Abstract
In this theoretical paper we consider the problem of accurately triangulating a scene plane. Rather than first triangulating a set of points and then fitting a plane to these points, we try to minimize the back-projection errors as functions of the plane parameters directly. As this is both geometrically and statistically meaningful our method performs better than the standard two step procedure. Furthermore, we show that the error residuals of this formulation are quasiconvex thereby making it very easy to solve using for example standard local optimization methods.
Chapter PDF
References
Bartoli, A., Sturm, P.: Constrained structure and motion from multiple uncalibrated views of a piecewise planar scene. International Journal of Computer Vision 52(1) (2003)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Boykov, Y., Veksler, O., Zabih, R.: Fast approximate energy minimization via graph cuts. IEEE Trans. Pattern Analysis and Machine Intelligence 23(11), 1222–1239 (2001)
Fischler, M.A., Bolles, R.C.: Random sample consensus: a paradigm for model fitting with application to image analysis and automated cartography. Commun. Assoc. Comp. Mach. 24, 381–395 (1981)
Furukawa, Y., Curless, B., Seitz, S.M., Szeliski, R.: Reconstructing building interiors from images. In: Proc. Int. Conf. on Computer Vision, Kyoto, Japan (2009)
Hartley, R., Seo, Y.: Verifying global minima for L2 minimization problems. In: Conf. Computer Vision and Pattern Recognition, Anchorage, USA (2008)
Hartley, R., Sturm, P.: Triangulation. Computer Vision and Image Understanding 68(2), 146–157 (1997)
Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision. Cambridge University Press, Cambridge (2004)
Kahl, F., Hartley, R.: Multiple view geometry under the L1-norm. IEEE Trans. Pattern Analysis and Machine Intelligence 30(9), 1603–1617 (2008)
Kanatani, K., Niitsuma, H.: Optimal two-view planar scene triangulation. In: Kimmel, R., Klette, R., Sugimoto, A. (eds.) ACCV 2010, Part II. LNCS, vol. 6493, pp. 242–253. Springer, Heidelberg (2011)
Kanazawa, Y., Kenichi, K.: Direct reconstruction of planar surfaces by stereo vision. IEICE Transactions on Information and Systems, E78-D(7) (1995)
Ke, Q., Kanade, T.: Quasiconvex optimization for robust geometric reconstruction. IEEE Trans. Pattern Analysis and Machine Intelligence 29(10), 1834–1847 (2007)
Kolmogorov, V., Zabih, R.: Multi-camera scene reconstruction via graph cuts. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2352, pp. 82–96. Springer, Heidelberg (2002)
Li, H.: A practical algorithm for L ∞ triangulation with outliers. In: Conf. Computer Vision and Pattern Recognition, Minneapolis, USA (2007)
Lowe, D.: Distinctive image features from scale-invariant keypoints. Int. Journal Computer Vision (2004)
Olsson, C., Enqvist, O., Kahl, F.: A polynomial-time bound for matching and registration with ouliers. In: Conf. Computer Vision and Pattern Recognition, Anchorage, USA (2008)
Olsson, C., Eriksson, A., Hartley, R.: Outlier removal using duality. In: Conf. Computer Vision and Pattern Recognition, San Francisco, USA (2010)
Olsson, C., Kahl, F.: Generalized convexity in multiple view geometry. J. Math. Imaging Vis. 38(1), 35–51 (2010)
Olsson, C., Kahl, F., Hartley, R.: Projective least-squares: Global solutions with local optimization. In: Conf. Computer Vision and Pattern Recognition, Miami, USA (2009)
Seo, Y., Hartley, R.: A fast method to minimize L ∞ error norm for geometric vision problems. In: Int. Conf. Computer Vision, Rio de Janeiro, Brazil (2007)
Sim, K., Hartley, R.: Removing outliers using the L ∞ -norm. In: Conf. Computer Vision and Pattern Recognition, New York City, USA, pp. 485–492 (2006)
Woodford, O., Torr, P., Reid, I., Fitzgibbon, A.: Global stereo reconstruction under second order smoothness priors. Proc. IEEE Trans. Pattern Analysis and Machine Intelligence 31, 2115–2128 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Olsson, C., Eriksson, A. (2011). Triangulating a Plane. In: Heyden, A., Kahl, F. (eds) Image Analysis. SCIA 2011. Lecture Notes in Computer Science, vol 6688. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21227-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-21227-7_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21226-0
Online ISBN: 978-3-642-21227-7
eBook Packages: Computer ScienceComputer Science (R0)