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]

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

$C^{-1} = \omega A^{-1} + (1-\omega) B^{-1} \, ,$

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

The weighting coefficient ω is chosen so to have reasonable properties as the sizes of A and B vary: such a consideration gives^[4]

$\omega= \frac{ \operatorname{trace} (B)}{\operatorname{trace}(A) +\operatorname{trace}(B)}.$

Corresponding formulae for the combination of more than two estimates are available.^[4]

References

1. ^ Sonia Marques (2007)"Covariance intersection algorithm for formation flying spacecraft navigation from RF measurements", 4 ISLAB workshop, 12 November 2007
2. ^ Simon J. Julier and Jeffrey K. Uhlmann (2007) "Using covariance intersection for SLAM" , Robotics and Autonomous Systems, 55,(7), 3-20
3. ^ Lingji Chen, Pablo O. Arambel and Raman K. Mehra (2002) "Fusion under unknown correlation - Covariance intersection as a special case", International Society of Information Fusion Conference 2002
4. ^ ^{a b} Niehsen, W., Bosch, R. (2002) Information Fusion based on Fast Covariance Intersection Filtering, International Society of Information Fusion Conference 2002