Exact simultaneous recovery of locations and structure from known orientations and corrupted point correspondences

TL;DR

This paper proves that the ShapeFit convex algorithm can exactly recover locations and structure in bipartite location problems with corrupted data, under certain probabilistic and deterministic conditions, and applies it to Structure from Motion.

Contribution

The paper provides a rigorous proof of exact recovery guarantees for ShapeFit in bipartite location problems with corrupted observations, extending its applicability to Structure from Motion.

Findings

ShapeFit recovers locations exactly with high probability under specified conditions.

The method tolerates a constant fraction of adversarial corruptions.

Application to Structure from Motion improves robustness and accuracy.

Abstract

Let $t_1,\ldots,t_{n_l} \in \mathbb{R}^d$ and $p_1,\ldots,p_{n_s} \in \mathbb{R}^d$ and consider the bipartite location recovery problem: given a subset of pairwise direction observations $\{(t_i - p_j) / \|t_i - p_j\|_2\}_{i,j \in [n_l] \times [n_s]}$, where a constant fraction of these observations are arbitrarily corrupted, find $\{t_i\}_{i \in [n_ll]}$ and $\{p_j\}_{j \in [n_s]}$ up to a global translation and scale. We study the recently introduced ShapeFit algorithm as a method for solving this bipartite location recovery problem. In this case, ShapeFit consists of a simple convex program over $d(n_l + n_s)$ real variables. We prove that this program recovers a set of $n_l+n_s$ i.i.d. Gaussian locations exactly and with high probability if the observations are given by a bipartite Erd\H{o}s-R\'{e}nyi graph, $d$ is large enough, and provided that at most a constant fraction of observations involving any particular location are adversarially corrupted. This recovery theorem is based on a set of deterministic conditions that we prove are sufficient for exact recovery. Finally, we propose a modified pipeline for the Structure for Motion problem, based on this bipartite location recovery problem.