Journal
JOURNAL OF MATHEMATICAL IMAGING AND VISION
Volume 43, Issue 1, Pages 50-71Publisher
SPRINGER
DOI: 10.1007/s10851-011-0283-1
Keywords
Mathematical morphology; Complete lattice; L-fuzzy sets; Interval-valued fuzzy sets; Atanassov's intuitionistic fuzzy sets; L-fuzzy mathematical morphology; L-fuzzy connectives; Inclusion measure; Duality; Negation; Adjunction
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Funding
- CNPq [309608/2009-0]
- FAPESP [2009/16284-2]
- Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP) [09/16284-2] Funding Source: FAPESP
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Mathematical morphology (MM) offers a wide range of tools for image processing and computer vision. MM was originally conceived for the processing of binary images and later extended to gray-scale morphology. Extensions of classical binary morphology to gray-scale morphology include approaches based on fuzzy set theory that give rise to fuzzy mathematical morphology (FMM). From a mathematical point of view, FMM relies on the fact that the class of all fuzzy sets over a certain universe forms a complete lattice. Recall that complete lattices provide for the most general framework in which MM can be conducted. The concept of L-fuzzy set generalizes not only the concept of fuzzy set but also the concepts of interval-valued fuzzy set and Atanassov's intuitionistic fuzzy set. In addition, the class of L-fuzzy sets forms a complete lattice whenever the underlying set L constitutes a complete lattice. Based on these observations, we develop a general approach towards L-fuzzy mathematical morphology in this paper. Our focus is in particular on the construction of connectives for interval-valued and intuitionistic fuzzy mathematical morphologies that arise as special, isomorphic cases of L-fuzzy MM. As an application of these ideas, we generate a combination of some well-known medical image reconstruction techniques in terms of interval-valued fuzzy image processing.
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