Some Results on Minimal Euclidean Reconstruction from Four Points

Abstract

Abstract. Methods for reconstruction and camera estimation from miminal data are often used to boot-strap robust (RANSAC and LMS) and optimal (bundle adjustment) structure and motion estimates. Minimal methods are known for projective reconstruction from two or more uncalibrated images, and for "5 point" relative orientation and Euclidean reconstruction from two calibrated parameters, but we know of no efficient minimal method for three or more calibrated cameras except the uniqueness proof by Holt and Netravali. We reformulate the problem of Euclidean reconstruction from minimal data of four points in three or more calibrated images, and develop a random rational simulation method to show some new results on this problem. In addition to an alternative proof of the uniqueness of the solutions in general cases, we further show that unknown coplanar configurations are not singular, but the true solution is a double root. The solution from a known coplanar configuration is also generally unique. Some especially symmetric point-camera configurations lead to multiple solutions, but only symmetry of points or the cameras gives a unique solution.

Four Points in Two or Three Calibrated Views: Theory and Practice

“…We apply the theory to create an efficient solution to the 3-view 4-point perspective pose problem (3v4p problem for short), which amounts to finding the relative poses of three calibrated perspective cameras given 4 corresponding point triplets. The 3v4p problem is notoriously difficult to solve, but has a unique solution in general (Holt and Netravali, 1995;Quan et al, 2003aQuan et al, , 2003b. It is in fact overconstrained by one, meaning that in general four point triplets can not be realised as the three calibrated images of four common world points.…”

Multi-view structure computation without explicitly estimating motion

“…Each triangle in this set will contribute one basic equation over three unknowns. If we collect enough of these basic equations into a polynomial system, then in theory it is possible to solve for all these unknown legs and unknown edges ( [23])-the latter is what we intend to solve. Two examples.…”

“…However, none of them is efficient even for moderate size problem (see [23,20] for discussions), due to their prohibitive complexity.…”

A Clever Elimination Strategy for Efficient Minimal Solvers

“…We exploit that many minimal problems in computer vi- Figure 1. An illustration of the two equations (17) and (18), which define the f+E+f problem, cut by six linear equations for six image point correspondences. sion lead to coupled sets of linear and polynomial equations where image measurements enter the linear equations only.…”