Enforcing consistency constraints in uncalibrated multiple homography estimation using latent variables

Abstract

Springer is a green publisher, as we allow self-archiving, but most importantly we are fully transparent about your rights. Abstract An approach is presented for estimating a set of interdependent homography matrices linked together by latent variables. The approach allows enforcement of all underlying consistency constraints while accounting for the arbitrariness of the scale of each individual matrix. The input data is assumed to be in the form of a set of homography matrices individually estimated from image data with no regard to the consistency constraints, appended by a set of error covariances, each characterising the uncertainty of a corresponding homography matrix. A statistically motivated cost function is introduced for upgrading, via optimisation, the input data to a set of homography matrices satisfying the constraints. The function is invariant to a change of any of the individual scales of the input matrices. The proposed approach is applied to the particular problem of estimating a set of homography matrices induced by multiple planes in the 3D scene between two views. An optimisation algorithm for this problem is developed that operates on natural underlying latent variables, with the use of those variables ensuring that all consistency constraints are satisfied. Experimental results indicate that the algorithm outperforms previous schemes proposed for the same task and is fully comparable in accuracy with the 'gold standard' bundle adjustment technique, rendering the whole approach both of practical and theoretical interest. With a view to practical application, it is shown that the proposed algorithm can be incorporated into the familiar random sampling and consensus technique, so that the resulting modified scheme is capable of robust fitting of fully consistent homographies to data with outliers. Publishing in a subscription-based journal

Robust multiple homography estimation: An ill-solved problem

“…However, just because the residual error for consistent homographies may be slightly higher does not mean that consistent homographies are less accurate. On the contrary, experiments conducted in [8,9,32] demonstrate that enforcing consistency results in substantial improvements in accuracy. The apparent contradiction can be resolved with recourse to the machine learning concepts of training error-in our case, the geometric error for the set of corresponding points on which the homography is estimated-and generalisation error-in our case, the geometric error evaluated on other corresponding points that belong to the same planar structure.…”

“…Extensive simulations on synthetic data reported in [5,8,9,32,36] have already demonstrated that enforcing consistency constraints leads to more accurate homography estimates. Therefore rather than repeat similar simulations, we instead chose to investigate whether compatibility constraints are violated on typical real-world datasets.…”

“…Following this path, Chojnacki et al [8,9] employed an appropriate parametrisation and a distinct cost function to develop an upgrade procedure based on unconstrained optimisation. Szpak et al [32] used the same parametrisation and the Sampson distance to develop an alternative technique with a sound statistical basis.…”

“…Furthermore, finding suitable initial values for the latent variables is a non-trivial task. The initialisation methods described by Chojnacki et al [8,9] and Szpak et al [32] are based on factorising a collection of homography matrices. The factorisation methods are sensitive to noise and are not robust to outliers.…”

Full Explicit Consistency Constraints in Uncalibrated Multiple Homography Estimation

“…Furthermore, finding suitable initial values for the latent variables is a non-trivial task. The initialisation methods utilised by Chojnacki et al [10,11] and Szpak et al [35] are based on factorising a collection of homography matrices. The factorisation procedure is described in detail in [11,Sect.…”