Edge jitter and bias in median filter
Harold G. Longbotham, Susmitha C. Baddipudi, Yu Miao · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1995
More and more attention has been given to the edge jitter that may occur when a robust nonlinear filter for edge restoration is applied. In this paper, we consider the possible edge jitter introduced by the median filter i.e., the bias behavior of the median filter at an edge both for edge location and for restoration of the constant signal on each side of the edge. A nonparametric method is proposed to evaluate the edge jitter (location only) of the nonlinear filters. Simulation shows that the method gives an efficient and effective way to study the performance of a nonlinear filter at edges. The second part of this paper concentrates on quantifying the bias behavior of the median filter in restoration of the constant signals on either side of the edges in one and two dimensional signals. Median filters preserve convex and concave edges, this is a well known fact. The behavior of the median filter at perfect edges which have been corrupted by noise has not been studied in detail. It is proven that the median filter actually `rearranges' the noisy perfect edges to concave or convex edges i.e., they do not preserve perfect edges perturbed by noise and, indeed, preserve the noise as well as the edges. It is argued in this paper that the median filter actually implements a max or min filter at the edges in one dimensional signals and in the two dimensional signals the output of the median is a value, the order of which is a function of the height and width of the window. Proofs in the second part of the paper question the credibility of the median filter as an edge preserving filter in the presence of noise, in that the median yields biased estimates of each constant region.