Self-dual morphological operators and filters

The median operator is a nonlinear image transformation celebrated for its noise cleaning capacities. It treats the foreground and background of an image identically, i.e., it is self-dual. Unfortunately, the median operator has one major drawback: it is not idempotent. Even worse, subsequent iterations of a given image may lead to oscillations. This paper describes a general method for the construction of morphological operators which are self-dual. This construction is based upon the concept of a switch operator. Subsequently, the paper treats a class of operators, the so-called activity-extensive operators, which have the intriguing property that every sequence of iterates of a given image is pointwise monotone and therefore convergent. The underlying concept is that of the activity ordering. Every increasing, self-dual operator can be modified in such a way that it becomes activity-extensive. The sequence of iterates of this modification converges to a self-dual morphological filter.