The median operator is a nonlinear (morphological) image transformation which has become very popular because it can suppress noise while preserving the edges. It treats the foreground and background of an image in an identical way that is, it is a self-dual operator. Unfortunately, the median operator lacks the idempotence property: it is not a morphological filter. This paper gives a complete characterization of morphological operators on discrete binary images which are increasing, translation invariant, and self-dual. Furthermore, it presents a general method for the modification of an increasing operator such that it becomes activity-extensive. Such modifications lead to idempotent operators under iteration. The general procedure is illustrated by giving several modifications of the 3 x 3 median operator.
IEEE International Conference on Image Processing
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Heijmans, H.J.A.M. (1994). Construction of self-dual morphological operators and modifications of the median. In Proceedings - International Conference on Image Processing, ICIP (pp. 492–496). doi:10.1109/ICIP.1994.413619