Let
be a Lipschitz
function associated to region
(we assume the existence
of such a )
such that
(4)
Thus, the region
is completely described by the level set function .
Let define the approximations
and
of Dirac
and Heaviside distributions with
(see the illustration )
(5)
(6)
From the above definitions we have
(7)
(8)
Dirac and Heaviside approximations
region and boundary definition by the use of
level set functions
As
,
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 :
(9)
The real parameters e_{i},
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.
Functions
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
(10)
where dt is the step in time. From (9) and (10) we
can see that we get a "natural" narrow band due to the operator
.