Skip to main content
Springer Nature Link
Log in

An Approach to the Vanishing Line Identification Based on Normalized Barycentric Coordinates

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

The vanishing line is useful information for recovering affine properties of the plane in computer vision. This paper describes how to determine analytically the vanishing line from a single perspective view of a plane containing the four points of known normalized barycentric coordinates in a general position, and further how to compute the vanishing line via the eigenvector representation. We also propose that the projectivity may be expressed directly and analytically from the vanishing line and three 3D–2D point correspondences. It is shown that plane affine properties may be computed and the metric may be recovered from known metric information, which includes an angle, two equal but unknown angles, and a length ratio of two non-parallel line segments, without using the image of the circular points as an intermediate step. The correctness and performance of the novel results are demonstrated by thorough testing on both synthetic and real data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Semple, J.G., Kneebone, G.T.: Algebraic projective geometry. Clarendon Press, Oxford (1952)

    MATH  Google Scholar 

  2. Hartley, R., Zisserman, A.: Multiple view geometry in computer vision. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  3. Rothwell, C., Zisserman, A., Mundy, J., Forsyth, D.: Efficient model library access by projectively invariant indexing functions. In: IEEE conference on computer vision and pattern recognition, Champaign, IL, pp. 109–114 (1992)

  4. Slama, C.: Manual of photogrammetry, 4th edn. American Society of Photogrammetry, Falls Church (1980)

    Google Scholar 

  5. Liebowitz, D., Zisserman, A.: Metric rectification for perspective images of planes. In: IEEE conference on computer vision and pattern recognition, Santa Barbara, CA, pp. 482–488 (1998)

  6. Koenderink, J.J., van Doorn, A.J.: Affine structure from motion. J. Opt. Soc. Am. A 8(2), 377–385 (1991)

    Article  Google Scholar 

  7. Faugeras, O.D.: Stratification of three-dimensional vision: projective, affine and metric representation. J. Opt. Soc. Am. A 12(3), 465–484 (1995)

    Article  Google Scholar 

  8. Collins, R.T., Beveridge, J.R.: Matching perspective views of coplanar structures using projective unwarping and similarity matching. In: IEEE conference on computer vision and pattern recognition, New York, NY, pp. 240–245 (1993)

  9. Criminisi, A., Reid, I., Zisserman, A.: Single view metrology. Int. J. Comput. Vis. 40(2), 123–148 (2000)

    Article  MATH  Google Scholar 

  10. Schaffalitzky, F., Zisserman, A.: Planar grouping for automatic detection of vanishing lines and points. Image Vis. Comput. 18, 647–658 (2000)

    Article  Google Scholar 

  11. Wang, G.H., Hu, Z.Y., Wu, F.C.: Single view based measurement on space planes. J. Comput. Sci. Technol. 19(3), 374–382 (2004)

    Article  MathSciNet  Google Scholar 

  12. Se, S.: Zebra-crossing detection for the partially sighted. In: IEEE conference on computer vision and pattern recognition, Hilton Head Island, SC, pp. 211–217 (2000)

  13. Möbius, A.F.: Der barycentrische Calcul. Verlag von Johann Ambrosius Barth, Leipzig (1827)

  14. Coxeter, H.S.M.: Introduction to geometry. Wiley, New York (1969)

    MATH  Google Scholar 

  15. Fauvel, J., Flood, R., Wilson, R.: Möbius and his band: mathematics and astronomy in nineteenth-century Germany. Oxford University Press, England (1993)

    MATH  Google Scholar 

  16. Floater, M.S., Hormann, K., Kós, G.: A general construction of barycentric coordinates over convex polygons. Adv. Comput. Math. 24(1–4), 311–331 (2006)

    Article  MATH  Google Scholar 

  17. Warren, J., Schaefer, S., Hirani, A.N., et al.: Barycentric coordinates for convex sets. Adv. Comput. Math. 27(3), 319–338 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rustamov, R.M.: Interpolated eigenfunctions for volumetric shape processing. Vis. Comput. 27(11), 951–961 (2011)

    Article  Google Scholar 

  19. Haralick, R.M.: Determining camera parameters from the perspective projection of a rectangle. Pattern Recognit. 22(3), 225–230 (1989)

    Article  MathSciNet  Google Scholar 

  20. Penna, M.A.: Determining camera parameters from the perspective projection of a quadrilateral. Pattern Recognit. 24(6), 533–541 (1991)

    Google Scholar 

  21. Horn, B.K.P., Hilden, H.M., Negahdaripour, S.: Closed-form solution of absolute orientation using orthonormal matrices. J. Opt. Soc. Am. A 5(7), 1127–1135 (1988)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author thanks the anonymous referees for valuable suggestions. This work was supported by the Fundamental Research Funds for the Central Universities under Grant No. 100405012.

Author information

Authors and Affiliations

  1. Department of Mathematics, Northeastern University, Shenyang , 110004, P. R. China

    Yang Guo

Authors
  1. Yang Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yang Guo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guo, Y. An Approach to the Vanishing Line Identification Based on Normalized Barycentric Coordinates. J Math Imaging Vis 50, 286–299 (2014). https://doi.org/10.1007/s10851-014-0499-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-014-0499-y

Keywords