Dual Representation of Gaussians

Moments parameterization with mean $\mu$ and covariance matrix $\Sigma$

\[p(x) = \det(2 \pi \Sigma)^{-\frac{1}{2}} \exp(-\frac{1}{2} (x-\mu)^T \Sigma^{-1} (x-\mu))\]

Canonical parameterization with information vector $\xi$ and information matrix $\Omega$

\[p(x) = \frac{\exp(-\frac{1}{2} \xi^T (\Omega^{-1})^T \xi)}{\det(2\pi \Omega^{-1})^{\frac{1}{2}}} \exp(-\frac{1}{2} x^T \Omega x + x^T \xi)\]

Moments to canonical

\[\Omega = \Sigma^{-1}\] \[\xi = \Sigma^{-1} \mu\]

Canonical to moments

\[\Sigma = \Omega^{-1}\] \[\mu = \Omega^{-1} \xi\]

Marginalization and conditioning

Marginalization is cheap for moment parameterization whereas conditioning is cheap for canonical parameterization; Contioning is expensive for moment parameterization whereas marginalization is expensive for canonical parameterization.

Extended Inofrmation Filter (EIF) SLAM

EIF vs. EKF

• Same expressiveness as the EKF
• Prediction step is more costly, Correction step is cheaper

Literature

Extended Information Filter, Thrun et al.: “Probabilistic Robotics”, Chapter 3.5

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