Order Statistics of RANSAC and Their Practical Application

• Published in: International journal of computer vision Vol. 111; no. 3; pp. 276 - 297
• Main Authors: İmre, Evren, Hilton, Adrian
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-02-2015; Springer; Springer Nature B.V
• Subjects: Analysis; Approximation; Artificial Intelligence; Bernoulli Hypothesis; Calibration; Cameras; Cascades; Computer Imaging; Computer Science; Computer simulation; Equivalence; Exact solutions; Generators; Hypotheses; Image Processing and Computer Vision; Image processing systems; Mathematical analysis; Monte Carlo method; Monte Carlo methods; Monte Carlo simulation; Noise; Pattern Recognition; Pattern Recognition and Graphics; Random variables; Regression analysis; Statistics; Studies; Vision; RANSAC; Camera calibration; Robust regression; Structure-from-motion
• ISSN: 0920-5691; 1573-1405
• DOI: 10.1007/s11263-014-0745-1

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,
δ
1
, among a number of hypotheses drawn from a finite set (
Θ
). We employ the rank of
δ
1
within
Θ
for the statistical characterisation of the output, present a closed-form expression for its exact probability mass function, and demonstrate that
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
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.