# Order Statistics of RANSAC and Their Practical Application

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## 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).

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