Likelihood Functions for Errors-in-Variables Models: Bias-free Local Estimation with Minimum Variance

Parameter estimation in the presence of noisy measurements characterizes a wide range of computer vision problems. Thus, many of them can be formulated as errors-in-variables (EIV) problems. In this paper we provide a closed form likelihood function to EIV problems with arbitrary covariance structure. Previous approaches either do not offer a closed form, are restricted in the structure of the covariance matrix, or involve nuisance parameters. By using such a likelihood function, we provide a theoretical justification for well established estimators of EIV models. Furthermore we provide two maximum likelihood estimators for EIV parameters, a straight forward extension of a well known estimator and a novel, local estimator, as well as confidence bounds by means of the Cramer Rao Lower Bound. We show their performance by numerical experiments on optical flow estimation, as it is well explored and understood in literature. The straight forward extension turned out to have oscillating behavior, while the novel, local one performs favorably with respect to other methods. For small motions, it even performs better than an excellent global optical flow algorithm on the majority of pixel locations