Fuzzy logic and mathematical morphology

In this report we investigate the general theory of grey-scale morphology within the framework of complete lattices and fuzzy logic. This includes grey-scale granulometries, hit-or-miss operators for grey-scale images, rank operators, and connected operators. We also show that the Matheron's representation theory does not hold for general grey-scale images and we present some results related to the representation theory. Besides these, in this report, we put forward a new approach to fuzzy morphology through the extension of infimum, supremum, and conjunction.