Covariance intersection

Covariance intersection is an algorithm for combining two or more estimates of state variables in a Kalman filter when the correlation between them is unknown.[1][2][3][4]

Specification

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 $\hat a$, $A$ and $\hat b$, $B$, but the cross correlation is not known. The covariance intersection update gives mean and covariance for c as

$C^{-1} = \omega A^{-1} + (1-\omega) B^{-1} \, ,$
$\hat c = C(\omega A^{-1} \hat a + (1-\omega)B^{-1} \hat b) \, .$

where ω is computed to minimize a selected norm, e.g., logdet or trace. While it is necessary to solve an optimization problem for higher dimensions, closed-form solutions exist for lower dimensions.[5] 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]

References

1. ^ a b Uhlmann, Jeffrey (1995). Dynamic Map Building and Localization: New Theoretical Foundations (Ph.D. thesis). University of Oxford.
2. ^ Marques, Sonia (12 November 2007). "Covariance intersection algorithm for formation flying spacecraft navigation from RF measurements". 4 ISLAB workshop.
3. ^ Julier, Simon J.; Uhlmann, Jeffrey K. (2007). "Using covariance intersection for SLAM". Robotics and Autonomous Systems 55 (7): 3–20. doi:10.1016/j.robot.2006.06.011. CiteSeerX: 10.1.1.106.8515.
4. ^ Chen, Lingji; Arambel, Pablo O.; Mehra, Raman K. (2002). "Fusion under unknown correlation - Covariance intersection as a special case". International Conference on Information Fusion 2002.
5. ^ Reinhardt, Marc; Noack, Benjamin; Hanebeck, Uwe D. (2012). "Closed-form Optimization of Covariance Intersection for Low-dimensional Matrices". International Conference on Information Fusion 2012.
6. ^ Uhlmann, Jeffrey (2003). Covariance Consistency Methods for Fault-Tolerant Distributed Data Fusion 4. Elsevier. pp. 201–215.