Items of information a and b are known and are to be fused into information item c. We know a and b have
mean/covariance , and , , but the cross
correlation is not known. The covariance intersection update gives mean and covariance for c as
where ω is computed to minimize a selected norm, e.g., the
trace, or the logarithm of the determinant. While it is necessary to solve an
optimization problem for higher
dimensions,
closed-form solutions exist for lower dimensions.[5]
Application
CI can be used in place of the conventional Kalman update equations to ensure that the resulting estimate is conservative, regardless of the correlation between the two estimates, with covariance strictly non-increasing according to the chosen measure. The use of a fixed measure is necessary for rigor to ensure that a sequence of updates does not cause the filtered
covariance to increase.[1][6]
Advantages
According to a recent survey paper [7] and,[8] the covariance intersection has the following advantages:
The identification and computation of the cross covariances are completely avoided.
It yields a consistent fused estimate, and thus a non-divergent filter is obtained.
The accuracy of the fused estimate outperforms each local one.
It gives a common upper bound of actual estimation
error variances, which has robustness with respect to unknown correlations.
It is widely believed that unknown
correlations exist in a diverse range of
multi-sensor fusion problems. Neglecting the effects of unknown correlations can result in severe performance degradation, and even divergence. As such, it has attracted and sustained the attention of researchers for decades. However, owing to its intricate, unknown nature, it is not easy to come up with a satisfying scheme to address fusion problems with unknown correlations. If we ignore the correlations, which is the so-called "naive fusion",[10] it may lead to filter divergence. To compensate this kind of divergence, a common sub-optimal approach is to artificially increase the system noise. However, this
heuristic requires considerable expertise and compromises the integrity of the Kalman filter framework.[11]
References
^
abUhlmann, Jeffrey (1995). Dynamic Map Building and Localization: New Theoretical Foundations (Ph.D. thesis). University of Oxford.
S2CID47808603.
^Wangyan Li, Zidong Wang, Guoliang Wei, Lifeng Ma, Jun Hu, and Derui Ding. "A Survey on Multi-Sensor Fusion and Consensus Filtering for Sensor Networks." Discrete Dynamics in Nature and Society, vol. 2015, Article ID 683701, 12 pages, 2015.
[1]
^Julier, S.; Uhlmann, J. (2001). Building a Million-Beacon Map. Proceedings of ISAM Conference on Intelligent Systems for Manufacturing.
doi:
10.1117/12.444158.
^Niehsen, W. (2002-07-01). "Information fusion based on fast covariance intersection filtering". Proceedings of the Fifth International Conference on Information Fusion. FUSION 2002. (IEEE Cat.No.02EX5997). Vol. 2. pp. 901–904 vol.2.
doi:
10.1109/ICIF.2002.1020907.
ISBN978-0-9721844-1-0.
S2CID122743543.