On the fitting of surfaces to data with covariances

We consider the problem of estimating parameters of a model described by an equation of special form. Specific models arise in the analysis of a wide class of computer vision problems, including conic fitting and estimation o f the fundamental matrix. We assume that noisy data are accompanied by (kno wn) covariance matrices characterizing the uncertainty of the measurements. A cost function is first obtained by considering a maximum-likelihood form ulation and applying certain necessary approximations that render the probl em tractable. A novel, Newton-like iterative scheme is then generated for d etermining a minimizer of the cost function. Unlike alternative approaches such as Sampson's method or the renormalization technique, the new scheme h as as its theoretical limit the minimizer of the cost function. Furthermore , the scheme is simply expressed, efficient, and unsurpassed as a general t echnique in our testing. An important feature of the method is that it can serve as a basis for conducting theoretical comparison of various estimatio n approaches.