Now we have an energy, which describes contour point candidates for each ray (external energy), and an energy, which connects the rays to get a smooth contour (internal energy). Then, similar to active contours we define a total energy E
Now, the contour extraction can be described as an energy minimization problem. Using the variational approach the Euler--Lagrange differential equation
must be solved. Again, this differential equation depends on a 1D function, in contrast to the same differential equation for active contours.
Before we discuss different kinds of judgement functions in
5, especially on
those, which can locate boundaries of textured regions,
some remarks must be done regarding
the reference point 
. As already mentioned, this point must
lie inside the object contour, but the position
may be arbitrary. We are interested in object tracking. Usually, a prediction
step can improve and stabilize tracking over time.
For such a prediction step, a unique position of the reference point 
would be of great advantage. In the following we always chose
the center of gravity of the contour
as the reference point. If equation (12)
does not hold for an actual
reference point and an extracted contour, we update the reference point
using equation (12), and restart the contour extraction
with the updated reference point's position.
As a result, for a certain contour a unique reference point can be computed.
