Now we want to relate the motion of the camera frame to the optical flow
pattern which it generates. For simplicity we will concentrate again only
on a single point with 3-D coordinates 
. Assume that the camera
moves with translational velocity 
and the
angular velocity 
and
the observed scene is stationary (see
Figure 5). This is referred to as egomotion.
Figure 5: Showing the motion of the camera relative to a point.
By differentiating the equation for a projective transformation, we find that the optical flow of a point caused by the motion of the camera is:
where 
gives the coordinates of the scene point corresponding
to the image at 
. This equation is complicated, but can be
understood better for the case of pure translation, that is, 
, in which case the flow field becomes:
It is of interest to note that 
at the point 
. This point is called the focus of
expansion. Using this, as the origin, that is defining 
, it follows that
The useful thing about this equation (3 is that it enables us to
determine the time to impact with an object at a distance 
,
away given by 
. You can imagine that this is important for
navigation and also your well being.
Another use of motion flow is in rendering . Here you take multiple 2-D views of a fixed scene and use it to reconstruct the 3-D shape of the object. An example of this is given in Figure (6).
Figure 6: 3-D reconstruction of a house from four photographs taken
from different locations (left) and the photograph of the house taken
from the same location (right). Figure taken from Russell and Norvig,
``Artificial Intelligence - A Modern Approach, Prentice Hall, 1995,
Figure 24.11, page 738
Fancier versions of this are due to Debevec, Taylor and Malik and are available here.
