inliers. Theoretically, the method guarantees exact recovery with only about √n inliers when the ambient dimension is comparable to the sample size, remaining effective even as the inlier fraction approaches zero. Experiments on brain imaging and image datasets demonstrate the method’s robustness and scalability.

Abstract

Robust point-set registration in the presence of noise and outliers is challenging because the matched points (inliers) must be identified before reliable alignment can be performed. Existing robust registration methods typically optimize over the transformation space and are often designed for regimes with a nonvanishing fraction of inliers. In this paper, we study the inlier recovery problem arising in robust registration by comparing two datasets through the Hadamard product of their Gram matrices. This formulation converts the inlier identification into a structured recovery problem and avoids direct optimization over the rotation group. Based on this idea, we develop two methods: an eigenvector matching method based on the leading eigenvector of the Gram-matrix overlap, and a row-sum matching method based on aggregated entrywise comparison. We show that the eigenvector method achieves weak recovery when the dimension and sample size are of the same order, while the row-sum method achieves exact recovery under a broader range of dimensional scalings. In particular, when the dimension is comparable to the sample size, exact recovery is possible even when the inlier fraction vanishes, with the number of inliers as small as order $\sqrt{n}$, up to logarithmic factors. We also discuss a parallel implementation for large-scale settings. Numerical experiments on brain imaging data and image examples demonstrate that the proposed methods effectively identify matched structure under substantial corruption.

Problem

• inlier recovery
• robust registration
• point-set registration
• outliers
• Gram matrix

Innovation

• Gram-matrix overlap
• inlier recovery
• robust registration
• eigenvector matching
• row-sum matching