International Journal of Computer Vision

, Volume 48, Issue 1, pp 53–67 | Cite as

On Projection Matrices \(\mathcal{P}^k \to \mathcal{P}^2 ,k = 3,...,6, \) and their Applications in Computer Vision

  • Lior Wolf
  • Amnon Shashua


Projection matrices from projective spaces \({\mathcal{P}}^3 {\text{ to }}{\mathcal{P}}^2 \) have long been used in multiple-view geometry to model the perspective projection created by the pin-hole camera. In this work we introduce higher-dimensional mappings \({\mathcal{P}}^k \to {\mathcal{P}}^2 ,k = 3,4,5,6\) for the representation of various applications in which the world we view is no longer rigid. We also describe the multi-view constraints from these new projection matrices (where k > 3) and methods for extracting the (non-rigid) structure and motion for each application.

dynamic structure from motion multiple view geometry multi-linear constraints 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Avidan, S. and Shashua, A. 2000. Trajectory triangulation: 3D reconstruction of moving points from a monocular image sequence. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(4):348–357.Google Scholar
  2. Barnabei, M., Brini, A., and Rota, G.C. 1985. On the exterior calculus of invariant theory. J. Alg., 96:120–160.Google Scholar
  3. Costeira, J. and Kanade, T. 1998. A multibody factorization method for independent moving objects. International Journal on Computer Vision, 29(3):159–179.Google Scholar
  4. Faugeras, O.D. 1993. Three-Dimensional Computer Vision: A Geometric Viewpoint. MIT Press: Reading, MA.Google Scholar
  5. Faugeras, O.D. and Mourrain, B. 1995. 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.Google Scholar
  6. Fitzgibbon, A.W. and Zisserman, A. 2000. Multibody structure and motion: 3–D reconstruction of independently moving object. In Proceedings of the European Conference on Computer Vision (ECCV), Dublin, Ireland, June 2000.Google Scholar
  7. Han, M. and Kanade, T. 2000. Reconstruction of a scene with multiple linearly moving objects. In Proc. of Computer Vision and Pattern Recognition, June 2000.Google Scholar
  8. Hartley, R.I. and Zisserman, A. 2000. Multiple View Geometry. Cambridge University Press: Cambridge, UK.Google Scholar
  9. Lucas, B.D. and Kanade, T. 1981. An iterative image registration technique with an application to stereo vision. In Proceedings IJCAI, Vancouver, Canada, 1981, pp. 674–679.Google Scholar
  10. Manning, R.A. and Dyer, C.R. 1999. Interpolating view and scene motion by dynamic view morphing. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Fort Collins, CO, June 1999, pp. 388–394.Google Scholar
  11. Open source computer vision library. Available at Scholar
  12. Shashua, A. and Wolf, L. 2000. Homography tensors: On algebraic entities that represent three views of static or moving planar points. In Proceedings of the European Conference on Computer Vision, Dublin, Ireland, June 2000.Google Scholar
  13. Wexler, Y. and Shashua, A. 2000. On the synthesis of dynamic scenes from reference views. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, South Carolina, June 2000.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Lior Wolf
  • Amnon Shashua

There are no affiliations available

Personalised recommendations