Geodesic
Algorithmic solution of the five-point pose problem based on the Cayley representation of rotation matrices
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 5 (2013) no. 2, pp. 31-38 Cet article a éte moissonné depuis la source Math-Net.Ru

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A new algorithmic solution to the five-point relative pose problem is introduced. Our approach is not connected with or based on the famous cubic constraint on an essential matrix. Instead, we use the Cayley representation of rotation matrices in order to obtain a polynomial system of equations from epipolar constraints. Solving that system, we directly obtain positional relationships and orientations of the cameras through the roots for a 10th degree polynomial.
Keywords: five-point pose problem; calibrated camera; epipolar constraints; Cayley representation. 
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E. V. Martyushev. Algorithmic solution of the five-point pose problem based on the Cayley representation of rotation matrices. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 5 (2013) no. 2, pp. 31-38. http://geodesic.mathdoc.fr/item/VYURM_2013_5_2_a3/

[1] E. Kruppa, “Zur Ermittlung eines Objektes aus zwei Perspektiven mit Innerer Orientierung”, Sitz.-Ber. Akad. Wiss., Wien, Math. Naturw. Kl. Abt. IIa, 122 (1913), 1939–1948 | Zbl

[2] M. Demazure, Sur Deux Problemes de Reconstruction, No RR-0882, INRIA, 1988, 29 pp.

[3] O. Faugeras, S. Maybank, “Motion from Point Matches: Multiplicity of Solutions”, International Journal of Computer Vision, 4 (1990), 225–246 | DOI

[4] A. Heyden, G. Sparr, “Reconstruction from Calibrated Camera — A New Proof of the Kruppa–Demazure Theorem”, Journal of Mathematical Imaging and Vision, 10 (1999), 1–20 | DOI | MR

[5] J. Philip, “A Non-Iterative Algorithm for Determining all Essential Matrices Corresponding to Five Point Pairs”, Photogrammetric Record, 15 (1996), 589–599 | DOI

[6] D. Nistér, “An Efficient Solution to the Five-Point Relative Pose Problem”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 26 (2004), 756–777 | DOI

[7] H. Li, R. Hartley, “Five-Point Motion Estimation Made Easy”, IEEE 18th Int. Conf. of Pattern Recognition (2006), 630–633

[8] Z. Kukelova, M. Bujnak, T. Pajdla, “Polynomial Eigenvalue Solutions to the 5-pt and 6-pt Relative Pose Problems”, Proceedings of the British Machine Conference (2008), 56.1–56.10

[9] H. Stewénius, C. Engels, D. Nistér, “Recent Developments on Direct Relative Orientation”, ISPRS Journal of Photogrammetry and Remote Sensing, 60 (2006), 284–294 | DOI

[10] A. Cayley, “Sur Quelques Propriétés des Déterminants Gauches”, J. Reine Angew. Math., 32 (1846), 119–123 | DOI | Zbl

[11] O. Faugeras, Three-Dimensional Computer Vision: A Geometric Viewpoint, MIT Press, 1993, 663 pp.

[12] R. Hartley, A. Zisserman, Multiple View Geometry in Computer Vision, Second Edition, Cambridge University Press, 2004, 655 pp. | MR | Zbl

[13] S. Maybank, Theory of Reconstruction from Image Motion, Springer-Verlag, 1993, 261 pp. | MR | Zbl

[14] D. G. Hook, P. R. McAree, “Using Sturm Sequences to Bracket Real Roots of Polynomial Equations”, Graphic Gems I, Sb. nauch. tr., ed. A. Glassner, Academic Press, 1990, 416–422 | DOI

[15] W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in C, Second Edition, Cambridge University Press, 1993, 1020 pp. | MR