Annular filters for binary images

A binary annular filter removes isolated points in the foreground and the background of an image. Here, the adjective “isolated” refers to an underlying adjacency relation between pixels, which may be different for foreground and background pixels. In this paper, annular filters are represented in terms of switch pairs. A switch pair consists of two operators which govern the removal of points from foreground and background, respectively. In the case of annular filters, switch pairs are completely determined by foreground and background adjacency. It is shown that a specific triangular condition in terms of both adjacencies is required to establish idempotence of the resulting annular filter. In the case of translation-invariant operators, an annular filter takes the form X 7! (X A) \ X [ (X B); where A and B are structuring elements satisfying some further conditions: when A \ B \ (A B) 6= ;; it is an (idempotent) morphological filter; when A [ B A B; it is a strong filter, and in this case it can be obtained by composing in either order the annular opening X 7! (X A) \ X and the annular closing X 7! X [ (X B):