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The fundamentals of fuzzy mathematical morphology, part 2: idempotence, convexity and decomposition

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Abstract
Fuzzy mathematical morphology is an alternative extension of binary mathematical morphology to gray-scale images. This paper discusses some of the more advanced properties of the fuzzy morphological operations. The possible extensivity of the fuzzy closing, anti-extensivity of the fuzzy opening and idempotence of the fuzzy closing and fuzzy opening are studied in detail. It is demonstrated that these properties only partially hold. On the other hand, it is shown that the fuzzy morphological operations satisfy the same translation invariance and have the same convexity properties as the binary morphological operations. Finally, the paper investigates the possible decomposition, by taking (strict) alpha-cuts, of the fuzzy morphological operations into binary morphological operations.
Keywords
extensivity, fuzzy mathematical morphology, anti-extensivity, idempotence, translation invariance, convexity, concavity, (strict) alpha-cuts, decomposition

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Chicago
De Baets, Bernard, Etienne Kerre, and Madan Gupta. 1995. “The Fundamentals of Fuzzy Mathematical Morphology, Part 2: Idempotence, Convexity and Decomposition.” International Journal of General Systems 23 (4): 307–322.
APA
De Baets, Bernard, Kerre, E., & Gupta, M. (1995). The fundamentals of fuzzy mathematical morphology, part 2: idempotence, convexity and decomposition. INTERNATIONAL JOURNAL OF GENERAL SYSTEMS, 23(4), 307–322.
Vancouver
1.
De Baets B, Kerre E, Gupta M. The fundamentals of fuzzy mathematical morphology, part 2: idempotence, convexity and decomposition. INTERNATIONAL JOURNAL OF GENERAL SYSTEMS. 1995;23(4):307–22.
MLA
De Baets, Bernard, Etienne Kerre, and Madan Gupta. “The Fundamentals of Fuzzy Mathematical Morphology, Part 2: Idempotence, Convexity and Decomposition.” INTERNATIONAL JOURNAL OF GENERAL SYSTEMS 23.4 (1995): 307–322. Print.
@article{186580,
  abstract     = {Fuzzy mathematical morphology is an alternative extension of binary mathematical morphology to gray-scale images. This paper discusses some of the more advanced properties of the fuzzy morphological operations. The possible extensivity of the fuzzy closing, anti-extensivity of the fuzzy opening and idempotence of the fuzzy closing and fuzzy opening are studied in detail. It is demonstrated that these properties only partially hold. On the other hand, it is shown that the fuzzy morphological operations satisfy the same translation invariance and have the same convexity properties as the binary morphological operations. Finally, the paper investigates the possible decomposition, by taking (strict) alpha-cuts, of the fuzzy morphological operations into binary morphological operations.},
  author       = {De Baets, Bernard and Kerre, Etienne and Gupta, Madan},
  issn         = {0308-1079},
  journal      = {INTERNATIONAL JOURNAL OF GENERAL SYSTEMS},
  language     = {eng},
  number       = {4},
  pages        = {307--322},
  title        = {The fundamentals of fuzzy mathematical morphology, part 2: idempotence, convexity and decomposition},
  url          = {http://dx.doi.org/10.1080/03081079508908045},
  volume       = {23},
  year         = {1995},
}

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