Geometrical Properties of Quasi-Perspective Projection

  • Guanghui Wang
  • Q. M. Jonathan Wu
Part of the Advances in Pattern Recognition book series (ACVPR)


The chapter investigates geometrical properties of quasi-perspective projection model in one and two-view geometry. The main results are as follows. (i) Quasi-perspective projection matrix has nine degrees of freedom, and the parallelism alongX andY directions in world system are preserved in images. (ii) Quasi-fundamental matrix can be simplified to a special form with only six degrees of freedom. The fundamental matrix is invariant to any non-singular projective transformation. (iii) Plane induced homography under quasi-perspective model can be simplified to a special form defined by six degrees of freedom. The quasi-homography may be recovered from two pairs of corresponding points with known fundamental matrix. (iv) Any two reconstructions in quasi-perspective space are defined up to a non-singular quasi-perspective transformation.


Fundamental Matrix Stereo Vision Camera Parameter Perspective Projection Epipolar Geometry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Cheng, C.M., Lai, S.H.: A consensus sampling technique for fast and robust model fitting. Pattern Recogn.42(7), 1318–1329 (2009) MATHCrossRefGoogle Scholar
  2. 2.
    Dellaert, F., Seitz, S.M., Thorpe, C.E., Thrun, S.: Structure from motion without correspondence. In: Proc. of IEEE Conference on Computer Vision and Pattern Recognition, pp. 2557–2564 (2000) Google Scholar
  3. 3.
    Faugeras, O.: Stratification of 3-D vision: projective, affine, and metric representations. J. Opt. Soc. Am. A12, 465–484 (1995) CrossRefGoogle Scholar
  4. 4.
    Fischler, M.A., Bolles, R.C.: Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography. Commun. ACM24(6), 381–395 (1981) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Guilbert, N., Bartoli, A., Heyden, A.: Affine approximation for direct batch recovery of Euclidean structure and motion from sparse data. Int. J. Comput. Vis.69(3), 317–333 (2006) CrossRefGoogle Scholar
  6. 6.
    Hartley, R.I.: In defense of the eight-point algorithm. IEEE Trans. Pattern Anal. Mach. Intell.19(6), 580–593 (1997) CrossRefGoogle Scholar
  7. 7.
    Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2004). ISBN: 0521540518 MATHCrossRefGoogle Scholar
  8. 8.
    Hu, M., McMenemy, K., Ferguson, S., Dodds, G., Yuan, B.: Epipolar geometry estimation based on evolutionary agents. Pattern Recogn.41(2), 575–591 (2008) MATHCrossRefGoogle Scholar
  9. 9.
    Lehmann, S., Bradley, A.P., Clarkson, I.V.L., Williams, J., Kootsookos, P.J.: Correspondence-free determination of the affine fundamental matrix. IEEE Trans. Pattern Anal. Mach. Intell.29(1), 82–97 (2007) CrossRefGoogle Scholar
  10. 10.
    Mendonça, P.R.S., Cipolla, R.: Analysis and computation of an affine trifocal tensor. In: Proc. of British Machine Vision Conference, pp. 125–133 (1998) Google Scholar
  11. 11.
    Mundy, J.L., Zisserman, A.: Geometric Invariance in Computer Vision. MIT Press, Cambridge (1992) Google Scholar
  12. 12.
    Oliensis, J., Hartley, R.: Iterative extensions of the Sturm/Triggs algorithm: Convergence and nonconvergence. IEEE Trans. Pattern Anal. Mach. Intell.29(12), 2217–2233 (2007) CrossRefGoogle Scholar
  13. 13.
    Torr, P.H.S., Zisserman, A., Maybank, S.J.: Robust detection of degenerate configurations while estimating the fundamental matrix. Comput. Vis. Image Underst.71(3), 312–333 (1998) CrossRefGoogle Scholar
  14. 14.
    Poelman, C., Kanade, T.: A paraperspective factorization method for shape and motion recovery. IEEE Trans. Pattern Anal. Mach. Intell.19(3), 206–218 (1997) CrossRefGoogle Scholar
  15. 15.
    Shapiro, L.S., Zisserman, A., Brady, M.: 3D motion recovery via affine epipolar geometry. Int. J. Comput. Vis.16(2), 147–182 (1995) CrossRefGoogle Scholar
  16. 16.
    Shimshoni, I., Basri, R., Rivlin, E.: A geometric interpretation of weak-perspective motion. IEEE Trans. Pattern Anal. Mach. Intell.21(3), 252–257 (1999) CrossRefGoogle Scholar
  17. 17.
    Wang, G.: A hybrid system for feature matching based on sift and epipolar constraints. Tech. Rep. Department of ECE, University of Windsor (2006) Google Scholar
  18. 18.
    Wang, G., Wu, J.: Quasi-perspective projection with applications to 3D factorization from uncalibrated image sequences. In: Proc. of IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8 (2008) Google Scholar
  19. 19.
    Wang, G., Wu, J.: Perspective 3D Euclidean reconstruction with varying camera parameters. IEEE Trans. Circuits Syst. Video Technol.19(12), 1793–1803 (2009) CrossRefGoogle Scholar
  20. 20.
    Wang, G., Wu, J.: The quasi-perspective model: Geometric properties and 3D reconstruction. Pattern Recogn.43(5), 1932–1942 (2010) MATHCrossRefGoogle Scholar
  21. 21.
    Wang, G., Wu, J.: Quasi-perspective projection model: Theory and application to structure and motion factorization from uncalibrated image sequences. Int. J. Comput. Vis.87(3), 213–234 (2010) CrossRefGoogle Scholar
  22. 22.
    Weng, J., Huang, T., Ahuja, N.: Motion and structure from two perspective views: Algorithms. IEEE Trans. Pattern Anal. Mach. Intell.11(5), 451–476 (1997) CrossRefGoogle Scholar
  23. 23.
    Wolf, L., Shashua, A.: Affine 3-D reconstruction from two projective images of independently translating planes. In: Proc. of International Conference on Computer Vision, pp. 238–244 (2001) Google Scholar
  24. 24.
    Zhang, Z.: A flexible new technique for camera calibration. IEEE Trans. Pattern Anal. Mach. Intell.22(11), 1330–1334 (2000) CrossRefGoogle Scholar
  25. 25.
    Zhang, Z., Anandan, P., Shum, H.Y.: What can be determined from a full and a weak perspective image? In: Proc. of International Conference on Computer Vision, pp. 680–687 (1999) Google Scholar
  26. 26.
    Zhang, Z., Kanade, T.: Determining the epipolar geometry and its uncertainty: A review. Int. J. Comput. Vis.27(2), 161–195 (1998) CrossRefGoogle Scholar
  27. 27.
    Zhang, Z., Xu, G.: A general expression of the fundamental matrix for both projective and affine cameras. In: Proc. of International Joint Conference on Artificial Intelligence, pp. 1502–1507 (1997) Google Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

There are no affiliations available

Personalised recommendations