A fuzzy mathematical morphology based on discrete t-norms: fundamentals and applications to image processing

In this paper, a new approach to fuzzy mathematical morphology based on discrete t-norms is studied. The discrete t-norms that have to be used in order to preserve the most usual algebraical and morphological properties, such as monotonicity, idempotence, scaling invariance, among others, are fully determined. In addition, the properties related to B-open and B-closed objects and the generalized idempotence are also studied. In fact, all properties satisfied by the approach based on continuous nilpotent t-norms hold in the discrete case. This is quite important since in practice we only work with discrete objects. In addition, it is proved that more discrete t-norms satisfying all the properties are available in this approach than in the continuous case, which reduces to the Łukasiewicz t-norm. This morphology based on discrete t-norms can be considered embedded in more general frameworks, such as L-fuzzy sets or quantale modules, but all these frameworks have been studied only from a theoretical point of view. Our main contribution is the practical application of this discrete approach to image processing. Experimental results on edge detection, noise removal and top-hat transformations for some discrete t-norms and their comparison with the corresponding ones obtained by the umbra approach and the continuous Łukasiewicz t-norm are included showing that this theory can be suitable to be used in a wide range of applications on image processing. In particular, a new edge detector based on the morphological gradient, non-maxima suppression and a hysteresis method is presented.