An information theoretic justification for covariance intersection and its generalization

Abstract

A technique for fusing Kalman filter information has been developed by Jeffrey Uhlmann, Simon Julier, and their associates that addresses the problems that arise from fusing correlated measurements. The researchers have named this technique "covariance intersection" and have presented papers on it at several robotics and control theory conferences. The technique is applicable to these research areas because robotic systems often have data flowing between multiple interconnected algorithms with no guarantee that the data flowing into any algorithm are independent. This paper shows that the covariance intersection technique is equivalent to the. log-linear combination of two Gaussian functions and is thus related to Chernoff information. Given this relationship, covariance intersection can be generalized to the fusion of any two probability density functions. A number of different optimization criteria for covariance intersection have been suggested, one being the minimization of the determinant of the fused covariance. This paper shows that this criterion is equivalent to the minimization of the Shannon information of the fused density function. This equivalence provides justification for the selection of the determinant criterion for many applications of covariance intersection. The relationship to Chernoff information suggests that this function may be useful-in other applications.