A Detail-Preserving Type-2 Fuzzy Logic Filter for Impulse NoiseRemoval from Digital Images

M. Tiilin YILDIRIM, Student Member, IEEE, Alper BASTURK, Member, IEEE, andM. Emin YUKSEL, Senior Member, IEEE

Abstract-A novel filtering operator based on type-2 fuzzylogic techniques is proposed for detail preserving restorationof impulse noise corrupted images. The performance of theproposed operator is tested for different test images corruptedat various noise densities and also compared with representativeconventional as well as state-of-the-art impulse noise removaloperators from the literature. Experimental results show thatthe proposed operator exhibits superior performance over thecompeting operators and is capable of efficiently suppressing thenoise in the image while at the same time effectively preservingthe useful information in the image.

I. INTRODUCTION

Digital images are often contaminated by impulse noiseduring image acquisition and/or transmission due to a num-ber of imperfections encountered in image sensors and com-munication channels. In most image processing applications,it is very important to remove the impulse noise from theimage because the performances of subsequent image pro-cessing tasks, such as edge detection, image segmentation,object recognition, etc., are severely degraded by noise.A large number of methods have been proposed to remove

impulse noise from digital images. The majority of thesemethods are based on order statistics filters, which utilizethe rank order information of the pixels contained in a givenfiltering window. The standard median filter [1] attempts toremove impulse noise by changing the center pixel of thefiltering window with the median of the pixels within thewindow. This approach provides a reasonable noise removalperformance but removes thin lines and blurs image detailseven at low noise densities. The weighted median filterand the center-weighted median filter [2]-[4], which aremodified median filters giving more weight to certain pixelsin the filtering window, have been proposed to avoid theinherent drawbacks of the standard median filter. These filtersdemonstrate better performance in preserving image detailsthan the median filter at the expense ofreduced noise removalperformance.

M. Tilin YILDIRIM is with the Department of Aircraft Electrics andElectronics, Civil Aviation School, Erciyes University, Kayseri, 38039,Turkey (phone: +90 352 4374901/41058; fax:+90 352 4375744; email:tulingerciyes.edu.tr).

Alper BA$TURK is with Digital Signal and Image Processing Laboratory,Department of Computer Engineering, Erciyes University, Kayseri, 38039,Turkey (phone: +90 352 4374901/32552; fax: +90 352 4375784; email:ab@erciyes.edu.tr).M. Emin YUKSEL is with Digital Signal and Image Processing Labo-

ratory, Department of Electronics Engineering, Erciyes University, Kayseri,38039, Turkey (phone: +90 352 4374901/32204; fax: +90 352 4375784;email: yukselgerciyes.edu.tr).

A number of methods [5]-[21] combine the median filterwith an impulse detector that aims to determine whetherthe center pixel of a given filtering window is corruptedor not. If the center pixel is classified by the impulsedetector as a corrupted pixel, its restored value is obtainedby processing the pixels in the filtering window by themedian filter. Otherwise, i.e. if the center pixel is classifiedas uncorrupted, it is left unchanged. Although this approachsignificantly enhances the performance of the median filterby reducing its distortion effects, its performance inherentlydepends on the performance of the impulse detector. As aconsequence, several different impulse detection approachesexploiting median filters [5]-[7], center-weighted medianfilters [8]-[1 1], boolean filters [12], edge detection kernels[13], homogeneity level information [14], statistical tests[15], classifier based methods [16], rule based methods [17],pixel counting methods [18] and soft computing methods[19]-[21] have been proposed.

In addition to the median based filters mentioned above,various types of mean filters are successfully utilized forimpulse noise removal from digital images [22]-[24]. Thesefilters usually exhibit good filtering performance at the costof increased computational complexity.

There are also a number of impulse noise filters based onsoft computing methodologies [25]-[29] as well as severalother nonlinear filters [30]-[35] that combine the desiredproperties of the above mentioned filters. These filters areusually more complex than the above mentioned median-and the mean-based filters, but they usually offer much betternoise suppression and detail preservation performance.

All of these methods more or less have the undesirableproperty of blurring image details and texture during filtering.This is due mainly to the uncertainty introduced by noise.As the density of the noise corrupting the image increases,it becomes more and more difficult for the filter to correctlydistinguish between the corrupted and the uncorrupted pixelsin the noisy input image. As a direct consequence of thisuncertainty, some of the corrupted pixels are left unfilteredleaving a considerable number of noisy pixels in the restoredoutput image, and some of the uncorrupted pixels are unnec-essarily filtered causing undesirable distortions and blurringeffects in the output image.

In recent years, there has been a growing interest in the ap-plications of type-2 fuzzy logic systems. Unlike conventional(type-1) fuzzy logic systems where membership functionsare scalar, the membership functions in type-2 FLSs are alsofuzzy and this extra degree of fuzziness provides a more

1-4244-1210-2/07/$25.00 C 2007 IEEE.

efficient way of handling uncertainty, which is inevitably en-countered in noisy environments. Hence, type-2 fuzzy logicsystems may be utilized to design efficient filtering operatorsexhibiting much better performance in noisy environmentsprovided that appropriate network structures and processingstrategies are employed [36].Based on these observations, we propose in this paper a

novel filtering operator based on type-2 fuzzy logic tech-niques for detail preserving restoration of impulse noisecorrupted images. The performance of the proposed operatoris tested at various noise densities and for different testimages, and also compared with representative conventionalas well as state-of-the-art impulse noise removal operators.Experimental results show that the proposed operator yieldssuperior performance over the competing operators and iscapable of efficiently suppressing the noise in the imagewhile at the same time effectively preserving the usefulinformation in the image such as thin lines, edges, finedetails, and texture.

II. METHOD

A. The Proposed Neuro-Fuzzy OperatorFigure- I shows the structure of the proposed impulse noise

removal operator. The operator is constructed by combiningfour type-2 NF filters, four defuzzifiers and a postprocessor.The operator processes the noisy pixels contained in itsfiltering window shown in Figure-2 and outputs the restoredvalue of the center pixel. All NF filters employed in thestructure of the operator are identical to each other andfunction as subfilters processing the horizontal, vertical,diagonal and the reverse diagonal pixel neighborhoods inthe filtering window, respectively. Hence, each of the fourNF filters accepts the center pixel and two of its appropriateneighboring pixels as input and then produces an output,which is a type-I interval fuzzy set representing the un-certainty interval for the restored value of the center pixel.The four output fuzzy sets coming from the four NF filtersare then fed to the corresponding defuzzifier blocks. Eachdefuzzifier defuzzifies the input fuzzy set and converts itinto a single scalar value. The four scalar values obtained atthe outputs of the four defuzzifiers represent four candidatesfor the restored value of the center pixel of the filteringwindow. These four candidate values are finally evaluated bythe postprocessor and converted into a single output value.The output of the postprocessor is also the output of theproposed filtering operator and represents the restored valueof the center pixel of the filtering window.

B. The Type-2 Neuro-Fuzzy FiltersEach NF filter employed in the structure of the proposed

impulse noise removal operator is a first order TSK type-2interval fuzzy inference system with 3-inputs and 1-output.The internal structures of the NF filters are identical to eachother. The input-output relationship of any of the NF filtersis as follows:

Let Xl, X2, X3 denote the inputs of the NF filter andY denote its output. Each combination of inputs and their

X(r,C+l Type=2 Y~

xX(r± ,c± TyeF Defuzzifier

(r, c- T

x, Defuzzifier

x(r- 1, c-1 NF

X (r+l, c-l NF

-o0

a)00

0

-,y(r, C)

Fig. 1. The structure of the proposed type-2 NF noise removal operatorwith four type-2 NF filters evaluating pixel neighborhoods in horizontal,vertical, left diagonal and right diagonal directions, respectively.

x(r-1, c-1)

x(r, c-1)

x(r+1,c-1)

x(r-1, c)

x(r, c)

x(r+1,c)

x(r-1, c+1)

x(r,c+ 1)

x(r+i, c + 1)

Fig. 2. Filtering window.

associated membership functions is represented by a rule inthe rule base of the NF filter. The rulebase contains a desirednumber of fuzzy rules, which are as follows:

1. if (Xl C Mul) and (X2 C M12) and (X3 C M13), thenRi = kllXl + kl2X2 + kl3X3 + k14

2. if (Xi e M21) and (X2 e M22) and (X3 e M23), thenR2 = k2lXl + k22X2 + k23X3 + k24

3. if (Xi e M31) and (X2 e M32) and (X3 e M33), thenR3 = k3lXl + k32X2 + k33X3 + kk4

i. if (Xi E Mci) and (X2 E Mi2) and (X3 C Mi3), thenRi = kilXl + ki2X2 + ki3X3 + ki4

N. if (Xi E MNi) and (X2 e MN2) and (X3 e MN3),then RN = kNlXl + kN2X2 + kN3X3 + kN4

where N is the number of fuzzy rules in the rulebase, Mdenotes the ith membership function of the jth input andRi denotes the output of the ith rule. The input membershipfunctions are type-2 interval Gaussian membership functionswith uncertain mean:

Tij e [Tij,Tij](1)

with i =1,2, , N and j 1,2, 3. Here, the parametersTij and rij are the mean and the standard deviation of thetype-2 interval Gaussian membership function Mij, respec-tively, and the interval [r,j, miji] denote the lower and theupper bounds of the uncertainty in the mean. A sample type-2 interval Gaussian membership function and its associatedfootprint of uncertainty (FOU) are illustrated in Figure-3.

Since the membership functions Mij are interval member-ship functions, the boundaries of their FOU are characterized

f

2

Mij (u) = expI U Tnij2 07ij

I.-..............................

0 4

mij mij

Fig. 3. FOU for Gaussian primary membership function with uncertainmean.

a bFig. 4. Training images: a) Original training image b) Noisy training image.

by their upper and lower membership functions, which aredefined as

u < rn.| [ 1 ( tl )2

Mij (u) = 1

exp[I (U )T2

and

Mii (u) =

exp [

exp [

1 (ua-m ii)

2 (7ijJ3

Uu- Tij A22 V (7ij J

image and can easily be generated in a computer. Duringthe training Figure-4 a and b are feeded as the target and theinput training images respectively.

The output of the NF filter is the weighted average of theindividual rule outputs:

N

E wiRiy i=l

N

E Wii=l

(4)

The weighting factor, wi, of the i rule is calculated byevaluating the membership expressions in the antecedent ofthe rule. This is accomplished by first converting the inputvalues to fuzzy membership values by utilizing the inputmembership functions Mij and then applying the "and"operator to these membership values. The "and" operatorcorresponds to the multiplication of the input membershipvalues:

Wi = Mil(X1) 4Mi2(X2) 4Mi3(X3) (5)Since the membership functions Mij in the antecedent

of the ith rule are type-2 interval membership functions,the weighting factor wi is a type-I interval set, i.e. wi =

[Wi, TWj], whose lower and upper boundaries are determinedby using the lower and the upper membership functionsdefined before:

Wi

WiU > Tnij

u >, mij +-m

u < mij+m2

where Mij and Mij are the upper and the lower membershipfunctions of the type-2 interval membership function Mij.

Based on these definitions, it is straightforward to observethat the parameters characterizing the membership functionsin the antecedent of the ith rule are rnj, Til, (7il, ni2, mi2,(7i2, mi3, mi3 and Oi3- Similarly, the parameters determiningthe consequent of the ith rule are ki1, ki2, ki3 and ki4.Therefore, there are 13 parameters determining the output ofthe ith rule. Since the total number of rules in the rulebaseis N, then the total number of parameters in the rulebase is13N. Each NF filter in the proposed operator has 30 rulesand 390 parameters. The optimal values of these parametersare determined by training by using the least squares op-

timization algorithm. Once the training is completed, thereis no need for further training. Figure-4 shows the trainingimages. The training image is a 64-by-64 pixel artificial

(6)Mil(Xl) Mi2(X2) Mi3(X3)Mil(Xl) * Mi2(X2) 4Mi3(X3)

where wi and Wi (i = 1, 2,... , N) are the lower and theupper boundaries of the interval weighting factor wi of theith rule.

After the weighting factors are obtained, the output Yof the NF filter can be found by calculating the weightedaverage of the individual rule outputs by using (4). Theoutput Y is also a type-I interval set, i.e. Y = [Y, Y], sincethe wi's in the above equation are type-I interval sets andRi's are scalars. The lower and the upper boundaries of Yare determined by using the iterative procedure proposed byKamik and Mendel [37].

The information presented in this subsection is relatedwith the input-output relationship of a first order TSK type-2 interval fuzzy logic system with 3-inputs and 1-output.Readers interested in details of TSK type-2 fuzzy inferencesystems as well as other type-2 fuzzy logic systems arereferred to an excellent book on this subject [36].

C. The DefuzzifierThe defuzzifier block takes the type-I interval fuzzy set

obtained at output of the corresponding NF filter as inputand converts it into a scalar value by performing centroiddefuzzification. Since the input set is a type-I interval fuzzyset, i.e. Y = [Y, Y], its centroid is equal to the center of theinterval:

D - (7)2

D. The Postprocessor

The postprocessor generates the final output of the pro-posed operator. It processes the four scalar values obtainedat the outputs of the four defuzzifiers and produces a singlescalar output, which represents the output of the proposedfilter. The operation of the postprocessor may be describedas follows:

Let D1, D2, D3, D4 denote the outputs of the four defuzzi-fiers. The postprocessor first sorts these values, such thatDI < DI < DI < DI, where D,D, D, DI representthe output values of the defuzzifiers after sorting. Next, thelowest (DI) and the highest (DI) of the four values arediscarded. Finally, the remaining two are averaged to obtainthe postprocessor output, which is also the output of theproposed operator:

(8)=2 3

E. Filtering of the Noisy Input Image

The proposed operator is applied recursively to noisyimages. The overall filtering procedure for the restorationof a noisy input image may be summarized as follows:

1) A 3-by-3 pixel filtering window is slided on the image.The window is started from the upper-left corner

of the image and moved sideways and progressivelydownwards in a raster scanning fashion.

2) For each filtering window position, the appropriate pix-els of the filtering window representing the horizontal,vertical, diagonal and the reverse diagonal neighbor-hoods of the center pixel are fed to the correspondingNF filters in the structure. Each NF filter individuallygenerates a type-I interval fuzzy set, which representsthe uncertainty interval of the restored value for thecenter pixel of the filtering window.

3) The outputs of the NF filters are fed to their cor-

responding defuzzifiers. The defuzzifiers process theinput type-I interval fuzzy sets coming from the NFfilters and output the centroid of their input sets.The scalar values obtained at the outputs of the fourdefuzzifiers are scalar values that represent four candi-dates for the restored value for the center pixel of thefiltering window.

4) The outputs of the four defuzzifiers are fed to the post-processor, which sorts these four candidates, discardsthe lowest and the highest values, and then outputsthe average of the remaining two values. The valueobtained at the output of the postprocessor is also theoutput value of the operator and represents the restoredvalue for the center pixel of the filtering window.

5) This procedure is repeated for all pixels of the noisyinput image.

III. RESULTS

The proposed impulse noise removal operator discussedin the previous section is implemented. The performance ofthe operator is tested under various noise conditions and on

c d

Fig. 5. Test images a) Baboon b) Boats c) Bridge d) Pentagon.

popular test images from the literature including Baboon,Boats, Bridge and Pentagon images which are shown inFigure-5. All test images are 8-bit gray level images. Theexperimental images used in the simulations are generatedby contaminating the original images by fixed valued im-pulse noise with an appropriate noise density depending on

the experiment. For comparison, the corrupted experimentalimages are also restored by using several conventional andstate-of-the-art impulse noise removal operators including theswitching median filter (SMF) [5], signal-dependent rank-ordered mean filter (SDROMF) [22], fuzzy filter (FF) [25],progressive switching median filter (PSMF) [6], multistatemedian filter (MSMF) [10], edge detecting median filter(EDMF) [13], adaptive fuzzy switching filter (AFSF) [32],alpha-trimmed mean-based filter (ATMBF) [24] and adaptivemedian filter with difference type noise detector (DNDAM)[17]. All operators including the proposed operator operateon a 3-by-3 filtering window.

Several experiments are performed to measure and com-

pare the noise suppression and detail preservation perfor-mances of all operators. The experiments are especiallydesigned to reveal the performances of the operators fordifferent image properties and noise conditions. The perfor-mances of all operators are also evaluated by using the meansquared error-MSE criterion, which is defined as

R C

MSE RCEE (s [r, c]r=l c=l

Here, s [r, c] and y [r, c] represent the original and therestored versions of a corrupted test image, respectively. MSEvalues calculated for the output images of all operators forthe Baboon, Boats, Bridge and Pentagon images corruptedby 25%, 50% and 75% impulse noise are also presented inTable-I. In Table-II, the average MSE values are presented. Itis seen that the proposed operator offers the best performance

a b

(9)

TABLE I

MSE VALUES FOR BABOON, BOATS, BRIDGE AND PENTAGON IMAGES CORRUPTED BY 25%, 50% AND 75% IMPULSE NOISE.

Baboon Boats Bridge PentagonOperator 25% 50% 75% 25% 50% 75% 25% 50% 75% 25% 50% 75%SMF 681 2625 8511 344 2209 8563 333 2347 8816 325 2116 7927SDROMF 587 1076 4081 249 789 4420 248 766 4488 228 575 3277FF 464 1012 3172 214 688 3134 215 726 3428 167 511 2510PSMF 536 880 3003 275 548 2612 202 461 2740 189 416 2206MSMF 847 3852 10269 525 3566 10482 534 3735 10780 542 3408 9736EDMF 460 1265 5188 270 967 5010 238 929 5108 223 867 4512AFSF 476 734 1849 187 466 1825 207 536 1862 212 450 1501ATMBF 706 2647 8522 343 2206 8559 337 2347 8815 337 2133 7938DNDAM 470 917 3048 403 997 2622 340 771 2762 270 516 2127PROPOSED 212 481 1026 115 403 1336 182 537 1560 110 343 925

regarding MSE criteria.Figure-6 shows the output images of all operators for the

Baboon image corrupted by impulse noise of 25% noisedensity for a visual evaluation of the noise removal and detailpreservation performances of the operators. It is observedfrom this figure that the operators efficiently suppressing thenoise (such as SDROMF, FF, PSMF, AFSF, DNDAM) fail topreserve image details. The output images of these operatorssuffer from considerable amount of blurring and distortion.On the other hand, the operators that are more successfulat preserving image details (such as SMF, MSMF, EDMF,ATMBF) fail to suppress noise efficiently. It is observed thatconsiderable amount of noisy pixels is still present in theoutput images of these filters.

The proposed type-2 NF noise removal operator, however,exhibits much better performance than the others. It is clearlyseen that the proposed operator successfully suppresses thenoise and preserves the useful image details. The differenceespecially in the detail preservation performance can easilybe observed by carefully looking at the appearance of theeyes and the hair around the mouth of the animal in theoutput images of all operators.As a final remark, it should be mentioned that the compu-

tational complexity of the proposed method is comparativelyhigher than the competing operators because it has a muchhigher number of internal parameters. However, this increasein complexity may be regarded as the price for the significantsuperiority in filtering performance over competing methods.

IV. CONCLUSIONS

A novel detail preserving type-2 fuzzy logic image filteringoperator for removing impulse noise from digital imagesis presented. The fundamental superiority of the proposedoperator over most other operators is that it efficiently re-moves impulse noise from digital images while successfullypreserving thin lines, edges, fine details and texture in theoriginal image. It is concluded that the proposed operatorcan be used as a powerful tool for efficient removal ofimpulse noise from digital images without distorting theuseful information within the image.

TABLE II

AVERAGE MSE VALUES FOR BABOON, BOATS, BRIDGE AND PENTAGON

IMAGES CORRUPTED BY 25%, 50% AND 75% IMPULSE NOISE.

AVERAGE OF FOUR IMAGES25% 50% 75% Total Average

SMF 421 2324 8454 3733SDROMF 328 802 4067 1732FF 265 734 3061 1353PSMF 301 576 2640 1172MSMF 612 3640 10317 4856EDMF 298 1007 4955 2086AFSF 271 547 1759 859ATMBF 431 2333 8459 3741DNDAM 371 800 2640 1270PROPOSED 155 441 1212 603

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