Homography estimation and heteroscedastic noise - a first order perturbation analysis

Abstract

It is well known that one can collect the coefficients of the homgraphies between two views into a large, rank deficient matrix. In principle, such an observation implies that one can refine the accuracy of the estimates of the homography coefficients by exploiting the rank constraint. However, the straightforward approach suggested by this observation is impractical because it requires many homographies and it also does not take into account correlations between the errors in the coefficients. In a companion paper [4], we show how to jointly estimate multiple (but a realistic number of) homographies over 2 views. By studying the special structure of the homography, we show that it is possible to calculate the dimension 4 subspace of the homographies from ≥ 3 planes (and, in principle, with even two planes). This contradicts what seems to be the accepted situation regarding the exploitation of the rank-4 constraint amongst homographies: that more than 4 planes are needed to calculate and exploit the dimension 4 subspace. Practical issues arise because the homography coefficients, before rank-constrained refinement, are themselves estimates whose noise covariances need to be characterised and accounted for. In this paper, we develop a statistical analysis allowing for estimation of the covariance matrices required for the calculation of low-rank “denoised” homographies