For any motion, the relation eq:ambiguity implies that the form of is :

where commutes with .

When the rotation component of the displacement is performed around an axis parallel to the basis axes of the camera, takes special forms :

If the rotation axis is ** parallel to the horizontal axis** of the camera :
eq:Sx
Q=(

) and S S_x (

)

If the rotation axis is ** parallel to the vertical axis** of the camera :

In practice, **r** is often negligible in comparison with **k** and we can consider that :
eq:Sy
S S_y (

)

Finally, if the rotation axis is ** orthogonal to the image plane**, is the identity and :
eq:Sz
S S_z (

)

We can observe that, in these 3 cases, the structure of is independent of any ambiguity in the real Jordan decomposition. It will be shown later that these cases correspond to critical motions for affine-to-Euclidean calibration.

Mon Dec 7 13:48:06 GMT 1998