Affine Arithmetic Based Estimation of Cue Distributions in Deformable Model Tracking

In this paper we describe a statistical method for the integration of an unlimited number of cues within a deformable model framework. We treat each cue as a random variable, each of which is the sum of a large number of local contributions with unknown probability distribution functions. Under the assumption that these distributions are independent, the overall distributions of the generalized cue forces can be approximated with multidimensional Gaussians, as per the central limit theorem. Estimating the covariance matrix of these Gaussian distributions, however, is difficult, because the probability distributions of the local contributions are unknown. We use affine arithmetic as a novel approach toward overcoming these difficulties. It lets us track and integrate the support of bounded distributions without having to know their actual probability distributions, and without having to make assumptions about their properties. We present a method for converting the resulting affine forms into the estimated Gaussian distributions of the generalized cue forces. This method scales well with the number of cues. We apply a Kalman filter as a maximum likelihood estimator to merge all Gaussian estimates of the cues into a single best fit Gaussian. Its mean is the deterministic result of the algorithm, and its covariance matrix provides a measure of the confidence in the result. We demonstrate in experiments how to apply this framework to improve the results of a face tracking system.