Active contours have been widely used in computer vision in the past eight years, especially in contour segmentation and tracking [3,6,14,19]. An active contour is a parametric function
defined in the 
image plane of an image 
. For closed
contours one gets 
. Each contour element 
has an
energy 
, with
being the internal energy. It describes the elasticity and stiffness of the contour. The second energy is
which is called the external or image energy, smoothed with a Gaussian filter
with variance 
.
The active contour has a total energy E
During the contour extraction one looks for a parametric function
which minimizes (4).
As a result, closed contours are
extracted, along which strong edges are found in the image plane.
An assumption for this result is, that the initial closed contour 
is placed near the contour which should be extracted. In the case where
an object can not be clearly localized by its contour, because the
negative edge strength (3)
along the contour does not correspond
to the minimum in the total energy, this approach fails. One example, for
which the external energy (3) would fail,
can be seen in 5. There is only a weak contrast between
the boundary of the circle in the middle of the image and the different
textured regions around it. However, the circle can be localized by
human inspection. For this, region based approaches
for active contours have be proposed [6,16].
They take into account image information not only at the contour element
itself, but also in a region around the contour element. This
region is divided into an inside and an outside region. The defined external
energy reaches its minimum, when the image information in the
outside region differs from the inside region. Due to the nature of
the region definition, the complexity is proportional to the
size of the two regions. Another problem is, in what direction the
two regions should be defined. [16] proposes
a depth adapting algorithm. This again increase the complexity of the search.
In the next section we propose a different representation of a closed contour in the image plane. In terms of active contours, we define an internal and an external energy. In the case of the internal energy we formally derive the description for the representation from the internal energy of active contours, which has been very successful in describing smooth contours. Then, we formulate the contour extraction as an optimization problem. However, the chosen representation allows a very efficient computation of energies, which are suited to localize boundaries between textured regions.
