Research Article Fuzzy Logic-Based Histogram Equalization ...downloads.hindawi.com/journals/mpe/2013/891864.pdf · Fuzzy Logic-Based Histogram Equalization for Image Contrast Enhancement

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 891864 10 pageshttpdxdoiorg1011552013891864

Research ArticleFuzzy Logic-Based Histogram Equalization forImage Contrast Enhancement

V Magudeeswaran1 and C G Ravichandran2

1 Department of Electronics and Communication Engineering PSNA College of Engineering and Technology DindigulTamil Nadu 624622 India

2 Ratnavel Subramaniam (RVS) College of Engineering and Technology Dindigul Tamil Nadu 624005 India

Correspondence should be addressed to V Magudeeswaran magudeeswarangmailcom

Received 23 May 2013 Revised 27 June 2013 Accepted 27 June 2013

Academic Editor Erik Cuevas

Copyright copy 2013 V Magudeeswaran and C G Ravichandran This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

Fuzzy logic-based histogram equalization (FHE) is proposed for image contrast enhancementThe FHE consists of two stages Firstfuzzy histogram is computed based on fuzzy set theory to handle the inexactness of gray level values in a better way compared toclassical crisp histograms In the second stage the fuzzy histogram is divided into two subhistograms based on the median value ofthe original image and then equalizes them independently to preserve image brightness The qualitative and quantitative analysesof proposed FHE algorithm are evaluated using two well-known parameters like average information contents (AIC) and naturalimage quality evaluator (NIQE) index for various images From the qualitative and quantitative measures it is interesting to seethat this proposed method provides optimum results by giving better contrast enhancement and preserving the local informationof the original image Experimental result shows that the proposed method can effectively and significantly eliminate washed-outappearance and adverse artifacts induced by several existing methods The proposed method has been tested using several imagesand gives better visual quality as compared to the conventional methods

1 Introduction

The main objective of an image enhancement is to bring outthe hidden image details or to increase the image contrastwith a new dynamic range Histogram equalization (HE) isone of the most popular techniques used for image contrastenhancement since HE is computationally fast and simple toimplement [1 2] HE performs its operation by remapping thegray levels of the image based on the probability distributionof the input gray levels However HE is rarely employed inconsumer electronic applications such as video surveillancedigital camera and television since HE tends to introducesome annoying artifacts and unnatural enhancement includ-ing intensity saturation effect [3] One of the reasons forthis problem is that HE normally changes the brightness ofthe image significantly and thus makes the output imagebecome saturated with very bright or dark intensity values

Hence brightness preserving is an important characteristicthat needs to be considered in order to enhance the image forconsumer electronic products

In order to overcome the limitations of HE and topreserve image brightness several brightness preserving his-togram equalization techniques have been proposed At firstKim proposed brightness preserving bi-Histogram equaliza-tion (BBHE) [4] BBHE divides the input image histograminto two parts based on the mean of the input image andthen each part is equalized independently Consequentlythe mean brightness is preserved because the original meanbrightness is retained Wan et al introduced dualistic sub-image histogram equalization (DSIHE) which is similar toBBHE except that the median of the input image is usedfor histogram partition instead of mean brightness [5] Chenand Ramli proposed minimum mean brightness error bi-histogram equalization (MMBEBHE) which is the extension

2 Mathematical Problems in Engineering

of BBHEmethod that provides maximal brightness preserva-tion [6] This algorithm finds the minimummean brightnesserror between the original and the enhanced image Itemploys the optimal point as the separating point insteadof the mean or median of the input image Though thesemethods can perform good contrast enhancement they alsocause more annoying side effects depending on the variationof gray level distribution in the histogram Recursive meanseparate HE (RMSHE) is another improved version of BBHE[7] This method is recursively separates the histogram intomulti-subhistograms instead of two subhistograms as in theBBHE Initially two subhistograms are created based on themean brightness of the original histogram Subsequentlythe mean brightness from the two subhistograms obtainedearlier are used as the second and third separating pointsin creating more sub-histograms In a similar fashion thealgorithm is executed recursively until the desired numbersof sub-histograms are met Then the HE approach is appliedindependently on each of the subhistograms The methodsdiscussed previously are based on dividing the originalhistogram into several sub-histograms by using either themedian or mean brightness Although the mean brightnessis well preserved by the aforementioned methods thesemethods cannot further expand the region of subhistogramthat is located near to theminimumormaximum value of thedynamic range However it is also not free from side effectssuch as washed-out appearance undesirable checkerboardeffects and significant change in image brightness

In order to deal with the problem mentioned aboveAbdullah-Al-Wadud et al introduced a dynamic histogramequalization (DHE) technique [8] DHE partitions the orig-inal histogram based on local minima However DHE doesnot consider the preserving of brightness For this purposeIbrahim and Kong proposed brightness preserving dynamichistogram equalization (BPDHE) [9]This method partitionsthe image histogram based on the local maxima of thesmoothed histogram It assigns a new dynamic range to eachpartition Finally the output intensity is normalized to makethe mean intensity of the resulting image equal to the inputone Although the BPDHE performs well inmean brightnesspreserving the ratio for brightness normalization plays animportant role A small ratio value leads to insignificantcontrast enhancement For large ratio (ie ratio value morethan 1) the final intensity value may exceed the maximumintensity value of the output dynamic range The exceedpixels will be quantized to the maximum intensity valueof gray levels and produce intensity saturation problem(in MATLAB environment) Brightness preserving dynamicfuzzy histogram equalization (BPDFHE) has been proposedby Sheet et al which is an enhanced version of BPDHE [10]The BPDFHE technique manipulates the image histogram insuch a way that no remapping of the histogram peaks takesplace while only redistribution of the gray-level values in thevalley portions between two consecutive peaks takes placeThe results using BPDFHE method show well-enhancedcontrast and little artifacts

To overcome the unwanted over enhancement and noiseamplifying the fuzzy logic-based histogram equalizationtechnique is proposed for both gray scale and color images

The proposed FHE method does not only preserve imagebrightness but also improves the local contrast of the originalimage First fuzzy histogram is computed using fuzzy settheory Second the fuzzy histogram is separated into twobased on the median value of the original image Finally theHE approach is applied independently on each subhistogramto improve the contrast

The remainder of this paper is organized as followsThe conventional histogram equalization is discussed inSection 2 and Section 3 presents the fuzzy image enhance-mentThe proposed algorithm for fuzzification and enhance-ment are presented in Sections 4 and 5 Simulation of the testimages and the qualitative and quantitative comparison of theresults are discussed in Sections 6 and 7 Finally Section 8concludes this paper

2 Histogram Equalization

Consider a digital image 119891 = 119891(119894 119895) which has the totalnumber of 119873 pixels with gray levels in the range [0 119871 minus 1]For a given image 119891 the probability density function 119875(119891119896) isdefined as

119875 (119891119896) =119899119896

119899(1)

for 119896 = 0 1 2 119871 minus 1 where 119899119896 represents the number oftimes that the gray level 119891119896 appears in the input image 119891 and119899 is the total number samples in the input image Note that119875(119891119896) is associated with the histogram of input image whichrepresents the number of pixels that have a specific intensity119891119896 In fact a plot of 119891119896 versus 119899119896 is called histogram of inputimage 119891 The respective cumulative density function is thendefined as

119862 (119891119896) =119871minus1

sum119895=0

119901 (119891119895) (2)

for 119896 = 0 1 2 119871 minus 1 Note that 119862(119891119871minus1) = 1 by definitionHistogram equalization is a method that maps the inputimage into entire dynamic range (1198910 119891119871minus1) by using thecumulative distribution function as a transform functionThat is let us define a transform function 119879(119891) based oncumulative density function as

119879 (119891) = 1198910 + (119891119871minus1 minus 1198910) 119862 (119891119896) (3)

Then the enhanced image of HE 119892 = 119892(119894 119895) can be expressedas

119892 (119894 119895) = 119879 (119891) = 119879 (119891 (119894 119895)) | forall119891 (119894 119895) isin 119891 (4)

where 119891 and 119892 are the original and enhanced images (119894 119895)are the 2D coordinates of the images and 119879 is the intensitytransformation function which maps the original image intothe entire dynamic range (1198910 119891119871minus1) However HE producesan undesirable checkerboard effects on enhanced imagesAnother problem of this method is that it also enhances thenoises in the input image along with the image features [11ndash19] In order to overcome these problems several methods

Mathematical Problems in Engineering 3

were proposed for certain purposes For example BBHE isproposed for preserving themean brightness of a given imagewhile enhancing the contrastThe concept will be introducedin the remainder of the section

Let 119891119898 be the mean of the image 119891 and assume that 119891119898 isin0 119871 minus 1 Based on input mean 119891119898 the image is decomposedinto two subimages 119891119871 and 119891119880 as

119891 = 119891119871 cup 119891119880 (5)

where

119891119871 = 119891 (119894 119895) | 119891 (119894 119895) le 119891119898 forall119891 (119894 119895) isin 119891

119891119880 = 119891 (119894 119895) | 119891 (119894 119895) gt 119891119898 forall119891 (119894 119895) isin 119891 (6)

Note that subimage119891119871 is composed of 1198910 1198911 119891119898 and thesubimage 119891119880 is composed of 119891119898+1 119891119898+2 119891119871minus1

Next define the respective probability density functionsof the subhistograms 119891119871 and 119891119880 as

119875119871 (119891119896) =119899119896119871119899119871

(7)

where 119896 = 0 1 119898 and

119875119880 (119891119896) =119899119896119880119899119880

(8)

where 119896 = 119898 + 1119898 + 2 119871 minus 1 in which 119899119896119880 and119899119896119871 represent the respective numbers of 119891119896 in 119891119871 and 119891119880and 119899119871 and 119899119880 are the total number of samples in 119891119871 and119891119880 respectively Note that 119875119871(119891119896) and 119875119880(119891119896) are associatedwith the histogram of the input image which representsthe number of pixels that have a specific intensity 119891119896 Therespective cumulative density functions for subhistograms119891119871and 119891119880 are then defined as

119862119871 (119891119896) =119898

sum119895=0

119901119871 (119891119895)

119862119880 (119891119896) =119871minus1

sum119895=119898+1

119901119880 (119891119895)

(9)

Let us define the following transform functions based oncumulative density functions as

119879119871 (119891119896) = 1198910 + (119891119898 minus 1198910) 119862119871 (119891119896)

119879119880 (119891119896) = 119891119898+1 + (119891119871minus1 minus 119891119898+1) 119862119880 (119891119896) (10)

Based on these transform functions the decomposed sub-images are equalized independently and the compositionof the resulting equalized sub-images which constitute theoutput image The overall mapping function is defined as

119892 (119894 119895) = 119879119871 (119891119896) cup 119879119880 (119891119896)

= 1198910 + (119891119898 minus 1198910) 119862119871 (119891119896) if 119891 le 119891119898119891119898+1 + (119891119871minus1 minus 119891119898+1) 119862119880 (119891119896) else

(11)

If we note that 0 le 119862119871(119891119896) 119862119880(119891119896) le 1 it is easy to see that119879119871(119891119896) equalizes the sub-image 119891119871 over the range (1198910 119891119898)whereas 119879119880(119891119896) equalizes the sub-image 119891119880 over the range(119891119898+1 119891119871minus1) As a consequence the input image119891 is equalizedover the entire dynamic range (1198910 119891119871minus1) with the constraintthat the samples less than the input mean are mapped to(1198910 119891119898) and the samples greater than the mean are mappedto (119891119898+1 119891119871minus1)

3 Fuzzy Image Enhancement

Fuzzy image enhancement is done by mapping image graylevel intensities into a fuzzy plane using membership func-tions the membership functions are modified for contrastenhancement and the fuzzy plane is mapped back to imagegray level intensities The aim is to generate an image ofhigher contrast than the original image by giving the largerweight to the gray levels that are closer to the mean graylevel of the image than to those that are farther from themean In recent years many researchers have applied fuzzyset theory to develop new techniques for image enhancementFuzzy set theory offers a mathematical framework for anew image understanding An image of size 119872 times 119873 pixelsand L gray levels can be considered as an array of fuzzysingletons each having a value of membership denoting itsdegree of brightness relative to some brightness levels with119892 = (0 1 2 119871 minus 1) [20] The fuzzy matrix corresponding tothis image can be written as

119865 =119872

⋃119898=1

119873

⋃119899=1

120583119898119899119892119898119899

with 120583119898119899 sube [0 1] (12)

where 119892119898119899 is the intensity of (119898 119899)th pixel and 120583119898119899 isits membership value Fuzzy image processing consists ofthree stages fuzzification (image coding) operations in themembership plane and defuzzification (decoding of results)Fuzzification does mean that we assign image with one ormore membership values regarding interesting properties(eg brightness edginess and homogeneity) After transfor-mation of image into the membership plane (fuzzification) asuitable fuzzy approachmodifies themembership values [21]To achieve modified gray levels the output of membershipplane should be decoded (defuzzification) It means thatthe membership values are retransformed into the gray-levelplane For instance to manage the grayness ambiguity thegray levels must be fuzzified in relation to the location ofimage histogram It means that each gray level is assignedto a degree of membership depending on its location inthe histogram Generally dark pixels are assigned as lowmembership values and bright pixels are assigned as highmembership values

31 Fuzzy ContrastBrightness Adaptation Contrast andbrightness are image qualities that are specially importantif the preprocessing is done for human perception [22]There are two different ways to adapt the image contrast orbrightness gray level modification with suitable function 119891

119892 = 119891 (119892) (13)


or histogram operations The performance of the firstapproach depends on the shape of the selected functionThe second approach modifies the image histogram Thehistogram equalization is one of the best known histogramtechniques The modified gray levels 119892 are computed by

119892 = 119891ℎ (119892) = (119871 minus 1)

119892

sum119894=0

ℎ (119894)

119872119873 (14)

where the right side should be round off in order to achieveinteger values for 119892 Fuzzy techniques use different tools offuzzy set to adjust image contrast minimization of fuzzinessequalization using fuzzy expected value hyperbolization 120582-enhancement and enhancement using fuzzy relations

4 Proposed Work

In the conventional histogram equalization method theremapping of the histogram peaks (local maxima) takesplace which leads to the introduction of undesirable artifactsand large change in mean image brightness Hence theproposed fuzzy logic-based histogram equalization is notonly preserves the image brightness but also improves thelocal contrast of the original image First fuzzy histogram iscreated using fuzzy logic to handle the inexactness of graylevel values in a better way and it is separated into two sub-histograms based on the median value of the original imageThen each histogram is assigned to a new dynamic rangeFinally the HE approach is applied independently on eachsub-histogram The FHE technique consists of the followingoperational stages

(i) image fuzzification and intensification(ii) fuzzy histogram computation(iii) histogram partitioning and equalization(iv) image defuzzification

The following subsections include the details of the stepsinvolved

41 Image Fuzzification and Intensification In image fuzzi-fication the gray level intensities are transformed to fuzzyplane whose value ranges between 0 and 1 An image 119891of size 119872 lowast 119873 and intensity level in the range (0 119871 minus 1)can be considered as a set of fuzzy singletons in the fuzzyset notation each with a membership function denotingthe degree of having some gray level The fuzzy matrix 119865corresponding to this image can be expressed as

119865 =

[[[[[[[[

[

1205831111989111

1205831211989112

12058311198731198911119873

1205832111989121

1205832211989122

12058321198731198912119873

sdot sdot sdot sdot sdot sdot sdot sdot sdot12058311987211198911198721

12058311987221198911198722

120583119872119873119891119872119873

]]]]]]]]

]

(15)

Here 120583119894119895 = 0 indicates dark and 120583119894119895 = 1 indicates brightAny intermediate value refers to the grade of maximum gray

level of the pixel A set consisting of all 120583119894119895 is called the fuzzyproperty plane of the image In order to reduce the amountof image fuzziness contrast intensification is applied to thefuzzy set119865 to generate another fuzzy set and themembershipfunction of which is expressed as

120583119865(119894119895) =

2 lowast (120583119894119895)2

0 le 120583119894119895 le 05

1 minus (2 lowast (1 minus 120583119894119895)2

) 05 lt 120583119894119895 le 1(16)

42 Fuzzy Histogram Computation To enhance the imagewe concentrate on contrast enhancement This is achievedby making dark pixel more darker and making bright pixelbrighter Hence fuzzy histogram is computed using (16) Afuzzy histogram is a sequence of real numbers ℎ(119894) 119894 isin(0 1 119871minus1) here ℎ(119894) is the frequency of occurrence of graylevels that are around 119894 By considering the gray value 119891(119894 119895)as a fuzzy number 120583119865(119894119895) the fuzzy histogram is computed as

119865 larr997888 ℎ (119894) +sum119894

sum119895

120583119865(119894119895) (17)

where 120583119865(119894119895) is the fuzzy membership function Fuzzy statis-tics is able to handle the inexactness of gray values in amuch better way compared to classical crisp histograms thusproducing a smooth histogram

43 Histogram Partition and Equalization Based on inputmedian 119872 the fuzzy histogram 119865 is decomposed into twosubhistograms 119865119871 and 119865119880 as

119865 = 119865119871 cup 119865119880 (18)where

119865119871 = 119865 (119894 119895) | 119865 (119894 119895) le 119872 forall119865 (119894 119895) isin 119865

119865119880 = 119865 (119894 119895) | 119865 (119894 119895) gt 119872 forall119865 (119894 119895) isin 119865 (19)

Next define the respective probability density functions ofthe sub histograms 119865119871 and 119865119880 as

119875119871 (119865119896) =119899119896119871119899119871

(20)

where 119896 = 0 1 119898 and

119875119880 (119865119896) =119899119896119880119899119880

(21)

where 119896 = 119898 + 1119898 + 2 119871 minus 1 in which 119899119896119880 and 119899119896119871represent the respective numbers of 119865119896 in 119865119871 and 119865119880 and119899119871 and 119899119880 are the total number of samples in 119865119871 and 119865119880respectively Note that 119875119871(119865119896) and 119875119880(119865119896) are associated withthe fuzzy histogram of the input image which representsthe number of pixels that have a specific intensity 119865119896 Therespective cumulative density functions for sub-histograms119865119871 and 119865119880 are then defined as

119862119871 (119865119896) =119898

sum119895=0

119901119871 (119865119895)

119862119880 (119865119896) =119871minus1

sum119895=119898+1

119901119880 (119865119895)

(22)



119879119871 (119865119896) = 1198650 + (119872 minus 1198650) 119862119871 (119865119896)

119879119880 (119865119896) = 119872 + 1 + (119865119871minus1 minus119872 + 1)119862119880 (119865119896) (23)

Based on these transform functions the decomposedsub-images are equalized independently and the composi-tion of the resulting equalized sub-images which constitutethe output image The output image 119892 = 119892(119894 119895) is expressedas

119892 (119894 119895) = 119879119871 (119865119871) cup 119879119880 (119865119880) (24)

where119879119871 (119865119871) = 119879119871 (119865 (119894 119895)) | forall119865 (119894 119895) isin 119865119871

119879119880 (119865119880) = 119879119880 (119865 (119894 119895)) | forall119865 (119894 119895) isin 119865119880 (25)

If we note that 0 le 119862119871(119865119896) 119862119880(119865119896) le 1 it is easy to seethat119879119871(119865119871) equalizes the sub-image119865119871 over the range (1198650119872)whereas 119879119880(119865119880) equalizes the sub-image 119865119880 over the range(119872+1119865119871minus1) As a consequence the input image F is equalizedover the entire dynamic range (1198650 119865119871minus1) with the constraintthat the samples less than the input median are mapped to(1198650119872) and the samples greater than the median are mappedto (119872+ 1 119865119871minus1)

44 Image Defuzzification Image defuzzification is theinverse of fuzzification The algorithm maps the fuzzy planeback to gray level intensities Finally the enhanced image119866(119894 119895) can be obtained by the following inverse transforma-tion

119866 (119894 119895) = 119879minus1 (119892 (119894 119895)) =119872

⋃119894=1

119873

⋃119895=1

119892 (119894 119895) lowast (119871 minus 1) (26)

where 119866(119894 119895) denotes the gray level of the (119894 119895)th pixel in theenhanced image and 119879minus1 denotes the inverse transformationof 119879 This brightness preserving procedure ensures that themean intensity of the image obtained after process is the sameas that of the input

5 Contrast Enhancement of Color Images

Most electronic equipments acquire and display the colorimages In this respect themethod of enhancing color imageswould be of better interest Hence the contrast enhancementcan be easily extended to color imagesThemost obvious wayto extend the proposed gray-scale contrast enhancement tocolor images is to apply the method to luminance componentonly and to preserve the chrominance components Thismethod produces better perceptible results as compared toother conventional methods

6 Image Quality Assessment

Image quality assessment (IQA) aims to use computationalmodels tomeasure the image quality consistentlywith subjec-tive evaluations IQA indices used in our evaluation include

entropy or average information contents (AIC) and naturalimage quality evaluator (NIQE) index for measuring imagequality

TheAIC is used tomeasure the content of an imagewherea higher value indicates an image with richer details Highervalue of the AIC indicates that more information is broughtout from the images NIQE is used for measuring imagequality Smaller NIQE indicates the better image quality Theaverage information contents or entropy is defined as

AIC = minus119871minus1

sum119896=0

119875 (119896) log119875 (119896) (27)

where 119875(119896) is the probability density function of the 119896th graylevel [23] Moreover natural image quality evaluator (NIQE)index is used formeasuring image quality [24 25] It ismainlybased on the construction of a ldquoquality awarerdquo collection ofstatistical features based on a simple and successful spacedomain natural scene statistic (NSS) model These featuresare derived from a corpus of natural undistorted imagesThequality of a given test image is then expressed as the distancebetween amultivariate gaussian (MVG) fit of theNSS featuresextracted from the test image and aMVGmodel of the qualityaware features extracted from the corpus of natural imagesThe NIQE score typically has a value between 0 and 100 (0represents the best quality 100 the worst)

7 Results and Discussion

In this section comparison between the proposed methodand several other conventional methods such as histogramequalization (HE) brightness preserving bi-histogramequalization (BBHE) minimum mean brightness error bi-histogram equalization (MMBEBHE) and brightnesspreserving dynamic fuzzy histogram equalization (BPDFHE)are presented A subjective assessment to compare thevisual quality of the images is carried out An Averageinformation contents (AIC) and natural image qualityevaluator (NIQE) index is used to assess the effectivenessof the proposed method The proposed method wastested with several gray scale and color images and hadbeen compared with other conventional methods HEBBHE MMBEBHE and BPDFHE The test images usedfor the experiments are available on the website httpwwwponomarenkoinfotid2008htm

Figure 1 shows the resulting images obtained by thevarious existing methods and proposed method for the grayscale imageThe histograms of all resultant images in Figure 1are illustrated in Figure 2 Figure 1(b) shows that HE providesa significant improvement in image contrast However italso amplifies the noise level of the images along with someartifacts and undesirable side effects such as washed-outappearance Figure 1(c) shows that the BBHE method pro-duces unnatural look and insignificant enhancement to theresultant image However it also has unnatural look becauseof over enhancement in brightness This can be proved fromthe result shown in the histogram of two methods shown inFigures 2(b) and 2(c) which are squeezed to left tail of thehistogram


(a) (b)

(c) (d)

(e) (f)

Figure 1 Simulation results of the gray scale image (Image Id I20) (a)Original image (b)HE-ed image (c) BBHE-ed image (d)MMBEBHE-ed image (e) BPDFHE-ed image and (f) proposed method

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(f)

Figure 2 Histogram of simulated gray scale image (Image Id I20) (a) Original histogram (b) HE-ed histogram (c) BBHE-ed histogram(d) MMBEBHE-ed histogram (e) BPDFHE-ed histogram and (f) proposed histogram


(a) (b)

(c) (d)

(e) (f)

Figure 3 Simulation results of the color image (Image Id I17) (a) Original image (b) HE-ed image (c) BBHE-ed image (d) MMBEBHE-edimage (e) BPDFHE-ed image and (f) Proposed Method

The results ofMMBEBHE andBPDFHE (Figures 1(d) and1(e)) show good contrast enhancement and they also causemore annoying side effects depending on the variation of graylevel distribution in the histogram The output histograms ofMMBEBHE and BPDFHE fail to achieve smooth distributionbetween low and high gray levels Thus the enhancementresults of MMBEBHE and BPDFHE are visually unpleasingThe results of HE BBHE MMBEBHE and BPDFHE showthat they do not prevent thewashed-out appearance in overallimage due to significant change in brightness The resultshows that the proposed method preserves the naturalness ofimage and also prevents the side effect due to the significantchange in brightness effectively By visually inspecting theimages on these figures we can clearly see that only theproposed method is able to generate natural looking imageand still offer contrast enhancement Since the output his-togramof proposedmethod achieves a smoother distributionbetween low and high gray values as shown in Figure 2(f)its result is more natural looking The histogram of FHE issuccessfully distributed evenly to its full dynamic range ascompared to other conventional methods Figure 3 shows theresulting images obtained by the various existing methodssuch as HE BBHE MMBEBHE BPDFHE and proposedmethod for the color image The performance of FHE isevaluated on various color images These images are shownin Figure 4 Figures 5 and 6 show the comparison of averageAIC and NIQE values between the proposed method andother conventional methods for various images Further thequalities of the test images which are enhanced using theaforementioned techniques that are measured in terms of

AIC and NIQE are given in Tables 1 and 2 According toTable 1 the FHE produces the highest AIC and thus becomesthe best method to bring out the details of the images Table 2shows the natural image quality evaluator index of testedimages From Table 2 it is found that FHE produces lowestNIQE values as compared to the other conventionalmethodsHence FHE produces enhanced images with natural lookingBased upon qualitative and quantitative analyses the pro-posed FHE method has been found effective in enhancingcontrasts of images in comparison to a few existing methodsIn addition fuzzy histogram equalized images uses fulldynamic range of the pixel values for maximum contrastTheperformance of the proposed method has been comparedwith five state-of-the-art methods both quantitatively andvisually Experimental result shows that the proposedmethodnot only outperforms in contrast enhancement but alsoprovides good visual representation in visual comparisonHence FHE gives better visual quality images

8 Conclusion

In this work Fuzzy logic-based histogram equalization ispresented for image contrast enhancement FHE uses fuzzystatistics of digital images to handle the inexactness ofgray level values in a better way compared to classicalcrisp histograms resulting in improved performance It canalso improve the color content brightness and contrastof an image automatically This algorithm was tested ondifferent gray scale and color images (they were extracted


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0

2000

0 50

1000

100 150 200 250

0

2000

0 50 100 150 200 250

0

2000

0 50 100 150 200 250

1000

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

(b)

Figure 4 Contrast enhancement results on the tested images (a) Original image with its histogram (b) proposed FHE-ed image with itshistogram

68979

70907

67065 66843 66171

71475

Aver

age A

IC v

alue

s

Orig

inal HE

BBH

E

MM

BEBH

E

BPD

FHE

Prop

osed

met

hod

Figure 5 Comparison of average AIC values

211712

241167

225678 225512 224592

205469

1819202122232425

Aver

age N

IQE

valu

es

Orig

inal HE

BBH

E

MM

BEBH

E

BPD

FHE

Prop

osed

met

hod

Figure 6 Comparison of average NIQE values


Table 1 Comparison of AIC values

Image ID Original HE BBHE MMBEBHE BPDFHE Proposed methodI03 71366 73239 69263 69541 69032 73794I04 70484 73835 68931 69294 67861 74470I05 65355 68877 63495 63735 61581 70169I06 74612 74275 73512 73435 71991 74481I07 71888 74265 70187 70751 69260 73796I12 71710 74952 69772 69376 68926 76427I17 60779 61511 58237 57842 57094 62738I19 75198 73572 73502 73543 73152 76406I20 56379 59485 53846 52187 54697 56576I21 72026 75061 69909 68722 68118 75895Average 68979 70907 67065 66843 66171 71475

Table 2 Comparison of NIQE values


from the TID2008 database) The qualitative and subjectiveenhancement performance of the proposed FHE algorithmwas evaluated and compared to the other conventionalmethods for different gray scale and color natural imagesThe performance of proposed FHE algorithm was evaluatedand compared in terms of AIC and NIQE indices Thesimulation results demonstrated that the proposed FHEalgorithm provided better results as compared to con-ventional methods such as histogram equalization (HE)brightness preserving bi-histogram equalization (BBHE)minimum mean brightness error bi-histogram equalization(MMBEBHE) and brightness preserving dynamic fuzzyhistogram equalization (BPDFHE) for different color imagesThe visual enhancement results of proposed FHE algorithmwere also better as compared to the other conventionalmethods for different gray scale and color natural imagesObserving from the simulation results obtained the FHEhas produced the best performance for both qualitative andquantitative evaluations Experimental results demonstratethat the proposed method can effectively and significantlyeliminate the washed-out appearance and adverse artifactsinduced by several existing methods This method is simpleand suitable for consumer electronic products

References

[1] X Su W Fang Q Shen and X Hao ldquoAn image enhancementmethod using the quantum-behaved particle swarm optimiza-tion with an adaptive strategyrdquo Mathematical Problems inEngineering vol 2013 Article ID 824787 13 pages 2013

[2] Y Yang J Zhang and X Huang ldquoAdaptive image enhancementalgorithm combining kernel regression and local homogeneityrdquoMathematical Problems in Engineering vol 2010 Article ID693532 14 pages 2010

[3] C G Ravichandran and V Magudeeswaran ldquoAn efficientmethod for contrast enhancement in still images using his-togram modification frameworkrdquo Journal of Computer Sciencevol 8 no 5 pp 775ndash779 2012

[4] Y Kim ldquoContrast enhancement using brightness preservingbi-histogram equalizationrdquo IEEE Transactions on ConsumerElectronics vol 43 no 1 pp 1ndash8 1997

[5] Y Wan Q Chen and B-M Zhang ldquoImage enhancementbased on equal area dualistic sub-image histogram equalizationmethodrdquo IEEE Transactions on Consumer Electronics vol 45no 1 pp 68ndash75 1999

[6] S D Chen and A R Ramli ldquoMinimum mean brightnesserror bi-histogramequalization in contrast enhancementrdquo IEEETransactions on Consumer Electronics vol 49 no 4 pp 1310ndash1319 2003


[7] S D Chen and A R Ramli ldquoContrast enhancement usingrecursive mean-separate histogram equalization for scalablebrightness preservationrdquo IEEE Transactions on Consumer Elec-tronics vol 49 no 4 pp 1301ndash1309 2003

[8] M Abdullah-Al-Wadud M H Kabir M A A Dewan and OChae ldquoA dynamic histogram equalization for image contrastenhancementrdquo IEEE Transactions on Consumer Electronics vol53 no 2 pp 593ndash600 2007

[9] N S P Kong andH Ibrahim ldquoColor image enhancement usingbrightness preserving dynamic histogram equalizationrdquo IEEETransactions on Consumer Electronics vol 54 no 4 2008

[10] D Sheet H Garud A Suveer M Mahadevappa and JChatterjee ldquoBrightness preserving dynamic fuzzy histogramequalizationrdquo IEEE Transactions on Consumer Electronics vol56 no 4 pp 2475ndash2480 2010

[11] H Kabir A Al-Wadud and O Chae ldquoBrightness preservingimage contrast enhancement using weighted mixture of globaland local transformation functionsrdquo International Arab Journalof Information Technology vol 7 no 4 pp 403ndash410 2010

[12] S-C Huang and H-Y Chien ldquoImage contrast enhancementfor preserving mean brightness without losing image featuresrdquoEngineering Applications of Artificial Intelligence vol 6 pp1487ndash1492 2013

[13] H Ibrahim and N S Pik Kong ldquoImage sharpening using sub-regions histogram equalizationrdquo IEEE Transactions on Con-sumer Electronics vol 55 no 2 pp 891ndash895 2009

[14] K Gunaseelan and E Seethalakshmi ldquoImage resolution andcontrst enhancement using singular value and discrete waveletdecompositionrdquo Journal of Scientific and Industrial Researchvol 72 pp 31ndash35 2013

[15] S H Lim N A M Isa C H Ooi and K K V Toh ldquoA newhistogram equalization method for digital image enhancementand brightness preservationrdquo Signal Image and Video Process-ing 2013

[16] A M Reza ldquoRealization of the contrast limited adaptivehistogram equalization (CLAHE) for real-time image enhance-mentrdquo Journal of VLSI Signal Processing Systems for SignalImage and Video Technology vol 38 no 1 pp 35ndash44 2004

[17] R C Gonzalez R E Woods and S L Eddins Digital ImageProcessing Prentice Hall Upper Saddle River NJ USA 2004

[18] S Chen and A Beghdadi ldquoNatural enhancement of colorimagerdquoEurasip Journal on Image andVideo Processing vol 2010Article ID 175203 30 pages 2010

[19] MM Riaz and Abdul Ghafoor ldquoThrough-wall image enhance-ment based on singular value decompositionrdquo InternationalJournal of Antennas and Propagation vol 2012 Article ID961829 20 pages 2012

[20] K Hasikin and N A M Isa ldquoAdaptive fuzzy contrast factorenhancement technique for low contrast and non-uniformillumination imagesrdquo Signal Image and Video Processing 2012

[21] N Y Suple and SM Kharad ldquoBasic approach to image contrastenhancement with fuzzy inference systemrdquo International Jour-nal of Scientific and Research Publications vol 3 no 6 2013

[22] E E Kerre and M Nachtegael Eds Fuzzy Techniques in ImageProcessing Physica Heidelberg Germany 2000

[23] Y Chang and C Chang ldquoA simple histogram modificationscheme for contrast enhancementrdquo IEEE Transactions on Con-sumer Electronics vol 56 no 2 pp 737ndash742 2010

[24] A K Moorthy and A C Bovik ldquoBlind image quality assess-ment from natural scene statistics to perceptual qualityrdquo IEEETransactions on Image Processing vol 20 no 12 pp 3350ndash33642011

[25] A Mittal R Soundararajan and A C Bovik ldquoMaking acompletely blind image quality analyzerrdquo IEEE Signal ProcessingLetters vol 20 no 3 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of


of BBHEmethod that provides maximal brightness preserva-tion [6] This algorithm finds the minimummean brightnesserror between the original and the enhanced image Itemploys the optimal point as the separating point insteadof the mean or median of the input image Though thesemethods can perform good contrast enhancement they alsocause more annoying side effects depending on the variationof gray level distribution in the histogram Recursive meanseparate HE (RMSHE) is another improved version of BBHE[7] This method is recursively separates the histogram intomulti-subhistograms instead of two subhistograms as in theBBHE Initially two subhistograms are created based on themean brightness of the original histogram Subsequentlythe mean brightness from the two subhistograms obtainedearlier are used as the second and third separating pointsin creating more sub-histograms In a similar fashion thealgorithm is executed recursively until the desired numbersof sub-histograms are met Then the HE approach is appliedindependently on each of the subhistograms The methodsdiscussed previously are based on dividing the originalhistogram into several sub-histograms by using either themedian or mean brightness Although the mean brightnessis well preserved by the aforementioned methods thesemethods cannot further expand the region of subhistogramthat is located near to theminimumormaximum value of thedynamic range However it is also not free from side effectssuch as washed-out appearance undesirable checkerboardeffects and significant change in image brightness

In order to deal with the problem mentioned aboveAbdullah-Al-Wadud et al introduced a dynamic histogramequalization (DHE) technique [8] DHE partitions the orig-inal histogram based on local minima However DHE doesnot consider the preserving of brightness For this purposeIbrahim and Kong proposed brightness preserving dynamichistogram equalization (BPDHE) [9]This method partitionsthe image histogram based on the local maxima of thesmoothed histogram It assigns a new dynamic range to eachpartition Finally the output intensity is normalized to makethe mean intensity of the resulting image equal to the inputone Although the BPDHE performs well inmean brightnesspreserving the ratio for brightness normalization plays animportant role A small ratio value leads to insignificantcontrast enhancement For large ratio (ie ratio value morethan 1) the final intensity value may exceed the maximumintensity value of the output dynamic range The exceedpixels will be quantized to the maximum intensity valueof gray levels and produce intensity saturation problem(in MATLAB environment) Brightness preserving dynamicfuzzy histogram equalization (BPDFHE) has been proposedby Sheet et al which is an enhanced version of BPDHE [10]The BPDFHE technique manipulates the image histogram insuch a way that no remapping of the histogram peaks takesplace while only redistribution of the gray-level values in thevalley portions between two consecutive peaks takes placeThe results using BPDFHE method show well-enhancedcontrast and little artifacts

To overcome the unwanted over enhancement and noiseamplifying the fuzzy logic-based histogram equalizationtechnique is proposed for both gray scale and color images

The proposed FHE method does not only preserve imagebrightness but also improves the local contrast of the originalimage First fuzzy histogram is computed using fuzzy settheory Second the fuzzy histogram is separated into twobased on the median value of the original image Finally theHE approach is applied independently on each subhistogramto improve the contrast

The remainder of this paper is organized as followsThe conventional histogram equalization is discussed inSection 2 and Section 3 presents the fuzzy image enhance-mentThe proposed algorithm for fuzzification and enhance-ment are presented in Sections 4 and 5 Simulation of the testimages and the qualitative and quantitative comparison of theresults are discussed in Sections 6 and 7 Finally Section 8concludes this paper

2 Histogram Equalization

Consider a digital image 119891 = 119891(119894 119895) which has the totalnumber of 119873 pixels with gray levels in the range [0 119871 minus 1]For a given image 119891 the probability density function 119875(119891119896) isdefined as

119875 (119891119896) =119899119896

119899(1)

for 119896 = 0 1 2 119871 minus 1 where 119899119896 represents the number oftimes that the gray level 119891119896 appears in the input image 119891 and119899 is the total number samples in the input image Note that119875(119891119896) is associated with the histogram of input image whichrepresents the number of pixels that have a specific intensity119891119896 In fact a plot of 119891119896 versus 119899119896 is called histogram of inputimage 119891 The respective cumulative density function is thendefined as

119862 (119891119896) =119871minus1

sum119895=0

119901 (119891119895) (2)

for 119896 = 0 1 2 119871 minus 1 Note that 119862(119891119871minus1) = 1 by definitionHistogram equalization is a method that maps the inputimage into entire dynamic range (1198910 119891119871minus1) by using thecumulative distribution function as a transform functionThat is let us define a transform function 119879(119891) based oncumulative density function as

119879 (119891) = 1198910 + (119891119871minus1 minus 1198910) 119862 (119891119896) (3)

Then the enhanced image of HE 119892 = 119892(119894 119895) can be expressedas

119892 (119894 119895) = 119879 (119891) = 119879 (119891 (119894 119895)) | forall119891 (119894 119895) isin 119891 (4)

where 119891 and 119892 are the original and enhanced images (119894 119895)are the 2D coordinates of the images and 119879 is the intensitytransformation function which maps the original image intothe entire dynamic range (1198910 119891119871minus1) However HE producesan undesirable checkerboard effects on enhanced imagesAnother problem of this method is that it also enhances thenoises in the input image along with the image features [11ndash19] In order to overcome these problems several methods




119891 = 119891119871 cup 119891119880 (5)

where

119891119871 = 119891 (119894 119895) | 119891 (119894 119895) le 119891119898 forall119891 (119894 119895) isin 119891

119891119880 = 119891 (119894 119895) | 119891 (119894 119895) gt 119891119898 forall119891 (119894 119895) isin 119891 (6)



119875119871 (119891119896) =119899119896119871119899119871

(7)

where 119896 = 0 1 119898 and

119875119880 (119891119896) =119899119896119880119899119880

(8)


119862119871 (119891119896) =119898

sum119895=0

119901119871 (119891119895)

119862119880 (119891119896) =119871minus1

sum119895=119898+1

119901119880 (119891119895)

(9)


119879119871 (119891119896) = 1198910 + (119891119898 minus 1198910) 119862119871 (119891119896)

119879119880 (119891119896) = 119891119898+1 + (119891119871minus1 minus 119891119898+1) 119862119880 (119891119896) (10)


119892 (119894 119895) = 119879119871 (119891119896) cup 119879119880 (119891119896)


(11)




119865 =119872

⋃119898=1

119873

⋃119899=1

120583119898119899119892119898119899

with 120583119898119899 sube [0 1] (12)



119892 = 119891 (119892) (13)



119892 = 119891ℎ (119892) = (119871 minus 1)

119892

sum119894=0

ℎ (119894)

119872119873 (14)


4 Proposed Work





119865 =

[[[[[[[[

[

1205831111989111

1205831211989112

12058311198731198911119873

1205832111989121

1205832211989122

12058321198731198912119873


12058311987221198911198722

120583119872119873119891119872119873

]]]]]]]]

]

(15)



120583119865(119894119895) =

2 lowast (120583119894119895)2

0 le 120583119894119895 le 05


) 05 lt 120583119894119895 le 1(16)


119865 larr997888 ℎ (119894) +sum119894

sum119895

120583119865(119894119895) (17)



119865 = 119865119871 cup 119865119880 (18)where

119865119871 = 119865 (119894 119895) | 119865 (119894 119895) le 119872 forall119865 (119894 119895) isin 119865

119865119880 = 119865 (119894 119895) | 119865 (119894 119895) gt 119872 forall119865 (119894 119895) isin 119865 (19)


119875119871 (119865119896) =119899119896119871119899119871

(20)

where 119896 = 0 1 119898 and

119875119880 (119865119896) =119899119896119880119899119880

(21)


119862119871 (119865119896) =119898

sum119895=0

119901119871 (119865119895)

119862119880 (119865119896) =119871minus1

sum119895=119898+1

119901119880 (119865119895)

(22)



119879119871 (119865119896) = 1198650 + (119872 minus 1198650) 119862119871 (119865119896)

119879119880 (119865119896) = 119872 + 1 + (119865119871minus1 minus119872 + 1)119862119880 (119865119896) (23)


119892 (119894 119895) = 119879119871 (119865119871) cup 119879119880 (119865119880) (24)


119879119880 (119865119880) = 119879119880 (119865 (119894 119895)) | forall119865 (119894 119895) isin 119865119880 (25)



119866 (119894 119895) = 119879minus1 (119892 (119894 119895)) =119872

⋃119894=1

119873

⋃119895=1

119892 (119894 119895) lowast (119871 minus 1) (26)









sum119896=0

119875 (119896) log119875 (119896) (27)






(a) (b)

(c) (d)

(e) (f)


8000

6000

4000

2000

0

Num

ber o

f pix

els

0 100 15050 200 250Gray levels

(a)

8000

6000

4000

2000

0

Num

ber o

f pix

els

Gray levels0 100 15050 200 250

(b)

8000

6000

4000

2000

0

Num

ber o

f pix

els

Gray levels0 100 15050 200 250

(c)

8000

6000

4000

2000

0

Num

ber o

f pix

els

Gray levels0 100 15050 200 250

(d)

8000

6000

4000

2000

0

Num

ber o

f pix

els

Gray levels0 100 15050 200 250

(e)

10000

15000

5000

0

Num

ber o

f pix

els

0 100 15050 200 250Gray levels

(f)



(a) (b)

(c) (d)

(e) (f)




8 Conclusion



0

100

0 50 100 150 200 250

0 50 100 150 200 2500

100

0

100

0 50 100 150 200 250

0

100

0 50

50

100 150 200 250

0

100

0 50 100 150 200 250

0

100

0 50

50

100 150 200 250

Gray levels

Gray levels

Num

ber o

f pix

elsN

umbe

r of p

ixels

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

(a)

0

2000

0 50 100 150 200 250

0

5000

0 50 100 150 200 250

0

2000

0 50

1000

100 150 200 250

0

2000

0 50

1000

100 150 200 250

0

2000

0 50 100 150 200 250

0

2000

0 50 100 150 200 250

1000

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

(b)


68979

70907

67065 66843 66171

71475

Aver

age A

IC v

alue

s

Orig

inal HE

BBH

E

MM

BEBH

E

BPD

FHE

Prop

osed

met

hod


211712

241167

225678 225512 224592

205469

1819202122232425

Aver

age N

IQE

valu

es

Orig

inal HE

BBH

E

MM

BEBH

E

BPD

FHE

Prop

osed

met

hod








References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119891 = 119891119871 cup 119891119880 (5)

where

119891119871 = 119891 (119894 119895) | 119891 (119894 119895) le 119891119898 forall119891 (119894 119895) isin 119891

119891119880 = 119891 (119894 119895) | 119891 (119894 119895) gt 119891119898 forall119891 (119894 119895) isin 119891 (6)



119875119871 (119891119896) =119899119896119871119899119871

(7)

where 119896 = 0 1 119898 and

119875119880 (119891119896) =119899119896119880119899119880

(8)


119862119871 (119891119896) =119898

sum119895=0

119901119871 (119891119895)

119862119880 (119891119896) =119871minus1

sum119895=119898+1

119901119880 (119891119895)

(9)


119879119871 (119891119896) = 1198910 + (119891119898 minus 1198910) 119862119871 (119891119896)

119879119880 (119891119896) = 119891119898+1 + (119891119871minus1 minus 119891119898+1) 119862119880 (119891119896) (10)


119892 (119894 119895) = 119879119871 (119891119896) cup 119879119880 (119891119896)


(11)




119865 =119872

⋃119898=1

119873

⋃119899=1

120583119898119899119892119898119899

with 120583119898119899 sube [0 1] (12)



119892 = 119891 (119892) (13)



119892 = 119891ℎ (119892) = (119871 minus 1)

119892

sum119894=0

ℎ (119894)

119872119873 (14)


4 Proposed Work





119865 =

[[[[[[[[

[

1205831111989111

1205831211989112

12058311198731198911119873

1205832111989121

1205832211989122

12058321198731198912119873


12058311987221198911198722

120583119872119873119891119872119873

]]]]]]]]

]

(15)



120583119865(119894119895) =

2 lowast (120583119894119895)2

0 le 120583119894119895 le 05


) 05 lt 120583119894119895 le 1(16)


119865 larr997888 ℎ (119894) +sum119894

sum119895

120583119865(119894119895) (17)



119865 = 119865119871 cup 119865119880 (18)where

119865119871 = 119865 (119894 119895) | 119865 (119894 119895) le 119872 forall119865 (119894 119895) isin 119865

119865119880 = 119865 (119894 119895) | 119865 (119894 119895) gt 119872 forall119865 (119894 119895) isin 119865 (19)


119875119871 (119865119896) =119899119896119871119899119871

(20)

where 119896 = 0 1 119898 and

119875119880 (119865119896) =119899119896119880119899119880

(21)


119862119871 (119865119896) =119898

sum119895=0

119901119871 (119865119895)

119862119880 (119865119896) =119871minus1

sum119895=119898+1

119901119880 (119865119895)

(22)



119879119871 (119865119896) = 1198650 + (119872 minus 1198650) 119862119871 (119865119896)

119879119880 (119865119896) = 119872 + 1 + (119865119871minus1 minus119872 + 1)119862119880 (119865119896) (23)


119892 (119894 119895) = 119879119871 (119865119871) cup 119879119880 (119865119880) (24)


119879119880 (119865119880) = 119879119880 (119865 (119894 119895)) | forall119865 (119894 119895) isin 119865119880 (25)



119866 (119894 119895) = 119879minus1 (119892 (119894 119895)) =119872

⋃119894=1

119873

⋃119895=1

119892 (119894 119895) lowast (119871 minus 1) (26)









sum119896=0

119875 (119896) log119875 (119896) (27)






(a) (b)

(c) (d)

(e) (f)


8000

6000

4000

2000

0

Num

ber o

f pix

els

0 100 15050 200 250Gray levels

(a)

8000

6000

4000

2000

0

Num

ber o

f pix

els

Gray levels0 100 15050 200 250

(b)

8000

6000

4000

2000

0

Num

ber o

f pix

els

Gray levels0 100 15050 200 250

(c)

8000

6000

4000

2000

0

Num

ber o

f pix

els

Gray levels0 100 15050 200 250

(d)

8000

6000

4000

2000

0

Num

ber o

f pix

els

Gray levels0 100 15050 200 250

(e)

10000

15000

5000

0

Num

ber o

f pix

els

0 100 15050 200 250Gray levels

(f)



(a) (b)

(c) (d)

(e) (f)




8 Conclusion



0

100

0 50 100 150 200 250

0 50 100 150 200 2500

100

0

100

0 50 100 150 200 250

0

100

0 50

50

100 150 200 250

0

100

0 50 100 150 200 250

0

100

0 50

50

100 150 200 250

Gray levels

Gray levels

Num

ber o

f pix

elsN

umbe

r of p

ixels

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

(a)

0

2000

0 50 100 150 200 250

0

5000

0 50 100 150 200 250

0

2000

0 50

1000

100 150 200 250

0

2000

0 50

1000

100 150 200 250

0

2000

0 50 100 150 200 250

0

2000

0 50 100 150 200 250

1000

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

Gray levels

Num

ber o

f pix

els

(b)


68979

70907

67065 66843 66171

71475

Aver

age A

IC v

alue

s

Orig

inal HE

BBH

E

MM

BEBH

E

BPD

FHE

Prop

osed

met

hod


211712

241167

225678 225512 224592

205469

1819202122232425

Aver

age N

IQE

valu

es

Orig

inal HE

BBH

E

MM

BEBH

E

BPD

FHE

Prop

osed

met

hod








References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119892 = 119891ℎ (119892) = (119871 minus 1)

119892

sum119894=0

ℎ (119894)

119872119873 (14)


4 Proposed Work





119865 =

[[[[[[[[

[

1205831111989111

1205831211989112

12058311198731198911119873

1205832111989121

1205832211989122

12058321198731198912119873


12058311987221198911198722

120583119872119873119891119872119873

]]]]]]]]

]

(15)



120583119865(119894119895) =

2 lowast (120583119894119895)2

0 le 120583119894119895 le 05


) 05 lt 120583119894119895 le 1(16)


119865 larr997888 ℎ (119894) +sum119894

sum119895

120583119865(119894119895) (17)



119865 = 119865119871 cup 119865119880 (18)where

119865119871 = 119865 (119894 119895) | 119865 (119894 119895) le 119872 forall119865 (119894 119895) isin 119865

119865119880 = 119865 (119894 119895) | 119865 (119894 119895) gt 119872 forall119865 (119894 119895) isin 119865 (19)


119875119871 (119865119896) =119899119896119871119899119871

(20)

where 119896 = 0 1 119898 and

119875119880 (119865119896) =119899119896119880119899119880

(21)


119862119871 (119865119896) =119898

sum119895=0

119901119871 (119865119895)

119862119880 (119865119896) =119871minus1

sum119895=119898+1

119901119880 (119865119895)

(22)



119879119871 (119865119896) = 1198650 + (119872 minus 1198650) 119862119871 (119865119896)

119879119880 (119865119896) = 119872 + 1 + (119865119871minus1 minus119872 + 1)119862119880 (119865119896) (23)


119892 (119894 119895) = 119879119871 (119865119871) cup 119879119880 (119865119880) (24)


119879119880 (119865119880) = 119879119880 (119865 (119894 119895)) | forall119865 (119894 119895) isin 119865119880 (25)



119866 (119894 119895) = 119879minus1 (119892 (119894 119895)) =119872

⋃119894=1

119873

⋃119895=1

119892 (119894 119895) lowast (119871 minus 1) (26)









sum119896=0

119875 (119896) log119875 (119896) (27)






(a) (b)

(c) (d)

(e) (f)


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8 Conclusion



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References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879119871 (119865119896) = 1198650 + (119872 minus 1198650) 119862119871 (119865119896)

119879119880 (119865119896) = 119872 + 1 + (119865119871minus1 minus119872 + 1)119862119880 (119865119896) (23)


119892 (119894 119895) = 119879119871 (119865119871) cup 119879119880 (119865119880) (24)


119879119880 (119865119880) = 119879119880 (119865 (119894 119895)) | forall119865 (119894 119895) isin 119865119880 (25)



119866 (119894 119895) = 119879minus1 (119892 (119894 119895)) =119872

⋃119894=1

119873

⋃119895=1

119892 (119894 119895) lowast (119871 minus 1) (26)









sum119896=0

119875 (119896) log119875 (119896) (27)






(a) (b)

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8 Conclusion



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References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)

(e) (f)


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(a) (b)

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8 Conclusion



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References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)

(e) (f)




8 Conclusion



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References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

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(a)

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211712

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225678 225512 224592

205469

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Aver

age N

IQE

valu

es

Orig

inal HE

BBH

E

MM

BEBH

E

BPD

FHE

Prop

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met

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References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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