IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 5, MAY 2006 1153

A Fuzzy Impulse Noise Detectionand Reduction Method

Stefan Schulte, Mike Nachtegael, Valérie De Witte, Dietrich Van der Weken, and Etienne E. Kerre

Abstract—Removing or reducing impulse noise is a very activeresearch area in image processing. In this paper we describe a newalgorithm that is especially developed for reducing all kinds of im-pulse noise: fuzzy impulse noise detection and reduction method(FIDRM). It can also be applied to images having a mixture of im-pulse noise and other types of noise. The result is an image quasiwithout (or with very little) impulse noise so that other filters canbe used afterwards. This nonlinear filtering technique contains twoseparated steps: an impulse noise detection step and a reductionstep that preserves edge sharpness. Based on the concept of fuzzygradient values, our detection method constructs a fuzzy set im-pulse noise. This fuzzy set is represented by a membership functionthat will be used by the filtering method, which is a fuzzy averagingof neighboring pixels. Experimental results show that FIDRM pro-vides a significant improvement on other existing filters. FIDRM isnot only very fast, but also very effective for reducing little as wellas very high impulse noise.

Index Terms—Fuzzy filter, image processing, impulse noise,membership functions, noise reduction.

I. INTRODUCTION

I N THE literature, several (fuzzy and nonfuzzy) filters havebeen studied for impulse noise reduction [1]–[5]. Impulse

noise is caused a.o. by errors in the data transmission generatedin noisy sensors or communication channels, or by errors duringthe data capture from digital cameras. Noise is usually quanti-fied by the percentage of pixels which are corrupted. Corruptedpixels are either set to the maximum value or have single bitsflipped over. In some cases, single pixels are set alternatively tozero or to the maximum value. This is the most common form ofimpulse noise and is called salt and pepper noise. Nevertheless,other types of impulse noise are possible as well.

In this work, we will present a new, faster, and more effi-cient noise reduction method for images corrupted with im-pulse noise. For clarity, we first give the definition of impulsenoise. Afterwards we introduce the new filter called fuzzy im-pulse noise detection and reduction method (FIDRM). The restof the paper is structured as follows. The details of the detectionphase are given in Section II. In Section III, the filtering step isdiscussed. Experimental results and conclusions are finally pre-sented in Sections IV and V.

Manuscript received October 22, 2004; revised April 29, 2005. This work wassupported by the GOA project 12.0515.03 of Ghent University. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Prof. Vicent Caselles.

The authors are with the Department of Applied Mathematics and Com-puter Science, Fuzziness and Uncertainty Modeling Research Unit, GhentUniversity, B-9000 Gent, Belgium (e-mail: stefan.schulte@ugent.be; eti-enne.kerre@ugent.be).

Digital Object Identifier 10.1109/TIP.2005.864179

A. Impulse Noise for Grayscale Images

A Monochrome (grayscale) digital image is often repre-sented by a two-dimensional array where an address de-fines a position in , called a pixel or picture element. In agrayscale (or graylevel) image, the only colors are shades ofgray. A “gray” color is one in which the red, green, and bluecomponents all have equal intensity in the RGB space, so it isonly necessary to specify one single intensity value for eachpixel, as opposed to the three intensities needed to specify apixel in a full color image. Often, the (grayscale) intensity isstored as an 8-bit integer giving 256 possible different shadesof gray going from black to white, which can be represented asa [0,255] integer interval. In this interval, we consider severalinteger values with and . If

denotes the pixel value of the (two-dimensional) imageat position , then we can model the occurrence of im-

pulse noise, for grayscale images, as

with probabilitywith probabilitywith probability

......with probability

(1)

where is the probability that a pixel is corrupted, and is thecorrupted image. The value (with ) indicatesthe probability that an original image pixel becomes . Ofcourse, the following restriction must be valid: .In the case of saturated impulse noise (salt and pepper noise),there are only two values and , which are the maximumand the minimum pixel value of the considered integer interval(in our case respectively 255 and 0). This definition of impulsenoise is a simplification of a more general noise model in whicha noisy pixel can take on arbitrary values in the dynamic rangeaccording to some underlying probability distribution. Let

and denote the luminance values of the originaland the noisy image, respectively, at position . Then, wehave

with probabilitywith probability

(2)

where is an identically distributed, independent randomprocess with an arbitrary underlying probability density func-tion. In this paper, we concentrate on the more common discreteimpulse noise model from (1). In case of random impulse noise(2), the global detection and reduction method discussed herecan be changed to a local method.

1057-7149/$20.00 © 2006 IEEE

1154 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 5, MAY 2006

Fig. 1. Neighborhood of a central pixel.

B. Fuzzy Impulse Noise Detection and Reduction Method

We introduce a new two step filter called FIDRM. This newfilter has two separated steps or phases: the detection phase andthe filtering phase. The detection phase uses fuzzy rules to de-termine whether a pixel is corrupted with impulse noise or not.When impulse noise is detected, we try to indicate the values

( with ). Moreover, some pa-rameters will be determined which will be passed to the filteringphase. After this detection, our fuzzy filtering technique focusesonly on the values, i.e., the filtering is concentrated on the realnoisy pixels.

II. FUZZY IMPULSE NOISE DETECTION

Our noise detection method uses fuzzy gradient values as in-troduced with the GOA filter [6], [7] to determine if a certainpixel is corrupted with impulse noise or not.

A. Fuzzy Gradient Values

For each pixel of the image (that is not a border pixel),we use a 3 3 neighborhood window as illustrated in Fig. 1.Each neighbor with respect to corresponds to one direction{NW = north west, N = north, NE = north east, W = west, E =east, SW = south west, S = south, SE = south east}. Each suchdirection with respect to can also be linked to a certainposition (also indicated in Fig. 1).

If we denote as the input image, then the gradientis defined as the difference

(3)

where the pair corresponds to one of the eight directionsand is called the center of the gradient. The eight gradientvalues (according to the eight different directions or neighbors)are called the basic gradient values. One such gradient valuewith respect to can be used to determine if a central pixelis corrupted with impulse noise or not, because if this gradient isquite large then it is a good indication that some noise is presentin the central pixel , but there are two cases in which thisconclusion is wrong.

Fig. 2. Involved centers for the calculation of the related gradient values in theNW-direction.

TABLE IINVOLVED GRADIENT VALUES TO CALCULATE THE FUZZY GRADIENT

1) If the central pixel is not noisy, but one of the neighborsis, then this can also cause large gradient values.

2) An edge in an image causes some kind of natural largegradient values.

To handle the first case, we use not only one gradient value,but eight different gradient values (according to the eight dif-ferent directions) to make a conclusion; to solve the secondcase, we use not one basic gradient for each direction, but onebasic and two related gradient values for each direction. Thetwo related gradient values in the same direction are deter-mined by the centers making a right angle with the directionof the first (basic) gradient. For example (Fig. 2), in the NW-di-rection [i.e., for ] we calculate the basic gra-dient value plus the two related gradient values

and . The twoextra gradient values are used for making the separation betweennoisy pixels and edge pixels: when all these gradients are large,then is considered to be not a noisy but an edge pixel. InTable I, we give an overview of the involved gradient values:each direction (column 1) corresponds to a position (Fig. 1)with respect to a central position. Column two gives the basicgradient for each direction; column three gives the two relatedgradients.

Finally, we define eight fuzzy gradient values for eachof the eight directions. These values indicate in which de-gree the central pixel can be seen as an impulse noisepixel. The fuzzy gradient value for direction

SCHULTE et al.: FUZZY IMPULSE NOISE DETECTION AND REDUCTION METHOD 1155

Fig. 3. Membership functions (a) SMALL, respectively, LARGE; (b) BIGNEGATIVE, respectively, BIG POSITIVE.

, is calculated bythe following fuzzy rule.

IF is large AND issmall

ORis large AND is

smallOR

is big positive ANDAND are big negative

ORis big negative AND

AND are big positiveTHEN is large

where is the basic gradient value and andare the two related gradient values for the direction

. Because “large,” “small,” “big negative,” and “big positive”are nondeterministic features, these terms can be represented asfuzzy sets [8]. Fuzzy sets can be represented by a membershipfunction. Examples of the membership functions LARGE (forthe fuzzy set large), SMALL (for the fuzzy set small), BIG POS-ITIVE (for the fuzzy set big positive), and BIG NEGATIVE (forthe fuzzy set big negative) are shown in Fig. 3.

The horizontal axis of these functions (Fig. 3) represents allthe possible gradient values (the universe ) and thevertical axis represents a membership degree . A mem-bership degree indicates the degree in which a certain gradientvalue matches the predicate (e.g., large). If a gradient value hasmembership degree one, for the fuzzy set large, it means that itis large for sure.

The fuzzy rule contains some conjunctions and disjunctions.In fuzzy logic triangular norms and co-norms are used to rep-resent conjunctions [9] (roughly the equivalent of AND oper-ators) and disjunctions (roughly the equivalent of OR opera-

Fig. 4. For (a) a noisy Lena image, we display (in white) all (b) j 5A(i; j)j 2 [0; 40], (c) j5 A(i; j)j 2 [40;125], and (d) j5 A(i; j)j 2[125;255].

tors). Some well-known triangular norms (together with theirdual co-norms) are the minimum (maximum) and the product(probabilistic sum) [10]. So, we can for example translate thesubrule “ is large AND is small” as

, whereand are the membership functions shown in

Fig. 3(a). These two membership functions depend on the twoparameters and . According to the following three observa-tions we can derive these values.

1) Gradient values for a given direction , which are situ-ated in the interval [0,40] are most likely nonedge andnonnoisy pixels but very fine detail pixels as illustratedin Fig. 4(b). So, these pixels can be categorized as noisefree without (or with very little) uncertainty, resulting ina zero membership degree in the fuzzy set impulse noise.

2) Gradient values for a given direction , which are situ-ated in the interval [40, 125] are most likely edge pixelsor noise pixels as shown in Fig. 4(c). Here, we have somekind of uncertainty which is expressed by the member-ship degrees in the fuzzy set impulse noise: These mem-bership degrees are now situated between zero and one.

3) Finally, gradient values for a given direction , which aresituated in the interval [125, 255] are most likely noisepixels as shown in Fig. 4(d). In this case, the membershipdegrees in the fuzzy set impulse noise are one.

Since we are searching for noise pixels, we chooseand , which are good choices for our method.The idea behind the usage of the fuzzy sets big positive and bignegative is that if the basic gradient and the two related gradi-ents are both large but have different signs then it is a good in-dication that noise is present. Therefore, we also use the fuzzysets big negative and big positive. We represent these fuzzy setsby membership functions pictured in Fig. 3(b). Gradient valuesaround zero are seen as more or less unsigned and gradientvalues above 15 or under 15 become significant to matchingthe feature big positive, respectively, big negative. Therefore,we choose , , and

1156 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 5, MAY 2006

Fig. 5. (a) Lena image corrupted with 5% impulse noise [(p ; p ; p ; p ; p ) = (0; 50;85;175;255)]. (b) The corresponding histogram of the detected noisypixels. (c) Lena image corrupted with 3% impulse noise [(p ; p ) = (25;250)] plus Gaussian noise with � = 5. (d) The corresponding histogram of the detectedimpulse noisy pixels.

. Finally, after choosing a T-norm (triangular norm)and a S-norm (triangular co-norm), we can calculate the acti-vation degree of such an “IF-THEN” rule. This activation de-gree also indicates the degree in which can be consideredas large. So the calculated activation degree will be used as amembership degree for in the fuzzy set large.

B. Detection Method

To decide if a central pixel (a nonborder pixel of course) is animpulse noise pixel, we use following (fuzzy) rule:

IF most of the eight are largeTHEN the central pixel is an im-pulse noise pixel.

We translate this rule by: if for a certain central pixel morethan half of the fuzzy gradient values (thus more than four) arepart of the support of the fuzzy set large [8], then we can con-clude that this pixel is an impulse noise pixel. The support of acertain fuzzy set is the crisp set of all points in the Universe ofDiscourse such that the membership function of is nonzero.

If a pixel is detected as an impulse noise pixel, thenwe store the corresponding grayscale value in a histogram [e.g.,Fig. 5(b) and (d)]. This histogram indicates the amount of noisedetections ( axis) for each possible grayscale value ( axis). Weuse this histogram (which we simply call the noise histogram) toinvestigate the presence of impulse noise. If the noise histogramcontains some peaks, then we conclude that the image contains

Fig. 6. Pseudo-code of the decision procedure.

impulse noise pixels; otherwise, we conclude that the image isfree of impulse noise. Fig. 5 shows two noise histograms fortwo images. The first image Fig. 5(a) is only corrupted withimpulse noise which results in a noise histogram containing onlypeaks [see Fig. 5(b)]. Fig. 5(c) is corrupted with a mixture ofimpulse noise and Gaussian noise. This mixture of noise willbe displayed in the noise histogram which now contains twoaccumulations around two extreme values.

Finally, this detection phase ends with the decision procedurepictured in Fig. 6. In this procedure, we decide that an image

is impulse noise-free if the maximum value (the maximalvalue) of the noise histogram (for image ) contains less than

of the total detections. is a threshold value,which should be situated in the interval [2.5;10] because when

SCHULTE et al.: FUZZY IMPULSE NOISE DETECTION AND REDUCTION METHOD 1157

Fig. 7. Pseudo-code of the “ImpulseNoise” function.

Fig. 8. Pseudo-code of the “ImpulseNoise&Other” function.

the maximum value is lower than 2.5% we cannot really speakof peaks. On the other hand, in images corrupted with mixedtypes of noise (including impulse noise) a threshold value above10% could lead to the wrong conclusion that there is no impulsenoise present.

In cases where impulse noise is detected, we distinguish twosubcases. In the first subcase, where images are only corruptedwith impulse noise, we perform the “ImpulseNoise” function(Fig. 7). If we call the grayscale value ( axis value) wherethe maximal value of the noise histogram is reached. Now, weexecute the “ImpulseNoise” function if the two closest grayscalevalues and both have very low values with re-spect to the value of grayscale value of the noise histogram(Fig. 6). In the other case, we take into account that the image iscorrupted with a mixture of impulse noise and other noise types(as for example Gaussian noise). In this second subcase we per-form the “ImpulseNoise&Other” function shown in Fig. 8.

The function “ImpulseNoise” (Fig. 7) determines the integervalues [used in (1)]. We determine max-imal five such integer values to make the filtering step morerobust against overfiltering. If an image is corrupted with im-pulse noise, with more than five such values than we restartour complete method when the filtering method has finished.As indicated in Fig. 7 we use a threshold value , whichis related to ( , and

). This threshold value is used to select the quan-tity of noise integers , that agrees with the amount of peaksin the histogram. The function “ImpulseNoise&Other” (Fig. 8)also determines the integer values , but herewe consider a maximal amount of two such integers (per execu-tion) because the presence of impulse noise mixed with another

Fig. 9. Membership function �(:) representing the fuzzy set “more or lessimpulse noise.”

kind of noise should be solved in several complete iterations.Besides the determination of the integer values we also cal-culate some parameters , which we will explainin the next section.

III. FILTERING PHASE

If the detection phase has detected impulse noise, then weperform the filtering step explained in this section. Otherwise,we leave the image unchanged. This section is structured asfollows. First we explain the calculation of the parameters

, which are used to construct the fuzzy set moreor less impulse noise. Afterward, we construct our first filteringiteration based on the membership function that representsthis fuzzy set. The explanation of the other iterations and somestop criteria finally terminate this section.

A. Configuration of the Parameters

As shown in line seven of the functions “ImpulseNoise”(Fig. 7) and “ImpulseNoise&Other” (Fig. 8), we calculate foreach integer four parameters . These param-eters are used to construct the fuzzy set more or less impulsenoise, which is represented by a proper membership function

. An example of such a membership function is illustratedin Fig. 9. The obtained membership function is a simplificationof the obtained noise histogram [e.g., Fig. 5(d)]. Therefore,we calculate or select the parameters for thefunction “ImpulseNoise&Other” by the following rules:

(4)

with , where is thelargest integer value smaller than the variance . These rules areused to approximate the noisy histogram [as in Fig. 5(d)]. Be-cause is unknown, we try to make an accurate estimation ofthis value first. Our noise estimation is based on the edge imageproduced by the Sobel operator [11]. For more information, werefer to the work of Zlokolica [12]. Other approaches are pos-sible as well (e.g., a statistical approach). When , thenwe restrict and to be 25 to prevent overfiltering.Otherwise, this large value could lead to an extreme widemembership function which would cause some kind of blurringof the image.

1158 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 5, MAY 2006

Fig. 10. Pseudo-code of the first filtering iteration.

Fig. 11. Pseudo-code of the e (recursive) filtering iteration.

B. First Iteration

The filtering step of the first iteration is given in the pseudo-code presented in Fig. 10. This method is based on the mem-bership function “more or less impulse noise” . The corre-sponding membership degree of a certain intensity valueis denoted as . A degree one (zero) indicates that theintensity value is noisy for sure (not noisy for sure). When thedegree is between one and zero, then there is some kind of un-certainty. We only filter pixels which are part of the support ofthe fuzzy set more or less impulse noise (pixels which have anonzero membership degree in this fuzzy set). Otherwise, weleave the pixel unchanged. As shown in line three of Fig. 10,we use, in the first iteration, a 3 3 window around the filteredpixel. In addition, we used the standard negation toexpress the membership degrees in the fuzzy set noise free [9],[10]. If the output image is the same as the input image ,then the filter method is called recursively; otherwise, we call itnonrecursively.

C. Next Iterations

After the first iteration, it is possible as a side effect (espe-cially with high initial impulse noise) that the impulse noiseis clustered around one or more pixels. To reduce these noisypixels, we will provide some more (recursive) iterations that arequite similar to the first one.

In each iteration, we use the modified image of the previousperformed iteration and a different window (indicated by and

in Fig. 11) around a given pixel. Fig. 12 shows the neighbor-hood windows used in the first, second, third, and fourth iter-ation. The changing window is used to avoid future clustering

Fig. 12. Used neighborhood window for the first, second, third, and fourthiteration.

and also speeds up the execution time. We always use a windowof nine pixels (inclusive the center). The window change alsoimplies the change of the term “nonborder pixel.” In theiteration nonborder pixels are defined as having left, right,

upper, and lower neighborhood pixels. The iter-ation is illustrated in Fig. 11. In addition to the different window,we also modify the membership function “more or less impulsenoise” [denoted as for the iteration] by changing theparameters as shown in (5)

(5)

Since this change will reduce the slope of the membership func-tion, and, therefore, also the amount of investigated pixels foran image , it will speed up the execution time. We reduce theslope of the function, because the amount of noisy pixels wasalready reduced in the previous iteration.

D. Stop Criteria

There are several techniques to improve the efficiency of thefiltering phase. We will focus on the stop criteria and the amountof pixels that must be scanned during an iteration. During thefirst iteration, we check every pixel. If the pixel value does notbelong to the support of the fuzzy set more or less impulse noisethen we do not change this pixel value, not only in this iterationneither in the other ones. By remembering only the positionsof pixels, whose pixel value is an element of the support of thefuzzy set more or less impulse noise, we can drastically reducethe scanning amount in the next iterations.

Of course, we also need some stop criteria. If we defineas the amount of pixel values which belong to the support of thefuzzy set more or less impulse noise in the iteration, then wecan apply the following stop criteria.

1) There are no pixel values in the support of the fuzzy setin any iteration .

2) is equal to . This indicates that the resultingpixels (pixels which still are element of the support of thefuzzy set more or less impulse noise) are not noisy evenwith the nonzero membership degrees.

SCHULTE et al.: FUZZY IMPULSE NOISE DETECTION AND REDUCTION METHOD 1159

TABLE IIPSNR AND EXECUTION TIME RESULTS FOR THE (512 � 512) MANDRILL IMAGE FOR DIFFERENT

IMPULSE NOISE LEVELS AND DIFFERENT FILTERS (MATLAB ENVIRONMENT)

3) Since hold we can define. When is (very) small, then we can

decide to stop too.

IV. SIMULATION RESULTS

In this section, we compare the efficiency and performanceof FIDRM for grayscale images with other well-known filtersfor impulse noise reduction. The first group of filters consistsof the FIRE filters (fuzzy inference rule by else-action filters).The idea behind these filters is that they try to calculate positiveand negative correction terms in order to express the degree ofnoise for a certain pixel. We distinguish three FIRE filters: thenormal FIRE [13] (fuzzy inference rule by else-action filters),the DS-FIRE [14] (dual step FIRE), and the PWLFIRE [15](piecewise linear FIRE). A second group of filters contains fil-ters which are extensions of classical median (denoted by MED)and weighted filters. We use the FMF [16], [17] (fuzzy me-dian filter), AWFM [18], [19] (adaptive weighted fuzzy mean)and the ATMAV [20] (asymmetrical triangular fuzzy filter withmoving average center). The FCF (fuzzy control-based) filtersconstitute a third group of filters. These filters correct a certaincentral pixel value according to some features of some lumi-nance (pixel values) differences between the central pixel valueand some neighbor pixel values. In the literature, we know thatthe IFCF [21] (iterative fuzzy control based filter), the MIFCF[21] (modified IFCF), the EIFCF [21] (extended IFCF), andthe SFCF [22] (smoothing fuzzy control based filter). Further-more, there exist many other types of filters, such as, e.g., thehistogram adaptive fuzzy filter which of course uses the his-togram of an image [23] (HAF) or the fuzzy similarity filter [24](FSB), where the local similarity between some patterns is used.Besides all these fuzzy filters, some other popular filters exist.We use the CWM [3] (center weighted median), the TSM [4]

(tri-state median filter), and the LUM [5] (lower-upper-middlefilter) for comparison.

As a measure of objective dissimilarity between a filteredimage and the original one, we use the mean square error (MSE)and the peak signal to noise ratio (PSNR) in decibels

(6)

(7)

where is the original image, is the filtered image ofsize , and is the maximum possible pixel value (with8-bit integer values the maximum will be 255). We used thewell-known Lena and Mandrill (Baboon) images of size, re-spectively, 256 256 and 512 512, to compare FIDRM withthe other filters. From Tables II and III, we see that FIDRM isone of the fastest algorithms, which also generates the smallestPSNR values. The IFCF filters are iterative filters, so the exe-cution time must be multiplied with the amount of iterations.For the IFCF filters, these amounts are indicated next to thetime. The FIDRM filter is iterative too, but here the executiontime shown for that filter is the total execution time. After thefirst iteration, the FIDRM filter reinvestigates only those pixelswhich belong to the support of the fuzzy set more or less im-pulse noise. This causes lower execution time for each furtheriteration. FIDRM performs very well even at high noise levels.As an illustration, Fig. 13 shows the results for a Mandrill imagecorrupted with 30% salt and pepper noise. One can observe thatthe proposed filter preserves edge sharpness and reduces many

1160 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 15, NO. 5, MAY 2006

TABLE IIIPSNR AND EXECUTION TIME RESULTS FOR THE (256 � 256) LENA IMAGE FOR DIFFERENT

IMPULSE NOISE LEVELS AND DIFFERENT FILTERS (MATLAB ENVIRONMENT)

Fig. 13. Restoration of a noisy (512� 512) Mandrill image. (a) The increasednoise-free Mandrill, (b) image contaminated with 50% impulse noise (salt andpepper noise), (c) FIDRM, (d) HAF, (e) ATMAV, (f) AWFM, (g) TSM (7 �7), (h) CWM (7 � 7), (i) basic MEDIAN (7 � 7), (j) DSFIRE, (k) LUM, and(l) IFCF.

artefacts in contrast to the other well performing algorithms asAWFM, HAF, and ATMAV.

In addition to the previous experiments, where only salt andpepper noise was considered, we also compared our method forother noise situations in Table IV. The first experiment (case1) shows the numerical results for images corrupted with im-pulse noise with . In thesecond experiment (case 2), we illustrate the filtering perfor-mance for images corrupted with a mixture of Gaussian noiseand salt and pepper noise. In this case the proposed method fil-ters out the impulse noise but leaves the Gaussian noise, so thata Gaussian denoising filter can be applied afterwards. Fig. 14confirms this by showing the visual results for some of thesefilters. Most of the compared filters can not deal well with themixture of Gaussian noise and impulse noise (the images tendto blur and they lose detailed information). In a last experiment(case 3), we want to apply FIDRM for an image corrupted withrandom impulse noise. We applied the FIDRM locally insteadof globally in order to handle this random impulse noise (i.e.,the proposed fuzzy detection and reduction method is appliedon different parts of the image, separately). As one can see inTable IV, FIDRM does not outperform the Median based fil-ters for random impulse noise. Nevertheless, the idea behind thisfilter (especially the detection method) can be used to develop amethod that can suppress random impulse noise very well. Ourfuture research will be concentrated on this issue and on the con-struction of fuzzy filtering methods for color images.

V. CONCLUSION

A new two step filter (FIDRM), which uses a fuzzy detec-tion and an iterative filtering algorithm, has been presented. Thisfilter is especially developed for reducing all kinds of impulsenoise (not only salt and pepper noise). Its main feature is that itleaves the pixels which are noise-free unchanged. Experimental

SCHULTE et al.: FUZZY IMPULSE NOISE DETECTION AND REDUCTION METHOD 1161

TABLE IVPSNR RESULTS FOR THE (256 � 256) LENA IMAGE FOR DIFFERENT IMPULSE NOISE CASES WITH CASE 1: IMPULSE NOISE WITH

(p ; p ; p ; p ) = (15; 25;225;250); CASE 2: GAUSSIAN NOISE WITH � = 5 MIXED WITH SALT AND PEPPER NOISE; CASE 3: RANDOM IMPULSE NOISE

Fig. 14. Restoration of a noisy (256 � 256) Lena image. (a) The increasednoise-free Lena, (b) image contaminated with � = 5 gaussian noise and 30%impulse noise (salt and pepper noise), (c) FIDRM, (d) ATMAV, (e) AWFM, (f)HAF, (g) CWM (7 � 7), (h) basic MED (5 � 5), and (i) EIFCF.

results show the feasibility of the new filter. A numerical mea-sure, such as the PSNR, and visual observations (Figs. 13 and14) show convincing results for grayscale images. Finally, thisnew method is easy to implement and has a very low executiontime.

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Stefan Schulte was born in Dortmund, Germany, in1979. He received the M.Sc. degree in computer sci-ence from the Ghent University, Ghent, Belgium, in2003, where he is currently pursuing the Ph.D. degreewith a thesis on fuzzy techniques in image processingand fuzzy filters for image noise removal with Prof.E. E. Kerre.

In September 2003, he joined the Department ofApplied Mathematics and Computer Science, GhentUniversity, where he is a member of the Fuzzinessand Uncertainty Modeling Research Unit.

Mike Nachtegael was born on February 16, 1976.He received the M.Sc. degree in mathematics and thePh.D. degree in mathematics on the topic of fuzzytechniques in image processing from the Universityof Ghent, Ghent, Belgium, in 1998 and May 2002,respectively.

In 2002, he joined the Department of AppliedMathematics and Computer Science, Ghent Uni-versity, where he is a member of the Fuzziness andUncertainty Modeling Research Unit. Currently, heholds the position of Doctor Assistant.

Valérie De Witte was born in Ghent, Belgium, in1981. She received the M.Sc. degree in mathematicsfrom Ghent University in 2003, where she is cur-rently pursuing the Ph.D. degree with a thesis onfuzzy mathematical morphology with Prof. E. E.Kerre.

In September 2003, she joined the Department ofApplied Mathematics and Computer Science, GhentUniversity, where she is a member of the Fuzzinessand Uncertainty Modeling Research Unit.

Dietrich Van der Weken was born in Beveren, Bel-gium, in 1978. He received the M.Sc. degree in math-ematics and the Ph.D. degree in mathematics on theuse and construction of similarity measures in imageprocessing from Ghent University, Ghent, Belgium,in 2000 and May 2004, respectively.

In September 2000, he joined the Department ofApplied Mathematics and Computer Science, GhentUniversity, where he is a member of the Fuzzinessand Uncertainty Modeling Research Unit. Currently,he is a Postdoctoral Researcher.

Etienne E. Kerre was born in Zele, Belgium, on May8, 1945. He received the M.Sc. and Ph.D. degreesin mathematics from Ghent University, Ghent, Bel-gium, in 1967 and 1970, respectively.

Since 1984, he has been a Lector, and since 1991,a Full Professor at Ghent University. In 1976, hefounded the Fuzziness and Uncertainty ModelingResearch Unit (FUM), Ghent University, and, sincethen, his research has been focused on the modelingof fuzziness and uncertainty and has resulted in agreat number of contributions in fuzzy set theory

and its various generalizations, as well as in evidence theory. The theoriesof fuzzy relational calculus and fuzzy mathematical structures owe a verygreat deal to him. Over the years, he has also been a promotor of 19 Ph.D.degrees on fuzzy set theory. His current research interests include fuzzy andintuitionistic fuzzy relations, fuzzy topology, and fuzzy image processing.He has authored or coauthored 11 books and more than 100 papers and hasappeared in international refereed journals.

Dr. Kerre has been an honorary chairman at various international conferences.He is a referee for more than 30 international scientific journals, as well asa member of the editorial boards of international journals and conferences onfuzzy set theory.