Image Processing Reference
where F i is a function that is determined according to the problem. The most com-
monly used are normalization functions or S functions [PAL 92] (which is equivalent
to considering that the lighter parts of the image have a high membership to the class),
functions Π (monomodal, they associate the class with a range of gray levels with
imprecise limits), or also multimodal functions.
These functions are often determined under supervision, but can also be learned,
for example, using automatic classification algorithms such as fuzzy C-means
[BEZ 81] or possibilistic C-means [KRI 93] (see, for example, [BEZ 99] for an
overview of fuzzy classification algorithms). The main drawback of fuzzy C-means
is that the membership functions have counter-intuitive forms: the class membership
values are non-decreasing with respect to the distance to the center of the class. This
problem is avoided with possibility C-means.
Other characteristics can be used to achieve this goal. For example, the set of
contours in an image can be defined by a spatial fuzzy set whose membership function
is a function of the image's gradient:
μ i ( x )= F i ∇
I ( x ) ,
where F is a decreasing function.
If specific object detectors are available, the membership functions of these objects
can be defined as functions of the response to these detectors (the case of contours
falls into this category). For example, a road detector can provide in a satellite image
a response whose amplitude increases with the membership to the road.
In the case of linguistic variables, the forms of membership functions and their
parameters are often defined by the user.
The spatial imprecision over the definition of the limits between classes (if the
membership functions are defined in the image space) can be introduced based on a
preliminary binary detection of the classes. A membership function is constructed as
equal to 1 inside the binary area at a certain distance from the edges, as equal to 0
outside this area at a certain distance from the edges and as decreasing between these
two limits. For example, an imprecision zone on the edge of the class can be modeled
as the zone included between the erosion and the dilatation of this object, since the
size of these operations depends on the spatial extension of the imprecision we wish to
represent. If R is the binary area we start with, E n ( R ) its erosion of size n and D m ( R )
its dilatation of size m , the fuzzy class membership function can be defined by:
μ ( x )=1if x
E n ( R ) ,
D m ( R ) C ,
μ ( x )= F d x, E n ( R ) otherwise
where F is a decreasing function of the distance from x to E n ( R ).
μ ( x )=0if x