Inlier Recovery for Robust Registration via Gram-Matrix Overlap

TL;DR

This paper introduces two novel methods for inlier recovery in robust point-set registration by leveraging Gram-matrix overlap, enabling effective identification of inliers even with high noise and outliers.

Contribution

It proposes eigenvector and row-sum based algorithms that improve inlier recovery, especially in high-dimensional and low-inlier-fraction regimes, without direct optimization over transformations.

Findings

Eigenvector method achieves weak recovery when dimension and sample size are comparable.

Row-sum method achieves exact recovery over a broader range of dimensions.

Exact recovery is possible with as few as order √n inliers when dimension is large.

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.