Image Processing Reference
In-Depth Information
where
μ
k
refers to the membership function to the class
k
resulting from the combi-
nation.
The quality of the decision is measured basically according to two criteria:
- the first involves the “crispness” of the decision: the maximum degree of mem-
bership (or more generally the one that corresponds to the decision) is compared to a
threshold, which is chosen depending on the applications (and possibly depending on
the chosen combination operator);
- the second involves the “discriminating” nature of the decision, which is evalu-
ated by comparing the two highest values.
If these criteria are not met for an element
x
, then this element is placed in a
rejection class, or reclassified according to other criteria, such as spatial criteria, for
example (see Chapter 9).
8.12. Application examples
In this section, we will illustrate fuzzy methods with two examples of multi-source
classification.
8.12.1.
Example in satellite imagery
For the first example, we go back to the SPOT images from the example in Chapter
6 (Figure 6.1). The classes that are considered are still cities or urban areas (class
C
1
), rivers (class
C
2
) and a class
C
3
encompassing all the other structures (mostly
vegetated areas). This example was discussed in [CHA 95].
First, a supervised learning phase is conducted based on the histograms condi-
tional to the classes, either simply by smoothing these histograms, or by minimizing
the distance to the histograms of parametric functions such as truncated Gaussian dis-
tributions or piecewise linear L-R functions (trapezoidal functions). This first phase is
illustrated in the first line of Figure 8.13.
However, these functions, denoted by
f
i
for learning the class
C
i
in the image
j
, have no satisfactory interpretation in terms of membership functions. Particularly,
the tails of the histogram correspond to gray levels that are rare in the images, but
whose corresponding points belong without ambiguity to the darkest class (the light-
est, respectively). The change from the functions
f
i
to the membership functions
μ
i
is done by a transformation such that:
μ
i
(
x
)=
λ
i
(
x
)
f
i
(
x
)
[8.107]
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