On the Generalized Essential Matrix Correction: An efficient solution to the problem and its applications

TL;DR

This paper introduces an efficient steepest descent algorithm for finding the closest generalized essential matrix, significantly reducing computational effort compared to traditional constrained optimization methods, with applications in pose estimation problems.

Contribution

It develops a novel unconstrained reformulation and an efficient steepest descent algorithm for the generalized essential matrix correction problem, which was previously unaddressed in literature.

Findings

The proposed method is faster than existing optimization techniques.

It effectively solves pose problems with synthetic and real data.

The approach simplifies the constrained optimization to an orthogonal constraint problem.

Abstract

This paper addresses the problem of finding the closest generalized essential matrix from a given $6\times 6$ matrix, with respect to the Frobenius norm. To the best of our knowledge, this nonlinear constrained optimization problem has not been addressed in the literature yet. Although it can be solved directly, it involves a large number of constraints, and any optimization method to solve it would require much computational effort. We start by deriving a couple of unconstrained formulations of the problem. After that, we convert the original problem into a new one, involving only orthogonal constraints, and propose an efficient algorithm of steepest descent-type to find its solution. To test the algorithms, we evaluate the methods with synthetic data and conclude that the proposed steepest descent-type approach is much faster than the direct application of general optimization techniques to the original formulation with 33 constraints and to the unconstrained ones. To further motivate the relevance of our method, we apply it in two pose problems (relative and absolute) using synthetic and real data.