GPSfM: Global Projective SFM Using Algebraic Constraints on Multi-View Fundamental Matrices

Abstract

This paper addresses the problem of recovering projective camera matrices from collections of fundamental matrices in multiview settings. We make two main contributions. First, given n 2 fundamental matrices computed for n images, we provide a complete algebraic characterization in the form of conditions that are both necessary and sufficient to enabling the recovery of camera matrices. These conditions are based on arranging the fundamental matrices as blocks in a single matrix, called the n-view fundamental matrix, and characterizing this matrix in terms of the signs of its eigenvalues and rank structures. Secondly, we propose a concrete algorithm for projective structure-frommotion that utilizes this characterization. Given a complete or partial collection of measured fundamental matrices, our method seeks camera matrices that minimize a global algebraic error for the measured fundamental matrices. In contrast to existing methods, our optimization, without any initialization, produces a consistent set of fundamental matrices that corresponds to a unique set of cameras (up to a choice of projective frame). Our experiments indicate that our method achieves state of the art performance in both accuracy and running time.

Necessary and Sufficient Polynomial Constraints on Compatible Triplets of Essential Matrices

“…for all distinct indices i, j, k ∈ {1, 2, 3}. Geometrically, constraints (12) mean that the epipoles e ij and e k j are matched and correspond to the projections of the jth camera center onto the ith and kth image plane respectively. The proof of compatibility of three fundamental matrices with noncollinear epipoles satisfying (12) can be found in [8].…”

A Hybrid Global Image Orientation Method for Simultaneously Estimating Global Rotations and Global Translations

“…Bourmaud et al (2014) derive the image pose parameters as a Lie group SE(3), they propose a generative model based on the formulation of a concentrated Gaussian distribution on the matrix Lie group and solve an iterated extended Kalman filter on that group to compute the elements of SE(3). Kasten et al (2019a) propose a method to globally recover the projection matrix of each image by using fundamental matrices of image pairs. However, as the projection matrix yields a projective reconstruction, information on interior orientation parameters cannot be introduced.…”

“…To identify the compatibility of each triplet, similar to Wang et al (2019b) and Kasten et al (2019a), we compute two triplet closure discrepancies with respect to relative rotations and translations, respectively. Given three relative rotations of a triplet, Rij, Rjk and Rki, RijRjkRki = I3×3 should hold.…”

Algebraic Characterization of Essential Matrices and Their Averaging in Multiview Settings

“…Note that any (consistent) n-view essential matrix is also a (consistent) n-view fundamental matrix. In [15] necessary and sufficient conditions for the consistency of the nview fundamental matrix were proved. The main theoretical contribution of [15] is summarized in Theorem 1.…”

“…To achieve this goal we investigate an object called the n-views essential matrix, which is obtained by stacking the n 2 essential matrices into a 3n × 3n matrix whose i, j'th 3 × 3 block is the essential matrix E ij relating the i'th and the j'th frames. We prove that, in addition to projective consistency (introduced in [15]), this matrix must have three pairs of eigenvalues each of the same magnitude but opposite signs, and its eigenvectors directly encode camera parameters.…”