Connected morphological operators and filters for binary images

Connected morphological operators act on the level of the flat zones of an image, i.e., the connected regions where the grey-level is constant. For binary images, the flat zones are the foreground and background grains (connected components) of the image. The flat zones constitute a partition of the underlying space. A connected operator is an operator that makes this partition coarser. A grain operator is a connected operator that is uniquely specified by two grain criteria, one for the foreground and one for the background components. This paper contains a brief discussion of connected operators and grain operators with an emphasis on connected filters, all in the context of binary images.