Heteroscedastic regression in computer vision: problems with bilinear constraint

We present an algorithm to estimate the parameters of a linear model in the presence of heteroscedastic noise, i.e., each data point having a different covariance matrix. The algorithm is motivated by the recovery of bilinear forms, one of the fundamental problems in computer vision which appears whenever the epipolar constraint is imposed, or a conic is fit to noisy data points. We employ the errors-in-variables (EIV) model and show why already at moderate noise levels most available methods fail to provide a satisfactory solution. The improved behavior of the new algorithm is due to two factors: taking into account the heteroscedastic nature of the errors arising from the linearization of the bilinear form, and the use of generalized singular value decomposition (GSVD) in the computations. The performance of the algorithm is compared with several methods proposed in the literature for ellipse fitting and estimation of the fundamental matrix. It is shown that the algorithm achieves the accuracy of nonlinear optimization techniques at much less computational cost.