Optimal Motion Estimation from Multiple Images by Normalized Epipolar Constraint

In this paper, we study the structure from motion problem as a constrained nonlinear least squares problem which minimizes the so called reprojection error subject to all constraints among multiple images. By converting this constrained optimization problem to an unconstrained one, we contend that multilinear constraints, when used for motion and structure estimation, need to be properly normalized, which makes them no longer tensors. We demonstrate this by using the bilinear epipolar constraints and show how they give rise to a multiview version of the (crossed) normalized epipolar constraint of two views 5]. Such a (crossed) normalized epipolar constraint serves as an optimal objective function for motion (and structure) estimation. This objective function further reveals certain statistic relationship between bilinear and trilinear constraints: Even the rectilinear motion can be correctly estimated by the normalized epipolar constraint as a limit of generic cases, hence trilinear constraints are not really necessary. Since the so obtained objective function is deened naturally on a product of Stiefel manifolds, we show how to use geometric optimization techniques 2] to minimize such a function. Simulation and experimental results are presented to evaluate the proposed algorithm and verify our claims.