Skip to main content
SpringerLink
Log in

Homography and Fundamental Matrix Estimation from Region Matches Using an Affine Error Metric

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

Abstract

Matching a pair of affine invariant regions between images results in estimation of the affine transformation between the regions. However, the parameters of the affine transformations are rarely used directly for matching images, mainly due to the lack of an appropriate error metric of the distance between them.

In this paper we derive a novel metric for measuring the distance between affine transformations: Given an image region, we show that minimization of this metric is equivalent to the minimization of the mean squared distance between affine transformations of a point, sampled uniformly on the image region. Moreover, the metric of the distance between affine transformations is equivalent to the l 2 norm of a linear transformation of the difference between the six parameters of the affine transformations. We employ the metric for estimating homographies and for estimating the fundamental matrix between images. We show that both homography estimation and fundamental matrix estimation methods, based on the proposed metric, are superior to current linear estimation methods as they provide better accuracy without increasing the computational complexity.

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

Access this article

Log in via an institution

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

Similar content being viewed by others

References

  1. Bentolila, J., Francos, J.M.: Affine consistency graphs for image representation and elastic matching. In: International Conference on Image Processing (2012)

    Google Scholar 

  2. Cho, M., Lee, J., Lee, K.M.: Feature correspondence and deformable object matching via agglomerative correspondence clustering. In: IEEE 12th International Conference on Computer Vision, 2009, pp. 1280–1287 (2009)

    Google Scholar 

  3. Chum, O., Matas, J., Stepan, O.: Epipolar geometry from three correspondences. In: Computer Vision Winter Workshop (2003)

    Google Scholar 

  4. Faugeras, O.: Three-Dimensional Computer Vision: A Geometric Viewpoint. Artificial Intelligence. MIT Press, New York (1993)

    Google Scholar 

  5. Ferrari, V., Tuytelaars, T., Van Gool, L.: Simultaneous object recognition and segmentation from single or multiple model views. Int. J. Comput. Vis. 67(2), 159–188 (2006)

    Article  Google Scholar 

  6. Harris, C., Stephens, M.: A combined corner and edge detection. In: Proceedings of the Fourth Alvey Vision Conference, pp. 147–151 (1988)

    Google Scholar 

  7. Hartley, R., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, New York (2003)

    Google Scholar 

  8. Hartley, R.I.: In defence of the 8-point algorithm. In: Proceedings of the Fifth International Conference on Computer Vision, ICCV ’95, p. 1064. IEEE Computer Society, Washington (1995)

    Chapter  Google Scholar 

  9. Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision Cambridge University Press, Cambridge (2000). ISBN:0521623049

    MATH  Google Scholar 

  10. Kannala, J., Brandt, S.: Quasi-dense wide baseline matching using match propagation. In: IEEE Conference on Computer Vision and Pattern Recognition, CVPR ’07, pp. 1–8 (2007)

    Chapter  Google Scholar 

  11. Lowe, D.G.: Distinctive image features from scale-invariant keypoints. Int. J. Comput. Vis. 60, 91–110 (2004)

    Article  Google Scholar 

  12. Matas, J.: Robust wide-baseline stereo from maximally stable extremal regions. Image Vis. Comput. 22(10), 761–767 (2004)

    Article  Google Scholar 

  13. Mikolajczyk, K., Schmid, C.: An affine invariant interest point detector. In: Proc. European Conf. Computer Vision, pp. 128–142. Springer, Berlin (2002)

    Google Scholar 

  14. Mikolajczyk, K., Schmid, C.: Scale & affine invariant interest point detectors. Int. J. Comput. Vis. 60(1), 63–86 (2004)

    Article  Google Scholar 

  15. Mikolajczyk, K., Tuytelaars, T., Schmid, C., Zisserman, A., Matas, J., Schaffalitzky, F., Kadir, T., Gool, L.V.: A comparison of affine region detectors. Int. J. Comput. Vis. 65, 2005 (2005)

    Article  Google Scholar 

  16. Moravec, H.: Obstacle avoidance and navigation in the real world by a seeing robot rover. Tech. report CMU-RI-TR-80-03. Robotics Institute, Carnegie Mellon University. CMU-RI-TR-80-03 (1980)

  17. Perdoch, M., Matas, J., Chum, O.: Epipolar geometry from two correspondences. In: Proceedings of the 18th International Conference on Pattern Recognition, vol. 04, ICPR ’06, pp. 215–219. IEEE Computer Society, Washington (2006)

    Google Scholar 

  18. Riggi, F., Toews, M., Arbel, T.: Fundamental matrix estimation via tip—transfer of invariant parameters. In: Proceedings of the 18th International Conference on Pattern Recognition, ICPR ’06, vol. 02, pp. 21–24. IEEE, Washington (2006)

    Google Scholar 

  19. Szeliski, R., Torr, P.: Geometrically constrained structure from motion: points on planes. In: 3D Structure from Multiple Images of Large-Scale Environments, pp. 171–186 (1998)

    Chapter  Google Scholar 

  20. Tola, E., Lepetit, V., Fua, P.: A fast local descriptor for dense matching. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 1–8 (2008)

    Google Scholar 

  21. Tola, E., Lepetit, V., Fua, P.: Daisy: an efficient dense descriptor applied to wide-baseline stereo. IEEE Trans. Pattern Anal. Mach. Intell. 32(5), 815–830 (2010)

    Article  Google Scholar 

  22. Tuytelaars, T., Gool, L.V.: Wide baseline stereo matching based on local, affinely invariant regions. In: Proc. BMVC, pp. 412–425 (2000)

    Google Scholar 

  23. Tuytelaars, T., Mikolajczyk, K.: Local invariant feature detectors: a survey. Found. Trends Comput. Graph. Vis. 3(3), 177–280 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Electrical & Computer Engineering Department, Ben Gurion University, P.O.B 653, Beer-Sheva, 84105, Israel

    Jacob Bentolila & Joseph M. Francos

Authors
  1. Jacob Bentolila
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Joseph M. Francos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jacob Bentolila.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bentolila, J., Francos, J.M. Homography and Fundamental Matrix Estimation from Region Matches Using an Affine Error Metric. J Math Imaging Vis 49, 481–491 (2014). https://doi.org/10.1007/s10851-013-0481-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-013-0481-0

Keywords

Search

Navigation