A Robust O(n) Solution to the Perspective-n-Point Problem

Abstract

We propose a noniterative solution for the Perspective-n-Point ({\rm P}n{\rm P}) problem, which can robustly retrieve the optimum by solving a seventh order polynomial. The central idea consists of three steps: 1) to divide the reference points into 3-point subsets in order to achieve a series of fourth order polynomials, 2) to compute the sum of the square of the polynomials so as to form a cost function, and 3) to find the roots of the derivative of the cost function in order to determine the optimum. The advantages of the proposed method are as follows: First, it can stably deal with the planar case, ordinary 3D case, and quasi-singular case, and it is as accurate as the state-of-the-art iterative algorithms with much less computational time. Second, it is the first noniterative {\rm P}n{\rm P} solution that can achieve more accurate results than the iterative algorithms when no redundant reference points can be used (n\le 5). Third, large-size point sets can be handled efficiently because its computational complexity is O(n).

Leveraging Feature Uncertainty in the PnP Problem

“…Subsequent works have improved the accuracy of the EPnP, still in O(n), by replacing the linearization with polynomial solvers. The most remarkable works along these lines are the Robust PnP (RPnP) [19], the Direct-Least-Squares (DLS) [11], the Accurate and Scalable PnP (ASPnP) [34] and the Optimal PnP (OPnP) [33].…”

“…We have compared our formulations against the most recent PnP approaches: the robust version of DLS [11], ASPnP [34], OPnP [33], RPnP [19], PPnP [8], EPnP + GN [18], SDP [30], EPPnP [5] and the LHM [21].…”

“…Finally, we chose the groundtruth translation t true as the centroid of the 3D reference points and we randomly generated a ground truth rotation matrix R true . As a metric errors we used the same as in [5,19,33]. The absolute error is measured in degrees between the R true and the estimated R as e rot (deg) = max 3 k=1 {acos(r k,true · r k ) × 180/π} where r k,true and r k are the k-th column of R true and R. The translation error is computed as e trans (%) = t true − t / t × 100.…”

“…For instance, the Efficient PnP (EPnP) [23] was the first closed-form solution to the problem with O(n) complexity and almost no loss of accuracy with respect to the most accurate iterative methods existing at the moment [21]. Subsequent works [11,19,33,34] have improved, also with O(n) complexity, the accuracy of the EPnP specially for the minimal cases with n = {3, 4, 5} correspondences. In addition, the flexibility of the linear formulation of the EPnP has allowed reformulations of the problem to also estimate the intrinsic parameters of an uncalibrated camera [26], or to integrate an algebraic outlier rejection criterion which does not require executing multiple RANSAC iterations [5].…”

Accurate and Linear Time Pose Estimation from Points and Lines

“…Early approaches were focused on solving the problem for the minimal cases with n = {3, 4, 5} [7,11,12,15,19,42]. The proliferation of feature point detectors [16,36] and descriptors [3,26,29,37,40] able to consistently retrieve many feature points per image, brought a series of new PnP algorithms that could efficiently handle arbitrarily large sets of points [9,13,18,24,25,27,30,38,45]. Amongst…”