be a Lipschitz
function associated to region 
(we assume the existence
of such a 
)
such that
is completely described by the level set function .
and
of Dirac
and Heaviside distributions with
(see the illustration )
|
|
| | |
,
the minimization of the following functional with respect to the set
,
if it exist, leads to a partition taking into account the
3 conditions exposed on the previous section, i.e. compound of homogeneous classes with smooth
boundaries :
and 
are some weighting
constants. The "data term" takes into account the Gaussian parameters
of the classes. The "partition condition" term penalizes the formation of vacuum
(pixels without any label) and overlapping regions (pixels with more
than one label) : this term is coupling the 's. The "smooth boundaries" term leads to
boundaries 
with minimal length.
are initialy set to signed distance functions whose zero
level set represents the boundary set .
We embed the minimization
of (9) into a dynamical scheme (t stands for the time index), and we get a system of K coupled PDE's
.
