RECURSIVE MEDIAN FILTERING WITH PARTIAL REPLACES Adrian Burian and Pauli Kuosmanen Signal Processing Laboratory, Digital Media Institute Tampere University of Technology PO Box 553, FIN-33101, Tampere, Finland E-mail: burian, pqo @cs.tut.® ABSTRACT To be able to ®lter the outmost input sample, when parts of the ®lter's window fall outsidethe input signal, the input sig- The median ®lter is a special case of nonlinear ®lter used for nal is appended to the required size by replicating the out- smoothing signals. Since the output of the median ®lter is most input sample as many times as needed. This is the non- always one of the input samples, it is conceivable that cer- recursive median ®lter. If the point X(n) is replaced with the tain signals could pass through the median ®lter unaltered. output of the median ®lter before shifting the window to the These signals, invariant to further passes of the median ®l- next position, we obtain the recursive median ®lter. The out- ter, de®ne the signature of a ®lter and are referred to as root put of the recursive ®lter is given by: signals. This represents the convergence property of median ®lters. The convergence behavior of different schemes of re- n= [Y n k ; :::; Y n 1;Xn; :::; X n + k ]: Y med cursive median ®lters, and algorithms for image processing (2) using these ®lters will be studied. Recursive median ®lters use also previously ®ltered val- ues as their inputs. In this case the ®ltering operation is per- 1. INTRODUCTION formed 'in-place', so that the output of the ®lter replaces the Since Tuckey suggested thestandard median ®lter for smooth- old input value, before the ®lter window is moved to the next ing statistical time series [1], this concept has been widely position. With the same amount of operations recursive ®l- studied. By repeating the median ®ltering, the root signal, ters can usually providebetter smoothingcapability than non- which is invariant to further ®ltering, is found. The exis- recursive ®lters, at the expense of increased distortion. Re- tence of root signals is a fundamental property of median ®l- cursive median ®lters have stronger noise attenuation capa- ter, used in characterizing these nonlinear ®lters. bilities than their nonrecursive version, and a faster conver- Frequency analysis and impulse response have no mean- gence. In [6] simple proofs of the convergence properties ing in median and recursive median ®ltering: the impulse re- of median ®lters and the idempotent property (reduction to sponse of a recursive median ®lter is zero. As a result, new a root after one pass) of recursive median ®lters are given. tools had to be developed to analyze and characterize the be- An upper bound of the number of ®lter passes for median L 2=2 havior of these nonlinear ®lters, deterministically and statis- ®ltering a ®nite length signal to a root is ,whereL tically [2], [3], [4]. By associating the nonlinear operation of is the length of the input sequence. This bound is indepen- median ®ltering with a linear cost function, in [5] was shown dent of the window width of the ®lter. A more tight bound L 2=[2k + 2] N =2k +1 that median ®ltering is an optimization process in which a is 3 ,where is the ®lter's two-term cost function is minimized. window width. To compute the outputof a 1-D median ®lter, an odd num- For image processing applications, two-dimensionalme- ber of sample values are ranked, and the median value is used dian ®lters have been used with success. In [7] a new ap- proach for designing the recursive median ®lter for image = as the ®lter output. If the ®lter's window length is N processing applications was introduced. The original sig- k +1 2 , the ®ltering procedure is given by: nal replaces the output of the previous pass in the middle of the operation window. The convergence of this improved n= [X n k ; :::; X n; :::;Xn + k ]; Y med (1) recursive median ®lter within a ®nite number of iterations was proven. This new scheme of recursive median ®lter pro- where X(n) and Y(n) are the input and the output sequences, vides an improved MSE performance over the standard re- respectively. It is reasonable to assume that the signal is of cursive median ®lter. In this paper we further investigate the 0 X L 1 ®nite length, consisting of samples from X to . possibility of using partial replaces of the old input value at the ®lter's output, before moving the window to the next po- 3. THRESHOLD DECOMPOSITION sition. We show that even better MSE performances could be attained by different recursive median ®ltering schemes. In [9] a powerful tool called threshold decomposition for an- Proofs of the convergence of these recursive median ®ltering alyzing rank order based ®lters was introduced. Using this schemes are also given. technique, the analysis of these ®lters is reduced to studying their effects on binary signals. The importance of the thresh- old decomposition arises from the fact that binary signalsare 2. CENTER WEIGHTED MEDIAN FILTERS much easier to analyze than multi-valued signals. X ng Threshold decomposition of a signal vector f M- An immediate generalization of the median ®lter and a ma- X i < valued, where the samples are integer-valued, 0 jor subclass of stack ®lters are the weighted median (WM) ;0 i< L M means decomposing it into M-1 binary sig- 2 M 1 ®lters [6], [8]. The standard median ®lter has a better noise 1 n;X n; :::; X n nal vectors X , according to the attenuation than any WM ®lter, regardless of the noise dis- following rule: tribution. But, in order to preserve small details, WM ®lters 1 X n m m can be the solution. m if = T X n = X (6) 0 =2k +1 The output of the WM ®lter of window size N otherwise. w ; :::w k k associated with the integer weights is given by: This thresholding scheme can be applied to any signal that is quantized to a ®nite number of arbitrary signals. The Y n= [w X n k ; ; k med originalmulti-valuedsignal samples X(n) can be reconstruc- w X 0; ;w X n + k ]; k 0 (3) ted from the threshold levels by adding them: M 1 where the symbol is used to denote duplication, i.e., X m n= X n: X (7) nx = x; ;x: =1 (4) m | {z } n times Applying a recursive median ®lter to an M-valued sig- nal is equivalent to decomposition the signal to M-1 binary Center weighted median (CWM) ®lters are a subclass of threshold signals, ®ltering each binary signal separately with WM ®lters which combines the simplicity of median ®lters the corresponding binary recursive median ®lter, and then with some of the design freedom of WM ®lters. For these ®l- reversing the decomposition. ters only the center sample in the window has a weight larger m g f X n g The binary sequence f0, 1 of is transferred m than one. All other weights are equal to one. The CWM m g f Z n g Z n= into f-1, 1 binary sequence of by ®lters are the simplest WM ®lters and the easiest to be de- m X n 1 2 . For the recursive median ®ltering of a binary signed and implemented. A CWM ®lter of window width m Z ng sequence f , the output of the ®lter is given by: =2k +1 N is de®ned as: +1 S n 0 m if p n= O (8) n = [X n k ; ;pX n; ;Xn + k ]: Y med 1 otherwise, (5) where k + p After the center is weighted, the ®lter is effectively 2 N X p =2m 1 long, with smaller than or equal to k (otherwise m Z n + j : n= S (9) the ®lter would be reduced to the identity ®lter). Different j =N p = m = values of p produce different CWM ®lters. When 1 the median ®lter is obtained, which has the convergence 4. DIFFERENT RECURSIVE MEDIAN FILTERING = k k>1 property. When m , it has been shown that when , SCHEMES the resulting 1-D ®lter is idempotent. A CWM ®lter is completely speci®ed by two parame- It is known that in the case of 2-D signals the recursive me- ters: the window size and the center weight. In general, the dian ®lters are not necessarily idempotent [3]. Thus, in order longer the window size of a CWM ®lter, the better noise at- to ®nd the root signal, it is necessary to apply the recursive tenuation ability the ®lter has. CWM ®lter can be designed median ®lter iteratively. For the recursive median ®lter, at to possess good noise attenuation and preserve small details. each iteration for every point of the image we have to com- In contrast to recursive median ®lters, which are idem- pute the output of the ®lter: potent, the recursive WM ®lters usually are not. All the re- m m m m O [O n= n k ; :::O n; :::; O n + k ]; cursive CWM ®lters corresponding to a WM ®lter make an med t t1 t1 t1 arbitrary input signal to converge to a root signal. (10) j 2 2k where the subscript t represents the iteration index. The re- Notice that cannot take values greater than ;other- 2=1 cursive median ®ltering is a sequential process and the noise wise, the recursive process became meaningless. If j in¯uence at his output will be accumulated. To alleviate such the only replacement takes place in the middle of the win- an undesirable effect it may be useful to encourage the ®lter dow of the ®lter.