Morphological pyramid with alternating sequential filters

The aim of this paper is to find a relationship between alternating sequential filters and the morphological sampling theorem developed by Haralick4. First, we show an alternative proof for opening and closing in the sampled and unsampled domain. This is done by using basis functions. This decomposition is used then to show the relationship of opening-closing in the sampled and unsampled domain. An upper and a lower bound, for the previous relationships, were found. Under certain circumstances, an equivalence is shown for opening-closing between the sampled and the unsampled domain. An extension to more complicated algorithms is also considered, namely ; union of openings and intersection of closings. The reason to consider such transformations is that in some applications we would like to eliminate pixels removed by individual openings (closings).