# Order Statistics of RANSAC and Their Practical Application

- 586 Downloads
- 7 Citations

## 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

- Arnold, B. C., Balakrishnan, N., & Nagaraja, H. N. (2008).
*A first course in order statistics*. New York: Wiley.CrossRefzbMATHGoogle Scholar - 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 - Chum, O., & Matas, J. (2008). Optimal randomized RANSAC.
*IEEE Transactions on Pattern Analysis and Machine Intelligence*,*30*(8), 1472–1482.CrossRefGoogle Scholar - Chum, O., Matas, J., & Kittler, J. (2003).
*Locally optimized RANSAC. Lecture notes in computer science*(Vol. 2781, pp. 236–243). Berlin: Springer.Google Scholar - CMP. (2013).
*Czech Technical University, Center for Machine Perception Datasets*. Retrieved February 6, 2013 from http://cmp.felk.cvut.cz/. - 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 - 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 - Hartley, R., & Zisserman, A. (2003).
*Multiple view geometry in computer vision*(2nd ed.). Cambridge, UK Cambridge University Press.Google Scholar - Hughes-Hallett, D., McCallum, W., Gleason, A. et al. (1998).
*Calculus: Single and multivariable*(4th ed.). Wiley.Google Scholar - 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 - 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 - Johnson, N. L., Kemp, A. W., & Kotz, S. (2005).
*Univariate discrete distributions*(3rd ed.). Hoboken, NJ: Wiley.Google Scholar - Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995).
*Continuous univariate distributions*(vol. 2). New York, NY: Wiley.Google Scholar - Lowe, D. G. (2004). Distinctive image features from scale-invariant keypoints.
*International Journal of Computer Vision*,*60*(2), 91–110.CrossRefGoogle Scholar - 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 - Pollefeys, M. (2013).
*Leuven Castle*. Retrieved February 6, 2013, from http://www.cs.unc.edu/marc/data/castlejpg.zip. - Powell, M. J. D. (1970). A hybrid method for nonlinear equations.
*Numerical Methods for Nonlinear Algebraic Equations*,*7*, 87–114.Google Scholar - 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 - 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 - 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 - 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 - VGG. (2013).
*Oxford Visual Geometry Group Datasets*. Retrieved February 6, 2013, from http://www.robots.ox.ac.uk/vgg/data. - Wald, A. (1945). Sequential tests of statistical hypotheses.
*The Annals of Mathematical Statistics*,*16*(2), 117–186.CrossRefzbMATHMathSciNetGoogle Scholar