The Five Points Pose Problem : A New and Accurate Solution Adapted to any Geometric Configuration

arXiv:0807.2047v3 [cs.CV] 16 Jul 2008 The Five Points Pose Problem : A New and Accurate Solution Adapted to any Geometric Configuration Mahzad Kalantari a,b,c, Franck Jungd, Jean-Pierre Guedonb,c and Nicolas Paparoditisa aMATIS Laboratory, Institut Geographique National 2, Avenue Pasteur. 94165 Saint-Mand´e Cedex - FRANCE - firstname.lastname@ign.fr b Institut Recherche Communications Cybern´etique de Nantes (IRCCyN) UMR CNRS 6597 1, rue de la No¨e BP 92101F-44321 Nantes Cedex 03 - FRANCE c Institut de Recherche sur les Sciences et Techniques de la Ville CNRS FR 2488 d DDE - Seine Maritime - firstname.lastname@equipement.gouv.fr Abstract. The goal of this paper is to estimate directly the rotation and translation between two stereoscopic images with the help of five homologous points. The methodology presented does not mix the ro- tation and translation parameters, which is comparably an important advantage over the methods using the well-known essential matrix. This results in correct behavior and accuracy for situations otherwise known as quite unfavorable, such as planar scenes, or panoramic sets of im- ages (with a null base length), while providing quite comparable results for more ”standard” cases. The resolution of the algebraic polynomials resulting from the modeling of the coplanarity constraint is made with the help of powerful algebraic solver tools (the Gr¨obner bases and the Rational Univariate Representation). 1 Introduction The determination of the relative orientation between two cameras with the help of homologous points is the basis of many applications in the domains of pho- togrammetry and computer vision. The configuration often called ”minimal case problem” takes the intrinsic parameters (i. e. the elements of calibration) of the camera as a priori known. Then only five points homologous are necessary to estimate the remaining three unknowns of rotation and two ones of translation (up to a scale factor). This problem has been dealt by many authors, and most of recent methods published provide a resolution based on the properties of the essential matrix. Even if its use simplifies remarkably the problem of the relative orientation, in some cases, due to the fact that all parameters of rotation and translation are mixed, this is the origin of geometric ambiguousnesses. So as to improve this point, we propose in this article a model that separates completely the rotation and the translation unknowns. We show that the major advan- tage of this model is that it allows to solve degenerate problems such as pure 2 Authors Suppressed Due to Excessive Length rotations (null translation). We use an algebraic modeling for the coplanarity constraint, through a system of polynomial equations. We solve them with the help of powerful algebraic solver tools, the Gr¨obner bases and the Rational Uni- variate Representation. So as to assess this new approach, three cases have been processed: a classical case, a planar scene, and a case where the base length is close to zero. We will see that the new method is still accurate even for the last two cases - quite unfavorable - configurations. We will also compare with the Stewenius’s algorithm and see that in planar scenes the new algorithm is more accurate. An evaluation on real scenes will finally be presented. 2 Historical background of the five points relative pose problem. It was for the first time demonstrated by Kruppa [1] in 1913 that the direct resolution of the relative orientation from 5 points in general contained at most 11 solutions. The described method consisted to find all intersections of two curves of degree 6. Unfortunately, one century ago, this method could not lead to a numerical implementation. Lately in [2], [3], [4], [5] it has been demonstrated that the number of solutions is in general equal to 10, including the complex solutions. Triggs [6] has provided a detailed version for a numeric implementation. Philip [7] presented in 1996 a solution using a polynomial of degree 13, and has proposed a numeric method to solve his system. The roots of his polynomial give directly the relative orientation. Philip has exploited the constraints on essential matrix. Philip’s ideas have been followed in 2004 by Nister [8] who has refined this algorithm, has obtained a 10th order polynomial and has given a numerical resolution using a Gauss-Jordan elimination. Since then, number of papers tried to give some improvements to the method of Nister, notably Stewenius [9] that has provided a polynomial resolution using the Gr¨obner bases. Many papers have proposed some modifications to the method of Nister in view of a numeric improvement [12], [13], or for a simplification of implementation [10], [11]. 3 Geometry review of Relative Orientation In this section we recall the various ways to present the geometry of relative orientation, that consist in the determination of the translation and the relative rotation between two images of a scene having a common informati

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