Order Statistics of RANSAC and Their Practical Application

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, $$\delta _1$$δ1, among a number of hypotheses drawn from a finite set ($$\Theta $$Θ). We employ the rank of $$\delta _1$$δ1 within $$\Theta $$Θ for the statistical characterisation of the output, present a closed-form expression for its exact probability mass function, and demonstrate that $$\beta $$β-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 $$\Theta $$Θ 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.