From reference frames to reference planes: Multi-view parallax geometry and applications

  • M. Irani
  • P. Anandan
  • D. Weinshall
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1407)


This paper presents a new framework for analyzing the geometry of multiple 3D scene points from multiple uncalibrated images, based on decomposing the projection of these points on the images into two stages: (i) the projection of the scene points onto a (real or virtual) physical reference planar surface in the scene; this creates a virtual “image” on the reference plane, and (ii) the re-projection of the virtual image onto the actual image plane of the camera. The positions of the virtual image points are directly related to the 3D locations of the scene points and the camera centers relative to the reference plane alone. All dependency on the internal camera calibration parameters and the orientation of the camera are folded into homographies relating each image plane to the reference plane.

Bi-linear and tri-linear constraints involving multiple points and views are given a concrete physical interpretation in terms of geometric relations on the physical reference plane. In particular, the possible dualities in the relations between scene points and camera centers are shown to have simple and symmetric mathematical forms. In contrast to the plane+parallax (p+p) representation, which also uses a reference plane, the approach described here removes the dependency on a reference image plane and extends the analysis to multiple views. This leads to simpler geometric relations and complete symmetry in multi-point multiview duality.

The simple and intuitive expressions derived in the reference-plane based formulation lead to useful applications in 3D scene analysis. In particular, simpler tri-focal constraints are derived that lead to simple methods for New View Synthesis. Moreover, the separation and compact packing of the unknown camera calibration and orientation into the 2D projection transformation (a homography) allows also partial reconstruction using partial calibration information.


Multi-point multi-view geometry uncalibrated images new view synthesis duality of cameras and scene points plane+parallax trilinearity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Avidan and A. Shashua. Novel view synthesis in tensor space. In IEEE Conference on Computer Vision and Pattern Recognition, pages 1034–1040, San-Juan, June 1997.Google Scholar
  2. 2.
    S. Carlsson. Duality of reconstruction and positioning from projective views. In Workshop on Representations of Visual Scenes, 1995.Google Scholar
  3. 3.
    H.S.M Coxeter, editor. Projective Geometry. Springer Verlag, 1987.Google Scholar
  4. 4.
    O.D. Faugeras. What can be seen in three dimensions with an uncalibrated stereo rig? In European Conference on Computer Vision, pages 563–578, Santa Margarita Ligure, May 1992.Google Scholar
  5. 5.
    O.D. Faugeras and B. Mourrain. On the geometry and algebra of the point and line correspondences between n images. In International Conference on Computer Vision, pages 951–956, Cambridge, MA, June 1995.Google Scholar
  6. 6.
    Olivier Faugeras. Three-Dimensional Computer Vision — A Geometric Viewpoint. MIT Press, Cambridge, MA, 1996.Google Scholar
  7. 7.
    Richard Hartley. Lines and poins in three views — a unified approach. In DARPA Image Understanding Workshop Proceedings, 1994.Google Scholar
  8. 8.
    Richard Hartley. Euclidean Reconstruction from Uncalibrated Views. In Applications of Invariance in Computer Vision, J.L. Mundy, D. Forsyth, and A. Zisserman (Eds.), Springer-Verlag, 1993.Google Scholar
  9. 9.
    M. Irani and P. Anandan. Parallax geometry of pairs of points for 3d scene analysis. In European Conference on Computer Vision, Cambridge, UK, April 1996.Google Scholar
  10. 10.
    M. Irani, B. Rousso, and S. Peleg. Computing occluding and transparent motions. International Journal of Computer Vision, 12(1):5–16, January 1994.CrossRefGoogle Scholar
  11. 11.
    M. Irani, B. Rousso, and P. Peleg. Recovery of ego-motion using region alignment. IEEE Trans. on Pattern Analysis and Machine Intelligence, 19(3):268–272, March 1997.CrossRefGoogle Scholar
  12. 12.
    R. Kumar, P. Anandan, and K. Hanna. Direct recovery of shape from multiple views: a parallax based approach. In Proc 12th ICPR, 1994.Google Scholar
  13. 13.
    H.C. Longuet-Higgins. A computer algorithm for reconstructing a scene from two projections. Nature, 293:133–135, 1981.CrossRefGoogle Scholar
  14. 14.
    R. Mohr. Accurate Projective Reconstruction In Applications of Invariance in Computer Vision, J.L. Mundy, D. Forsyth, and A. Zisserman, (Eds.), Springer-Verlag, 1993.Google Scholar
  15. 15.
    A. Shashua. Algebraic functions for recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17:779–789, 1995.CrossRefGoogle Scholar
  16. 16.
    A. Shashua and N. Navab. Relative affine structure: Theory and application to 3d reconstruction from perspective views. In IEEE Conference on Computer Vision and Pattern Recognition, pages 483–489, Seattle, Wa., June 1994.Google Scholar
  17. 17.
    A. Shashua and P. Ananadan Trilinear Constraints revisited: generalized trilinear constraints and the tensor brightness constraint. IUW, Feb. 1996.Google Scholar
  18. 18.
    P.H.S. Torr. Motion Segmentation and Outlier Detection. PhD Thesis: Report No. OUEL 1987/93, Univ. of Oxford, UK, 1993.Google Scholar
  19. 19.
    M. Spetsakis and J. Aloimonos. A unified theory of structure from motion. DARPA Image Understanding Workshop, pp.271–283, Pittsburgh, PA, 1990.Google Scholar
  20. 20.
    D. Weinshall, M. Werman, and A. Shashua. Shape descriptors: Bilinear, trilinear and quadlinear relations for multi-point geometry, and linear projective reconstruction algorithms. In Workshop on Representations of Visual Scenes, 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • M. Irani
    • 1
  • P. Anandan
    • 2
  • D. Weinshall
    • 3
  1. 1.Dept. of Applied Math and CSThe Weizmann Inst. of ScienceRehovotIsrael
  2. 2.Microsoft ResearchRedmondUSA
  3. 3.Institute of Computer ScienceHebrew UniversityJerusalemIsrael

Personalised recommendations