Essential Matrix Estimation using Convex Relaxations in Orthogonal Space

Arman Karimian, Roberto Tron; Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), 2023, pp. 17142-17152

We introduce a novel method to estimate the essential matrix for two-view Structure from Motion (SfM). We show that every 3 by 3 essential matrix can be embedded in a 4 by 4 rotation, having its bottom right entry fixed to zero; we call the latter the quintessential matrix. This embedding leads to rich relations with the space of 4-D rotations, quaternions, and the classical twisted-pair ambiguity in two-view SfM. We use this structure to derive a succession of semidefinite relaxations that require fewer parameters than the existing non-minimal solvers and yield faster convergence with certifiable optimality. We then exploit the low-rank geometry of these relaxations to reduce them to an equivalent optimization on a Riemannian manifold and solve them via the Riemannian Staircase method. The experimental evaluation confirms that our algorithm always finds the globally optimal solution and outperforms the existing non-minimal methods. We make our implementations open source.