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1 April 2003 Combining belief functions and fuzzy membership functions
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In several practical applications of data fusion and more precisely in object identification problems, we need to combine imperfect information coming from different sources (sensors, humans, etc.), the resulting uncertainty being naturally of different kinds. In particular, one information could naturally been expressed by a membership function while the other could best be represented by a belief function. Usually, information modeled in the fuzzy sets formalism (by a membership function) concerns attributes like speed, length, or Radar Cross Section whose domains of definition are continuous. However, the object identification problem refers to a discrete and finite framework (the number of objects in the data base is finite and known). This implies thus a natural but unavoidable change of domain. To be able to respect the intrinsic characteristic of uncertainty arising from the different sources and fuse it in order to identify an object among a list of possible ones in the data base, we need (1) to use a unified framework where both fuzzy sets and belief functions can be expressed, (2) to respect the natural discretization of the membership function through the change of domain (from attribute domain to frame of discernment). In this paper, we propose to represent both fuzzy sets and belief function by random sets. While the link between belief functions and random sets is direct, transforming fuzzy sets into random sets involves the use of α-cuts for the construction of the focal elements. This transformation usually generates a large number of focal elements often unmanageable in a fusion process. We propose a way to reduce the number of focal elements based on some parameters like the desired number of focal elements, the acceptable distance from the approximated random set to the original discrete one, or the acceptable loss of information.
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Mihai Cristian Florea, Anne-Laure Jousselme, Dominic Grenier, and Eloi Bosse "Combining belief functions and fuzzy membership functions", Proc. SPIE 5099, Multisensor, Multisource Information Fusion: Architectures, Algorithms, and Applications 2003, (1 April 2003);

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