Abstract
The problem of recovering scene structure and camera motion from images has a number of inherent ambiguities. In this paper, configurations of points and cameras are analyzed for which the image points alone are insufficient to recover the scene geometry uniquely. Such configurations are said to be critical. For two views, it is well-known that a configuration is critical only if the two camera centres and all points lie on a ruled quadric. However, this is only a necessary condition. We give a complete characterization of the critical surfaces for two calibrated cameras and any number of points. Both algebraic and geometric characterizations of such surfaces are given. The existence of critical sets for n-view projective reconstruction has recently been reported in the literature. We show that there are critical sets for n-view Euclidean reconstruction as well. For example, it is shown that for any placement of three calibrated cameras, there always exists a critical set consisting of any number of points on a fourth-degree curve.
Chapter PDF
Similar content being viewed by others
References
T. Buchanan. Critical sets for 3d reconstruction using lines. In G. Sandini, editor, European Conf. Computer Vision, pages 730–738, Santa Margherita Ligure, Italy, 1992. Springer-Verlag.
S. Carlsson. Duality of reconstruction and positioning from projective views. In IEEE Workshop on Representation of Visual Scenes, pages 85–92, Cambridge Ma, USA, 1995.
R. Hartley. Estimation of relative camera positions for uncalibrated cameras. In European Conf. Computer Vision, pages 579–587, Santa Margherita Ligure, Italy, 1992. Springer-Verlag.
R. Hartley. Ambiguous configurations for 3-view projective reconstruction. In European Conf. Computer Vision, volume I, pages 922–935, Dublin, Ireland, 2000.
R. Hartley and G. Debunne. Dualizing scene reconstruction algorithms. In 3D Structure from Multiple Image of Large-Scale Environments, European Workshop, SMILE, pages 14–31, Freiburg, Germany, 1998.
F. Kahl, R. Hartley, and K. Aström. Critical configurations for N-view projective reconstruction. In Conf. Computer Vision and Pattern Recognition, volume II, pages 158–163, Hawaii, USA, 2001.
J. Krames. Zur Ermittlung eines Objectes aus zwei Perspectiven (Ein Beitrag zur Theorie der gefährlichen Örter). Monatsh. Math. Phys., 49:327–354, 1940.
S. Maybank. Theory of Reconstruction from Image Motion. Springer-Verlag, Berlin, Heidelberg, New York, 1993.
S. Maybank and A. Shashua. Ambiguity in reconstruction from images of six points. In Int. Conf. Computer Vision, pages 703–708, Mumbai, India, 1998.
J. G. Semple and G. T. Kneebone. Algebraic Projective Geometry. Clarendon Press, Oxford, 1952.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kahl, F., Hartley, R. (2002). Critical Curves and Surfaces for Euclidean Reconstruction. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds) Computer Vision — ECCV 2002. ECCV 2002. Lecture Notes in Computer Science, vol 2351. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47967-8_30
Download citation
DOI: https://doi.org/10.1007/3-540-47967-8_30
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43744-4
Online ISBN: 978-3-540-47967-3
eBook Packages: Springer Book Archive