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Recursive Median Filtering with Partial Replaces

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Recursive Median Filtering with Partial Replaces 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.
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