Closed-form optimization of covariance intersection for low-dimensional matrices

The fusion under unknown correlations is an important technique in sensor-network information processing as the cross-correlations between different estimates remain often unknown to the nodes. Covariance intersection is a widespread and efficient algorithm to fuse estimates under such uncertain conditions. Although different optimization criteria have been developed, the trace or determinant minimization of the fused covariance matrix seems to be most meaningful. However, this minimization requires numeric solutions of a convex optimization problem. We derive an algorithm to reduce this nonlinear optimization to the well-known polynomial root-finding problem. This allows us to present closed-form solutions for the determinant criterion when the dimension of the occurring covariance matrices is at most 4 and for the trace criterion when the dimension of the covariance matrices is at most 3. We demonstrate the effectiveness of the approach by means of a speed evaluation.