morillas_fuzzy peer groups for reducing mixed gaussian-impulse noise from color images

1452 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 18, NO. 7, JULY 2009

Fuzzy Peer Groups for Reducing MixedGaussian-Impulse Noise From Color Images

Samuel Morillas, Valentín Gregori, and Antonio Hervás

Abstract—The peer group of an image pixel is a pixel similarity-based concept which has been successfully used to devise imagedenoising methods. However, since it is difficult to define the pixelsimilarity in a crisp way, we propose to represent this similarity infuzzy terms. In this paper, we introduce the fuzzy peer group con-cept, which extends the peer group concept in the fuzzy setting. Afuzzy peer group will be defined as a fuzzy set that takes a peer groupas support set and where the membership degree of each peer groupmember will be given by its fuzzy similarity with respect to the pixelunder processing. The fuzzy peer group of each image pixel will bedetermined by means of a novel fuzzy logic-based procedure. Weuse the fuzzy peer group concept to design a two-step color imagefilter cascading a fuzzy rule-based switching impulse noise filter bya fuzzy average filtering over the fuzzy peer group. Both steps usethe same fuzzy peer group, which leads to computational savings.The proposed filter is able to efficiently suppress both Gaussiannoise and impulse noise, as well as mixed Gaussian-impulse noise.Experimental results are provided to show that the proposed filterachieves a promising performance.

Index Terms—Color image denoising, fuzzy logic, fuzzy metrics,fuzzy sets, peer group, vector filter.

I. INTRODUCTION

D IGITAL images are often corrupted by noise during theiracquisition and transmission. A fundamental problem in

image processing is to effectively suppress noise while keepingintact the desired image features such as edges, textures, and finedetails. In particular, two common sources of noise are the socalled additive Gaussian noise and impulse noise which are in-troduced during the acquisition and transmission processes, re-spectively [1]–[3]. Noisy images can be found in many today’simaging applications. TV images are corrupted because of at-mospheric interference and imperfections in the image recep-tion. Noise is also introduced in digital artworks when scan-ning damaged surfaces of the originals. Digital cameras may

Manuscript received February 08, 2008; revised March 09, 2009. First pub-lished May 12, 2009; current version published June 12, 2009. S. Morillas wassupported in part by the Generalitat Valenciana under Grant GVPRE/2008/257and in part by the Universidad Politécnica de Valencia under grant PrimerosProyectos de Investigación 2008/3202. V. Gregori was supported in part by theSpanish Ministry of Education and Science under Grant MTM 2006-14925-C02-01. The associate editor coordinating the review of this manuscript and ap-proving it for publication was Prof. Mario A. T. Figueiredo.

S. Morillas and A. Hervás are with the Faculty of Computer Science, Uni-versidad Politécnica de Valencia, Department of Applied Mathematics, 46022Valencia, Spain (e-mail: [email protected]; [email protected]).

V. Gregori is with the EPS Gandia, Department of Applied Mathematics, Uni-versidad Politécnica de Valencia, 46730 Grao de Gandia, Spain (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIP.2009.2019305

introduce noise because of CCD sensor malfunction, electronicinterference or flaws in data transmission. cDNA microarrayimage data contains imperfections due to both source and de-tector noise in microarray technology, etc. In the past years,many methods have been introduced in the literature to removeeither Gaussian or impulse noise. However, not all methods areable to deal with images which are simultaneously corruptedwith a mixture of Gaussian and impulse noise.

Most filters for Gaussian noise suppression are designed totake advantage of the zero-mean property of the noise and tryto suppress it by locally averaging pixel channel values. Clas-sical linear filters, such as the Arithmetic Mean Filter or theGaussian Filter [1], smooth noise but blur edges significantly.To approach this problem, many nonlinear methods have beenrecently proposed, for instance: the bilateral filter [4], [5], theanisotropic diffusion [7], the chromatic filter [8], or the soft-switching methods in [9] and [10], which motivate other fuzzymethods as the fuzzy directional derivative filter [11], the fuzzybilateral filter [12], the fuzzy noise reduction method [13], or thefuzzy-switching filter [14]. The aim of these methods is to detectedges and details by means of local statistics and smooth themless than the rest of the image to better preserve their sharpness.However, these methods commonly identify impulses as detailsor edges to be preserved, and, therefore, they are not able to re-duce them.

On the other hand, it is noted that earlier filters for impulsenoise are based on the theory of robust statistics because im-pulses are identified with outlier data, and, therefore, robuststatistics allow us to appropriately determine noise-free sam-ples and remove outliers. Filters of this family are, for instance,the popular median filter [15], the vector median filter [16],the vector directional filter [17], the directional-distance filter[18], and the HSV vector median filter [19], among others [2],[3], [20]. These filters are efficient in reducing impulse noisebut their signal-preserving capability is deficient because thefiltering operation is applied to each image pixel regardlesswhether it is noisy or not. To overcome this drawback, severaladaptive filters have been recently introduced. These filters maybe classified into the following categories: switching filters[21]–[56], filters using weighting coefficients [57]–[64], fuzzyfilters [65]–[74], and neuro-fuzzy filters [75]–[77]. However,many of these techniques select the appropriate noise-freeoutput from the input samples, and, therefore, they are notuseful to remove Gaussian noise because in such a case thereare no noise-free samples.

According to the above, the filter design is a challenging taskfor mixed Gaussian-impulse noise removal. A possible solutionis to apply two consecutive filters to remove first impulse noise

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MORILLAS et al.: FUZZY PEER GROUPS FOR REDUCING MIXED GAUSSIAN-IMPULSE NOISE FROM COLOR IMAGES 1453

and then Gaussian noise, or vice versa. However, the applicationof two filters could dramatically decrease the computational ef-ficiency of the method which implies that this solution couldnot be practical for real applications. Therefore, it is more inter-esting to devise specific filters to remove mixed noise. To date, afew methods in the literature are able to approach this problemefficiently. The Peer Group Averaging (PGA) technique pre-sented in [44]–[47] removes mixed noise by combining a statis-tical method for impulse noise detection and replacement withan averaging operation to smooth out Gaussian noise. The Tri-lateral Filter (TF) [6] is based on the well-known BilateralFilter [4] to smooth Gaussian noise but including an impulsedetector to be also able to reject impulse noise. The AdaptiveNearest Neighbor Filter (ANNF) and its variants [69], [70] use aweighted averaging where the weights are computed accordingto robust measures so that impulses that receive lower weightsare reduced. The Fuzzy Vector Median Filter (FVMF) [67] per-forms a weighted averaging where the weight of each pixel iscomputed according to its similarity to the robust vector me-dian. Another important family of filters are the partition basedfilters [78], [79] that classify each pixel to be processed into sev-eral signal activity categories which, in turn, are associated toappropriate processing methods. Other filters follow a regular-ization approach [80]–[86] based on the minimization of appro-priate energy functions by means of Partial Differential Equa-tions (PDEs). Wavelet theory has also been used to design imagefiltering methods [87]–[92] and the combination of collabora-tive and wavelet filtering is proposed in [93], [94]. In addition,other methods based on Principal Component Analysis (PCA)[95]–[98] have been studied.

The motivation of the method in this paper is the so-calledpeer group concept introduced in [44]–[47] and further studiedin [48]–[52]. The peer group of a given pixel is a set constitutedby this pixel and those of its neighbors which are similar to it.However, the similarity between two color pixels is not easilyexpressed in a crisp way, and, therefore, in this work, we pro-pose to use a fuzzy representation. This leads us to introduce thefuzzy peer group concept which we use to devise a novel filteringprocedure. The method presented in this paper is based on wellestablished concepts. It uses fuzzy metrics [99], [100], whichhave been proven to be efficient and effective for noise detection[41]–[43], [48], [52], but, in this case, fuzzy metrics are appliedto build the fuzzy peer groups. The proposed method is based onthe consecutive application of a fuzzy rule-based switching im-pulse noise filter and a fuzzy average filtering. Both steps use thesame fuzzy peer group, which leads to computational savings.This filter differs from previous peer group methods [44]–[52]because (i) fuzzy peer groups are represented as fuzzy sets in-stead of crisp sets used in [44]–[52], (ii) it employs a novel fuzzymethod first to determine the fuzzy peer group members and thento assign their corresponding membership degrees, (iii) it usesfuzzy rules to detect impulse noise pixels, and (iv) it performsa fuzzy weighted averaging to generate the output. Hence, thecombination of these fuzzy components is the main novelty ofthe proposed method. Experimental results will show that theproposed filtering technique exhibits competitive results withrespect to other state-of-the-art methods.

The paper is organized as follows. Section II describes thepeer group concept and defines the novel fuzzy peer group con-cept. The new filtering procedure is introduced in Section III.Section IV presents some experimental results and comparisons.Finally, conclusions are presented in Section V.

II. PEER GROUPS AND FUZZY PEER GROUPS

A. Peer Groups

Let denote the image to be processed, denote the centralpixel of the processing sliding window of size

, and let denote thepixels in the neighborhood of . Each pixel is represented asa 3-component vector comprising its R, G, and B components,i.e., . We have chosen the vector approachwhich is suitable for color image processing since it takes intoaccount the correlation among the color image channels [1], [2],[3], [16].

The peer group of an image pixel, roughly speaking, is de-fined as the set of its neighbor pixels which are similar to it.There are several ways of determining this set. One of them,introduced in [51] and used in [48] and [52], is based on theusage of a distance threshold to decide whether a pixel belongsto the peer group or not. So, the peer group of , denoted

, is the set , wheredenotes the Euclidean norm and . Obviously,

. In this approach, when the cardinalityis called a peer group of mem-

bers. This cardinality has been used in [48], [51], and [52] todecide whether is free of impulse noise or not. Notice thatbecause of the usage of a threshold based decision, the peergroup is defined in a crisp way. However, since the similaritybetween two color pixels is an imprecise concept, this approachdoes not provide a completely satisfactory representation of thepeer groups, and, therefore, we propose to represent this con-cept using fuzzy theory.

From another point of view, the definition of peer group givenin [44]–[47] is based on the ordering of the pixel neighbors withrespect to its similarity to the central pixel, as follows. Letbe an appropriate similarity measure between two color vec-tors.1 The color vectors in the processingwindow are sorted in descending order with respect to its sim-ilarity to the central pixel . That is, the ordering of thevectors of the processing window results in an ordered set ofthe elements or the colour vectors defined as follows:

(1)

such that

(2)

where, obviously, .Then, according to the definition of peer group given in

[44]–[47], for convenience, we denote by the peer groupof members associated with . This peer group is aset constituted by and its most similar neighbors, i.e.,

1The most commonly used similarity measures for multichannel image pro-cessing are listed in [1] and [2].



(3)

The choice of the number of members of the peer group is amain issue within this approach. The works in [44]–[47] proposeto use the Fisher’s linear discriminant (FLD) to solve this task.This method provides the best partition of the input set into twosubsets so that it includes neighbors in the peer group, andexcludes the rest. Unfortunately, this approach does not workproperly when the input set contains either only one cluster ormore than two clusters of data. In our context, for instance, whenprocessing an homogeneous noise-free area of the image onlyone cluster of data should be observed and, however, the FLDwill always partition the data into two subsets. On the otherhand, when three (or more) clusters of data are observed, whichmay occur for instance in an image area including impulse noiseand edges, the partition given by the FLD does not necessarilyleads to the desired peer group, as shown later.

According to the previous discussion, it is essential to per-form an appropriate construction of the peer groups, which in-volves to accurately determine the number of peer group mem-bers. This information can be used to decide whether is freeof impulse noise. Also, the peer group members may be usedto smooth the Gaussian noise from . In the approach in thispaper, we propose a more appropriate fuzzy logic-based methodto determine the peer group of an image pixel that we will callfuzzy peer group.

B. Determining the Fuzzy Peer Group

In the approach presented in this paper, we propose to use afuzzy similarity function, , as the function above which, fol-lowing the above terminology, is given by

(4)

where denotes the Euclidean norm and is aparameter which will be discussed later. We have chosen thisfunction because it is a fuzzy metric according to the conceptgiven by George and Veeramani, [99], [100], and this classof fuzzy metrics has been proven to be appropriate for fuzzycolor image processing [41]–[43], [48], [52]. Notice that now

takes values in[0,1] and that if and onlyif . Again, as discussed above, the color vectors

are sorted in a descending order with respect toits similarity to , which results in an ordered set de-fined as follows: such that

, where.

In order to establish our concept of fuzzy peer group we willdefine two fuzzy sets on the ordered set of pixels . Firstly,we consider the proposition “ is similar to .” Since itis a vague proposition, it can be represented by a fuzzy set

which, in turn, is given by a membership function [101]. Clearly, is a

monotonically decreasing function onthat computes the certainty of the proposition “ is similar to

” for each . Secondly, we define the accumulated simi-larity for , denoted , as

(5)

Obviously, . Moreover, takes thevalue if and only if , whichagrees with the cardinality of . Therefore, the larger valuethat can take on is , and

. Now, we consider the vague proposition“ is large” defined onand represented by the fuzzy set . For computing itscertainty, we will use a fuzzy membership function, , on

. Instead of using some of the standardand well-known membership functions, in this work we preferto define as a function of by means of a custom mem-bership function defined on that fulfills the followingrequirements.

i) for the minimum possible value of, that is, .

ii) for the maximum possible value of, that is, .

iii) In addition, it is known by previous works [48], [51], [52]that in order to distinguish between noisy and noise-freepixels, peer group cardinality differences are more sig-nificant when observed for low values of cardinality thanfor high values. To represent this, we prefer our member-ship function to be more sensitive in the low value rangethan in the high value range, and, therefore, we deviseso that the derivative of should be a strictly decreasingfunction.

We aim to devise the function using the simplest expres-sion possible. Since a linear function cannot be used because ofcondition (iii) above, the simplest function that can be used is asecond degree polynomial, that is, a quadratic function. Takinginto account the above requirements, the quadratic function ob-tained is expressed as

(see Fig. 1). Notice that the advantage of this customfunction with respect to most standard membership functions isthat it has no adjusting parameters. Finally, the function isgiven by

(6)

C. Best Number of Members for a Fuzzy Peer Group

According to the above, the peer group of a given pixel is con-stituted by that pixel and those neighbor pixels which are similarto it. Then, the best number of members of a peer group for agiven pixel will be determined by choosing so that all similarpixels are included in the set and the rest of the pixels are not. Inother words, if a pixel has similar neighbors, the best numberof members for its peer group is , and vice versa. For this, we



Fig. 1. Illustration of the membership function in (6) used to compute the cer-tainty of “� �� is large.”

introduce a novel method based on fuzzy logic to determine .Our proposal is based on determining as the value for which

is the largest set that contains only similar pixels. Sincethis is an imprecise statement, it can be represented using fuzzytheory as follows. Notice that for representsthe degree to which the furthest pixel in is similar to .On the other hand, represents the degree to whichthe accumulated similarity for is large, which get larger as

is larger. So, the best number of members ofwill be the value of maximizingthe certainty of the following fuzzy rule.

Fuzzy Rule 1: Determining the certainty of to be the bestnumber of members for

IF “ is similar to ” and “ is large”THEN “the certainty of to be the best number of members

is high”.The certainty of Fuzzy Rule 1 is computed for each

and the value which maximizes the certainty isselected as the best number of members of . That is,

, where denotes thecertainty of the Fuzzy Rule 1 for . The certainty of “ issimilar to ” is given by the function , and the certaintyof “ is large” is given by . We use the productt-norm as the conjunction operator so that, since no defuzzy-fication is needed, . Also,note that the reasoning behind Fuzzy Rule 1 can be interpretedin such a way that for each value of it checks whether theincrement from to is sufficientlylarge to include in the fuzzy peer group. Two samplesof performance of the Fuzzy Rule 1 are depicted in Figs. 2–3.In these samples, it is shown that the proposed method canwork well regardless of the number of clusters in the particularneighborhood and, as a result, it overcomes the shortcomingsexhibited by the Fisher’s linear discriminant approach.

Once we obtain the best number of members for , wedefine the fuzzy peer group of as the fuzzy set definedon the set and given by the membershipfunction , i.e., the membership valueof to the fuzzy set corresponds with the degree towhich “ is similar to ”.

III. NEW COLOR IMAGE DENOISING TECHNIQUE

In this section, we investigate the usefulness of fuzzy peergroups for color image denoising. We propose a color image

Fig. 2. Sample of performance of the Fuzzy Rule 1 in an homogeneous re-gion contaminated with Gaussian noise of standard deviation equal to 10 (a)(taken from the Parrots Image). The values � �� (c), � �� (d),and � �� (b) are given for each � � � . The Fuzzy Rule 1 determines�� as the best number of members. In this same case, the Fisher’s lineardiscriminant [44]–[47] determines � � �, missing, unfortunately, three sim-ilar neighbors.

Fig. 3. Sample of performance of the Fuzzy Rule 1 in an edge contaminatedwith Gaussian noise of standard deviation equal to 10 and impulse noise(left-up corner) (a) (taken from the Parrots Image). The values � �� (c),� �� (d), and � �� (b) are given for each � � � . The FuzzyRule 1 determines �� as the best number of members, and, so, the impulseis excluded from the fuzzy peer group. In this same case, the Fisher’s lineardiscriminant [44]–[47] determines � � �, including the impulse in the peergroup.

filter for suppression of mixed Gaussian-impulse noise which isbased on the fuzzy peer group concept and that we name FuzzyPeer Group Averaging Filter (FPGA). As commented in the in-troductory section, pixel averaging allows to remove Gaussiannoise because of the zero-mean property of this noise. How-ever, in order to avoid impulse noise perturbing this operationthe impulse noise in the image must be reduced first. There-fore, we propose a filter performing in two steps, namely, (i)impulse noise detection and reduction and (ii) Gaussian noise



Fig. 4. Diagram of the filtering process applied to each image pixel.

smoothing, so that both steps follow a fuzzy approach that usesthe information on the fuzzy peer groups, which is the main nov-elty introduced by the method. To reduce the impulse noise wepropose a fuzzy rule based procedure which uses the fuzzy peergroup concept. For Gaussian noise smoothing, we use a fuzzyaveraging among the members of the fuzzy peer group of thepixel under processing. Fig. 4 shows a diagram of the processapplied over each image pixel. The following sections detail thetwo steps of the proposed filter.

A. Impulse Noise Detection and Reduction

An impulse noise pixel can be defined as a pixel which is sig-nificantly different from its pixels neighbors. Conversely, an im-pulse noise-free pixel should have some neighbors quite similarto it. According to the above, we can formulate this conditionin terms of fuzzy peer groups as follows: a pixel is free ofimpulse noise if for the fuzzy peer group it is satisfiedthat “ is large” and “ is similar to ”. The fol-lowing Fuzzy Rule 2 represents this condition:

Fuzzy Rule 2: Determining the certainty of the pixel to befree of impulse noise

IF “ is large” and “ is similar to ”THEN “ is free of impulse noise”.To compute the certainty of the Fuzzy Rule 2 (which is

denoted by ) we perform analogously that for theFuzzy Rule 1. That is, the certainty of “ is large”is given by , according to (6), and the certainty of “is similar to ” is given by the function . Finally, weuse the product t-norm as the conjunction operator so that

. Indeed, notice that. This implies that no additional com-

putation is needed since this certainty is already computed.Now, we use to detect and replace impulses ac-cording to threshold-based rule shown in (7), at the bottomof the page, where is a threshold parameter with values in[0,1] whose importance will be discussed later. This procedure

constitutes a switching filter between the identity operation andthe VMF operation, which is used for being the most robustand well-known vector filter. However, other robust filteringstructures could be applied, as well.

B. Gaussian Noise Smoothing Procedure

The second step of the proposed method concerns theGaussian noise smoothing task. As mentioned above, we pro-pose to perform a weighted averaging operation among colorvectors. So, to smooth the pixel we use the members of

where the weighting coefficient for each color vector is itsmembership degree to the fuzzy peer group as follows:

(8)

Notice that, unlike other smoothing filters based on weightingcoefficients, such as those in [4], [12], and [67], the set ofneighbor pixels involved in the proposed smoothing procedureis restricted to the members of the fuzzy peer group, whichimplies that only similar pixels are used. This approach shouldperform a better edge and detail preservation than those non-restricted approaches since nonsimilar color vectors out of thefuzzy peer group do not perturb the averaging.

C. Computational Analysis

In the following, analogously to [42], [48], [51], and [52], weanalyze the computational complexity of the proposed methodby computing the average number of vector distance calcula-tions (in this case fuzzy distances) performed per pixel, sincethe vector distance calculation is the most costly operation per-formed by the method. Such an analysis allows us to comparethe computational efficiency of our method with that of othermethods.

Initially, for an window, distances (1), (2) arecomputed. In addition, if the pixel is an impulse, the VMF op-eration has to be performed. This implies the computation of

additional distances (7). The average numberof distances computed per pixel is given by the expression

(9)

where denotes the probability of a pixel to be considered asimpulse by the method. For example, if and ,the average number of distance calculations is 11.6, which is thesame that for the PGA filter [44]–[47]. If we compare with otherpeer group-based methods, we have that, in analogous condi-tions, the average number of distance calculations for the PeerGroup method in [51] is 7.0, for the Fast Peer Group method in[48] is 4.5, and for the Iterative Peer Group Filter in [52] is 11.2.It can be seen that the cost of the proposed method is a littlehigher than that of these previous peer group-based methods.

(7)



Fig. 5. Test images: (a) Lena, (b) Flower [102], (c) Motorbikes [102], (d) Parrots, and (e)–(h) details taken for the experiments.

Fig. 6. Performance in terms of PSNR of the proposed method as a function ofthe � parameter performing over the images Flower (blue color) and Lena (redcolor) contaminated with 5% (solid lines), 15% (dashed lines), and 25% (dottedlines) of impulse noise.

TABLE ISUGGESTED SETTINGS FOR � PARAMETER

However, it is necessary to point out that these previous methodswere designed to remove only impulse noise whereas the pro-posed method is able to suppress both impulse and Gaussiannoise. Finally, we can see that the method is quite computation-ally efficient if we compare it with the popular VMF [16] thatcomputes, under the same conditions, a fixed number of 36 dis-tances per pixel.

IV. EXPERIMENTAL RESULTS

The test images Lena, Flower [102], Motorbikes [102], andParrots in Fig. 5 have been used to evaluate the performanceof the proposed filter. In particular, a detail of each imagehas been used in order to better appreciate the performancedifferences among different parameter settings and filteringmethods. These images have been corrupted with Gaussianand/or impulse noise. For Gaussian noise we have used the

Fig. 7. Performance in terms of PSNR of the proposed method as a functionof the � parameter performing over the images Flower (blue color) and Lena(red color) contaminated with Gaussian noise with � equal to 15 (solid lines),25 (dashed lines), and 30 (dotted lines).

TABLE IISUGGESTED SETTINGS FOR THE � PARAMETER

classical white additive Gaussian model [1] contaminatingindependently each color image channel where the standarddeviation of the Gaussian distribution represents the noiseintensity. On the other hand, the two most common impulsenoise models assume that the impulse is either an extreme valuein the signal range or a random uniformly distributed valuewithin the signal range. These models are known as fixed-valueand random-value impulse noise, respectively. Since the re-moval of fixed-value noise has been extensively studied in theliterature and there have been several methods developed andable to suppress this noise effectively, in this paper we focuson the uncorrelated random-value case following the definitionin [1]–[3]. From now on, we will denote the probability ofimpulse appearance as .

The filter performance is assessed by taking into account boththe noise suppression and the detail preserving capabilities of



Fig. 8. Filter outputs for visual comparison: (a) Parrots image, (b) image corrupted with � � � Gaussian and � � �� impulse noise and outputs from (c) IPGF,(d) GRF, (e) PBF, and (f) FPGA.

Fig. 9. Filter outputs for visual comparison: (a) Lena image, (b) image corrupted with � � �� Gaussian and � � �� impulse noise and outputs from (c) CWF,(d) GRF, (e) PGA, and (f) FPGA.

the filter. To this end, we have used the Mean Absolute Error(MAE), the Peak Signal to Noise Ratio (PSNR), and the Normal-ized Color Difference (NCD) that measure the detail preservingcapability, the noise suppression capability, and the color preser-vation ability, respectively. The definitions of these objectivequality measures can be found in [1]–[3].

A. Adjustment of Filter Parameters

In order to choose the appropriate adjustment of the filter pa-rameters in (7) and in (4) we have analyzed the filter per-formance in terms of PSNR as a function of and for theimages Lena [Fig. 5(e)] and Flower [Fig. 5(f)], that have beencontaminated with different densities of Gaussian and impulsenoise. The images Motorbikes [Fig. 5(g)] and Parrots [Fig. 5(h)]are used afterwards for validation.

First, we analyze the PSNR performance of FPGA as a func-tion of the parameter, which is closely related to the success-fulness of the impulse noise detection. We have observed thatwhen the percentage of impulses is low (lower than 10%) theoptimal performance is obtained for values of in [0.05, 0.10].This means that the method only replaces pixels for which thecertainty to be free of impulse noise is very low. However, asthe number of impulses in the image increases, the uncertaintyin distinguishing between noisy and noise-free pixels also in-creases. This implies that in order to obtain a robust performancethe value of should be increased. Indeed, it is observed thatthe PSNR optimal value increases as the impulse noise in theimage is higher. These results are shown in Fig. 6. In general, wehave observed that for impulse noise percentages in [5,30] thevalue of can be proportionally set in the interval [0.05,0.25].Otherwise, settings able to obtain suboptimal performance aresuggested in Table I.



Fig. 10. Filter outputs for visual comparison: (a) Motorbikes image, (b) image corrupted with � � �� Gaussian and � � �� impulse noise and outputs from(c) ANNF, (d) FVMF, (e) PGA, and (f) FPGA.

Fig. 11. Filter outputs for visual comparison: (a) Flower image, (b) image corrupted with � � �� Gaussian and � � �� impulse noise and outputs from(c) ANNF, (d) FVMF, (e) PGA, and (f) FPGA.

Second, we observe the FPGA performance as a function ofthe parameter, which is used to measure the fuzzy similaritybetween color vectors in (4). According to the filter design, the

parameter is very important in the Gaussian noise reductionstep since it is the only parameter involved in the weighted av-eraging defined in (8). By observing the PSNR performance asa function of both the parameter and the standard deviation

of Gaussian noise, we have seen that when is low (lowerthan 10) optimal PSNR performance is obtained for values of

about 50. However, as the Gaussian noise in the image be-comes more significant, the uncertainty in deciding whether twocolor vectors are similar or not, increases. This fact should bereflected in the observed fuzzy similarities and, to this end, the

parameter should be increased. Indeed, for higher Gaussiannoise, PSNR optimal performance is obtained also for highervalues of . These results are shown in Fig. 7. Analogously,for the parameter, we conclude that for in [5, 30] the value

of can be fixed proportionally within the interval [50, 175].Also, we suggest in Table II suboptimal settings for .

Since it is interesting from the user/application point of viewto automatically set the value of the filter parameters, we pro-pose the following procedure to decide whether the percentageof impulse noise in the image and the standard deviation ofthe Gaussian noise are Low, Medium, or High. Then, andcan be set according to the suboptimal settings in Tables I and II.To estimate the percentage of impulse noise we use the methodproposed in [39], [40], and [51]: This method consists of de-tecting as impulses all those pixels that do not have a peer groupin a 3 3 neighborhood including at least 2 neighbours withinEuclidean distance lower or equal to (typically, ).The standard deviation of the Gaussian noise can be esti-mated as the average of the differences observed between eachimpulse-free pixel and its peer group members. Notice that thisestimation is not influenced by edges in the image since we only



TABLE IIICOMPARISON OF THE PERFORMANCE MEASURED IN TERMS OF MAE, PSNR, AND NCD ��

USING THE LENA IMAGE CONTAMINATED WITH DIFFERENT DENSITIES OF MIXED NOISE

TABLE IVCOMPARISON OF THE PERFORMANCE MEASURED IN TERMS OF MAE, PSNR, AND NCD ��

USING THE FLOWER IMAGE CONTAMINATED WITH DIFFERENT DENSITIES OF MIXED NOISE

TABLE VCOMPARISON OF THE PERFORMANCE MEASURED IN TERMS OF MAE, PSNR, AND NCD ��

USING THE MOTORBIKES IMAGE CONTAMINATED WITH DIFFERENT DENSITIES OF MIXED NOISE

consider the deviation with respect to peer group members. Inthis way, we can compute a rough estimate of the noise in theimage that is sufficient to set and accordingly to Tables Iand II.

B. Performance Assessment

The performance of the FPGA filter is compared with a se-ries of state-of-the-art filters including methods able to reduceboth Gaussian and impulse noise: Adaptive Nearest NeighborFilter (ANNF) [69], [70], Trilateral Filter (TF) [6],2 Fuzzy

2Applied in a componentwise manner

Vector Median Filter (FVMF) [67], Peer Group Averaging(PGA) [44]–[47], Graph Regularization Filter (GRF) [82], andPartition Based Filter (PBF) [78]; filters specially designedto remove Gaussian noise: Fuzzy Wavelet Denoising method(FWD) [87], Collaborative Wavelet Filter (CWF) [93], [94],and Color Regularization Filter (CRF) [80]; and one filterspecifically designed to remove impulse noise: Iterative PeerGroup Filter (IPGF) [52]. All filters have been applied on a3 3 filter window in an iterative fashion except the PBF whichis applied recursively but not iteratively. For each method, theparameter setting advised by the respective authors has been



TABLE VICOMPARISON OF THE PERFORMANCE MEASURED IN TERMS OF MAE, PSNR, AND NCD ��

USING THE PARROTS IMAGE CONTAMINATED WITH DIFFERENT DENSITIES OF MIXED NOISE

TABLE VIICOMPARISON OF THE PERFORMANCE MEASURED IN TERMS OF MAE, PSNR, AND NCD ��

USING THE LENA IMAGE CONTAMINATED WITH DIFFERENT DENSITIES OF GAUSSIAN NOISE

TABLE VIIICOMPARISON OF THE PERFORMANCE MEASURED IN TERMS OF MAE, PSNR, AND NCD ��

USING THE PARROTS IMAGE CONTAMINATED WITH DIFFERENT DENSITIES OF GAUSSIAN NOISE

employed, tuning experimentally when necessary. For theproposed FPGA filter we have used the suggested settings inTables I and II.

First, we have made an experimental comparison using im-ages corrupted with mixtures of Gaussian and impulse noise.Experimental results in Tables III–VI show that the proposedmethod exhibits the best performance in almost all cases interms of the MAE, the PSNR, and the NCD quality measures.This means that the proposed method achieves a better noisereduction as well as it better preserves image details. By in-specting the denoised images in Figs. 8–11, it can be concludedthat FPGA also outperforms the other filters from the visualpoint of view. The following points may be stressed.

• (i) The IPGF is only able to obtain a superior performancefor low Gaussian noise, since this method can only re-move impulses. On the other hand, the FWD, CWF andCRF only achieve a good performance when smoothingGaussian noise, whereas they are not able to completelyremove impulses.

• (ii) Both FVMF and ANNF seem to smooth the imagestoo much in the denoising process. This may be due tothe fact that these methods modify all pixels in the image,regardless whether they are impulses or not, which leadsfinally to a low detail preservation.

• (iii) TF only provides good performance for low densi-ties of noise. This fact may be explained because TF per-



TABLE IXCOMPARISON OF THE PERFORMANCE MEASURED IN TERMS OF MAE, PSNR, AND NCD ��

USING THE LENA IMAGE CONTAMINATED WITH DIFFERENT DENSITIES OF IMPULSE NOISE

TABLE XCOMPARISON OF THE PERFORMANCE MEASURED IN TERMS OF MAE, PSNR, AND NCD ��

USING THE PARROTS IMAGE CONTAMINATED WITH DIFFERENT DENSITIES OF IMPULSE NOISE

forms in a componentwise fashion and it does not take intoaccount the correlation among the color channels, whichis necessary to achieve a superior performance, speciallywhen the noise in the image is high.

• (iv) PBF exhibits a performance superior than the formermethods, specially for low noise. It can preserve well edgesand details while removing noise. But, for higher levelsof noise the performance is a little lower since it seemsthat not all small impulses and Gaussian noise are reduced.Probably, the results could be improved a little in this senseif the filter was applied iteratively.

• (v) PGA exhibits a better performance than ANNF andFVMF since it is able to remove most impulses and betterpreserve image details. However, in some cases, the noisesuppression ability is not good enough. This is most prob-ably because the peer groups do not contain the appropriatenumber of members which implies, for instance, the dis-ability to reduce small impulses in Lena’s face [Fig. 9(e)]or to properly smooth Gaussian noise in Figs. 10(e) and11(e).

• (vi) GRF also exhibits a better performance than ANNFand FVMF, specially when the noise is low. However, itsperformance decreases for higher noise, where the gener-ated output images seem to be a little blurred.

• (vii) It can be concluded that FPGA achieves the bestoverall results mainly because by means of the novelfuzzy procedure it is able to properly build the fuzzy peergroups, which allows to accurately identify and reduce

impulses as well as to efficiently smooth Gaussian noisewhile preserving the quality of image edges and details.

Second, we have also assessed the proposed method when fil-tering images which were corrupted only with Gaussian noiseand only with impulse noise. The results of this assessment areshown in Tables VII–X. In the case of Gaussian noise, except forthe methods specifically designed to smooth this kind of noise(FWD, CWF, and CRF), and the GRF whose parameters havebeen set in these cases to optimize performance for Gaussiannoise smoothing, the proposed method performs better than therest of the filters. This supports the usefulness of the method toprocess images corrupted only with Gaussian noise. Two sam-ples of performance for visual comparison are shown in Figs. 12and 13, where the performance of the proposed method is com-pared to the one achieving the best performance in these cases. Itcan be seen that FPGA is able to reduce Gaussian noise withoutblurring image edges. However, it tends to sharpen a little someedges. For images corrupted only with impulse noise, the onlymethods performing better in general than FPGA are the IPGF,which is specifically designed to remove impulse noise, and thePBF, that, in these cases, has been specifically trained to re-move impulses. In overall, the FPGA exhibits a better perfor-mance than the rest, which means that the method is able toremove impulses without significantly degrading the quality ofthe noise-free parts of the image. This is also substantiated bythe two examples given in Figs. 14 and 15, where the output ofFPGA is compared with the IPGF.



Fig. 12. Filter outputs for visual comparison: (a) Parrots image, (b) image cor-rupted with � � �� Gaussian noise and outputs from (c) CRF, and (d) FPGA.

Fig. 13. Filter outputs for visual comparison: (a) Lena image, (b) image cor-rupted with � � �� Gaussian noise and outputs from (c) CRF, and (d) FPGA.

V. CONCLUSION

First in this paper, we introduced the concept of fuzzy peergroup for a color image pixel which extends the concept of peergroup in the fuzzy setting. This novel concept aims to representthe set of all pixel neighbors to a given pixel which are similarto it. Since the similarity between color pixels is an impreciseconcept, we have represented it using fuzzy similarities. Thus,fuzzy peer groups are built as fuzzy sets where the membershipdegree of the neighbor pixels depends on their fuzzy similaritywith respect to the pixel under processing. For this, we have in-troduced a method based on fuzzy logic that builds the fuzzy peergroup of a color image pixel by first determining the membersof the fuzzy peer group and then assigning their correspondingmembership degrees. The proposed method is able to accuratelydetermine the fuzzy peer group of any color image pixel over-coming shortcomings of previous peer group approaches.

Fig. 14. Filter outputs for visual comparison: (a) Parrots image, (b) image cor-rupted with � � �� impulse noise and outputs from (c) IPGF, and (d) FPGA.

Fig. 15. Filter outputs for visual comparison: (a) Lena image, (b) image cor-rupted with � � �� impulse noise and outputs from (c) IPGF, and (d) FPGA.

Second, we have used fuzzy peer groups to define a two-stepcolor image filter cascading a fuzzy rule-based switching im-pulse noise filter by a fuzzy average filtering. Both steps use thesame fuzzy peer group, which leads to computational savings.Experimental results have shown that the proposed method isable to reduce mixed Gaussian-impulse noise exhibiting animproved performance with respect to state-of-the-art methodsmainly because of its ability to properly determine the fuzzypeer groups. Also, the proposed method is competitive whenreducing noise from images which are corrupted only withGaussian noise and only with impulse noise.

ACKNOWLEDGMENT

The authors would like to thank the reviewers and the Asso-ciate Editor Dr. M. A. T. Figueiredo for their valuable commentsand suggestions, which have been very useful in improving the



technical content and the presentation of the paper. The authorswould also like to thank Dr. Z. Ma, Dr. H. Ren Wu, and Dr. S.Bougleux for the support of useful material and information.

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Samuel Morillas was born in Granada, Spain, in1979. He received the M.Sc. degree in computer sci-ence from the Universidad de Granada in 2002 andthe Ph.D. degree from the Universidad Politécnicade Valencia, Spain, in 2007.

He is an Assistant Professor of the Department ofApplied Mathematics, Faculty of Computer Science,Universidad Politécnica de Valencia. He is a memberof the Research Centre in Graphics Technology andhis research interests include fuzzy metrics, fuzzysets, nonlinear image and video processing, and

medical imaging.

Valentín Gregori was born in Rafelcofer, Spain, in1952. In 1975, he received the degree in mathematicsfrom the Universidad de Valencia, Spain, and thePh.D. degree in mathematics in 1985.

Since 2000, he has been a Full Professor with theDepartment of Applied Mathematics, UniversidadPolitécnica de Valencia. In his early years of re-search, he was dedicated to general topology. Overthe last 15 years, his main research interest has beenin the field of fuzzy topology, especially in fuzzymetrics. He has published more than 50 papers in

international journals.Dr. Gregori is a member of the Instituto de Matematica Pura y Aplicada

(IMPA) and he is an associate editor of the international journal Applied Gen-eral Topology.

Antonio Hervás received the M.S. degree in math-ematics from the Universidad de Valencia, Spain, in1985, and the Ph.D. degree in mathematics from theUniversidad Politécnica de Valencia, Spain, in 1998.

His current research interests include Riccati ma-trix differential equations and applications, image re-construction, and aspects related with e-learning andnew technology applications to higher education. Heis an author of papers in these areas. He is a Full Pro-fessor with the Department of Applied Mathematics,e Universidad Politécnica de Valencia, where he has

been Vice President for the last eight years.Dr. Hervás is a member of the Spanish Society of Applied Mathematics,

SEMA.


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