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International Journal of Computer Vision

, Volume 111, Issue 3, pp 276–297 | Cite as

Order Statistics of RANSAC and Their Practical Application

  • Evren İmreEmail author
  • Adrian Hilton
Article

Abstract

For statistical analysis purposes, RANSAC is usually treated as a Bernoulli process: each hypothesis is a Bernoulli trial with the outcome outlier-free/contaminated; a run is a sequence of such trials. However, this model only covers the special case where all outlier-free hypotheses are equally good, e.g. generated from noise-free data. In this paper, we explore a more general model which obviates the noise-free data assumption: we consider RANSAC a random process returning the best hypothesis, \(\delta _1\), among a number of hypotheses drawn from a finite set (\(\Theta \)). We employ the rank of \(\delta _1\) within \(\Theta \) for the statistical characterisation of the output, present a closed-form expression for its exact probability mass function, and demonstrate that \(\beta \)-distribution is a good approximation thereof. This characterisation leads to two novel termination criteria, which indicate the number of iterations to come arbitrarily close to the global minimum in \(\Theta \) with a specified probability. We also establish the conditions defining when a RANSAC process is statistically equivalent to a cascade of shorter RANSAC processes. These conditions justify a RANSAC scheme with dedicated stages to handle the outliers and the noise separately. We demonstrate the validity of the developed theory via Monte-Carlo simulations and real data experiments on a number of common geometry estimation problems. We conclude that a two-stage RANSAC process offers similar performance guarantees at a much lower cost than the equivalent one-stage process, and that a cascaded set-up has a better performance than LO-RANSAC, without the added complexity of a nested RANSAC implementation.

Keywords

RANSAC Robust regression Camera calibration Structure-from-motion 

Notes

Acknowledgments

This work is supported by the Technology Strategy Board(TSB) projects i3Dlive: interactive 3D methods for live-action media (TP/11/CII/6/I/AJ307D), “SYMMM: Synchronising Multimodal Movie Metadata”(11702-76150), and the European Commission ICT-7th Framework Program project “IMPART: Intelligent Management Platform for Advanced Real-Time Media Processes” (316564).

References

  1. Arnold, B. C., Balakrishnan, N., & Nagaraja, H. N. (2008). A first course in order statistics. New York: Wiley.CrossRefzbMATHGoogle Scholar
  2. Chum, O., & Matas, J. (2005). Matching with PROSAC-progressive sample consensus. In Proceedings of the 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR) (pp. 220–226).Google Scholar
  3. Chum, O., & Matas, J. (2008). Optimal randomized RANSAC. IEEE Transactions on Pattern Analysis and Machine Intelligence, 30(8), 1472–1482.CrossRefGoogle Scholar
  4. Chum, O., Matas, J., & Kittler, J. (2003). Locally optimized RANSAC. Lecture notes in computer science (Vol. 2781, pp. 236–243). Berlin: Springer.Google Scholar
  5. CMP. (2013). Czech Technical University, Center for Machine Perception Datasets. Retrieved February 6, 2013 from http://cmp.felk.cvut.cz/.
  6. Fischler, M. A., & Bolles, R. C. (1981). Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography. Communications of the ACM, 24(6), 381–395.CrossRefMathSciNetGoogle Scholar
  7. Haralick, R. M., Lee, C.-N., Ottenberg, K., & Nolle, M. (1991). Analysis and the solutions of the three point perspective pose estimation problem. Proceedings CVPR’91 (pp. 592–598).Google Scholar
  8. Hartley, R., & Zisserman, A. (2003). Multiple view geometry in computer vision (2nd ed.). Cambridge, UK Cambridge University Press.Google Scholar
  9. Hughes-Hallett, D., McCallum, W., Gleason, A. et al. (1998). Calculus: Single and multivariable (4th ed.). Wiley.Google Scholar
  10. İmre, E., Guillemaut, J.-Y., & Hilton, A. (2010). Moving camera registration for multiple camera setups in dynamic scenes. In Proceedings of the 21st British Machine Vision Conference (BMVC) (pp. 1–12).Google Scholar
  11. İmre, .E., Guillemaut, J.-Y., & Hilton, A. (2011). Calibration of nodal and free-moving cameras in dynamic scenes for post-production. In Proceedings-2011 International Conference on 3D Imaging, Modeling, Processing, Visualization and Transmission (3DIMPVT) (pp. 260–267).Google Scholar
  12. Johnson, N. L., Kemp, A. W., & Kotz, S. (2005). Univariate discrete distributions (3rd ed.). Hoboken, NJ: Wiley.Google Scholar
  13. Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions (vol. 2). New York, NY: Wiley.Google Scholar
  14. Lowe, D. G. (2004). Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision, 60(2), 91–110.CrossRefGoogle Scholar
  15. Nistér, D. (2003). Preemptive RANSAC for live structure and motion estimation. In Proceedings of the Ninth IEEE International Conference on Computer Vision (ICCV) (pp. 199–206).Google Scholar
  16. Pollefeys, M. (2013). Leuven Castle. Retrieved February 6, 2013, from http://www.cs.unc.edu/marc/data/castlejpg.zip.
  17. Powell, M. J. D. (1970). A hybrid method for nonlinear equations. Numerical Methods for Nonlinear Algebraic Equations, 7, 87–114.Google Scholar
  18. Raguram, R., Frahm, J., & Pollefeys, M. (2009). Exploiting uncertainty in random sample consensus. In Proceedings of the 12th IEEE International Conference on Computer Vision (ICCV) (pp. 2074–2081).Google Scholar
  19. Tordoff, B., & Murray, D. W. (2005). Guided-MLESAC: Faster image transform estimation by using matching priors. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27(10), 1523–1535. Google Scholar
  20. Torr, P. H. S., & Zisserman, A. (2000). MLESAC: A new robust estimator with application to estimating image geometry. Computer Vision and Image Understanding, 78(1), 138–156.CrossRefGoogle Scholar
  21. Tran, Q.-H., Chin, T.-J., Carneiro, G., Brown, M. S., & Suter, D. (2012). In defence of RANSAC for outlier rejection in deformable registration (pp. 274–287).Google Scholar
  22. VGG. (2013). Oxford Visual Geometry Group Datasets. Retrieved February 6, 2013, from http://www.robots.ox.ac.uk/vgg/data.
  23. Wald, A. (1945). Sequential tests of statistical hypotheses. The Annals of Mathematical Statistics, 16(2), 117–186.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.CVSSPUniversity of SurreyGuildfordUK

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