Automatic Generator of Minimal Problem Solvers

Kúkelová, Zuzana; Bujnak, Martin; Pajdla, Tomáš

doi:10.1007/978-3-540-88690-7_23

Cited by 187 publications

(281 citation statements)

References 18 publications

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“…Most noticeably, Kukelova et al [21] introduced a novel automatic generator of polynomial system solvers, which is also applicable to the multivariate polynomial systems arising from Case-1 and Case-2.…”

Section: Solving Multivariate Polynomial Systemsmentioning

confidence: 99%

ASPnP: An Accurate and Scalable Solution to the Perspective-n-Point Problem

Zheng

Sugimoto

Okutomi

2013

IEICE Trans. Inf. & Syst.

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Yinqiang ZHENG†a) , Nonmember, Shigeki SUGIMOTO †b) , and Masatoshi OKUTOMI †c) , Members SUMMARYWe propose an accurate and scalable solution to the perspective-n-point problem, referred to as ASPnP. Our main idea is to estimate the orientation and position parameters by directly minimizing a properly defined algebraic error. By using a novel quaternion representation of the rotation, our solution is immune to any parametrization degeneracy. To obtain the global optimum, we use the Gröbner basis technique to solve the polynomial system derived from the first-order optimality condition. The main advantages of our proposed solution lie in accuracy and scalability. Extensive experiment results, with both synthetic and real data, demonstrate that our proposed solution has better accuracy than the stateof-the-art noniterative solutions. More importantly, by exploiting vectorization operations, the computational cost of our ASPnP solution is almost constant, independent of the number of point correspondences n in the wide range from 4 to 1000. In our experiment settings, the ASPnP solution takes about 4 milliseconds, thus best suited for real-time applications with a drastically varying number of 3D-to-2D point correspondences * . key words: pose estimation, perspective-n-point, Gröbner basis, global optimum IntroductionThe perspective-n-point problem (PnP) aims to estimate the orientation and position, or rotation and translation, of a calibrated perspective camera by using n known 3D reference points and their corresponding 2D image projections. It has widespread applications in robot localization [1], hand-eye calibration [2], augmented reality [3] and so on. Considering these various application scenarios, a desirable PnP solution should be accurate, efficient and generally applicable, i.e., capable of handling both planar and non-planar cases with either a few or even hundreds of 3D-to-2D correspondences. Related WorksAs the minimal case, P3P (n = 3) has been thoroughly investigated in the literature [4], [5]. However, it has at most four possible solutions, and this multiplicity makes a P3P solver very sensitive to noise. In practice, it is usually used together with some robust estimation methods, like RANSAC [6], to remove outliers.To improve robustness to noise, cases of four and more than four correspondences should be considered. There are some specialized algorithms [7] restricted to the slightly redundant n = 4 (P4P) or n = 5 (P5P) cases only. However, the application of these specialized P4P and P5P solutions is limited, since the number of point correspondences might differ from one frame to the other, even in a specific application scenario. As a result, a large body of existing works have tried to improve flexibility, thus applicable to the general n ≥ 4 cases. Quan and Lan [8] proposed a linear solution by combining all the constraints from three-point subsets, whose computational complexity is O(n 5 ). Ansar and Daniilidis [9] developed linear solutions, but with complexity O(n 8 ), for general PnP ...

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Section: Solving Multivariate Polynomial Systemsmentioning

confidence: 99%

ASPnP: An Accurate and Scalable Solution to the Perspective-n-Point Problem

Zheng

Sugimoto

Okutomi

2013

IEICE Trans. Inf. & Syst.

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“…From the figure we see that all algorithms behaved almost the same way. However, 6pt polyeig solution provided more precise focal length estimation than existing 3 elimination [23] as well as 1-elimination Gröbner basis method [15]. deviation σ did not cause observable change in relative behaviour of the algorithms, therefore we do not plot these results.…”

Section: Methodsmentioning

confidence: 99%

“…A Gröbner basis solver with single G-J elimination of a 31 × 46 matrix which was generated using automatic generator has been presented in [15].…”

Section: Six Point Problemmentioning

confidence: 99%

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Polynomial Eigenvalue Solutions to the 5-pt and 6-pt Relative Pose Problems

Kúkelová

Bujnak

Pajdla

2008

Procedings of the British Machine Vision Conference 2008

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In this paper we provide new fast and simple solutions to two important minimal problems in computer vision, the five-point relative pose problem and the six-point focal length problem. We show that these two problems can easily be formulated as polynomial eigenvalue problems of degree three and two and solved using standard efficient numerical algorithms. Our solutions are somewhat more stable than state-of-the-art solutions by Nister and Stewenius and are in some sense more straightforward and easier to implement since polynomial eigenvalue problems are well studied with many efficient and robust algorithms available. The quality of the solvers is demonstrated in experiments 1 .

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“…The derivation of the exact solution is out of the scope of this paper, however the interested reader is refered to Groebner basis methods [11]. As argued before, there are nine degrees of freedom inF and so there can be no solution based on less than nine points.…”

Section: Single-sided Radial Fundamental Matrix Estimationmentioning

confidence: 99%

Unknown Radial Distortion Centers in Multiple View Geometry Problems

Brito

Angst

Köser

et al. 2013

Computer Vision – ACCV 2012

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Abstract. The radial undistortion model proposed by Fitzgibbon and the radial fundamental matrix were early steps to extend classical epipolar geometry to distorted cameras. Later minimal solvers have been proposed to find relative pose and radial distortion, given point correspondences between images. However, a big drawback of all these approaches is that they require the distortion center to be exactly known. In this paper we show how the distortion center can be absorbed into a new radial fundamental matrix. This new formulation is much more practical in reality as it allows also digital zoom, cropped images and camera-lens systems where the distortion center does not exactly coincide with the image center. In particular we start from the setting where only one of the two images contains radial distortion, analyze the structure of the particular radial fundamental matrix and show that the technique also generalizes to other linear multi-view relationships like trifocal tensor and homography. For the new radial fundamental matrix we propose different estimation algorithms from 9,10 and 11 points. We show how to extract the epipoles and prove the practical applicability on several epipolar geometry image pairs with strong distortion that -to the best of our knowledge -no other existing algorithm can handle properly.

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Automatic Generator of Minimal Problem Solvers

Cited by 187 publications

References 18 publications

ASP<i>n</i>P: An Accurate and Scalable Solution to the Perspective-<i>n</i>-Point Problem

ASP<i>n</i>P: An Accurate and Scalable Solution to the Perspective-<i>n</i>-Point Problem

Polynomial Eigenvalue Solutions to the 5-pt and 6-pt Relative Pose Problems

Unknown Radial Distortion Centers in Multiple View Geometry Problems

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