Order Statistics of RANSAC and Their Practical Application

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.

Appendix: RANSAC and Nonlinear Minimisation

The existing literature evaluates RANSAC as a standalone estimator. However, in practice, RANSAC is often employed as a component within larger pipelines, for generating an initial estimate for a subsequent nonlinear optimisation stage (Hartley and Zisserman 2003). When used for this purpose, RANSAC only needs to find a hypothesis somewhere in a good basin; the nonlinear optimisation stage can take the solution to the local minimum. This raises an interesting question: how is the final, post-optimisation performance affected by the choice of RANSAC variant?

As a motivating example, we repeated the experiments in Sect. 6.1 for P2P, P3P, H4 and F7, with the parameters $\sigma _f^2=1, \sigma _s^2=2.5\times 10^{-5}$, and $\rho =0.5$. However, in each case, RANSAC was followed by a refinement via Powell’s dog leg (PDL) algorithm (Powell 1970). The median validation errors after the application of PDL are reported in Table 4. The results indicate that PDL acts as an equaliser, mostly eliminating the performance differences observed in Sect. 6. However, a poorer initial estimate means more PDL iterations. Therefore, using B-RANSAC instead of, for instance, TS-RANSAC involves a trade-off between RANSAC and PDL iterations. Since the computational cost of the minimal solvers utilised in this paper are relatively cheap, compared to the operations involved in nonlinear minimisation (e.g. computation of the Jacobian for each correspondence), such a trade-off is not advisable. However, for more expensive hypothesis generators, it may be beneficial to consider RANSAC and the nonlinear optimisation stage jointly.

Table 4 Median validation error after nonlinear minimisation