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19
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 335 (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.
Theory and Practice of Projective Rectification
, 1998
"... This paper gives a new method for image rectification, the process of resampling pairs of stereo images taken from widely differing viewpoints in order to produce a pair of "matched epipolar projections". These are projections in which the epipolar lines run parallel with the xaxis and co ..."
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Cited by 81 (0 self)
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This paper gives a new method for image rectification, the process of resampling pairs of stereo images taken from widely differing viewpoints in order to produce a pair of "matched epipolar projections". These are projections in which the epipolar lines run parallel with the xaxis and consequently, disparities between the images are in the x direction only. The method is based on an examination of the fundamental matrix of LonguetHiggins which describes the epipolar geometry of the image pair. The approach taken is consistent with that recently advocated by Faugeras ([1]) of avoiding camera calibration. The paper uses methods of projective geometry to determine a pair of 2D projective transformations to be applied to the two images in order to match the epipolar lines. The advantages include the simplicity of the 2D projective transformation which allows very fast resampling as well as subsequent simplification in the identification of matched points and scene reconstruction. 1 In...
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 67 (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
Invariants of Six Points and Projective Reconstruction from Three Uncalibrated Images
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1995
"... There are three projective invariants of a set of six points in general position in space. It is well known that these invariants cannot be recovered from one image, however an invariant relationship does exist between space invariants and image invariants. This invariant relationship is first deriv ..."
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Cited by 47 (16 self)
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There are three projective invariants of a set of six points in general position in space. It is well known that these invariants cannot be recovered from one image, however an invariant relationship does exist between space invariants and image invariants. This invariant relationship is first derived for a single image. Then this invariant relationship is used to derive the space invariants, when multiple images are available. This paper establishes that the minimum number of images for computing these invariants is three, and the computation of invariants of six points from three images can have as many as three solutions. Algorithms are presented for computing these invariants in closed form. The accuracy and stability with respect to image noise, selection of the triplets of images and distance between viewing positions are studied both through real and simulated images. Applications of these invariants are also presented. Both the results of Faugeras [1] and Hartley et al. [2] for...
Epipole and Fundamental Matrix Estimation Using the Virtual Parallax Property
, 1995
"... This paper addresses the problem of computing the fundamental matrix which describes a geometric relationship between a pair of stereo images : the epipolar geometry. In the uncalibrated case, epipolar geometry captures all the 3D information available from the scene. It is of a central importance f ..."
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Cited by 37 (1 self)
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This paper addresses the problem of computing the fundamental matrix which describes a geometric relationship between a pair of stereo images : the epipolar geometry. In the uncalibrated case, epipolar geometry captures all the 3D information available from the scene. It is of a central importance for problems such as 3D reconstruction, selfcalibration and feature tracking. Hence, the computation of the fundamental matrix is of great interest. The existing methods [10] uses two steps : a linear step followed by a non linear one. But the linear step gives rarely a close form solution for the fundamental matrix resulting in more iterations for the non linear step which is not guaranteed to converge to the correct solution. In this paper, a novel method based on virtual parallax is proposed. The problem is formulated differently, instead of computing directely the 3 \Theta 3 fundamental matrix, we compute a homography with one epipole position, and show that this is equivalent to compute...
The Projective Geometry of the Gale Transform
, 1998
"... The Gale transform, an involution on sets of points in projective space, appears in a multitude of guises, in subjects as diverse as optimization, coding theory, thetafunctions, and recently in our proof that certain general sets of points fail to satisfy the minimal free resolution conjecture (see ..."
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Cited by 24 (5 self)
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The Gale transform, an involution on sets of points in projective space, appears in a multitude of guises, in subjects as diverse as optimization, coding theory, thetafunctions, and recently in our proof that certain general sets of points fail to satisfy the minimal free resolution conjecture (see EisenbudPopescu [1996]). In this paper we reexamine the Gale transform in the light of modern algebraic geometry. We give a more general definition, in the context of finite (locally) Gorenstein subschemes. We put in modern form a number of the more remarkable examples discovered in the past, and we add new constructions and connections to other areas of algebraic geometry. We generalize Goppa’s theorem in coding theory and we give new applications to Castelnuovo theory. We give
On the Determination of Epipoles Using CrossRatios
 CVGIP: Image Understanding
, 1998
"... We study the problem of computing the position of the epipoles in a pair of uncalibrated images. The approach, which is based on the invariance of the crossratio by the epipolar transformation, exploits algebraic constraints obtained from point correspondences and provides a solution in which only ..."
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Cited by 6 (0 self)
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We study the problem of computing the position of the epipoles in a pair of uncalibrated images. The approach, which is based on the invariance of the crossratio by the epipolar transformation, exploits algebraic constraints obtained from point correspondences and provides a solution in which only the epipoles are involved. This is in opposition to the methods based on the computation of the fundamental matrix. These notions are first presented as well as the new epipolar ordering constraint. Three families of methods are successively considered: the first uses statistics on closedform solutions provided by the socalled Sturm method, the second intersect plane cubics through deterministic procedures, and the third is based on nonlinear minimizations of a difference of crossratios. We discuss the shortcomings of each and show, using numerous experimental comparisons, that there is a tradeoff between elegance and robustness to noise. The crossratio based methods do not turn out to ...
Some Results on Minimal Euclidean Reconstruction from Four Points
, 2006
"... Methods for reconstruction and camera estimation from miminal data are often used to bootstrap 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 ” rel ..."
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Cited by 3 (0 self)
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Methods for reconstruction and camera estimation from miminal data are often used to bootstrap 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 pointcamera configurations lead to multiple solutions, but only symmetry of points or the cameras gives a unique solution.
A Historical Survey of Geometric Computer Vision
, 2011
"... Abstract. This short paper accompanies an invited lecture on a historical survey of geometric computer vision problems. It presents some early works on imagebased 3D modeling, multiview geometry, and structurefrommotion, from the last three centuries. Some of these are relatively well known to ph ..."
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Cited by 1 (0 self)
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Abstract. This short paper accompanies an invited lecture on a historical survey of geometric computer vision problems. It presents some early works on imagebased 3D modeling, multiview geometry, and structurefrommotion, from the last three centuries. Some of these are relatively well known to photogrammetrists and computer vision researchers whereas others seem to have been largely forgotten or overlooked. This paper gives a very brief summary of an ongoing historical study.