The dual representation of gray-scale morphological filters

One of the classic results of mathematical morphology is the filter-representation theorem of G. Matheron (1975) for black-and-white images. The theorem states that any morphological filter can be represented as a union of erosions by elements in the filter´s kernel. In its dual form, it states that the erosion representation can be replaced by an intersection of dilations by elements of the dual filter´s kernel. Here, the dual-form of the gray-scale representation is derived in terms of a minimum of dilations by elements in the dual filter´s kernel