On Errors-In-Variables Regression with Arbitrary Covariance and its Application to Optical Flow Estimation

Linear inverse problems in computer vision, including motion estimation, shape fitting and image reconstruction, give rise to parameter estimation problems with highly cor-related errors in variables. Established total least squares methods estimate the most likely corrections A ̂ and b ̂ to a given data matrix [A, b] perturbed by additive Gaussian noise, such that there exists a solution y with [A + Â, b + b̂]y = 0. In practice, regression imposes a more restric-tive constraint namely the existence of a solution x with [A + Â]x = [b + b̂]. In addition, more complicated corre-lations arise canonically from the use of linear filters. We, therefore, propose a maximum likelihood estimator for re-gression in the general case of arbitrary positive definite covariance matrices. We show that Â, b ̂ and x can be found simultaneously by the unconstrained minimization of a mul-tivariate polynomial which can, in principle, be carried out by means of a Gröbner basis. Results for plane fitting and optical flow computation indicate the superiority of the pro-posed method. 1