Fuzzy Measure Theory
Sugeno (1974) introduced the theory of fuzzy measures and fuzzy integrals.
A fuzzy measure g over a set X (the universe of discource with the subsets
E, F, ...) satifies the following conditions (X is finite):
A fuzzy measure is a Sugeno measure (or a -fuzzy measure) if it satifies
the following additional condition for some > -1:
The value of can be calculated regarding to the condition g(X)=1:
Example for calculation of Sugeno measure
Consider the set X={a, b, c}. The fuzzy density values are given as follows:
The value of can be calculated by solving the following equation:
The solutions are  ={-16.8,
1}. Regarding to the condition >
-1, we receive =1 as only
solution. The Sugeno measure can be constructed as follows:
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E
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Sugeno measure
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{a}
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g({a}) = 0.3
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{b}
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g({b}) = 0.4
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{c}
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g({c}) = 0.1
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{a, b}
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g({a, b}) = g({a}) + g({b})+
g({a}) g({b}) = 0.82
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{a, c}
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g({a, c}) = g({a}) + g({c})+
g({a}) g({c}) = 0.43
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{b, c}
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g({b, c}) = g({b}) + g({c})+
g({b}) g({c}) = 0.54
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{a, b, c}
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g({a, b, c}) = g(X) = 1
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Fuzzy Integral
The fuzzy integral (in the literature also called Sugeno integral) can
be regarded as an aggregation operator. Let X be a set of elements (e.g.
features, sensors, classifiers). Let h: X-->[0,1]. h(x) denotes the
confidence value delivered by element x (e.g. the class membership of
data determined by a specific classifier). The fuzzy integral of h over
E (a subst of X) with respect to the fuzzy measurre g can be calculated
as follows:
with
In image processing we have always finite sets of elements X={x1, x2,
..., xn}. If the elements are sorted so that h(xi) is descending function
the fuzzy integral can be calcultaed as follows:
with
Example for calculation of Sugeno integral
Assuming h is given as follows:
The Sugeno integral with respect to Sugeno measure defined above can be
calculated as follows:
Applications in image processing and pattern recognition
1) Data Classification (see the papers of Michel Grabisch)
2) Image Segmentation (see the papers of H.Tahani and Jim Keller)
3) Fusion of different classifiers (see the papers of Jim Keller and Paul
Gader)
4) Fusion of different images (see the book of Hamid Tizhoosh)
5) Fusion of different filters
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