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Reading this paper on visual odometry, where they have used a bearing vector to parameterize the features. I am having a hard time understanding what the state propagation equation for the bearing vector term means :

Bearing vector and inverse depth parametrization

The vector N is not mentioned in the equations, so its not very clear what it does. Would really appreciate if someone would help me understand it :)

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The text says:

[...] where $N^T (\mu)$ linearly projects a 3D vector onto the 2D tangent space around the bearing vector $\mu$ [...]

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  • $\begingroup$ Yes, but I am not able to get an intuitive understanding of what that means, plus how the equation comes up with those three terms, i.e. relating bearing vector with w and v and w(u,i) $\endgroup$ – rookie Mar 4 '16 at 14:53
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    $\begingroup$ Maybe you could rephrase your question accordingly. $\endgroup$ – Jakob Mar 8 '16 at 9:42
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It appears to me that the notation represents taking an arbitrary 3-vector, applying a rotation matrix to the 3-vector based on the bearing 3-vector, and then removing one of the dimensions of the result. If the arbitrary 3-vector and the bearing 3-vector are the same, then the result of the rotation should be [ 1 0 0 ], and the first coordinate would be removed, since the projection of that vector onto the plane orthogonal to it is zero. For an arbitrary 3-vector, you get the components of the projection onto a plane that is normal to the bearing 3-vector.

As to how to compute the rotation matrix based on the bearing 3-vector, I would defer to the following:

https://math.stackexchange.com/questions/114512/how-to-find-the-orthonormal-transformation-that-will-rotate-a-vector-to-the-x-ax

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