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I read many sources about kalman filter, yet no about the other approach to filtering, where canonical parametrization instead of moments parametrization is used.

What is the difference?


Other questions:

  1. Using IF I can forget KF,but have to remember that prediction is more complicated link

  2. How can I imagine uncertainty matrix turning into an ellipse? (generally I see, area is uncertainty, but I mean boundaries)

  3. Simple addition of information in IF was possible only under assumption that each sensor read a different object? (hence no association problem, which I posted here

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    $\begingroup$ You might pick up the "Probabilistic Robotics" Book from Thrun et. al. to learn about the information filter, especially in the context of robotics. If you have specific questions, feel free to ask them here :) $\endgroup$ – Jakob Apr 9 '13 at 17:44
  • $\begingroup$ I think he had a good question: What's the difference? So I answered that. It's good for reference, I think, and also because Thrun's book isn't that great and is expensive. $\endgroup$ – Josh Vander Hook Apr 10 '13 at 4:35
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They are exactly the same. Information matricies (aka precision matricies) are the inverse of covariance matricies. Follow this. The covariance update $$P_{+} = (I-KH)P$$ can be expanded by the definition of $K$ to be

$$ P_{+} = P - KHP $$ $$ P_{+} = P - PH^T (HPH^T+R)^{-1} HP $$

Now apply the matrix inversion lemma, and we have:

$$ P_{+} = P - PH^T (HPH^T+R)^{-1} HP $$ $$ P_{+} = (P^{-1} + H^TR^{-1}H)^{-1} $$

Which implies: $$ P_{+}^{-1} = P^{-1} + H^TR^{-1}H $$

The term $P^{-1}$ is called the prior information, $H^TR^{-1}H$ is the sensor information (inverse of sensor variance), and this gives us $P^{-1}_+$, which is the posterior information.

I'm glossing over the actual state estimate, but it's straightforward. The best intro I've seen on this is not Thrun's book, but Ben Grocholsky's PhD thesis. (Just the intro material). It's called Information Theoretic Control of Multiple Sensor Platforms. Here's a link.


EDITS

To answer the questions posed.

  1. It is not more complicated to predict, it is more computationally costly, since you must invert the $n \times n$ covariance matrix to get the true state output.

  2. To view the ellipse from a covariance matrix, just note that the covariance matrix has a nice Singular Value Decomposition. The square root of the eigenvalues of the ellipse, or the square root of the singular values of the ellipse, will define the principal axes of the ellipse.

  3. No, addition of information depends only on the assumption of independence of measurement noise. If you want to use two information filters to track two objects, that's fine. Or if you want to use an IF to track two objects, that's also fine. All you need is the correct association of measurements, so that you know which part of the state (object 1 or object 2) to update.

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  • $\begingroup$ nice explanation in thesis! For some questions: 1.Using IF I can forget KF,but have to remember that prediction is more complicated link 2. How can I imagine uncertainty matrix turning into an ellipse? (generally I see, area is uncertainty, but I mean boundaries) 3. Simple addition of information in IF was possible only under assumption that each sensor read a different object? (hence no association problem, which I posted here $\endgroup$ – josh131 Apr 10 '13 at 12:09
  • $\begingroup$ I addressed your questions. I also added them to your original question, since that's the preferred way of asking related questions. $\endgroup$ – Josh Vander Hook Apr 10 '13 at 14:02
  • $\begingroup$ Perfect, and sorry didn't think of updating the main question $\endgroup$ – josh131 Apr 11 '13 at 8:10
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The difference between kalman filter and information filter arise in there Gaussian belief representation. In kalman filter Gaussian belief represented by their moments(mean and covariance). Information filters represent Gaussians in their canonical representation, which is comprised of an information matrix and an information vector.

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Just a short note- There is a significant difference between information filters and Klaman filters. Although they are mathematically similar (inverses of each other), marginalization is simple in Kalman filters and complicated in information filters. However, smoothing is simpler in information filters while it is complicated in Kalman filters. Since modern state estimation techniques tend to employ smoothing to mitigate the effects of non-linearities, enhance precision and allow loop closures information filters are now on the rise.

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