IEEE SIGNAL PROCESSING LElTERS, VOL. 1, NO. 6. JUNE 1994 ~ 95 Parametric Analysis of Weighted Order Statistics Filters Ruikang Yang, Moncef Gabbouj, and Pao-Ta Yu Abstract-In this letter, we shall study the convergence prop- erties of weighted order statistics filters. Based on a set of parameters, weighted order statistics filters are divided into five categories making their convergence properties easily un- derstood. It will be shown that any symmetric weighted order statistics filters will make the input sequence converge to a root or oscillate in a cycle of period 2. This result is significant since a restriction imposed by an earlier research is eliminated making the result applicable for the whole class of symmetric weighted order statistics filters. A condition to guarantee convergence of symmetric weighted order statistics filters will be derived. I. INTRODUCTION EIGHTED order statistics (WOS) filters, including me- dian and weighted median filters as their special cases, belong to the class of stack filters. They have been studied extensively and successfully applied in many applications such as image processing, due mainly to their good performance for edge preservation and excellent suppression of impulsive noise [ 1]-[4]. Since weighted order statistics filters are nonlinear operations, their behavior and performance are assessed based on their statistical and deterministic properties. The former deals with, e.g., output distributions, and noise attenuation capability and has been the subject of several papers [3], [SI. The latter is related to the concept of root signals and deterministic convergence and is the focus of this letter. One of the contributions in this letter is to introduce a set of parameters to characterize the convergence properties of WOS filters. Based on the parameters, WOS filters can be classified into five categories, all of them except one possess convergence property. Study is then focused on the remaining category. It is shown that the WOS filters in this category are type-3 stack filters. A condition under which WOS filters will make any input signal converge to a root in a finite number of filtering is derived. Another contribution of the letter is the elimination of the restrictions imposed by Wendt [2]. In his paper, Wendt showed that in order to make a symmetric threshold logic function converge or oscillate in a cycle of 2, the threshold logic function must preserve the roots of the standard median filters. Manuscript received April 4, 1994; revised April 25, 1994. The associate editor coordinating the review of this letter and approving it for publication was Prof. V. Mathews. R. Yang is with the Audio-visual Processing Laboratory, Nokia Research Center, Tampere, Finland. M. Gabbouj is with the Signal Processing Laboratory, Tampere University of Technology, Finland. P.-T. Yu is with the Institute of Computer Science and Information Engineering, National Chung Cheng University, Taiwan. IEEE Log Number 9402922. Unfortunately, many WOS filters are excluded as they do not satisfy this condition. We shall show, in this letter, that this restriction is not necessary and can be eliminated. In other words, any symmetric threshold logic function will make an input converge to a root or oscillate in a cycle of 2. 11. CLASSIFICATION OF WOS FILTERS The weighted order statistics filter can be defined in the Definition 1: In the real domain, the output of a WOS fitler following way. with window size N = R + L + 1 and weight vector W w= (W-L'..,"J,''.,WR) is given by Y, = T : thlargest[W-L 0 X,-L, . . . WO Oxt,.",WROXz+R] (1) where T is called the threshold of the WOS filter and the 0 denotes duplication n times - nOX =X:..,X. The weights W, can be real numbers. Usually, we select R=L=K. The filtering procedure can be stated as follows: Sort the samples inside the filter, and add up the corresponding weights from the upper end of the sorted list until the sum just exceeds the threshold T; the output of the WOS filter is the sample corresponding to the last weight added. We shall concentrate on the convergence behavior of WOS filters in the binary domain, due to the threshold decomposition According to previous research on convergence properties of stack filters, the center weight WO of a WOS filter plays an important role to govem the convergence behavior of the WOS filter. This motivates us to introduce two parameters (a and p) to represent the center weight and the sum of the rest of the weights, respectively, i.e. property [a K a=~o and p= w,. (2) z=-K We define the threshold T as a linear function of a parameter t T =g(t) = (1 - t)a + t/?: for -m < t < m. (3) 1070-9908/94$04.00 0 1994 IEEE