Order Statistic-Based Nonlinear Filters: Stack Filters and Weighted Median Filters Edward J. Coyle +, Moncef Gabbouj $, and Jisang Yo0 ' tSchool of Electrical Engineering Purdue University West Lafayette, Indiana 47907, USA #Signal Processing Laboratory Tampcre University of Technology P.O. Box 553 SF-33101 Tampere, FINLAND Abstract Since the introduction of the median filter by John Tukey in 1971, many important classes of order statistic-based nonlinear filters have bccn developed. In this papcr we review some rcccnt results obtained for the two filter classes known as stack filters and weighted median filters. The highlights include new results on optimal filter design and fast training algorithms. . 1. Introduction Linear filters have long bcen the primary tool for signal and image processing. They arc easy to implement and analyze and, pcrhaps most importantly, the linear filter which minimizes the mean squared error criterion can usually be found in closed form. Furthermore, they are optimal among the class of all filtering operations when the noise being considered is additive and Gaus- sian. Unfortunately, a small deviation from this Gaussian assumption sometimes leads to a severe deterioration in the performance of linear filters. In the many applications in which non-Gaussian noise arises, linear methods have thus proven to be inadequate for signal smoothing and/or noise reduction. One such case occurs in the presence of speckle noise. Other types of non-Gaussian and/or signal dependent noise also cause prob- lems. We believe that these cases occur more frequently than not; therefore, linear methods are not completely satisfactory when dealing with real signals and noise rather than simply com- puter simulations. The obvious answer to this problem is to use a filter that is not linear. There are, however, many classes of non-linear filters, and the task of choosing the right class is itself a challenge. Each class of filters is good in certain applica- tions. The user could consult a look-up table to dctcrminc which filter or class of filters best fits the problem at hand. Onc filter that would certainly appear in any such catalogue would be the median filter. The median filter. or "running median" as it was called in the first publication in which it appeared [Tuk], consists of a window, usually of odd width, which is stepped one sample at a time along a signal. At each position of the window, the sample values inside are ranked according to their magnitude and the middle element in this ranking is defined to be the output. Typically, the window is assumed to have width 2N+1 where N is any positive integer. Suppose that the window is centered on the k'th sample in the input sequence and that the 2N+l points in the window, in time-order, are specified by the vector zN,k = (Xk-N, xk-N+1, * * xk, 7 xk+N). We want to find Yk, where yk = med(Xk-N, + , Xk+N) = mCd(?N,k) which is the output of the median filter when the T-3.1