Abstract
Given the projection of a sufficient number of points it is possible to algebraically eliminate the camera parameters and obtain view-invariant functions of image coordinates and space coordinates. These single view invariants have been introduced in the past, however, they are not as well understood as their dual multi-view tensors. In this paper we revisit the dual tensors (bilinear, trilinear and quadlinear), both the general and the reference-plane reduced version, and describe the complete set of synthetic constraints, properties of the tensor slices, reprojection equations, non-linear constraints and reconstruction formulas. We then apply some of the new results, such as the dual reprojection equations, for multi-view point tracking under occlusions.
Chapter PDF
References
N. Canterakis. A minimal set of constraints for the trifocal tensor. In Proceedings of the European Conference on Computer Vision, Dublin, Ireland, June 2000.
S. Carlsson. Duality of reconstruction and positioning from projective views. In Proceedings of the workshop on Scene Representations, Cambridge, MA., June 1995.
S. Carlsson and D. Weinshall. Dual computation of projective shape and camera positions from multiple images. International Journal of Computer Vision, 27(3), 1998.
A. Criminisi, I. Reid, and A. Zisserman. Duality, rigidity and planar parallax. In Proceedings of the European Conference on Computer Vision, Frieburg, Germany, 1998. Springer, LNCS 1407.
O.D. Faugeras and B. Mourrain. On the geometry and algebra of the point and line correspondences between N images. In Proceedings of the International Conference on Computer Vision, Cambridge, MA, June 1995.
W. Fulton and J. Harris. Representation Theory. Springer, 1991.
R.I. Hartley. Lines and points in three views and the trifocal tensor. International Journal of Computer Vision, 22(2):125–140, 1997.
R.I. Hartley and A. Zisserman. Multiple View Geometry. Cambridge University Press, 2000.
A. Heyden. Reconstruction from image sequences by means of relative depths. In Proceedings of the International Conference on Computer Vision, pages 1058–1063, Cambridge, MA, June 1995.
A. Heyden. A common framework for multiple view tensors. In Proceedings of the European Conference on Computer Vision, pages 3–19, Freiburg, Germany, June 1998.
M. Irani and P. Anandan. Parallax geometry of pairs of points for 3D scene analysis. In Proceedings of the European Conference on Computer Vision, LNCS 1064, pages 17–30, Cambridge, UK, April 1996. Springer-Verlag.
M. Irani, P. Anandan, and D. Weinshall. From reference frames to reference planes: Multiview parallax geometry and applications. In Proceedings of the European Conference on Computer Vision, Frieburg, Germany, 1998. Springer, LNCS 1407.
D.W. Jacobs. Space efficient 3D model indexing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 439–444, 1992.
P. Meer, D. Mintz, D. Kim and A. Rosenfeld. Robust regression methods for computer vision: A review. International Journal of Computer Vision 6(1), 1991.
Open source computer vision library http://www.intel.com/research/mrl/research/cvlib/
A. Shashua and M. Werman. Trilinearity of three perspective views and its associated tensor. In Proceedings of the International Conference on Computer Vision, June 1995.
A. Shashua and Lior Wolf. On the structure and properties of the quadrifocal tensor. In Proceedings of the European Conference on Computer Vision, Dublin, Ireland, June 2000.
C. Rother and S. Carlsson. Linear Multi View Reconstruction and Camera Recovery. In Proceedings of the International Conference on Computer Vision, Vancouver, Canada, July 2001.
G. Stein and A. Shashua. On degeneracy of linear reconstruction from three views: Linear line complex and applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 21(3):244–251, 1999.
B. Triggs. Matching constraints and the joint image. In Proceedings of the International Conference on Computer Vision, pages 338–343, Cambridge, MA, June 1995.
D. Weinshall, M. Werman, and A. Shashua. Duality of multi-point and multi-frame geometry: Fundamental shape matrices and tensors. In Proceedings of the European Conference on Computer Vision, LNCS 1065, pages 217–227, Cambridge, UK, April 1996. Springer-Verlag.
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
Levin, A., Shashua, A. (2002). Revisiting Single-View Shape Tensors: Theory and Applications. 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_27
Download citation
DOI: https://doi.org/10.1007/3-540-47967-8_27
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