Fusion under unknown correlation - covariance intersection as a special case

This paper addresses the problem of fusing several random variables (RVs) with unknown correlations. A family of upper bounds on the resulting covariance matrix is given, and is shown to contain the upper bound offered by the covariance intersection (CI) algorithm proposed by Julier and Uhlmann (2000). For trace minimization, the optimal one in this family is better than the one obtained by CI except in some cases where they are equal. It is further proved that the best pair of combination gains that minimizes the above optimal-trace-in-the-family coincides with the one associated with the best value of omega in CL. Thus, the CI Algorithm provides a convenient one-dimensional parameterization for the optimal solution in the n-square dimensional space. The results are also extended to multiple RVs and partial estimates.