Fuzzy Hit-or-Miss Transform Using the Fuzzy Mathematical Morphology Based on T-norms

Abstract

Abstract. The extension of the Hit-or-Miss transform (HMT) to grey-level images is difficult due to the problem of defining the complement of an image in this context. Thus, several extensions have been proposed in the literature avoiding the use of the complement. However, in the fuzzy framework, the complement is well-established by means of a fuzzy negation and the binary HMT can be extended preserving its geometrical interpretation. In this paper, we extend the binary HMT to a fuzzy HMT (FHMT) using the mathematical morphology based on t-norms. Some properties of this operator are studied and some initial experimental results are presented proving the potential of the FHMT in shape recognition and pattern matching.

On a functional equation related to the distributivity of fuzzy implications over triangular norms and conorms

“…Moreover, we shall not only find the general form of a function f , but we shall also prove that functions m 1 and m 2 must satisfy some properties, if we want the equation (2) to have some nontrivial solutions f . We believe that the results obtained in this article are not only theoretical, but they can be used in the future also in fuzzy control and approximate reasoning or in other theories like fuzzy mathematical morphology (see [12] or [14]), where solutions of functional equations play an important role.…”