Pattern Recognition Letters 11 (1990) 557-560 August 1990 North-Holland Convergence properties of recursive filter and neural network rank-order Akira ASANO, Wei ZHANG, Kazuyoshi ITOH and Yoshiki ICHIOKA Department of Applied Physics, Faculty of Engineering, Osaka University, Yamadaoka 2-1, Suita, Osaka 565, Japan Received 28 December 1989 Revised 23 January 1990 Abstract: It is demonstrated that all signals of arbitrary dimension converge to the root signals by iterative operations of the recursive rank-order filter under certain conditions. We utilize the relation between rank-order filters and Hopfield's neural network model. Key words: Rank-order filter, convergence property, Hopfield's neural network model. 1. Introduction Rank-order filtering, including median filtering, is a nonlinear filtering technique for smoothing signals. They recently have attracted much atten- tion because of their edge-preserving property and their effectivity for eliminating impulsive noise. The rank-order filter has a filter window of finite extent. It locates the center of the window on each pixel of the input sample. It takes the pixel values included in its filter window, and replaces the center pixel value with the rank-order estimate of the pixel values in the window. There are two types of operations of rank-order filters; the nonrecur- sive filter and the recursive one. The pixel values that the nonrecursive one takes are all from the in- put signal. The recursive one also takes the result of the filtering procedure at a position in the win- dow if the procedure at the position has already been carried out. It is well known that all one-dimensional finite extent signals converge to the root signals by itera- tive operations of nonrecursive and recursive rank- order filters (Gallagher and Wise, 1981; Bednar and Watt, 1984). The root signal is the signal that is invariant to additional filtering passes. Since root signals indicate important characteristics of the filters, they have been extensively investigated (Huang, 1981). In case of higher dimensional signals, the existence of the convergence property is intuitively believed. However, a rigorous proof has not been achieved, Only proofs for some varia- tions of the median filters have been given. Nodes and Gallagher (1983) have shown that the non- recursive separable median filter on two-dimen- sional signals has the convergence property with rare exceptions. The separable median filter, pro- posed by Narendra (1981), is a variation of the me- dian filter whose one operation is divided into two phases; first the horizontal 1-D median filtering, and second the vertical one. McLoughlin and Arce (1987) have shown the convergence property of the recursive separable median filter. Moreover, we have shown that the nearest neighbor median (NNM) filter of several types of parameters has the convergence property on multi-dimensional signals (Asano et al., in press). The NNM filter, proposed by Itoh et al. (1988), is also a variation of the me- dian filter. The NNM filter outputs the median of the neighbor ordered values of the center pixel in the window. In this paper, we prove under certain conditions 0167-8655/90/$03.50 © 1990 -- Elsevier Science Publishers B.V. (North-Holland) 557