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(9) 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004 
  
1 Lama c»? 
fá(&. y) == FF € 2 iS ) (10) 
2no;,(s) 
Thus, by generalizing the approach defined in Caillol ef al. 
[5], the pixel density is given by the following formula : 
K—-1 K 
p(ys) A s (ys ES > f Tufaeus)de. (101) 
$21 izl jze4rl 
where 7; and 7;; correspond to, respectively, the a priori proba- 
bility of the pure and the fuzzy classes. 
With the bayesian theory, we can describe the a posteriori 
probability for each class : 
- for the set of pure classes : 
T; fiys) 
P. = 0;/ Y, = Ys) = 
D(ys) 
(12) 
- for the set of fuzzy classes : 
1 
f Aufl, Ys )de 
P Tg = 0 zv Y Im Us = 
( fil ) PlYs) 
(13) 
From the definition of the above probabilities we can esti- 
mate the unknown parameters : ;, T;j, Tij, Mi, m;, 0;, 0; from 
a sample of X. À full description of the FSEM algorithm can be 
found in [7]. 
FSEM applied to multispectral data 
The fuzzy statistical analysis described previously is defined 
for only one spectral band. However the data on which we work 
are are made of several, say N, bands. Those N images have 
to be analysed, so in this context the unobservable random field 
can ne represented by X" — (XN es. 
The introduction of the multidimensionnal property of the 
data increases the algorithmic complexity of the problem. The 
generalization of the equations (12,13) to NV bands is not trivial. 
The solution is to set a simplifying hypothesis in order to make 
the algorithm practically realizable. The most used hypothesis 
is the conditional independence which stipulates that, knowing 
the class 0;, the joint density of two variables y! and y? is the 
product of the densities of each variable : 
ful y2) 9 fi(yl).fi(y2) (14) 
This hypothesis can be reinforce by applying a principal com- 
ponent analysis (PCA) which reduces the inter-band correlation 
and decrease the number of spectral bands. The equation (11) is 
written for a number N of spectral bands in the following way : 
K—1 K 
p(y) u + f mig ise yl de (15) 
jl jm 
The simplification of equation 15 with the independence 
conditional hypothesis result in the following formula : 
K-—1 K 
K N 
Pa?) - 3m I Eso Y: fe sut 
i=) ns] 
2=1;j=itl n=} 
(16) 
FUZZY STATISTICAL METHOD FOR MASS FUNCTION 
INITIALIZATION 
Mathematically, we can define non-normalized masses for all 
the simple et composed hypotheses as follows : 
"m - TTA (ys) (17) 
n=l 
l N 
M0, U0, = / Il Faj (€, Ys )de (18) 
0 n=l 
where fi(ys) and f;;(e, ys) are the conditional densities des- 
cribed in the previous section. 
APPLICATION TO REMOTE SENSING IMAGES 
Data set 
The studied zone is the region of “Grand Lake”, located in the 
area of Gooze Bay, Labrador. It is mainly composed of different 
forest densities and clear cuts. The last cartographic update rea- 
lized in 1988. 
We use the PCA on the LANDSAT image to reduce the data 
to three bands containing 95% of the information. We also com- 
pute the Tasseled cap images, to extract the brightness, green- 
ness and wetness informations, see figure 1.b to 1.d. We also 
have auxiliary information relative to the altitude of the studied 
area (fig.1.e). All this complementary and redundant informa- 
tion have to be extracted in a rigorous way and the orthogonal 
sum of DS is used to combine them. 
The extraction of the information is carried out by the FSEM 
algorithm from PCA data and Tasseled Cap images. The pure 
and fuzzy densities extracted are used in the initialization pro- 
cess to compute simple and composed hypotheses of the eviden- 
tial theory. We apply a sober filter on the altitude information, 
it results in a map containing the slope information of the area. 
We use this information to initialize the mass functions accor- 
ding to a slope threshold above which confidence for simple and 
composed hypotheses “water” or “boggy” is weak. 
Results analysis 
A simple probabilistic unsupervised classification based on 
the SEM algorithm gives 55 % of good classification (fig.2.a). 
The contribution of DS fusion initialized by the FSEM algo- 
rithm with the Tasseled Cap transformation can improve the 
classification quality in particular for boggy and vegetation 
classes. The rate of classification is 61 % (fig.2.b). The contribu- 
tion of slope information (fig.2.c) removes some natural artefact 
related to LANDSAT TM data acquisition. In fact, the shadow 
of some clouds is classified like “water” or “boggy” with the 
SEM algorithm. Some pixels classified as water or boggy are on 
high slope areas which is not realistic. Those pixels are in fact 
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