11.8 Applications of Fuzzy Set Theory

Fuzzy set theory, to treat fuzziness in data, was proposed by Zadeh in 1965. In Fuzzy set theory the membership grade can be taken as a value intermediate between 0 and 1 although in the normal case of set theory membership the grade can be taken only as 0 or 1.

Figure 11.8.1 shows a comparison between the normal case of set theory and fuzzy set theory. The function of the membership grade is called its "membership function" in Fuzzy theory. The membership function will be defined by the user in consideration of the fuzziness.

In remote sensing it is often not easy to delineate the boundary between two different classes. For example, there are transitive vegetation or mixed vegetation between forest and grass land. In such cases as unclearly defined class boundaries, Fuzzy set theory can be usefully applied, in a qualitative sense.

The following shows how the maximum likelihood method with Fuzzy set theory. Let the membership function be Mf() of class k (k=1,n), the likelihood Lf of fuzzy class f can be defined as follows.

Fuzzy set theory can be also extended to clustering. Figure 11.8.2 shows an example of land cover classification using Fuzzy set theory. In this classification, the concrete structure (code 90), with clearly defined characteristics, was first classified using the ordinary maximum likelihood method, while the loosely defined urban classes were classified by the fuzzy based maximum likelihood method.


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