Median Filtering Convergence And Cascading Filtering

The median filter was proposed by J.W. Tukey in 1971 as a smoothing device for discrete signals. In particular, he noted the filter to be particularly effective for suppresing impluse noise while simultaneousely preserving signal edges often containing significant imformation. It has since then been widely used in picture processing, speech processing and other areas of signal processing. B.J. Justusson[19] and S.G. Tyan[14] first studied statistic properties of median filters and deterministic properties of median filters, respectively. But the theory of median filters has been poorly developed compared with the theory of linear filters.Let Z be the set of integers and k a fixed positive integer.Let x = {x(n)}nî—€ be a real sequence. For each integer n, we denote by x^(n) the median value of the following 2k + 1 numbers:By this permuting operation, the sequence x ?{x(n}} is transferred into a new one which is called the median filter of a; with window 2k + l. If we operate the median filter on x^ (with window 2k + l) once again, another one x^ ?{x^\n)}n^z is obtained. Generally we denote by the resulting sequence of x through p times of median filers. If x^ = x, x is called a root; if x^ ^ x and there exists a s > 2 such that x^ = x, x is called a recurrent sequence (with s times);S.G. Tyan established the following two resultes in [14].Proposition 1 If {x(n)}n is a root, then either of the following is true:(i) any segment, with length fc + 1 in {x(n)}nez is monotone;(ii) any segment with length k + 1 in {x(n)}nz is not monotone.AbstractThus, according to the above proposition, the roots are divided into two categories: a root satisfying (i) is called a root of category I; a root satisfying (ii) is called a root of category II.Proposition 2 Any root of category II is a binary sequence.If, for each n e Z, lim x(n) = r(n) is a real number, we say that the median niters of a; is convergent, denoted by x^ -?r(p ??oo), where r = {r(n)}nez.If x = {x(n)}nî—€ is a finite sequence, that is, there are two integers n\ and nz such that ni < ni and x(n] is constant for n < n\ and n < ri2, then x will be transformed into root in finite passes of median filter ([15], [29]). For general sequences, it is not true. An example is given in [31], in which the median filters are not convergent. Some classes of sequence in which their median filters are convergent are given in([12],[31]). X.W.Zhou ([31], [32]) and J. Brandt[35] study independtly a class of sequences in which their median filters are not convergent (recurrent sequences).Thus a very interesting problem is: what kind of properties do x have as p ?> oo? In charpter 1, we obtain a fundamental result,!. e.,The first innovation: For any x = {x(n)}nZi both {xp}p>\ and {a;'2p are convergent. Let o;(2p) -> a, x^2p~1^ ?>?/3. Then(1) If a = (3, then x is convergent.(2) Ifa 7 (3, then both a and / are recurrent sequences. Moreover, aFurther we study the relationship between convergence of median filters and local convergence, which is the followingThe second innovation: Suppose that x = {x(n)}nz- [x}p>\ is convergentif and only if there exists a no 6 Z such that for each n 6 [no, no + Ik ?2],lim x^(n) exists and is finite. Moreover, 2k ?l(k > 4), the length of thesegment, is optimal.Based on the above results we obtain criteria of convergence of median filters and study convergence of median filters of pertubed sequences.J. Brandt[24], D. Eberly[32] and X.W. Zhou[33]study the properties of roots of category II by deferent methods, respectively. In this paper we present a new property of roots of category II: any root of category is completely determined by any of its segments with length 2fc ?1, morover the length of segments, 2k ?1, is optimal.We also prove that any of recurrent sequences is periodic.Symmetric weighted median filters are nonlinear niters and generalizations to median filters and symmetric center weighted median filters. In charpter 3, we study convergence of...