zbMATH — the first resource for mathematics

Some sets of indistinguishability operators as multiresolution families. (English) Zbl 1390.68645
Summary: Multiresolution is a general mathematical concept that allows us to study a property by means of several changes of resolution. From a fixed resolution, a coarser projection can be calculated and then the changes between a finer resolution and a coarser one can be studied. That information can give a good knowledge about the problem under consideration. Also using multiresolution techniques it is possible to present information with a higher or a lower detail, given a way to get the adequate granularity or abstraction for a context.{
}The granularity of a system can be obtained or modeled by the use of indistinguishability operators. In this work the relation between indistinguishability operators and multiresolution theory is studied and several methods to build families of indistinguishability operators with multiresolution capacities are given.

MSC:
 68T37 Reasoning under uncertainty in the context of artificial intelligence 03E72 Theory of fuzzy sets, etc.
Full Text:
References:
 [1] Belohlavek, R.; Vychodil, V., Algebras with fuzzy equalities, Fuzzy Sets Syst., 157, 1, 161-201, (2006) · Zbl 1087.08005 [2] Bezdek, J.; Harris, J., Fuzzy partitions and relations: an axiomatic basis for clustering, Fuzzy Sets Syst., 1, 112-127, (1978) · Zbl 0442.68093 [3] Boixader, D.; Jacas, J.; Recasens, J., Fuzzy equivalence relations: advanced material, (Dubois, D.; Prade, H., Fundamentals of Fuzzy Sets, (2000), Kluwer), 261-290 · Zbl 0987.03047 [4] Chui, C., An introduction to wavelets, (1992), Academic Press · Zbl 0925.42016 [5] Daubechies, I., Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, (1992), SIAM · Zbl 0776.42018 [6] de Boor, C., A practical guide to splines, (1978), Springer Verlag · Zbl 0406.41003 [7] de Soto, A. R.; Recasens, J., Modelling a linguistic variable as a family of hierarchical family of partitions inducided by an indistinguishability operator, Fuzzy Sets Syst., 121, 3, 57-67, (2001) [8] Dyba, M.; Novák, V., Eq-logics: non-commutative fuzzy logics based on fuzzy equality, Fuzzy Sets Syst., 172, 1, 13-32, (2011) · Zbl 1229.03027 [9] Fang, X.; Zhang, H.; Zhou, J., Fast window fusion using fuzzy equivalence relation, Pattern Recogn. Lett., 34, 6, 670-677, (2013) [10] Hájek, P., Metamathematics of fuzzy logic, (1998), Kluwer Dordrecht · Zbl 0937.03030 [11] Harten, A., Multiresolution representation of data: a general framework, SIAM J. Numer. Anal., 33, 3, 1205-1256, (1996) · Zbl 0861.65130 [12] Harten, A., Multiresolution representation in unstructured meshes, SIAM J. Numer. Anal., 35, 6, 2128-2146, (1998) · Zbl 0933.65110 [13] U. Höhle, Fuzzy equalities and indistinguishability, in: Proceedings of EUFIT, vol. 1, Aachen, Germany, 1993, pp. 358-363. [14] Jacas, J.; Recasens, J., Aggregation operators based on indistinguishability operators, Int. J. Intell. Syst., 21, 8, 857-873, (2006) · Zbl 1112.68122 [15] Jawerth, B.; Sweldens, W., An overview of wavelet based multiresolution analyses, SIAM Rev., 36, 3, 377-412, (1994), overview.ps · Zbl 0803.42016 [16] Jayaram, B.; Mesiar, R., I-fuzzy equivalence relations and i-fuzzy partitions, Inf. Sci., 179, 9, 1278-1297, (2009) · Zbl 1216.03060 [17] Klement, E. P.; Mesiar, R.; Pap, E., Triangular norms, (2000), Kluwer Dordrecht · Zbl 0972.03002 [18] Kruse, R.; Gebhardt, J. E.; Klowon, F., Foundations of fuzzy systems, (1994), John Wiley & Sons, Inc · Zbl 0843.68109 [19] Ling, C. M., Representation of associative functions, Publ. Math. Debrecen, 12, 189-212, (1965) · Zbl 0137.26401 [20] Luukka, P., Similarity classifier using similarities based on modified probabilistic equivalence relations, Knowl.-Based Syst., 22, 1, 57-62, (2009) [21] Mallat, S., Multiresolution approximations and wavelet orthogonal bases of $$L^2(\mathbb{R}$$), Trans. Am. Math. Soc., 315, 69-87, (1989) · Zbl 0686.42018 [22] Mallat, S. G., A theory for multiresolution signal decomposition: the wavelet representation, IEEE Trans. Pattern Anal. Mach. Intell., 11, 7, 674-693, (1989) · Zbl 0709.94650 [23] Mattioli, G.; Recasens, J., Structural analysis of indistinguishability operators and related concepts, Inf. Sci., 241, 0, 85-100, (2013), · Zbl 1321.03067 [24] Meyer, Y., Wavelets. algorithms and applications, (1993), SIAM [25] Meystel, A., Multiresolutional hierarchical decision support systems, IEEE Trans. Syst. Man Cybern. Part C: Appl. Rev., 33, 1, 86-101, (2003) [26] Millman, R. S.; Parker, G. D., Geometry: A metric approach with models, (1990), Springer [27] Moser, B., On the compactness of admissible transformations of fuzzy partitions in terms of t-equivalence relations, Fuzzy Sets Syst., 160, 1, 95-106, (2009) · Zbl 1183.03052 [28] Recasens, J., Indistinguishability operators, Modelling Fuzzy Equalities and Fuzzy Equivalence Relations, Studies in Fuzziness and Soft Computing, (2010), Springer · Zbl 0946.03064 [29] H.A. Simon, The architecture of complexity, in: Proceedings of the American Philosophical Society, 1962, pp. 467-482. [30] Sweldens, W.; Schröder, P., Building your own wavelets at home, (Wavelets in the Geosciences, (2000), Springer), 72-107 · Zbl 0963.65153 [31] Trillas, E., Sobre funciones de negación en la teoría de conjuntos difusos, Stochastica III, 1, 47-60, (1979), in Spanish [32] E. Trillas, E. Castineira, A. Pradera, On the equivalence between distances and T-indistinguishabilities, in: Proceedings of EUSFLAT-ESTYLF Joint Conference 1999, Palma de Mallorca, Spain, 1999, pp. 239-242. [33] E. Trillas, L. Valverde, An inquiry on indistinguishability operators, in: H.J. Skala, S. Termini, E. Trillas (Eds.), Aspects of Vagueness, Reidel, 1984, pp. 231-256. · Zbl 0564.03027 [34] F. Truchetet, O. Laligant, Industrial applications of the wavelet and multi-resolution-based signal/image processing: a review, 2007 . [35] Valverde, L., On the structure of F-indistinguishability operators, Fuzzy Sets Syst., 17, 3, 313-328, (1985) · Zbl 0609.04002 [36] Wang, Y.-J., A clustering method based on fuzzy equivalence relation for customer relationship management, Expert Syst. Appl., 37, 9, 6421-6428, (2010) [37] Zadeh, L., Similarity relations and fuzzy orderings, Inf. Sci., 3, 2, 177-200, (1971), · Zbl 0218.02058 [38] Zadeh, L., Computing with words, principal concepts and ideas, Studies in Fuzziness and Soft Computing, vol. 277, (2012), Springer · Zbl 1267.68238
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.