Connected morphological operators for binary images

This paper presents a comprehensive discussion on connected morphological operators for binary images. Introducing a connectivity on the underlying space, every image induces a partition of the space in foreground and background components. A connected operator is an operator that coarsens this partition for every input image. A connected operator is called a grain operator if it has the following `local property': the value of the output image at a given point $x$ is exclusively determined by the zone of the partition of the input image that contains $x$. Every grain operator is uniquely specified by two grain criteria, one for the foreground and one for the background components. A well-known criterion is the area criterion demanding that the area of a zone is not below a given threshold. The second part of the paper is devoted to connected filters and grain filters. It is shown that alternating sequential filters resulting from grain openings and closings are strong filters and obey a strong absorption property, two properties that do not hold in the classical non-connected case.