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41
An Efficient Solution to the FivePoint Relative Pose Problem
, 2004
"... An efficient algorithmic solution to the classical fivepoint relative pose problem is presented. The problem is to find the possible solutions for relative camera pose between two calibrated views given five corresponding points. The algorithm consists of computing the coefficients of a tenth degre ..."
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Cited by 303 (11 self)
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An efficient algorithmic solution to the classical fivepoint relative pose problem is presented. The problem is to find the possible solutions for relative camera pose between two calibrated views given five corresponding points. The algorithm consists of computing the coefficients of a tenth degree polynomial in closed form and subsequently finding its roots. It is the first algorithm well suited for numerical implementation that also corresponds to the inherent complexity of the problem. We investigate the numerical precision of the algorithm. We also study its performance under noise in minimal as well as overdetermined cases. The performance is compared to that of the well known 8 and 7point methods and a 6point scheme. The algorithm is used in a robust hypothesizeandtest framework to estimate structure and motion in realtime with low delay. The realtime system uses solely visual input and has been demonstrated at major conferences.
SelfCalibration of a Moving Camera From Point Correspondences and Fundamental Matrices
, 1997
"... . We address the problem of estimating threedimensional motion, and structure from motion with an uncalibrated moving camera. We show that point correspondences between three images, and the fundamental matrices computed from these point correspondences, are sufficient to recover the internal orien ..."
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Cited by 99 (2 self)
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. We address the problem of estimating threedimensional motion, and structure from motion with an uncalibrated moving camera. We show that point correspondences between three images, and the fundamental matrices computed from these point correspondences, are sufficient to recover the internal orientation of the camera (its calibration), the motion parameters, and to compute coherent perspective projection matrices which enable us to reconstruct 3D structure up to a similarity. In contrast with other methods, no calibration object with a known 3D shape is needed, and no limitations are put upon the unknown motions to be performed or the parameters to be recovered, as long as they define a projective camera. The theory of the method, which is based on the constraint that the observed points are part of a static scene, thus allowing us to link the intrinsic parameters and the fundamental matrix via the absolute conic, is first detailed. Several algorithms are then presented, and their ...
A Gröbner free alternative for polynomial system solving
 Journal of Complexity
, 2001
"... Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic ..."
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Cited by 82 (16 self)
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Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton’s iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation
Recent Developments on Direct Relative Orientation
, 2006
"... This paper presents a novel version of the fivepoint relative orientation algorithm given in Nister (2004). The name of the algorithm arises from the fact that it can operate even on the minimal five point correspondences required for a finite number of solutions to relative orientation. For the mi ..."
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Cited by 61 (0 self)
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This paper presents a novel version of the fivepoint relative orientation algorithm given in Nister (2004). The name of the algorithm arises from the fact that it can operate even on the minimal five point correspondences required for a finite number of solutions to relative orientation. For the minimal five correspondences the algorithm returns up to ten real solutions. The algorithm can also operate on many points. Like the previous version of the fivepoint algorithm, our method can operate correctly even in the face of critical surfaces, including planar and ruled quadric scenes. The paper
Stratification of 3D vision: Projective, affine, and metric representations
"... In this article we provide a conceptual framework in which to think of the relationships between the threedimensional structure of the physical space and the geometric properties of a set of cameras which provide pictures from which measurements can be made. We usually think of the physical space a ..."
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Cited by 50 (5 self)
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In this article we provide a conceptual framework in which to think of the relationships between the threedimensional structure of the physical space and the geometric properties of a set of cameras which provide pictures from which measurements can be made. We usually think of the physical space as being embedded in a threedimensional euclidean space where measurements of lengths and angles do make sense. It turns out that for artificial systems, such as robots, this is not a mandatory viewpoint and that it is sometimes sufficient to think of the physical space as being embedded in an affine or even projective space. The question then arises of how to relate these models to image measurements and to geometric properties of sets of cameras. We show that in the case of two cameras, a stereo rig, the projective structure of the world can be recovered as soon as the epipolar geometry of the stereo rig is known and that this geometry is summarized by a single 3 3 matrix, which we called the fundamental matrix [1, 2]. The affine structure can then be recovered if we add to this information a projective transformation between the two images which is induced by the plane at infinity. Finally, the euclidean structure (up to a similitude) can be recovered if we add to these two elements the knowledge of two conics (one for each camera) which are the images of the absolute conic, a circle of radius p;1 in the plane at in nity. In all three cases we showhowthe threedimensional information can be recovered directly from the images without explicitely reconstructing the scene structure. This defines a natural hierarchy of geometric structures, a set of three strata, that we overlay onthephysical world and which we show to be recoverable by simple procedures relying on two items, the physical space itself together with possibly, but not necessarily, some a priori information about it, and some voluntary motions of the set of cameras.
Kruppa's equations derived from the fundamental matrix
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1997
"... Abstract — The purpose of this note is to give a specific form for Kruppa’s equations in terms of the Fundamental matrix. Kruppa’s equations can be written explicitly in terms of the singular value decomposition (SVD) of the fundamental matrix. ..."
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Cited by 50 (0 self)
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Abstract — The purpose of this note is to give a specific form for Kruppa’s equations in terms of the Fundamental matrix. Kruppa’s equations can be written explicitly in terms of the singular value decomposition (SVD) of the fundamental matrix.
Uncalibrated Euclidean reconstruction: a review
, 1999
"... This paper provides a review on techniques for computing a threedimensional model of a scene from a single moving camera, with unconstrained motion and unknown parameters. In the classical approach, called autocalibration or selfcalibration, camera motion and parameters are recovered first, using ..."
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Cited by 35 (8 self)
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This paper provides a review on techniques for computing a threedimensional model of a scene from a single moving camera, with unconstrained motion and unknown parameters. In the classical approach, called autocalibration or selfcalibration, camera motion and parameters are recovered first, using rigidity; then structure is easily computed. Recently, new methods based on the idea of stratification have been proposed. They upgrade the projective structure, achievable from correspondences only, to the Euclidean structure, by exploiting all the available constraints.
Critical motions for autocalibration when some intrinsic parameters can vary
 J. Math. Imaging Vision
, 2000
"... Abstract. Autocalibration is the recovery of the full camera geometry and Euclidean scene structure from several images of an unknown 3D scene, using rigidity constraints and partial knowledge of the camera intrinsic parameters. It fails for certain special classes of camera motion. This paper deri ..."
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Cited by 29 (3 self)
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Abstract. Autocalibration is the recovery of the full camera geometry and Euclidean scene structure from several images of an unknown 3D scene, using rigidity constraints and partial knowledge of the camera intrinsic parameters. It fails for certain special classes of camera motion. This paper derives necessary and sufficient conditions for unique autocalibration, for several practically important cases where some of the intrinsic parameters are known (e.g. skew, aspect ratio) and others can vary (e.g. focal length). We introduce a novel subgroup condition on the camera calibration matrix, which helps to systematize this sort of autocalibration problem. We show that for subgroup constraints, criticality is independent of the exact values of the intrinsic parameters and depends only on the camera motion. We study such critical motions for arbitrary numbers of images under the following constraints: vanishing skew, known aspect ratio and full internal calibration modulo unknown focal lengths. We give explicit, geometric descriptions for most of the singular cases. For example, in the case of unknown focal lengths, the only critical motions are: (i) arbitrary rotations about the optical axis and translations, (ii) arbitrary rotations about at most two centres, (iii) forwardlooking motions along an ellipse and/or a corresponding hyperbola in an orthogonal plane. Some practically important special cases are also analyzed in more detail.
A Minimal Solution for Relative Pose with Unknown Focal Length
"... Assume that we have two perspective images with known intrinsic parameters except for an unknown common focal length. It is a minimally constrained problem to find the relative orientation between the two images given six corresponding points. We present an efficient solution to the problem and show ..."
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Cited by 24 (1 self)
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Assume that we have two perspective images with known intrinsic parameters except for an unknown common focal length. It is a minimally constrained problem to find the relative orientation between the two images given six corresponding points. We present an efficient solution to the problem and show that there are 15 solutions in general (including complex solutions). To the best of our knowledge this was a previously unsolved problem. The solutions
A survey of camera selfcalibration
 Proceedings of the IEEE Conference on Advanced Video and Signal Based Surveillance
, 2003
"... This paper surveys the developments of the last 10 years in the area of camera selfcalibration. In order to solve this problem, researches have used the camera intrinsic constraints separately and in conjunction with the camera motion constraints or the scene constraints. Most of the selfcalibrati ..."
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Cited by 20 (0 self)
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This paper surveys the developments of the last 10 years in the area of camera selfcalibration. In order to solve this problem, researches have used the camera intrinsic constraints separately and in conjunction with the camera motion constraints or the scene constraints. Most of the selfcalibration algorithms are concerned with unknown but constant intrinsic camera parameters. Recently, camera selfcalibration in the case of varying intrinsic camera parameters was also studied. We present the basic theories behind the different selfcalibration techniques and discuss the ideas behind most of the selfcalibration algorithms. 1.