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making it impractical to use methods based on thoroughly counting the possibilities
when the value of l goes beyond a few units. However, the number of possibilities
is greatly reduced by the monotonicity constraint imposed on the operator [THO 89].
Among the more interesting methods in this field, we can mention those based on
entropy for optimizing the fusion operator [DES 99]. This leads to an optimal fusion
rule expressed as a weighted sum of local decisions, that can be compared to a thresh-
old which is a function of false alarm and detection probabilities of the various local
sensors, a priori probabilities and costs. Methods based on entropy can also be used
on several levels: for choosing the most relevant sensors, for optimizing local sensors
(on each sensor's level) and for optimizing the fusion operator.
6.8. An example of Bayesian fusion in satellite imagery
In this section, we will illustrate Bayesian fusion by a simple example of multi-
source classification in satellite imagery, in which fusion is performed on pixels, based
on the information of the gray levels. This example was discussed in [CHA 95]. Fig-
ure 6.1 shows an example of six images to fuse. These are SPOT images in the XS
multi-band spectral mode in green (XS1), red (XS2) and near infrared (XS3), with a
sampling increment of 20 meters, registered in a common frame of reference (to allow
us to perform fusion on the pixel level).
The classes considered are cities or urban areas (class C 1 ), rivers (class C 2 ) and a
class C 3 for every other structure (mostly vegetated areas).
Since the main characteristic of the cities in these images is their texture, the three
initial images are completed with three texture images obtained by using an algorithm
for estimating the parameters of a Gaussian Markovian field [DES 93]. These texture
images are also shown in Figure 6.1.
Conditional probabilities are learned using a histogram of the gray levels. These
estimates can be smoothed with Parzen windows, for example. Figure 6.2 illustrates
the results of the learning process in one of the images. In practice, in order to avoid
unjustified hypotheses of independence, the joint probability of the three XS channels
conditionally to the classes is estimated, and likewise for the three texture images.
One of the difficulties of the Bayesian method is the a priori estimation of proba-
bilities. If they are set according to the proportion of classes found in the images, this
leads to a strong decrease in the probability of poorly represented classes (see Figure
6.2, on the right) thus making them very difficult to detect. The method chosen here
consists of making these estimations when there is little conflict between the classes
and to take uniform a priori probabilities when the statistical properties of the classes
indicate a strong conflict.
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