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FUZZYMAR
FUZZYMAR - Fuzzy Metrics and Indistinguishability Operators: Applications to Robotics
PGC2018-095709-B-C21 (2019-2022)
Fuzzy metrics are considered as an appropriate mathematical tool of measurement when such a measurement yields a degree of similarity
relative to the value of a parameter. On the other side, indistinguishability operators provide a degree of such a similarity extending the
notion of equivalence class to the fuzzy context. In the literature, one can find that the development of both mathematical frameworks is
carried out independently and assuming that both notions are unrelated, since fuzzy metrics involve a parameter in its formulation and
indistinguishability operators do not. The concept of modular indistinguishability operator, which involves a parameter in its formulation,
has been introduced recently. This new kind of operator has allowed unifying under the same framework the aforesaid apparently
unrelated fuzzy notions of measuring. This opens a wide range of possibilities which are totally unexplored from both the theoretical and
applied viewpoints.
The main theoretical target of the project is to study what properties of fuzzy metrics and indistinguishability operators
remain valid in the general framework of modular indistinguishability operators and, thus, try to make clear the differences with the results
coming from the two classical frameworks. The extension to the modular context of all topological results for fuzzy metrics and all results of
structure and representation for indistinguishability operators will be addressed. Moreover, this viewpoint will help us transfer, to the
indistinguishability context, typical results that only could be stated in the fuzzy metric context, and vice versa. Moreover, applications of
the new developed results to fixed point theory are expected to be given in such a way that fixed point results in the context of fuzzy
metrics will be retrieved as a particular case. Furthermore, relationships between aggregation theory, generalized metric structures and
modular indistinguishability operators are also an objective of the theoretical approach of the project.
Regarding the application of the theoretical results, the FUZZYMAR project aims at developing new techniques and algorithms that involve the use of
modular indistinguishability operators as similarity measures in: (1) addressing two recurring problems in general robotics, namely robust
model fitting and (visual) descriptors comparison and matching, both with also application to computer vision and
pattern recognition, and (2) modelling multi-robot systems adopting swarm techniques, to deepen in the analysis of their performance and
accordingly devise better coordinated task execution strategies.
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