OWA filters: A Robust Filtering Method and itsApplication to Color Images

A. Basua, M. Nachtegaelb

aFormerly with the Indian Statistical InstituteMachine Intelligence Unit, Kolkata, India

email:basu.aryabrata@gmail.com

bGhent University, Dept. of Applied Mathematics and Computer ScienceFuzziness and Uncertainty Modelling Research Unit

Krijgslaan 281 - S9, 9000 Gent, Belgiumemail: Mike.Nachtegael@UGent.be

Abstract— Bilateral filtering provides a scheme for non-iterative edge-preserving smoothing, but the results could bestrongly affected by the presence of outliers. In this paper wedevelop a robust bilateral filter for color images, and in orderto achieve this we propose to improve the bilateral filteringtechnique [13] by using Ordered Weighted Averaging operators.We adopt a fuzzy logic based approach: if the filtering isconsidered as a weighted averaging, then each filter is associatedwith a fuzzy set and the membership values of these fuzzy setsrepresent the weights. In this context, the bilateral filter is aconjunction of two fuzzy sets in the case of grayscale images:one in the spatial domain and one in a photometric domain.Applied to color images, we propose to extend the conjunction tothree fuzzy sets: one in the spatial domain, one in the brightnessdomain and one in the chromatic domain. Taking into accountthe robustness of rank filters, we propose to define an OWAfilter in order to obtain robust adaptive filters in brightnessand chromaticity. The robustness and performance of the filteris illustrated with several experiments, revealing its ability toremove different types of noise in the presence of outliers, whilepreserving edges. The noise types considered are impulse noiseand a combination of Gaussian noise with ”salt and pepper”noise types.

I. INTRODUCTION

The restoration of images from degraded ones is a classicallow level problem in image processing. The development ofnew acquisition devices for multi-component images increasesthe risks of system errors and the presence of noise. In thiscontext, the goal of filtering methods is to enhance the qualityof these images by suppressing noise and outliers.

Color images give an example of such complex multi-component images. They are obtained by three spectral ac-quisition channels leading to three components: red, greenand blue. At this low level of image processing, this paperproposes a new filtering scheme applied to color images. Thisscheme is adaptive to noise and robust against outliers, andboth of these aspects are important: in many applications thenoise types and the nature of the outliers that are present in thecorrupted image is not known. Of course, in cases where the

noise type is known, we will be inclined to use noise specificfilters rather than this robust filter.

In the field of image processing, many filtering methodsare proposed. The classical convolution filters like the meanfilter or the Gaussian filter reduce the noise by smoothing. Butthese filters do not preserve the details and edges in images,and they are not adapted to the presence of outliers. Rankfilters [12], such as the median filter, are more robust andsuppress some outliers while filtering, but also suffer from thedisadvantage that they do not always preserve small detailsand shapes in images. In order to deal with this problem,several methods that try to preserve details while smoothinghave been developed. For instance, the anisotropic diffusionmethod [10] allows us to decrease the amount of noise whilepreserving the edges in images. Such results are obtained bysmoothing along the edges and avoiding to diffuse across them.Bilateral filtering [13] is another classical method based on theconjunction of spatial and photometric filtering. Unfortunately,these anisotropic methods are strongly affected by outliers.Moreover, these methods require to determine two parametervalues in order to separate the noise from the edges. The clas-sical anisotropic diffusion uses both a contrast parameter and aresolution parameter [15]. The bilateral filtering [13] dependson both the spatial geometric closeness and the photometricsimilarity between neighbouring pixels. Other techniques likethe mean shift [1] have also two parameters controlling theresolution in the spatial and photometric domains. The tuningof these two parameters is not always obvious and remains adrawback.

The present work addresses an improvement of the bilateralfilter [13] by using fuzzy logic operators to gain robustnesswithout any assumption on any noise feature or outlier pres-ence. Bilateral filtering is based on a weighted averaging of thelocal neighborhood samples [3]. The weights are the productsof two values. The first one is computed in the spatial domainusing the Euclidean distance between the center sample andits neighbors. The other one is computed in the photometric

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domain using the Euclidean distance between the photometricvalues (or vectors) of the center sample and its neighbors.Note that the photometric domain is a RGB color space inthis paper. Now let us briefly describe the way we use toimprove the bilateral filter.

We define our version of bilateral filtering in the frameworkof fuzzy logic. Firstly, we consider that our bilateral filteris the conjunction of two fuzzy filters: a spatial filter anda photometric filter. We propose to use classical conjunctionoperators (t-norms [2]) to define the collaboration betweenseveral filters. Secondly, we consider that the filtered values arethe results of a fuzzy aggregation procedure [2]. We proposeto improve the robustness using Ordered Weighted Averaging(OWA) operators [18], resulting in what we will call OWAfilters. The aggregation with OWA operators is a weightedaveraging procedure, but the weights depend on the rank ofsimilarities between data instead of the similarities themselves.Consequently, this OWA aggregation derives advantage fromthe robustness of rank approaches. Finally, we apply thisfiltering scheme to color images by separating the colors intotwo photometric features: brightness and chromaticity. In thatway we obtain a trilateral filter by conjunction of three filters:a classical Gaussian filter in the spatial domain, an OWA filterin the brightness domain and an OWA filter in the chromaticdomain. The application to color images demonstrates thefilter’s ability to effectively remove noise in presence ofoutliers. We also compare our filtering scheme with threeclassical filtering methods, namely Gaussian filtering (i.e. onlyspatial filtering), the median filtering (i.e. only photometricfiltering), and the bilateral filtering [13] (i.e. both spatial andphotometric filtering).

The paper is organized as follows. Section 2 describesthe construction of our proposed filtering scheme. We reviewthe bilateral filter using conjunction operators, we describespatial filters in the framework of fuzzy logic, and we proposecolor filtering using OWA operators. Section 3 is devotedto experiments and results. Finally, Section 4 concludes thispaper with some remarks and directions for future research.For the convenience of readers that are not familiar with fuzzyset theory, some basic notions are summarized in an appendixat the end of the paper.

II. CONSTRUCTION OF THE PROPOSED FILTER

A. Bilateral filtering with conjunction operators

This section describes the bilateral filtering [13] in theframework of fuzzy logic. Let us briefly recall the bilateralfiltering method. The filtered value I(x) of a pixel x is con-sidered as a weighted average of values I(y) where y are thepixels belonging to a neighborhood of x. This neighborhoodN(x) is generally a n×n spatial window centered on x wheren is the width in pixel unit. The filtered values are defined by:

I ′(x) =

∑y∈N(x) w(x, y) · I(y)∑

y∈N(x) w(x, y), (1)

where w(x, y) are the weights applied to y in N(x). In thecase of multi-component images, the one-dimensional values

of I(y) and I(x) are replaced by vectors. In the bilateralfiltering scheme, we have w(x, y) = ws(x, y) ·wp(x, y) wherews(x, y) is the spatial weight and wp(x, y) is the photometricweight. Tomasi and Manduchi [13] proposed to use:

ws(x, y) = exp(−d2

s(x, y)2σ2

s

),

where ds(x, y) is the Euclidean distance in the spatial domainbetween the pixels x and y, and σs is geometric spreadfactor in the spatial domain. Similarly, using dp(x, y) as theEuclidean distance in a photometric domain between the pixelsx and y, and σp as the photometric spread factor, they define:

wp(x, y) = exp

(−d2

p(x, y)2σ2

p

).

Note that the bilateral filter uses two parameters (σs and σp)with empirically selected values. In this paper, we rescale thecoefficients ws(x, y) and wp(x, y) to values between 0 and 1.In that way we can consider these coefficients as membershipvalues of two fuzzy sets in the image field. The supports ofthese fuzzy subsets are N(x), i.e., the membership valuesare 0 outside N(x). The product of two membership values(ws(x, y) · wp(x, y)) corresponds to a conjunction operator,thus the final weights of the bilateral filter are the result ofa conjunction of two fuzzy sets. Such a conjunction can beobtained using any other t-norm as conjunction operator [18].The most classical conjunction operator is the minimum, andwe choose this standard operator in this paper to replace theproduct for computing the weights of filters (the minimum isthe largest t-norm and results in ’maximal’ combination ofweights; many other t-norms can be used as well).

The approach by using fuzzy aggregation operators allowsus to consider the bilateral filter as a multi-criteria conjunctionoperator. Thus we can generalize the bilateral filter by aggre-gating various weights and by suppressing any limitation tospatial and photometric weights. In this paper, we propose tofilter color images by defining three weights: the first weightws(x, y) uses the spatial coordinates of pixels, the secondweight wbright(x, y) uses the brightness of pixels and the lastweight wchroma(x, y) uses the chromaticity. In that way, ourfilter becomes the conjunction of three fuzzy sets, and thefinal weight is computed by:

w(x, y) = min(ws(x, y), wbright(x, y), wchroma(x, y)

).

In the next subsections we describe the computation of thesethree weights.

B. Spatial filtering

Tomasi and Manduchi [13] used the Gaussian filter whenthey introduced their bilateral filter, and so shall we. Sucha filter can be considered as a weighted averaging procedurewhere the weights are given by a Gaussian distribution. In thispaper, we consider that the weights are membership values ofa fuzzy set in the image field.

Note that we could use any classical convolution filter in theplace of the Gaussian filter as soon as a fuzzy set representsthis filter in the image field (in the spatial domain).

C. Color filtering

The photometric domain of color images is the RGB space[11]. Tomasi and Manduchi [13] do not take into accountany color perception. Then the weights wp(x, y) are directlycomputed using an Euclidean distance in the RGB space.In this paper, we improve their approach by separating thebrightness and the chromaticity which are the two mainfeatures of colors [17]. According to our fuzzy approach offiltering, we have to define two fuzzy sets: one for brightnessfiltering and the other one for chromaticity filtering. Thus theweight wp(x, y) is obtained by the aggregation of two weightswbright(x, y) and wchroma(x, y).

1) OWA filter in the brightness domain: We define thebrightness H1 by [16]:

H1 =R + G + B

3.

The weight wbright(x, y) of each pixel y lying in a spatialwindow centered on x depends on the brightness similaritybetween the pixels x and y. These weights are membershipvalues of a fuzzy neighborhood of x which is defined ac-cording to the brightness similarity between x and y (withouttaking into account the spatial coordinates of y). In this paper,we propose a new approach to define filters which remainrobust, even in presence of outliers.

Indeed, in a photometric domain, the bilateral filteringuses the Euclidean distance to define similarities betweenpixels. Unfortunately, this approach is not robust because thedistances could be strongly affected by outliers. We proposeanother way to define such similarities. Taking into accountthe robustness of rank filters, we propose that the ranks ofdistances take the place of the distances themselves. Accordingto their ranks we can give a weight to each pixel, i.e., theweights are assigned to ranks and not to the distances. Ifthe rank is not changed, then a large distance or a smalldistance from x to y does not modify the weight assigned toy. Consequently, these weights depending on ranks can dealwith outliers: the outliers which correspond to large distancescould not corrupt filtered values. This approach allows tobuild new filters by rank-based weighted averaging in anyphotometric domain. This filtering method corresponds to theOrdered Weighted Averaging (OWA) proposed by Yager [18],[19] in the field of fuzzy aggregation. Therefore these filtersare called OWA filters in this paper. Note that these filters arealso associated with fuzzy sets where the membership valuesare the weights depending on ranks.

We propose to set the coefficients of such a filter using thefollowing principle. The first coefficient is associated with thesmallest distance, which corresponds to the distance betweenthe pixel x and itself. Of course, this smallest distance is null.We set this first coefficient to zero. In this filtering scheme,the color vector of x cannot affect the filtered color becausethe weight of x is equal to zero. If x itself is an outlier, thenthe filter can remove this outlier. Except for the first filtercoefficient (rank equal to 0), the other ones are between 0 and1. In this paper, the value of 1 is assigned to pixels whose

rank is less or equal to 6. The others ones are set to 0. Anormalization procedure is applied in order to have the sumof all weights equal to one. Thus, we obtain a new schemefor building adaptive filters which are more robust than theclassical ones since they are more efficient in presence ofoutliers. Moreover, as quoted before, setting the coefficientof x to zero may enhance the efficiency of the filter w.r.t.outliers. Note that the median filter (and most of the rankfilters) can also exclude the value of x when computing filteredvalues (or vectors). Also note that this robust filter is evenadaptive because we have no tuning parameter depending onthe distance itself.

We apply this OWA filter approach in the brightness domain.The distances dbright(x, y) between the brightness values ofthe center sample x and its neighbors y are computed by:

dbright(x, y) = |H1(x) − H1(y)|.These distances are sorted by value and the ranks are set

from 0 to N −1 where N is the number of pixel samples. Theweights wbright(x, y) are assigned to each pixel y in a 3 × 3window centered on x as described previously.

2) OWA filter in the chromaticity domain: The same OWAfilter is applied in the chromaticity domain. We define thechromaticity space using two components H2 and H3 [16]where H2 is associated with the opposition Red-Green andH3 is associated with the opposition Blue-Yellow. Thus wedefine:

H2 = R − G,

H3 = B − R + G

2.

The linear transformation from RGB to H1 H2 H3 sepa-rates the color space into the brightness space (H1) andthe chromaticity space (H2,H3). In this chromaticity space,we use the ranks of Euclidean distances between the points(H2(y),H3(y)) and (H2(x),H3(x)) to build the OWAfilter. The assignment of the chromaticity weight values(wchroma(x, x) = 0) follows the same rules as the brightnessweights. Pixels of rank 6 or less are weighted by 1 whereasthe others are weighted by 0. Each weight is then divided bythe sum of all weights.

III. EXPERIMENTS AND RESULTS

In this section, we compare our OWA filter with theGaussian filter, the median filter and the bilateral filter. Thespatial neighborhood is the same for all the filters and isequal to a 3 × 3 window, as commonly used in experiments.The photometric spread parameter σp of the bilateral filteris set to almost twice the value of the standard deviation σof a hypothetic Gaussian noise deduced from the differencebetween the original image and the degraded image. TheGaussian filter and the median filter are applied separatelyon the three color bands (red, green and blue).

The filtered images are compared to the original ones usingthe peak signal to noise ratio (PSNR), expressed in dB:

PSNR = 20 · log(

255√MSE

),

where MSE stands for the Mean Square Error between thedegraded image and the original one.

The nature of the restoration problem depends on the kindof noise corrupting the images. Unfortunately the type isgenerally unknown, but robust filters allow us to deal withany kind of noise. Table I shows the average PSNR obtainedby the considered filters on a set of 25 images. The kind ofnoise applied to the images is quoted in the first column.

TABLE I

AVERAGE PSNR OBTAINED ON A SET OF 25 IMAGES.

PSNR Gaussian Median Bilateral OWA

Gaussianσ = 5 26.3 31.2 36.7 32.1σ = 10 26.0 29.7 31.7 30.3Impulse

5% 24.1 31.3 20.8 31.810% 22.5 30.5 17.8 30.020% 20.0 27.9 15.2 26.4

The bilateral filter outperforms the other filters whenGaussian noise is present, whereas the median filter obtainsalmost the best results in presence of impulse noise. It confirmsthe fact that bilateral filters are well suited for Gaussian noiseand median filters are well suited for impulse noise. But wenotice that the OWA filter is always sorted second, whichin a way is always advantageous because in practice we arenot always aware of the noise types present in the corruptedimage. This means that the OWA filter is robust enough toface different kinds of noise without a priori knowledge.

To outline the properties of the OWA filter, we present someresults obtained with four common color images in imageprocessing (peppers, house, Lena and mandrill). Original im-ages are corrupted by an additive Gaussian noise combinedwith a ”salt and pepper” noise:

I ′(x, y) = f (I(x, y) + ηg(x, y)) ,

where ηg(x, y) corresponds to a zero mean Gaussian distribu-tion with a variance σ2 and f(x) simulates a ”salt and pepper”noise with a probability p by:

f(x) ={

x with probability 1 − pηu(x) with probability p

,

where ηu(x) is a uniform distribution in the interval [0, 255].The amount of noise is chosen to be different depending on

the color band. Such experiment make it much more difficultto choose parameters such as the photometric spread parameterin the bilateral filter. This may reveal the adaptive ability ofthe filters. The parameters of noises applied to each color bandare resumed in Table II. The results for the four images arequoted in Table III.

As expected, the bilateral filter obtains the worse results inthe presence of outliers. Indeed, when the central pixel is anoutlier (a ”salt or pepper” pixel), then the distances with itsneighbors in the photometric domain tend to be high and theresulting weights are almost equal to 0 except for the centralpixel. The degraded image is therefore not affected by the

TABLE II

INTENSITY OF ADDED NOISE DEPENDING ON THE COLOR BAND.

Noise Gaussian Salt and Pepper

red band σ = 5 p = 10%green band σ = 5 p = 10%blue band σ = 5 p = 10%

TABLE III

RESULTS EXPRESSED IN DB OBTAINED USING VARIOUS FILTERS.

PSNR peppers house Lena mandrill

Degraded image 19.0 19.6 17.8 18.7(estimated σ) (28.3) (26.3) (33.7) (29.9)

Gaussian 25.5 29.1 23.7 24.3Median 25.3 32.1 28.0 22.8Bilateral 20.8 22.0 19.3 20.3

(σp) (60) (60) (70) (70)OWA filter 30.2 30.4 26.9 23.7

bilateral filtering. The result image looks very similar to thedegraded one (see Figure 3). Of course, one can improve theresults by increasing the spread factor σp in the photometricspace, but this will not outperform the Gaussian filter resultssince the bilateral filter will tend in fact to a Gaussian filter(the weights in the photometric domain will all tend to 1).

By setting the weight of the central pixel to 0 in theOWA filter, the “salt and pepper” pixels are almost removed.However, when the percentage of outliers is greater than 10%,the performance of the OWA filter is degraded compared to themedian filter; some outliers are not removed. Thus we considerthat OWA filter is less robust than the median filter when avery large amount of outliers are feared. This also explainswhy the median filters has a slightly better performance forthe House and Lena image in Table III.

The use of the ranks to determine the weights in a chromaticspace should yet prevent the outliers to affect the neighbor-hood. One can notice, in the case of the median filter, thatthe outliers may interfere on the results in the neighborhooddepending on the context. For instance, on Figure 1 whichrepresents a zoom on a region of the peppers image, somewrong colors appear on the limit between the brown and thered regions. These colors have almost the same hue than theoutliers which lie in the neighborhood.

Moreover, the contrast and details are better preserved usingOWA filters than median filters (Figure 2). The whole imagehas a less smoothed aspect. Figure 2 exemplifies that the brightreflect in the eye is more preserved using OWA filter. TheOWA filter offers less smoothing of areas near the Iris and theSclera of the Lena Image. The same is being observed with theroof gutter of the house image. The gutter contrast is somewhatless smoothed and enhanced with the OWA filter than withthe median filter. This shows that the presence of outlierscan affect the results in the neighborhood with a medianfilter, which is not the case with the OWA filter. This mayexplain why the OWA filter sometimes slightly outperformsthe median filter, despite the fact that some sets of outliers

Fig. 1. Effects of outliers on the filtering process: top left = zoom in thePepper image, top right = corrupted image, bottom left = application of themedian filter, bottom right = application of the proposed OWA filter.

are not suppressed, and thus we obtain a robust adaptive filterin the brightness and chromaticity domain, which is a majorimprovement over the bilateral filter in reference [13].

IV. CONCLUDING REMARKS AND FUTURE WORK

This paper proposes a new version of the bilateral filter byusing OWA operators. This approach of filtering, using fuzzylogic, allows to design robust and adaptive filters. More sophis-ticated filtering strategies are proposed in literature designingfilter by fuzzy techniques [5]. However, these techniques aredeveloped for higher levels of image processing using complexdecision models [14] which require a priori knowledge ofthe images. Our goal in this paper was to propose low leveladaptive methods without any assumption on the images.

The ability of fuzzy aggregation methods allows us todevelop specific filters for color images. Unlike Tomasi andManduchi [13], we take into account some elements of colorperception by separating RGB vectors into brightness andchromaticity vectors. Consequently the similarities betweencolors are not evaluated by Euclidean distances in the RGBspace. The results show the improvement of the color filteringwith our method when outliers are feared.

The low level filtering is developed for obtaining a fastprocessing method. A future work will consist in developingthis kind of fuzzy adaptive filters for image sequences takinginto account the time component for improving the quality ofcolor images.

To optimize the adaptive filtering a step further, futureresearch should focus on more advanced variations of OWAoperators, like for instance ”Induced Ordered Weighted Aver-aging” operators (IOWA) [20] or ”Weighted OWA Operator”(WOWA) [6].

Fig. 2. Zoom on the Lena image (right eye) and on the House image (roofgutter).

In a worst case scenario, where presence of outliers inhuge amounts is feared, one could/should focus on a morespecialized treatment of OWA operators, like the maximalRenya entropy factor [8] and the solution equivalence ofminimax disparity and minimum variance problems [7] forOWA operators in the framework of fuzzy sets.

APPENDIX: BASIC NOTIONS FROM FUZZY SET THEORY

A fuzzy set A in a universe U is characterized by a U −[0, 1] mapping, that associates with every element u of U amembership degree A(u) [21]. The value A(u) expresses a“degree of belonging to” or a “degree of satisfying a property”,which enables us to work with imprecise information. The

Fig. 3. Comparative filtering results with a combination of Gaussian noise and salt and pepper noise.

underlying logic of fuzzy set theory is commonly referred toas fuzzy logic. Since their introduction, fuzzy set theory andfuzzy logic have given rise to many real-world applications,also in the field of image processing [4], [9].

Classical set operations such as intersection and union caneasily be extended to fuzzy set theory by using fuzzy logicaloperators. For example, the intersection of two fuzzy setsA and B in a universe U is defined by (A ∩T B)(u) =T (A(u), B(u)), for all u in the universe U . In case of intersec-tion, the operator T in this expression has to be a conjunctor,or more specifically a t-norm. A [0, 1]2 − [0, 1] mapping Tis called a conjunctor if T (0, 0) = T (1, 0) = T (0, 1) = 0,T (1, 1) = 1, and if it has increasing partial mappings (note thatthis is a straightforward extension of the Boolean conjunction).It is called a t-norm if it is also commutative, associativeand satisfies (∀a ∈ [0, 1])(T (1, a) = T (a, 1) = a). Popularexamples are TM (a, b) = min(a, b), TP (a, b) = a · b andTW (a, b) = max(0, a + b − 1), with (a, b) ∈ [0, 1]2. Similardefinitions and operators can be introduced for union, setinclusion and set complement.

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