On the Generalized Essential Matrix Correction: An Efficient Solution to the Problem and Its Applications
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On the Generalized Essential Matrix Correction: An Efficient Solution to the Problem and Its Applications Pedro Miraldo1,2
· João R. Cardoso3
Received: 4 August 2018 / Accepted: 29 April 2020 © The Author(s) 2020
Abstract This paper addresses the problem of finding the closest generalized essential matrix from a given 6 × 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. Keywords Generalized essential matrix · General camera models · Pose estimation · Steepest descent type · Orthogonal constraints
1 Introduction The epipolar constraint is one of the fundamental geometry constraints in computer vision. It relates the rigid transformation between two cameras with different external parameters [13,21] and correspondences between points in the two images. It is one of the most common tools for scene reconstruction, known as passive techniques, i.e., two cameras looking at the same scene from different points of view. The epipolar constraint has been used in many other applications, such as visual odometry [34]. For many years, authors focused on perspective cameras to build this stereo pair [13]; see Fig. 1a. However, these
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Pedro Miraldo [email protected] João R. Cardoso [email protected]
1
LARSyS, Institute for Systems and Robotics (ISR/IST), Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal
2
Division of Decision and Control Systems, KTH Royal Institute of Technology, Stockholm, Sweden
3
Coimbra Polytechnic Institute (ISEC) and Center for Mathematics, University of Coimbra, Coimbra, Portugal
cameras have, among several disadvantages, a limited field of view. To overcome this, some authors have developed new camera systems. Special emphasis has been given to omnidirectional cameras, which get a larger field of view from a combination of perspective cameras with mirrors and/or fisheye lenses [23,31,32], or multi-perspective camera systems [15,17]. Most of these devices are non-central (see [42]). Other types of imaging sensors have been proposed. Nevertheless, the perspective camera model cannot model most of them due to their physica
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