Relationship between restricted dissimilarity functions, restricted equivalence functions and normal EN-functions: Image thresholding invariant

In this paper, we present the definition of restricted dissimilarity function. This definition arises from the concepts of dissimilarity and equivalence function. We analyze the relation there is between restricted dissimilarity functions, restricted equivalence functions (see [Bustince, H., Barrenechea, E., Pagola, M., 2006. Restricted equivalence functions. Fuzzy Sets Syst. 157, 2333–2346]) and normal EN-functions. We present characterization theorems from implication operators and automorphisms. Next, by aggregating restricted dissimilarity functions in a special way, we construct distance measures of Liu, proximity measures of Fan et al. and fuzzy entropies. We also study diverse interrelations between the above-mentioned concepts. These interrelations enable us to prove that under certain conditions, the threshold of an image calculated with the algorithm of Huang and Wang [Huang, L.K., Wang, M.J., 1995. Image thresholding by minimizing the measure of fuzziness. Pattern Recognit. 28 (1), 41–51], with the methods of Forero [Forero, M.G., 2003. Fuzzy thresholding and histogram analysis. In: Nachtegael, M., Van der Weken, D., Van de Ville, D., Kerre, E.E. (Eds.), Fuzzy Filters for Image Processing. Springer, pp. 129–152] or with the algorithms developed in [Bustince, H., Barrenechea, E., Pagola, M., 2007. Image thresholding using restricted equivalence functions and maximizing the measures of similarity. Fuzzy Sets Syst. 158, 496–516] is always the same, that is, it remains invariant.