Abstract
We prove that the 8-point algorithm always fails to reconstruct a unique fundamental matrix F independent on the camera positions, when its inputs are image point configurations that are perspective projections of the intersection of three quadrics in \(\mathbb {P}^3\). This generalizes a multiple results of the degeneracies of the 8-point algorithm. We give an algorithm that improves the 7- and 8-point algorithm in such a pathological situation. Additionally, we analyze the regions of focal point positions where a reconstruction of F is possible at all, when the world points are the vertices of a combinatorial cube in \(\mathbb {R}^3\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agarwal, S., Lee, H.-L., Sturmfels, B., Thomas, R.R.: On the existence of epipolar matrices. Int. J. Comput. Vis. 121(3), 403–415 (2017)
Brown, K.: Skew octagon solving the Thomson problem of eight points. http://mathpages.com/home/kmath005/elec/elec8.htm (2017). Accessed 13 Aug 2018
Bugarin, F., Bartoli, A., Henrion, D., Lasserre, J.-B., Orteu, J.-J., Sentenac, T.: Rank-constrained fundamental matrix estimation by polynomial global optimization versus the eight-point algorithm. J. Math. Imaging Vis. 53(1), 42–60 (2015)
Feng, C.L., Hung, Y.S.: A robust method for estimating the fundamental matrix. DICTA, pp. 633–642 (2003)
Hartley, R., Kahl, F.: Critical configurations for projective reconstruction from multiple views. Int. J. Comput. Vis. 71(1), 5–47 (2007)
Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge University Press, Cambridge (2003)
Henrion, D., Lasserre, J.-B., Löfberg, J.: Gloptipoly 3: moments, optimization and semidefinite programming. Optim. Methods Softw. 24(4–5), 761–779 (2009)
Krames, J.: Zur Ermittlung eines Objektes aus zwei Perspektiven.(Ein Beitrag zur Theorie der “gefährlichen Örter”.). Mon. Math. Phys. 49(1), 327–354 (1941)
Maybank, S.: Theory of Reconstruction from Image Motion. Springer, Berlin (2012)
Philip, J.: Estimation three-dimensional motion of rigid objects from noisy observations. IEEE Trans. Pattern Anal. Mach. Intell. 13(1), 61–66 (1991)
Philip, J.: Critical point configurations of the 5-, 6-, 7-, and 8-point algorithms for relative orientation. Technical report TRITA-MAT-MA (1998)
Turnbull, H.W.: On the vector algebra of eight associated points of three quadric surfaces. Math. Proc. Camb. Philos. Soc. 22(4), 481–487 (1925)
Turnbull, H., Young, A.: The linear invariants of ten quaternary quadrics. Trans. Camb. Phil. Soc. 23, 265–301 (1926)
Yi-Hong, W., Lan, T., Zhan-Yi, H.: Degeneracy from twisted cubic under two views. J. Comput. Sci. Technol. 25(5), 916–924 (2010)
Zhang, Z., Deriche, R., Faugeras, O., Luong, Q.-T.: A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry. Artif. intell. 78(1–2), 87–119 (1995)
Acknowledgements
We would like to thank Michael Joswig for his guidance and Fredrik Kahl for our correspondences about critical configurations.
Funding
Funding was provided by Deutsche Forschungsgemeinschaft (Grant No. SFB-TRR 195).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Wagner, A. Degeneracy of the Intersection of Three Quadrics. J Math Imaging Vis 61, 352–358 (2019). https://doi.org/10.1007/s10851-018-0841-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10851-018-0841-x