Rank filters in digital image processing

Rank filters operating on images assign the k th value of the gray levels from the window consisting of M pixels arranged according to their value to the center point of the window. The special cases k = 1, k = M (MIN and MAX filter) and k = (M + 1)/2 (medium filter), which have already been applied in image processing, are investigated in systematic connection with all rank filters. Some of their properties can be formulated analytically. They commute with monotonic transforms of the gray scale. In the one-dimensional case—also valid for line-like structures in images—the output functions of monotonic input functions can be calculated directly. The alternating application of MIN and MAX filters leads, if repeated more than once, to the same result as a single application. The application of the rank filters to a set of test images shows that there is no simple way to describe their action on the spectrum by means of a transfer or autocorrelation function. In particular the smoothing of the median filter cannot be described in terms of a low-pass filter, but rather by the reduction of the mean local variance. As shown on real and statistical model images, rank filters smooth less than linear filters, but preserve edges.