Abstract. Key Words: Image Filters, Fuzzy Filters, Order Statistics Filters, Rank Ordered Mean Filters, Channel Noise. 1.

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1 Fuzzy Weighted Rank Ordered Mean (FWROM) Filters for Mixed Noise Suppression from Images S. Meher, G. Panda, B. Majhi 3, M.R. Meher 4,,4 Department of Eletronis and I.E., National Institute of Tehnology, Rourkela (ORISSA). 3 Department of Computer S. and Engg., National Institute of Tehnology, Rourkela (ORISSA). {sukadev_meher@yahoo.om, gpanda@nitrkl.a.in, bmajhi@nitrkl.a.in, manasmeher@yahoo.om} Abstrat Some novel nonlinear filters are proposed for suppression of mixed Gaussian and salt-and-pepper noise from digital images. These filters are based on loal order statistis taken from the neighborhood of the pixel to be filtered. The performane of Rank Ordered Mean (ROM) filters is improved by assoiating Fuzzy weights with the order statistis. Triangular and Gaussian Fuzzy membership funtions are hosen and are seen to be quite effetive in improving the filter performane. The proposed FWROM filters show superior performanes as ompared to other standard filters suh as Moving Average (MAV), simple Rank Ordered Mean (ROM), and the fuzzy filter: TMAV [4]. Key Words: Image Filters, Fuzzy Filters, Order Statistis Filters, Rank Ordered Mean Filters, Channel Noise.. Introdution An image gets ontaminated with random noise during aquisition, transmission and retrieval proesses. Here, we will restrit our disussion to the noise during transmission, i.e. hannel noise. Further, we onfine ourselves to moderately good hannels, e.g able television systems. Usually, the noise is modeled as additive white Gaussian noise (AWGN). But in pratial situations, the noise value may be very high positive or very high negative at some random time instants, though the probability of suh ourrenes is very low, resulting in saturated values of some of the pixels in the image, either at the lowest level or at the highest level. This gives rise to a saturating impulsive noise that is modeled as saltand-pepper (S&P). It is also known as fixed valued impulse noise. In pratie, the noise in a ommuniation hannel is best depited with a ombination of AWGN and S&P. It may be noted that salt-and-pepper (minimum and maximum values) pixels are very few in an image if the hannel is not very highly noisy. This is so beause when the random noise in a good ommuniation hannel is modeled as Gaussian with low or moderate value of the variane (sine the hannel is not very highly noisy), the probability of having a positive max or negative max value is very low. This justifies the assumption we take that the noise in the hannel is best modeled by AWGN plus a low density (< %) S&P. There are various types of -D filters, reported in literature, to suppress suh noises. Linear filters those are based on Fourier Transform do not perform well; but are very simple to design. There is a wide range of nonlinear filters, whih perform well in ase of Gaussian noise, but their performane is very poor in presene of impulsive noise and vie versa. On the other hand, there are some filters that suppress noise very effetively, but they distort the edges as well, thereby giving rise to a blurring effet. Suh filters are not desirable. The moving average (MAV) filter removes Gaussian noise quite effetively but its performane is very poor in ase of impulsive noise. On the other hand, the median filter (MED) is a very good andidate for removal of impulsive noise. But it does not perform well if the image is orrupted with Gaussian noise. The MAV and MED are the two best-known filters that belong to a lass known as order statistis (OS) filters []. There are many other members of this ategory, e.g. min, max, et. The rank ordered mean (ROM) [] is another OS filter that is used for removing both AWGN and S&P impulsive noise. But it neither exels in suppressing AWGN nor in, S&P. Therefore, it needs further investigations to modify the ROM filter so that it will perform very well in the presene of both types of noise. In this paper, we deal with this problem and suggest some novel filters for this purpose. In the next setion, we disuss the basis of a ROM filter and weighted ROM (WROM) filter[5]. In setion-3, we desribe a fuzzy system and try to develop fuzzy WROM (FWROM) filters. In setion-4, simulation study is arried out to assess the performane of FWROM filters and ompare the same with that of other standard filters. Finally, onlusions are drawn in Setion-5.. ROM and WROM Filters The weighted rank ordered mean (WROM) filter is a modifiation to the simple ROM that performs slightly better than the latter. Its output Y( i, is given by,

2 ( i = F[ ( i, ] = { ( x) } Y, X ϕ () where, X ( i,, the input image, is a -D array of gray sale values of pixels ranging from 0 to B - for a B-bit system or, equivalently, from 0 to in the normalized sale; F(.) is the filter funtion operated on X; ϕ () is the sub-funtion, the true WROM, that is applied to a small region of the image (alled a window), x, entered at a pixel x(i, to get an estimate of the original pixel x ( i, N ˆ0. This funtion is defined as: ( ) = = k x(k) k ϕ x () k where, x is the k th order statistis of x ; ( k ) w k is the weight assoiated with ( k ) x ; N is the size of order statistis taken (N=MM neighborhood). The order statistis is a -D array. Let it be a olumn vetor represented by x ( ). Let the weight be represented by another olumn vetor w. Then, () is modified as: ( x) w T x ( ) ( ) ϕ = = x i, j (3) If the weight vetor is normalized, i.e. =, then the WROM formula [5], in matrix representation, is given by: w T x (4) ( ) = x ˆ0 ( i, where, is a matrix multipliation operation. Now the main issue is to hoose a suitable weight vetor w that operates on the ordered input vetor x ( ) so that we get a good quality output image. There are many solutions to this problem. One of them is intuitive: if we take a 33 window, then we may selet the middle three x (4), x (5) and x (6) or middle five, x (4),, x (7) order statistis and give equal weighting fator to eah, i.e. w=[/3 /3 /3] T or w=[/5 /5 /5 /5 /5] T. These are two simple ROM filters that we all ROM(3,3) or ROM(3,5). Similarly we an have ROM(5,), ROM(5,3), et. Here, we adopt a filter nomenlature format for all OS filters: FILTER_NAME(z,z); z representing the square window-size: zz, and z, the number of ordered statistis atually taken for omputing the filter output. On the other hand, the MAV filters will be named MAV33 or MAV55 depending on the window-size taken. ˆ0 If we take different weights, i.e. w m not neessarily equal to w n for m n, then the filter beomes a WROM filter [5]. This type of filter is different from L-filters [] that onsider all the ordered statistis of the neighborhood though the weights are different. We may design the weight values intuitively for better performane. Suh simple intuitive deision works out to be fruitful to some extent to suppress both types of noise simultaneously sine it is a ompromise between two extreme OS filters: the moving average filter and the median filter that perform well in ase of AWGN and S&P respetively. Can we further improve its performane by modifying the weights? The answer is yes. There are many methods to modify the weights. One solution stems out from adaptive filtering theory where the weight vetor w is updated using some optimization algorithm say a Least Mean Squared error (LMS) algorithm []. This is a very omputation intensive operation. Also it needs a referene signal (image) or a very good quality (noise free) image frame that we are not supposed to have all the time. The other method that we propose here is based on fuzzy logi. 3. Fuzzy Filters Prior to developing some effiient fuzzy filters, let us disuss the fundamentals of a fuzzy system. A fuzzy system is represented by fuzzy variables that are members of a fuzzy set. A fuzzy set is a generalization of a lassial set based on the onept of partial membership. Let F be a fuzzy set defined on universe of disourse U. The fuzzy set is desribed by the membership µ F (u) that maps U to the real interval [0,] i.e. the membership µ varying from 0 to : a membership of value 0 signifying the fat that the element u Є U does not belong to the set F; a membership of value telling us that the element u Є U belongs to the set F with full ertainty; a membership of any other value from 0 to representing the element u to be a partial member of the set F. Fuzzy sets are identified by linguisti labels e.g. low, medium, high, very high, tall, very tall, ool, hot, very hot, et. The knowledge of a human expert an very well be implemented, in an engineering system, by using fuzzy rules. Fuzzy image filters are already proposed by many researhers for suppressing various types of noises [3],[4]. Simple fuzzy moving average (TMAV) filters are proposed [4] using triangular membership funtion as shown in Fig.. The membership equals zero at some minimum and maximum gray values of the pixels in the neighborhood of the enter pixel under onsideration.

3 µ The Proposed Filters: The basi struture of the proposed filters is shown in Fig. (a). The order statistis of the 33 neighborhood of the pixel x(i,, to be filtered, are taken into onsideration. Only three/five mid ordered statistis are the atual inputs for WROM (3,3)/WROM (3,5) filters. The filter weights are omputed from fuzzy membership funtions; two fuzzy membership funtions: triangular and Gaussian are proposed in the paper. First we disuss FWROM filter using triangular fuzzy membership funtion and all it Filter. Filter : The triangular fuzzy membership funtion ( x) WROM (3,5) is given by: µ for 0, x x() ( x x() ), x() x x() µ F ( x) = (5) ( x(8) x), x x(8) x(8) 0, x x(8) ( x ) (8) where ( + x =. ) Similarly, for WROM(3,3), the triangular fuzzy membership funtion may be defined replaing x() with x(3) and, (8) x min x with x(7) in (5). This type of fuzzy membership funtion is self-tunable sine the base of the triangle varies depending on the loal ordered statistis. This is why we don t need any further fuzzy rule base. The membership µ F (x (i) ) represents how lose the ordered statistis x (i) is to the enter value,. And, this is the fuzzy weight w (i) we assoiate with x (i) to ompute the fuzzy weighted rank ordered mean (FWROM) filter output. The rest part of the filter, aggregation and normalization is self-explanatory. We name suh fuzzy weighted filters as FWROM-T(3,5) and FWROM-T(3,3). Similarly we an define suh filters for a 55 window, e.g. FWROM- T(5,), FWROM-T(5,3), et. Next, the development of the seond filter is disussed. = (x min + x max )/ x max Fig.. Fuzzy Membership for an image filter. x F Filter : This filter uses a Gaussian fuzzy membership funtion. In all other respets, it is same as Filter. The Gaussian fuzzy membership funtion is given by: ( x( i ) ) σ µ = (6) F ( x ) ( i) e where σ = the standard deviation of the seleted ordered statistis x ( ), e.g. x () to x (8), or to x (7) for our Gaussian fuzzy weighted ROM filters: FWROM-G(3,7) or FWROM-G(3,5) respetively. Similarly, we an define suh filters for a 55 window, e.g. FWROM-G(5,), FWROM-G(5,3), et. 4. Simulation and Results Extensive omputer simulations were arried out to assess the performane of our proposed filters and to ompare the same with the standard MAV, MED and ROM filters, and the fuzzy TMAV filter. A seleted slie of size 0000 pixels (fae portion) from the standard Lena image is taken as the referene image. Gaussian noise with variane, varying from 0.03 to 0.09, is added to this image. Fixed valued impulse noise (salt-and-pepper) of density % is further applied to this image to simulate a mixed noise ondition. We take MSE as the performane measure. The performanes of the various filters obtained from simulations are listed in Table-. Table-: Performane Comparison of FWROM-T and FWROM-G with other standard filters (% salt-and-pepper noise alongwith Gaussian noise of variane, σ ) Filter MSE σ =0.03 σ =0.05 σ =0.07 σ =0.09 ROM(3,3) ROM(3,5) ROM(5,) ROM(5,3) MAV MAV TMAV FWROM-T (3,5) FWROM-T (5,) FWROM-T (5,3) FWROM-G (3,5) FWROM-G (5,) FWROM-G (5,3) Next, we orrupt the Lena fae image with Gaussian noise of variane, σ =0. alongwith salt-and-pepper of density %. The output images, for this noisy image input, of some standard filters and our proposed filters are shown in Fig. 3. We observe, by visual inspetion, that: (i) the performane of MAV33 is very poor; (ii) TMAV performs slightly well; and (iii) our filters FWROM- T(3,5) and FWROM-G(3,5) perform muh better than all other filters. 3

4 x () x () x (4) x (5) x (6) x (7) x (4) x (5) x (6) x (7) w 3 w 4 w 5 w 6 w 7 Fuzzy Filter ( i x, 0 x (8) x (9) µ µ a b w 3 = µ( ) x () x (8) x ( ) Fig (a) Struture of Fuzzy Weighted Rank Ordered Mean Filter; (b) Triangular fuzzy membership assoiated with the order statistis; () Gaussian Fuzzy membership. x ( ) Original Lena fae Noisy Lena fae MAV33 filter TMAV filter FWROM-T (3,5) filter filter FWROM-G (3,5) filter filter a b d e f Fig. 3. Images for visual performane omparison: (a) the original Lena fae image; (b) noisy Lena fae with Gaussian noise(σ =0.) and and salt-and-pepper noise(%); (),(d),(e)and (f) the various filter output images. 4

5 Mean squared error -----> x Performane Evaluation: MSE vs noise variane MAV3x3 TMAV FWROM-G(3,5) FWROM-T(3,5) Gaussian noise in variane -----> Fig. 4. Plot of Mean Squared Error for MAV33, TMAV, FWROM-G(3,5), FWROM-T(3,5) filters. The Mean Squared Errors (MSE) for the filters: MAV33, TMAV, FWROM-G(3,5) and FWROM-T(3,5) are plotted in Fig. 4. We observe that both the filters: FWROM-G(3,5) and FWROM-T(3,5) show almost the same performane and they are quite superior to the other standard filters. Referenes [] I.Pitas and A.N. Venetsanopoulos, Order Statistis in Digital Image Proessing, Proeedings of IEEE Vol.80,no., pp.893-9, Deember 99. [] C. Ktropoulos and I. Pitas, Adaptive LMS L-Filters for noise suppression in Images, IEEE Trans. Image Proessing, Vol.5, no., pp , Deember 996. [3] Russo, Reent Advanes in Fuzzy Tehniques for Image Enhanement, IEEE Trans. Instrumentation and Measurement, 998 Vol 47, no.6, pp , Deember, 998. [4] H.K. Kwan and Y. Cai, Fuzzy Filter for Image Filtering, Proeedings of Ciruits and Systems, MWSCAS-00, The th Midwest Symposium, Vol.3 pp.iii , August 00. [5] S.Meher, G.Panda, B.Majhi and M.R.Meher, Weighted Order Statistis Filters, Proeedings on National Conferene, APSC-004, pp.67-73, 6 th 7 th Nov Conlusion We feel that the filters: FWROM-T(3,5) and FWROM- G(3,5) show muh superior performanes, for suppressing mixed noise, over the other standard ROM filters, the MAV filters, and the fuzzy filter, TMAV. The filter: FWROM-T(3,5) performs better than the filter: FWROM- G(3,5) under low noise variane (σ < 0.07) and slightly degrades at high noise variane (σ > 0.07); the performane measure, MSE for both the filters being the same at σ = But the Gaussian membership funtion is muh more omputation-intensive as ompared to the triangular membership funtion. Therefore, the filter FWROM-T(3,5) will be preferred to FWROM-G(3,5) for real time appliations, e.g. in able television systems. 5