Discrete t-norms in a fuzzy mathematical morphology: Algebraic properties and experimental results

In this paper, a new approach to fuzzy mathematical morphology based on discrete t-norms is studied. It is proved that the most usual algebraic and morphological properties are preserved, such as, duality, monotonicity, interaction with union and intersection, invariance under translating and scaling, local knowledge property, extensitivity, idempotence, and many others. In fact, all properties satisfied by the approach based on nilpotent t-norms hold in the discrete case. This is quite important since in practice we only work with discrete objects. Experimental results for some discrete t-norms are included. They are compared with classical morphological algorithms based on the Łukasiewicz t-norm and the umbra approach, and with the fuzzy approach based on idempotent uninorms, proving that they are suitable to be used in edge detection.