To localize contours, the idea is that we can identify a
point of the contour on each
ray 
by the parameter
where the 
is a judgement function, which has small values
at those 
, where a contour point is most likely.
Two examples of such a function 
are
with 
,
which are similar to the gradient energy of active contours in the image plane.
The step, which lead to (8),
is motivated by the assumption, that an edge in 2D can also be found
by a gradient search in the corresponding 1D signal. Of course, edges which
are in the direction 
from the reference point 
cannot be found on
the ray 
by 
.
The experiments in 6 will show, that this case is not relevant in practice.
Having the optimal value for 
,
the contour point 
in the 2D image plane can easily be computed by
with 
What are the results of this new representation up to now?
, i.e.
we always know where the contour point n
can be found, which corresponds to the direction 
. For
this we only have to look from the reference point in direction 
.
Thus, no crossings can occur in the contour.
rays, on which contour point
candidates are searched for. In 3,
the extracted contour of an object and in
3 the function 
are shown.
One can observe, that the function 
is smooth for
the angles which correspond to the correctly extracted contour (
).
Then, an error can be seen, both in the extracted contour and in the
function 
.
For 
the
function 
is not smooth, because a wrong contour has been extracted.
This is no surprise. Looking at equation (8) one can see, that
up to now, the contour points are calculated without taking into
account neighboring contour elements. Thus, we need to introduce some
linkage between neighboring contour points to take into consideration
that normally contours are coherent in space, i.e. that contours are smooth.
This energy makes the difference to [18], where also a radial representation
but no internal energy is used.
A usual approach to connect neighboring contour points together is to introduce an internal energy similar to the active contour approach.
Figure 3: (a): 2D contour extracted by active rays.
(b): 1D function 
of the corresponding 2D contour.
An internal energy which handles the above mentioned demands is
In the appendix we will show how this energy can be directly derived from the internal energy of active contours by substituting the representation (9) into (2). One important aspect of this internal energy term is, that this energy only depends on a 1D function, too, in contrast to active contours, where the internal energy depends on a 2D function. This results again in a reduction of the complexity of the following optimization algorithms.
