Image Algebra and Rank-order Filters

The unifying role that is one of the attractions of image algebra is distinctly less obvious when we come to include median filters and their many close relatives (weighted median filters, rank-order filters, weighted rankorder filters). The convolutional filters, which are linear, and the morphological filters, which are not, fit in naturally and indeed have a pleasingly similar appearance in the notation of image algebra. The non-linear operation of finding the median of a set of grey-level values (or selecting any other member of a set of ranked values) is not straightforward, however, since the task of rearranging the values in ascending order is iterative, requiring a sequence of comparisons. In this connection, a template generated by the original image to be enhanced and the window representing the zone around any given pixel, inside which the median is to be taken, proves to be very useful; this is the simplest example of the family of templates generated by suitable combination of an image and a template. The nature of such templates is the main subject of this paper.