Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Understanding the Behavior of SFM Algorithms: A Geometric Approach

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

We put forth in this paper a geometrically motivated motion error analysis which is capable of supporting investigation of global effect such as inherent ambiguities. This is in contrast with the usual statistical kinds of motion error analyses which can only deal with local effect such as noise perturbations, and where much of the results regarding global ambiguities are empirical in nature. The error expression that we derive allows us to predict the exact conditions likely to cause ambiguities and how these ambiguities vary with motion types such as lateral or forward motion. Given the erroneous 3-D motion estimates caused by the inherent ambiguities, it is also important to study the behavior of the resultant distortion in depth recovered under different motion-scene configurations. Such an investigation may alert us to the occurrence of ambiguities under different conditions and be more careful in picking the solution. Our formulation, though geometrically motivated, was also put to use in modeling the effect of noise and in revealing the strong influence of feature distribution. Experiments on both synthetic and real image sequences were conducted to verify the various theoretical predictions.

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

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  • Adiv, G. 1985. Determining 3-D motion and structure from optical flow generated by several moving objects. IEEE Trans. PAMI, 7:384–401.

    Google Scholar 

  • Adiv, G. 1989. Inherent ambiguities in recovering 3-D motion and structure from a noisy flow field. IEEE Trans. PAMI, 11:477–489.

    Google Scholar 

  • Brooks, M.J., Chojnacki, W., and Baumela, L. 1997. Determining the egomotion of an uncalibrated camera from instantaneous optical flow. Journal of the Optical Society of America A, 14(10):2670– 2677.

    Google Scholar 

  • Brooks, M.J., Chojnacki, W., Hengel, A.V.D., and Baumela, L. 1998. Robust techniques for the estimation of structure from motion in the uncalibrated case. In Proc. Conf. ECCV, pp. 283–295.

  • Cheong, L.-F., Fermüller, C., and Aloimonos, Y. 1998. Effects of errors in the viewing geometry on shape estimation. Computer Vision and Image Understanding, 71(3):356–372.

    Google Scholar 

  • Cheong, L.-F. and Ng, K. 1999. Geometry of distorted visual space and cremona transformation. International Journal of Computer Vision, 32(2):195–212.

    Google Scholar 

  • Cheong, L.-F. and Peh, C.H. 2000. Characterizing depth distortion due to calibration uncertainty. In Proc. Conf. ECCV, Dublin, Ireland, vol. I, pp. 664–677.

    Google Scholar 

  • Cheong, L.-F. and Xiang, T. 2001. Characterizing depth distortion under different generic motions. International Journal of Computer Vision, 44(3):199–217.

    Google Scholar 

  • Chiuso, A., Brockett, R., and Soatto, S. 2000. Optimal structure from motion: Local ambiguities and global estimates. International Journal of Computer Vision, 39(3):195–228.

    Google Scholar 

  • Daniilidis, K. and Spetsakis, M.E. 1997. Understanding noise sensitivity in structure from motion. In Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, Y. Aloimonos (Ed.), Lawrence Erlbaum: Hillsdale, NJ.

    Google Scholar 

  • Dutta, R. and Snyder, M.A. 1990. Robustness of correspondencebased structure from motion. In Proc. Int. Conf. on Computer Vision, Osaka, Japan, pp. 106–110.

  • Fermüller, C. and Aloimonos, Y. 2000. Observability of 3D motion. International Journal of Computer Vision, 37(1):43–63.

    Google Scholar 

  • Fermüller, C., Shulman, D., and Aloimonos, Y. 2001. The statistics of optical flow. Computer Vision and Image Understanding, 82:1– 32.

    Google Scholar 

  • Grossmann, E. and Victor, J.S. 2000. Uncertainty analysis of 3D reconstruction from uncalibrated views. Image and Vision Computing, 18(9):686–696.

    Google Scholar 

  • Heeger, D.J. and Jepson, A.D. 1992. Subspace methods for recovering rigid motion I: Algorithm and implementation. International Journal of Computer Vision, 7:95–117.

    Google Scholar 

  • Hoffman, D.D. 1998. Visual intelligence, W.W. Norton and Company, Inc.

  • Horn, B.K.P. 1987. Motion fields are hardly ever ambiguous. International Journal of Computer Vision, 1:259–274.

    Google Scholar 

  • Horn, B.K.P. 1990. Relative orientation. International Journal of Computer Vision, 4:59–78.

    Google Scholar 

  • Kahl, F. 1995. Critical motions and ambiguous Euclidean reconstructions in auto-calibration. In Proc. Int. Conf. on Computer Vision, pp. 469–475.

  • Kanatani, K. 1993. 3-D interpretation of optical flowby renormalization. International Journal of Computer Vision, 11(3):267–282.

    Google Scholar 

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

    Google Scholar 

  • Koenderink, J.J. and van Doorn, A.J. 1995. Relief: Pictorial and otherwise. Image and Vision Computing, 13(5):321–334.

    Google Scholar 

  • Longuet-Higgins, H.C. 1981. A computer algorithm for reconstruction of a scene from two projections. Nature, 293:133– 135.

    Google Scholar 

  • Longuet-Higgins, H.C. 1984. The visual ambiguity of a moving plane. Proc. R. Soc. Lond., B233:165–175.

    Google Scholar 

  • Lucas, B.D. 1984. Generalized image matching by the method of differences. Ph.D. Dissertation, Carnegie-Mellon University.

  • Ma, Y., Koŝecká, J., and Sastry, S. 2000. Linear differential algorithm for motion recovery:Ageometric approach. International Journal of Computer Vision, 36(1):71–89.

    Google Scholar 

  • Ma, Y., Koŝecká, J., and Sastry, S. 2001. Optimization criteria and geometric algorithms for motion and structure estimation. International Journal of Computer Vision, 44(3):219–249.

    Google Scholar 

  • Maybank, S.J. 1993. Theory of Reconstruction from Image Motion. Springer: Berlin.

    Google Scholar 

  • Negahdaripour, S. Critical surface pairs and triplets. International Journal of Computer Vision, 3:293–312.

  • Oliensis, J. 2000a. A critique of structure-from-motion algorithms. Computer Vision and Image Understanding, 80:172–214.

    Google Scholar 

  • Oliensis, J. 2000b. A new structure from motion ambiguity. IEEE Trans. PAMI, 22(7):685–700.

    Google Scholar 

  • Oliensis, J. 2001. The error surface for structure from motion.NECI Technical Report, available at http://www.neci.nec.com/ homepages/oliensis/.

  • Prazdny, K. 1980. Egomotion and relative depth map from optical flow. Biological Cybernetics, 36:87–102.

    Google Scholar 

  • Rieger, J.H. and Lawton, D.T. 1985. Processing differential image motion. J. Optic. Soc. Am. A, 2:354–359.

    Google Scholar 

  • Soatto, S. and Brockett, R. 1998. Optimal structure from motion: Local ambiguities and global estimates. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition, pp. 282–288.

  • Spetsakis, M.E. 1994. Models of statistical visual motion estimation. CVGIP, 60:300–312.

    Google Scholar 

  • Sturm, P. 1997. Critical motion sequences for monocular selfcalibration and uncalibrated Euclidean reconstruction. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition, pp. 1100–1105.

  • Szeliski, R. and Kang, S.B. 1997. Shape ambiguities in structure from motion. IEEE Trans. PAMI, 19(5):506–512.

    Google Scholar 

  • Weng, J., Huang, T.S., and Ahuja, N. 1991. Motion and Structure from Image Sequences, Springer-Verlag: Berlin.

    Google Scholar 

  • Xiang, T. 2001. Understanding the behavior of structure from motion algorithms: A geometric approach. Ph. D. Dissertation, National University of Singapore.

  • Young, G.S. and Chellapa, R. 1992. Statistical analysis of inherent ambiguities in recovering 3-D motion from a noisy flow field. IEEE Trans. PAMI, 14:995–1013.

    Google Scholar 

  • Zhang, Z. 1998. Understanding the relationship between the optimization criteria in two-view motion analysis. In Proc. Conf. ICCV, pp. 772–777.

  • Zhang, T. and Tomasi, C. 1999. Fast, robust, and consistent camera motion estimation. In Proc. IEEE Conf. on Computer Vision and Pattern Recognition, pp. 164–170.

Download references

Author information

Authors and Affiliations

  1. Electrical and Computer Engineering Department, National University of Singapore, 10 Kent Ridge Crescent, Singapore, 119260

    Tao Xiang & Loong-Fah Cheong

Authors
  1. Tao Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Loong-Fah Cheong
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xiang, T., Cheong, LF. Understanding the Behavior of SFM Algorithms: A Geometric Approach. International Journal of Computer Vision 51, 111–137 (2003). https://doi.org/10.1023/A:1021627622971

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021627622971