International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004
that it is impossible to dissociate the set of assigned hypotheses.
It is the main advantage but also the principal difficulty of the
DS method. Indeed, there is no generic method to define a mass
value on a single or a composed hypothesis.
Evidence combination
The greatest advantage of DS theory is the robustness of its
way of combining information coming from various sources
with the DS orthogonal rule. For instance, let us denote two
mass distributions zn; and ra from two sources. Then, the DS
combination can be represented by the following orthogonal
rule :
mi (B1 )ma(B2)
(m, © m2)(A) = SM TTF Ki (4)
K= NS my (B,)ma(B2) (5)
B,nBa=®
K is considered as a normalization factor and is interpreted as
a measure of conflict between the various sources. In addition,
it is a representation of the empty set mass function. Thus, the
larger K, the more the sources conflict and the less sense their
combination has. If IX = 1, then the sources are totally contra-
dictory.
Decision making
Unlike the bayesian theory, where the decision criterion is of-
ten the maximum of likelihood, the DS theory gives many so-
lutions to take a decision. The are several ways to decide which
is the most reliable hypothesis from single or unions of proposi-
tions.
The decision making rules that are the generally used are :
maximum of belief, maximum of plausibility or compromises
between them. We can find an exhaustive list of decision crite-
rion in [4].
MASS FUNCTION DEFINITION
Initialization methods for the mass function in the DS theory
are various and depend on the considered application framework
considered. According to the method applied for the initializa-
tion process, we can set with a correctly way the mass functions
were selected, it is possible to more or less correctly translate the
various aspects of uncertainty and the inaccuracy. There are cur-
rently two main categories of applications for the mass function
initialization. The first one is based on probabilistic methods and
lead to method like the consonant, partially consonant or diso-
nant distributions [4][8]. The drawback of these methods is that
they are defined in an empirical way for the composed propo-
sitions. They do not take into account fuzzy data. The second
mass function initialization category is based on fuzzy analysis
[2][12]. These techniques use the membership functions as mass
function, but they do not respect highly incertain information as
the presence of noise on image for example. The originality of
this project is to use a fuzzy statistical algorithm based on the
FSEM (Fuzzy Stochastic Estimation Maximization) in order to
better characterize the concepts of uncertainty and inaccuracy in
the mass function definition.
FUZZY STATISTICAL CLASSIFICATION METHOD
Principle of the fuzzy SEM algorithm
Classification processes of remote sensing images do not re-
present the complex reality of a studied area. In fact, let us consi-
der the problem of segmenting a satellite image into two classes :
“water” and “vegetation”. There may be some pixels with only
vegetation or water, but others, as in a boggy area, in which wa-
ter and vegetation are simultaneously present. In the first case,
the pixel will be called a pure pixel and in the second case, it
will be called a mixed pixel.
Some people proposed solutions to resolve this problem,
Caillol ef al.[5] introduced fuzzy data in some classical statical
models. To counter this drawback, some algorithms introduce a
fuzzy version of statistical modelling. Thus, parameter estima-
tion stochastic algorithms have been modified to take fuzzy data
into account such as the Expectation Maximisation (EM), Sto-
chastic Expectation Maximization(SEM) and Iterative Conditio-
nal Estimation (ICE) algorithm. All theses algorithms only ap-
ply for the case of a maximum of two pure classes. Estimation
algorithms become too complex if the number of pure classes
increases.
In the following section, we propose to generalize the Fuzzy
SEM (FSEM) for K classes with the following hypothesis :
a fuzzy class or a mixed class cannot be composed of more
_ than two "pure" classes [7]. This hypothesis is widely ob-
served in pratice, like “water-vegetation”, “trees-house” and
“agricultural-vegetation” fuzzy classes, whereas “trees-houses-
agricultural” mixed classes or others are fairly unusual.
A complete description of the FSEM algorithm can be find in
[7]. Let us define the unobservable random field X = (X,)ses
taking its values in a finite set of classes ©. We denote Y =
(Ys)ses the observed random field which is a corrupted version
of X, and S be the set of pixels 9 — (1, ..., n].
Let us define two independant gaussian random variables y;
and y; associated with the two pure classes 0; and 0;. The
gaussian densities f; and f; are defined by the distributions
R(m;,c;) and N(m;,o;) respectively. The fuzzy density defi-
ned for a fuzzy class between 0; and 0; can be simply obtained
by the following linear relation [10] :
Ya Fey + (I c), € lla (6)
where & is the mixture coefficient between 0; and 0;.
In this context, we show that the fuzzy distribution is a gaus-
sian density N(mij (&), Cij (£)) :
mae) e (1— em; 4 em; (7)
c7, (s) = (1-5) 0; + eig? (8)
Finally, the pure distribution density is defined by :
1 Eros my?
hy) = e "t (9)
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and the fuzzy distribution density :
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