Globally Optimal Estimates for Geometric Reconstruction Problems

Kahl, Fredrik; Henrion, Didier

doi:10.1007/s11263-006-0015-y

Cited by 82 publications

(52 citation statements)

References 32 publications

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“…In fact, due to the polynomial nature of the constraints created by point correspondences, finite ambiguities almost certainly do exist in minimal cases. Performing reliable orientation recovery with very few point correspondences is thus a major challenge that we intend to tackle by applying recent developments in global optimization of general computer vision problems [7,8,18] to the particular problem studied in this paper. …”

Section: Limitations and Future Workmentioning

confidence: 99%

Structure from Motion with Known Camera Positions

Carceroni

Kumar

Daniilidis

2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition - Volume 1 (CVPR'06)

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The wide availability of GPS sensors is changing the landscape in the applications of structure from motion techniques for localization. In this paper, we study the problem of estimating camera orientations from multiple views, given the positions of the viewpoints in a world coordinate system and a set of point correspondences across the views. Given three or more views, the above problem has a finite number of solutions for three or more point correspondences. Given six or more views, the problem has a finite number of solutions for just two or more points. In the three-view case, we show the necessary and sufficient conditions for the three essential matrices to be consistent with a set of known baselines. We also introduce a method to recover the absolute orientations of three views in world coordinates from their essential matrices. To refine these estimates we perform a least-squares minimization on the group cross product SO(3) × SO(3) × SO(3). We report experiments on synthetic data and on data from the ICCV2005 Computer Vision Contest. ©2006 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

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Section: Limitations and Future Workmentioning

confidence: 99%

Structure from Motion with Known Camera Positions

Carceroni

Kumar

Daniilidis

2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition - Volume 1 (CVPR'06)

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“…Iterative strategies have been proposed to overcome this drawback [10], and modern optimization approaches exist that take into account the rank-two constraint at some computational expense [2,11,17]. In contrast, we propose two closed-form solutions for the direct computation of a fundamental matrix satisfying the rank-two constraint.…”

Section: Overviewmentioning

confidence: 99%

Singular Vector Methods for Fundamental Matrix Computation

Espuny

Monasse

2014

Image and Video Technology

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Abstract. The normalized eight-point algorithm is broadly used for the computation of the fundamental matrix between two images given a set of correspondences. However, it performs poorly for low-size datasets due to the way in which the rank-two constraint is imposed on the fundamental matrix. We propose two new algorithms to enforce the rank-two constraint on the fundamental matrix in closed form. The first one restricts the projection on the manifold of fundamental matrices along the most favorable direction with respect to algebraic error. Its complexity is akin to the classical seven point algorithm. The second algorithm relaxes the search to the best plane with respect to the algebraic error. The minimization of this error amounts to finding the intersection of two bivariate cubic polynomial curves. These methods are based on the minimization of the algebraic error and perform equally well for large datasets. However, we show through synthetic and real experiments that the proposed algorithms compare favorably with the normalized eight-point algorithm for low-size datasets.

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“…The theory of convex linear matrix inequality (LMI) relaxations (Lasserre 2001) is used in (Kahl and Henrion 2005) to find global solutions to several optimization problems in multiview geometry, while (Chandraker et al 2007a) discusses a direct method for autocalibration using the same techniques. Triangulation and resectioning are solved with a certificate of optimality using convex relaxation techniques for fractional programs in (Agarwal et al 2006).…”

Section: Previous Workmentioning

confidence: 99%

“…Note that the cost function in (12) is a polynomial and some recent work in computer vision (Kahl and Henrion 2005;Chandraker et al 2007a) exploits convex linear matrix inequality (LMI) relaxations to achieve global optimality in polynomial programs. However, this is a degree 8 polynomial in three variables, which is far beyond what presentday solvers can handle (Henrion and Lasserre 2003;Prajna et al 2002).…”

Section: Problem Formulationmentioning

confidence: 99%

Globally Optimal Algorithms for Stratified Autocalibration

Chandraker,

Agarwal,

Kriegman

et al. 2009

Int J Comput Vis

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We present practical algorithms for stratified autocalibration with theoretical guarantees of global optimality. Given a projective reconstruction, we first upgrade it to affine by estimating the position of the plane at infinity. The plane at infinity is computed by globally minimizing a least squares formulation of the modulus constraints. In the second stage, this affine reconstruction is upgraded to a metric one by globally minimizing the infinite homography relation to compute the dual image of the absolute conic (DIAC). The positive semidefiniteness of the DIAC is explicitly enforced as part of the optimization process, rather than as a post-processing step.For each stage, we construct and minimize tight convex relaxations of the highly non-convex objective functions in a branch and bound optimization framework. We exploit the inherent problem structure to restrict the search space for the DIAC and the plane at infinity to a small, fixed number of branching dimensions, independent of the number of views. Chirality constraints are incorporated into our convex relaxations to automatically select an initial region which is guaranteed to contain the global minimum. Experimental evidence of the accuracy, speed and scalability of our algorithm is presented on synthetic and real data.

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Globally Optimal Estimates for Geometric Reconstruction Problems

Cited by 82 publications

References 32 publications

Structure from Motion with Known Camera Positions

Structure from Motion with Known Camera Positions

Singular Vector Methods for Fundamental Matrix Computation

Globally Optimal Algorithms for Stratified Autocalibration

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