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1 Univerity of Warwick intitutional repoitory: Thi paper i made available online in accordance with publiher policie. Pleae croll down to view the document itelf. Pleae refer to the repoitory record for thi item and our policy information available from the repoitory home page for further information. To ee the final verion of thi paper pleae viit the publiher webite. Acce to the publihed verion may require a ubcription. Author: Celik, T. and Tjahjadi, T. Article Title: Automatic Image Equalization and Contrat Enhancement Uing Gauian Mixture Modeling Year of publication: 212 Link to publihed article: Publiher tatement: 212 IEEE. Peronal ue of thi material i permitted. Permiion from IEEE mut be obtained for all other ue, in any current or future media, including reprinting/republihing thi material for advertiing or promotional purpoe, creating new collective work, for reale or reditribution to erver or lit, or reue of any copyrighted component of thi work in other work. Citation: Celik, T and Tjahjadi, T Automatic Image Equalization and Contrat Enhancement Uing Gauian Mixture Modeling. Tranaction on Image Proceing, IEEE, Vol. 21, No. 1, pp

2 Automatic Image Equalization and Contrat 1 Enhancement Uing Gauian Mixture Modelling Turgay Celik and Tardi Tjahjadi Senior Member, IEEE Abtract In thi paper, we propoe an adaptive image equalization algorithm which automatically enhance the contrat in an input image. The algorithm ue Gauian mixture model GMM to model the image grey-level ditribution, and the interection point of the Gauian component in the model are ued to partition the dynamic range of the image into input grey-level interval. The contrat equalized image i generated by tranforming the pixel grey level in each input interval to the appropriate output grey-level interval according to the dominant Gauian component and cumulative ditribution function CDF of the input interval. To take account of human perception the Gauian component with mall variance are weighted with maller value than the Gauian component with larger variance, and the grey-level ditribution i alo ued to weight the component in the mapping of the input interval to the output interval. Experimental reult how that the propoed algorithm produce better or comparable enhanced image than everal tate-of-the-art algorithm. Unlike the other algorithm, the propoed algorithm i free of parameter etting for a given dynamic range of the enhanced image and can be applied to a wide range of image type. Index Term Contrat enhancement, hitogram equalization, normal ditribution, Gauian mixture modelling, hitogram partition. I. INTRODUCTION The objective of an image enhancement technique i to bring out hidden image detail, or to increae the contrat of an image with low dynamic range [1]. Such a technique produce an output image that ubjectively look better than the original image by increaing the grey level difference i.e., the contrat among object and background. Numerou enhancement technique have been introduced and thee can be divided into three group: 1 technique that decompoe an image into high and low frequency ignal for manipulation [2], [3]; 2 tranform-baed technique [4]; and 3 hitogram modification technique [5] [16]. Technique in the firt two group often ue multicale analyi to decompoe the image into different frequency band and enhance it deired global and local frequencie [2] [4]. Thee technique are computationally complex but enable global and local contrat enhancement imultaneouly by tranforming the ignal in the appropriate band or cale. Furthermore they require appropriate parameter etting which might otherwie reult in image degradation. For example, the centre-urround Retinex [2] algorithm wa developed to attain lightne and colour contancy for machine viion application. The contancy refer to the reilience of perceived colour and lightne to patial and pectral illumination variation. The benefit of the Retinex algorithm include dynamic range compreion and colour independence from the patial ditribution of the cene Thi work wa upported by the Warwick Univerity Vice Chancellor Scholarhip. Turgay Celik and Tardi Tjahjadi are with the School of Engineering, Univerity of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, United Kingdom. Turgay Celik t.celik@warwick.ac.uk; celikturgay@gmail.com, Tardi Tjahjadi t.tjahjadi@warwick.ac.uk.

3 2 illumination. However, thi algorithm can reult in halo artefact, epecially in boundarie between large uniform region. Alo, a greying out can occur, in which the cene tend to change to middle grey. Among the three group the third group received the mot attention due to their traightforward and intuitive implementation qualitie. Linear contrat tretching LCS and global hitogram equalization GHE are two widely utilized method for global image enhancement [1]. The former linearly adjut the dynamic range of an image, and the latter ue an input-to-output mapping obtained from the cumulative ditribution function CDF which i the integral of the image hitogram. Since the contrat gain i proportional to the height of the hitogram, grey level with large pixel population are expanded to a larger range of grey level while other grey-level range with fewer pixel are compreed to maller range. Although GHE can efficiently utilize diplay intenitie, it tend to over-enhance the image contrat if there are high peak in the hitogram, often reulting in a harh and noiy appearance of the output image. LCS and GHE are imple tranformation but they do not alway produce good reult, epecially for image with large patial variation in contrat. In addition, GHE ha the undeired effect of over-emphaizing any noie in an image. In order to overcome the aforementioned problem, local hitogram equalization LHE baed enhancement technique have been propoed, e.g., [5], [6]. The LHE method [6] ue a mall window that lide through every image pixel equentially and only pixel within the current poition of the window are hitogram equalized, and the grey-level mapping for enhancement i done only for the centre pixel of the window. Thu, it utilie local information. However, LHE ometime caue overenhancement in ome portion of the image and enhance any noie in the input image along with the image feature. Furthermore, LHE baed method produce undeirable checkerboard effect. Hitogram Specification HS [1] i a method that ue a deired hitogram to modify the expected output image hitogram. However, pecifying the output hitogram i not a traightforward tak a it varie from image to image. The Dynamic Hitogram Specification DHS [7] generate the pecified hitogram dynamically from the input image. In order to retain the original hitogram feature, DHS extract the differential information from the input hitogram and incorporate extra parameter to control the enhancement uch a the image original and the reultant gain control value. However, the degree of enhancement achievable i not ignificant. Some reearche have alo focued on improving hitogram equalization baed contrat enhancement uch a mean preerving bi-hitogram equalization BBHE [8], equal area dualitic ub-image hitogram equalization DSIHE [9] and minimum mean brightne error bi-hitogram equalization MMBEBHE [1]. BBHE firt divide the image hitogram into two part with the average grey level of the input image pixel a the eparation intenity. The two hitogram are then independently equalized. The method attempt to olve the brightne preervation problem. DSIHE ue entropy for hitogram eparation. MMBEBHE i the extenion of BBHE that provide maximal brightne preervation. Although thee method can achieve good contrat enhancement, they alo generate annoying ide effect depending on the variation in the grey-level ditribution [7]. Recurive Mean-Separate Hitogram Equalization RMSHE [11] i another improvement of BBHE. However, it i alo not free from ide effect. Dynamic hitogram equalization DHE [12] firt mooth the input hitogram by uing a one dimenional moothing filter. The moothed hitogram i partitioned into ub-hitogram baed on the local minima. Prior to equalizing the ubhitogram, each ub-hitogram i mapped into a new dynamic range. The mapping i a function of the number of pixel in each ub-hitogram, thu a ub-hitogram with a larger number of pixel will occupy a bigger portion of the dynamic range. However, DHE doe not place any contraint on maintaining the mean brightne of the image. Furthermore, everal parameter are ued that require appropriate etting for different image. Optimiation technique have alo been employed for contrat enhancement. The method brightne preerving hitogram

4 3 equaliation with maximum entropy BPHEME [13] define the ideal hitogram to have maximum entropy with brightne preervation. The target hitogram which ha the maximum differential entropy under the mean brightne contraint i obtained uing variational approach [13]. BPHEME i deigned to achieve maximum entropy. Although entropy maximiation correpond to contrat tretching to ome extent, it i not a traightforward conequence and doe not definitely lead to contrat enhancement [14]. In flattet hitogram pecification with accurate brightne preervation FHSABP [14], convex optimiation i ued to tranform the image hitogram into the flattet hitogram, ubject to a mean brightne contraint. An exact hitogram pecification method i ued to preerve the image brightne. However, when the grey level of the input image are equally ditributed, FHSABP behave very imilar to GHE. Furthermore, it i deigned to preerve the average brightne which may produce low contrat reult when the average brightne i either too low or too high. In hitogram modification framework HMF which i baed on hitogram equalization, contrat enhancement i treated a an optimization problem that minimize a cot function [15]. Penalty term are introduced in the optimiation in order to handle noie and black/white tretching. HMF can achieve different level of contrat enhancement, through the ue of different adaptive parameter. Thee parameter have to be manually tuned according to the image content to achieve high contrat. In order to deign a parameter free contrat enhancement method, genetic algorithm GA i employed to find a target hitogram which maximize a contrat meaure baed on edge information [16]. We call thi method contrat enhancement baed on GA CEBGA. CEBGA uffer from the drawback of GA baed method, namely dependency on initialization and convergence to a local optimum. Furthermore, the mapping to the target hitogram i cored by only maximum contrat which i meaured according to average edge trength etimated from the gradient information. Thu, CEBGA may produce reult which are not patially mooth. Finally, the convergence time i proportional to the number of ditinct grey level of the input image. The aforementioned technique may create problem when enhancing a equence of image, or when the hitogram ha pike, or when a natural looking enhanced image i required. In thi paper, we propoe an adaptive image equalization algorithm which i effective in term of improving viual quality of different type of input image. Image with low-contrat are automatically improved in term of an increae in dynamic range. Image with ufficiently high contrat are alo improved but not a much. The algorithm further enhance the colour quality of the input image in term of colour conitency, higher contrat between foreground and background object, larger dynamic range and more detail in image content. The propoed algorithm i free from parameter etting. Intead the pixel value of an input image are modelled uing Gauian mixture model GMM. The interection point of the Gauian in the model are ued in partitioning the dynamic range of the input image into input grey-level interval. The grey level of the pixel in each input interval are tranformed according to the dominant Gauian component and the CDF of the interval to obtain the contrat equalized image. The paper i organized a follow. Section II preent the propoed automatic image equalization algorithm and the neceary background related to the GMM fit of the input image data. Section III preent the ubjective and quantitative comparion of the propoed method with everal tate-of-the-art enhancement technique. Section IV conclude the paper. II. PROPOSED ALGORITHM Let u conider an input image, X = {xi, j 1 i H, 1 j W }, of ize H W pixel, where xi, j R. Let u aume that X ha a dynamic range of [x d, x u ] where xi, j [x d, x u ]. The main objective of the propoed algorithm i to generate an enhanced image, Y = {y i, j 1 i H, 1 j W }, which ha a better viual quality with repect to X. The dynamic range of Y can be tretched or tightened into the interval [y d, y u ], where y i, j [y d, y u ], y d < y u and y d, y u R.

5 4 A. Modelling The Gauian mixture model GMM can model any data ditribution in term of a linear mixture of different Gauian ditribution with different parameter. Each of the Gauian component ha a different mean, tandard deviation and proportion or weight in the mixture model. A Gauian component with a low tandard deviation and a large weight repreent compact data with a dene ditribution around the mean value of the component. When the tandard deviation become larger, the data i dipered about it mean value. The human eye i not enitive to mall variation around dene data, but i more enitive to widely cattered fluctuation. Thu, in order to increae the contrat while retaining image detail, dene data with low tandard deviation hould be dipered while cattered data with high tandard deviation hould be compacted. Thi operation hould be done o that the grey-level ditribution i retained. In order to achieve thi, we ue GMM to partition the ditribution of the input image into a mixture of different Gauian component. The grey-level ditribution p x, where x X, of the input image X can be modelled a a denity function compoed of a linear combination of N function uing the GMM [17], i.e., p x = N P w n px w n, 1 n=1 where p x w n i the nth component denity, and P w n i the prior probability of the data point generated from component w n of the mixture. The component denity function are contrained to be Gauian ditribution function, i.e., 1 p x w n = exp x µ w n 2 2πσ 2 wn 2σw 2, 2 n where µ wn and σ 2 w n are repectively the mean and the variance of the nth component. Each of the Gauian ditribution function atifie the contraint and the prior probabilitie are choen to atify the contraint p x w n dx = 1, 3 N P w n = 1 and P w n 1. 4 n=1 A GMM i completely pecified by it parameter θ = { } P w n,µ wn, σw 2 N n. The etimation of the probability ditribution n=1 function PDF of an input image data x reduce to finding the appropriate value of θ. In order to etimate θ, maximum likelihood etimation MLE technique uch a the expectation maximization EM algorithm [18] have been widely ued. Auming the data point X = {x 1, x 2,...,x H W } are independent, the likelihood of the data X i computed by LX; θ = H W k=1 p x k ; θ 5 given the ditribution or, more pecifically, given the ditribution parameter θ. The goal of the etimation i to find ˆθ that maximize the likelihood, i.e., ˆθ = argmaxlx; θ. 6 θ Intead of maximizing thi function directly, the log-likelihood L X; θ = lnlx; θ H W k=1 p x k ; θ i ued becaue it i analytically eaier to handle. The EM algorithm tart from an initial gue θ for the ditribution parameter and the log-likelihood i guaranteed to increae on each iteration until it converge. The convergence lead to a local or global maximum, but it can alo lead to ingular

6 Hitogram Pw 1 px w 1 Pw 2 px w 2 Pw 3 px w 3 Pw 4 px w 4 px GMM px x a b Fig. 1. a A grey-level image; and b it hitogram and GMM fit. etimate, which i true particularly for Gauian mixture ditribution with arbitrary covariance matrice. The initialization i one of the problem of the EM algorithm. The election of θ partly determine where the algorithm converge or hit the boundary of the parameter pace to produce ingular, meaningle reult. Furthermore, the EM algorithm require the uer to et the number of component, and the number i fixed during the etimation proce. The Figueiredo-Jain FJ algorithm [19] which i an improved variant of the EM algorithm overcome major weaknee of the baic EM algorithm. The FJ algorithm adjut the number of component during etimation by annihilating component that are not upported by the data. It avoid the boundary when it annihilate component that are becoming ingular. It i alo allowed to tart with an arbitrarily large number of component, which addree the initialization of the EM algorithm. The initial guee for component mean can be ditributed into the whole pace occupied by the training ample, even etting one component for every ingle training ample. Due to it advantage over EM algorithm, in thi work we adopt the FJ algorithm for parameter etimation. Fig. 1a and b repectively illutrate an input image and it hitogram together with it GMM fit. The hitogram i modelled uing four Gauian component, i.e., N = 4. The cloe match between the hitogram hown a rectangular vertical bar and GMM fit hown a olid black line i obtained uing FJ algorithm. There are three main grey tone in the input image correponding to the tank, it hadow and the image background. The other grey-level tone are ditributed around the three main tone. However, FJ algorithm reult in four Gauian component N = 4 for the mixture model. Thi i becaue the grey tone with the highet average grey value correponding to the image background ha a deviation too large for a ingle Gauian component to repreent it. Thu it i repreented by two Gauian component, i.e., w 3 and w 4 a hown in Fig. 1b. All interection point between Gauian component that fall within the dynamic range of the input image are denoted by yellow circle, and ignificant interection point that are ued in dynamic range repreentation are denoted by black circle. There i only one dominant Gauian component between two interection point, which adequately repreent the data within thi grey-level interval. For intance, the range of the input data within the interval of [35, 9] i repreented by Gauian

7 6 TABLE I THE NUMERICAL VALUES OF INTERSECTION POINTS DENOTED BY YELLOW CIRCLES IN FIG. 1B BETWEEN COMPONENTS OF GMM FIT TO THE GREY-LEVEL IMAGE SHOWN IN FIG. 1A. GMM Component w 1 w 2 w 3 w 4 w , , , w , , , w , , , w , , , component w 1 hown a olid blue line. Thu the data within each interval i repreented by a ingle Gauian component which i dominant with repect to the other component. The dynamic range of the input image i repreented by the union of all interval. B. Partitioning The ignificant interection point are elected from all the poible interection between the Gauian component. The interection point between two Gauian component w m and w n are found by olving P w m px w m = P w n px w n, 7 or equivalently x µ w m 2 2σw 2 + x µ w n 2 P wn σ wm m 2σw 2 = ln, 8 n P w m σ wn which reult in ax 2 + bx + c =, 9 where a = σ 2 w m σ 2 w n, b = 2 µwm σ 2 w n µ wn σ 2 w m, c = µ 2 w n σw 2 m µ 2 w m σw 2 n 2σ 2 P wm σw 2 wn n ln P w m The econd order parametric equation Eq. 9 ha two root, i.e., x 1 σ wm σ wn m,n = b + b 2 4ac, x 2 m,n = b b 2 4ac. 1 2a 2a In Fig. 1b all interection point between GMM component are denoted by yellow circle. The numerical value of the interection point determined uing Eq. 1 are hown in Table I. Table I i ymmetric, i.e., the interection point between the component w 1 and w 2 are the ame a the interection point between component w 2 and w 1. The interection point of two component are independent of the order of the component. All poible interection point that are within the dynamic range of the image are detected. The leftmot interection point between component w 1 and w 2 i at which i not within the dynamic range of the input image, thu it could not be conidered. In order to allow combination of interection point to cover only the entire dynamic range of the input image a further proce i needed..

8 7 The total number of interection point calculated i N N 1. The ignificant interection point x d, where d {1,...,D}, D N N 1, are elected among all interection point. For a given interection point x k m,n, where k = {1, 2}, between Gauian component w m and w n it i elected a a ignificant interection point if and only if it i a real number in the dynamic range of the input image, i.e., x k m,n X, and the Gauian component w m and w n contain the maximum value in the mixture for the point x k m,n, i.e., where w k {w m, w n }. P w m p P w m p x k m,n w m x k m,n w m = P w n p > P w k p x k m,n w n x k m,n w k, 11, 12 The ignificant interection point are orted in acending order of their value and are partitioned into grey-level interval [ ] [ ] [ ] to cover the entire dynamic range of X, i.e., x x = x l, x 1 x 1, x 2 x D, x r. The leftmot ignificant interection point x l i elected a the value of x for which x l = x, F x T h HW, F x < T h HW, 13 where the minimum ditance between two conecutive number i, e.g., = 1 in the cae of 8-bit input image X conidered in thi work, F x i the CDF of x, and T h i the minimum number of pixel which will be excluded from the tail of the greylevel ditribution of x. To conider all pixel grey value of X we et T h = 1. Similarly, the rightmot ignificant interection point x r i elected by conidering the tail of the grey-level ditribution of x for which x r = x, 1 F x T h HW, 1 F x + < T h HW. 14 [ ] The ignificant interection point that fall outide of the interval x l, x r are ignored ince they are the interection point [ ] between two Gauian component that fall outide the dynamic range of X, and x i updated a x = x 1, x 2,...,x K with x 1 < x 2 <... < x K, where K i the maximum number of ignificant interection point. In Fig. 1 the ix ignificant interection point are denoted by black circle, and the range of x cover the entire dynamic range of X. The CDF of x i F x = = = x p x dx = N P w n n=1 N n=1 It can be calculated uing the cloed form expreion x x n=1 x µwn 2σwn P w n F x = N P w n px w n dx 1 exp x µ w n 2 2πσ 2 wn 2σw 2 dx n 1 π exp t 2 dt. 15 N P w n F wn x, 16 n=1 where F wn x i the CDF of Gauian component w n, and uing the definition of the error function erfx alo called the x µwn Gau error function [2] it i computed a F wn x = β 2σwn, where β x i computed in term of error function [2] a follow: 1 + erfx /2, iff x β x = 17 1 erf x /2, otherwie,

9 8 where the numerical value of erfx are tabulated in [2]. The function β x i invertible, i.e., for a given β x = a, x = β 1 a exit. The conecutive pair of ignificant interection point are ued to partition the dynamic range of X into ubinterval, i.e., [ ] [ ] [ ] [ ] [ ] [x d, x u ] = x 1, x 2 x 2, x 3 x K 2, x K 1 x K 1, x K. The ubinterval x k, x k+1 i repreented by a Gauian component w k which i dominant with repect to the other Gauian component in the ubinterval. The dominant [ ] Gauian component i found by conidering the a poteriori probability of each component in the interval x k, x k+1 which i equivalent to the area under the Gauian component, i.e., [ w k = argmax F wi w i x k+1 F wi x k ]. 18 C. Mapping [ The interval x k ], x k+1, where k = 1, 2,...,K 1, in x i mapped onto the dynamic range of the output image Y. In the mapping, each interval cover a certain range which i proportional to a weight α k, where α k [, 1], which i calculated by conidering two figure of merit imultaneouly: 1 the rate of the total number of pixel that fall into the interval [ ], x k+1 ; and 2 the tandard deviation of the dominant Gauian component w k, i.e., x k α k = σ wk γ N i=1 σ w i γ [ F K 1 i=1 x k+1 [ F ] F x k ]. 19 F The firt term adjut the brightne of the equalized image, and γ [, 1] i brightne contant in thi paper γ =.5 i ued. The lower the value of γ, the brighter i the output image. The econd term in Eq. 19 i related to the grey-level ditribution and i ued to retain the overall content of the data in the interval. Eq. 19 maintain a balance between the data ditribution and variance of the data in a certain interval. Since the human eye i more enitive to udden change in widely cattered data and le enitive to mooth change in denely cattered data, Eq. 19 give larger weight to widely cattered x i+1 data larger variance, and vice vera. [ ] Uing α k, the input interval, x k+1 i mapped onto the output interval [ y k, y k+1] according to x k x i k 1 y k = y d + y u y d α i, 2 i=1 y k+1 = y k + α k y u y d. The above mapping guarantee that the output dynamic range i covered by the mapping, i.e., [y d, y u ] = [ y 1 = y d, y 2] [ y 2, y 3] [ y K 1, y K = y u ]. In the final mapping of pixel value from the input interval onto the output interval, the CDF of the ditribution in the output interval i preerved. Let Gauian ditribution w k with parameter µ wk and σ 2 w k repreent the Gauian component w k in the range [ y k, y k+1]. The parameter µ wk and σ 2 w k are found by olving the following equation imultaneouly F wk F wk x k x k+1 [ o that the area under the Gauian ditribution w k between = F wk y k, 21 = F wk y k+1, 22 x k ], x k+1 i equal to the area under the Gauian ditribution

10 9 w k in the interval [ y k, y k+1]. Uing Eq. 17 together with equation Eq. 21 and Eq. 22, one can write x k µ wk y k µ wk β = β 23 2σwk 2σwk x k+1 µ wk y k+1 µ wk β = β, 24 2σwk 2σwk which i equivalent to x k µ wk = yk µ wk 2σwk 2σwk 25 µ wk = yk+1 µ wk. 26 2σwk x k+1 2σwk Uing equation Eq. 25 and Eq. 26, the parameter of Gauian ditribution w k are computed a follow: x k µw k y k+1 y k x k+1 µ wk µ wk = 27 x k µ wk 1 x k+1 µ wk y k µ wk [ The mapping of x to y, where x x k ditribution w k and Gauian ditribution w k σ wk = σ x k wk. 28 µ wk ], x k+1 and y [ y k, y k+1], i achieved by keeping the CDF of Gauian equal, i.e., x µwk β β 2σwk β y µwk 2σwk β x k µ wk = 2σwk y k µ wk 2σwk, 29 where uing the equality in Eq. 23, x µwk y µwk β = β 2σwk 2σwk x µ w k 2σwk = y µ w k 2σwk reult in the following mapping of given x to the correponding y according to the Gauian ditribution w k and w k x µwk y = σ wk + µ wk. 3 σ wk The final mapping from x to y i achieved by conidering all Gauian component in the GMM to retain the pixel ditribution in input and output interval equal. Uing the uperpoition of ditribution together with Eq. 3 one can find N x µwi y = σ wi + µ wi P wi. 31 i=1 σ wi Fig. 2a, b and c repectively how the input image, equalized image uing the propoed algorithm where the dynamic range of the output image i [y d, y u ] = [, 255], and the mapping between input image data point x and equalized output image data point y are according to Eq. 31. Fig. 2c how that a different mapping i applied to a different input greylevel interval. Fig. 2b how that the propoed algorithm increae the brightne of the input image while keeping the high contrat between object boundarie. The input image in the econd row of Fig. 2a ha only fifteen different grey level, thu it i difficult to oberve the image feature. The propoed algorithm linearly tranform the grey level a hown in Fig. 2c o that the image feature are eaily dicernable in Fig. 2b.

11 y grey level x grey level 25 2 y grey level x grey level a b c Fig. 2. output image. a A grey-level input image X; b The equalized output image Y uing the propoed algorithm; and c The data mapping between the input and One approach to extend the grey-cale contrat enhancement to colour image i to apply the method to their luminance component only and preerve the chrominance component. Another i to multiply the chrominance value with the ratio of their input and output luminance value to preerve the hue. The former approach i employed in thi paper where an input RGB image i tranformed to CIE L a b colour pace [1] and the luminance component L i proceed for contrat enhancement. The invere tranformation i then applied to obtain the contrat enhanced RGB image. III. EXPERIMENTAL RESULTS A dataet compriing tandard tet image from [21] [24] i ued to evaluate and compare the propoed algorithm with our implementation of GHE [1], BPHEME [13], FHSABP [14], CEBGA [16] and the weighted hitogram approximation of HMF [15]. GHE, BPHEME, FHSABP and CEBGA are free of parameter election but HMF require parameter tuning which i manually elected according the input tet image. It i worth to note that for FHSABP method exact hitogram pecification i ued [14] to achieve high degree of brightne preervation between input and output image. The tet image how wide variation in term of average image intenity and contrat. Thu they are uitable for meauring the trength of a contrat enhancement algorithm under different circumtance. An output image i conidered to have been enhanced over the input image if it enable the image detail to be better perceived. An aement of image enhancement i not an eay tak a an improved perception i difficult to quantify. Neverthele, in practice it i deirable to have both quantitative and ubjective aement. It i therefore neceary to etablih a bai which

12 11 define a good meaure of enhancement. We ue abolute mean brightne error AMBE [1], dicrete entropy DE [25], and edge baed contrat meaure EBCM [26] a quantitative meaure. For colour image, the contrat enhancement i quantified by computing thee meaure on their luminance channel L only. AMBE i the abolute difference between the mean value of an input image X and output image Y, i.e., AMBE X,Y = MB X MB Y, 32 where MB X and MB Y are the mean brightne value of X and Y, repectively. The lower the value of AMBE, the better i the brightne preervation. The dicrete entropy DE of an image X meaure it content, where a higher value indicate an image with richer detail. It i defined a 255 DE X = p x i log p x i, 33 i= where p x i i the probability of pixel intenity x i which i etimated from the normalized hitogram. The edge baed contrat meaure EBCM i baed on the obervation that the human perception mechanim are very enitive to contour or edge [26]. The grey level correponding to object frontier i obtained by computing the average value of the pixel grey level weighted by their edge value. The contrat c i, j for a pixel of an image X located at i, j i thu defined a c i, j = xi, j e i, j xi, j + e i, j, where the mean edge grey level i e i, j = k,l Ni,j / g k, lxk, l k,l Ni,j g k, l, N i, j i the et of all neighbouring pixel of pixel i, j, and g k, l i the edge value at pixel k, l. Without lo of generality we employ 3 3 neighbourhood, and g k, l i computed uing the magnitude of the gradient which i etimated from horizontal and vertical Sobel operator [1]. EBCM for image X i thu computed a the average contrat value, i.e., EBCM X = H W i=1 j=1 / c i, j H W. 34 It i expected that for an output image Y of an input image X, the contrat i improved when EBCM Y EBCM X. A. Qualitative Aement 1 Grey-Scale Image: Some contrat enhancement reult on grey-cale image are hown in Fig. 3, Fig. 4, Fig. 5 and Fig. 6. The mapping function ued are hown in Fig. 7 a-d, repectively. The input image in Fig. 3 how a firework diplay [21], and comprie very bright and dark object. GHE ha increaed the overall brightne of the image, but the increae in contrat i not ignificant and the wahout effect i apparent. Both the darker and brighter region become even brighter. Thi i verified by the mapping function in Fig. 7a which map input grey-level to output grey-level 15. BPHEME and FHSABP preerve the input image average brightne value of 18. Thi reult in the output image with very low brightne, and thu the contrat enhancement i not noticeable. The mapping function verifie thi obervation where the low output brightne and non-linear mapping from input to output are apparent. It i alo clear from the mapping function that BPHEME perform almot one to one mapping when it i compared with the mapping function of FHSABP. Thi i due to the fact that BPHEME i deigned to achieve brightne preervation a well a maximum

13 12 a b c d e f g Fig. 3. Contrat enhancement reult for image Firework: a original image; b GHE; c BPHEME; d FHSABP; e HMF; f CEBGA; and g propoed. a b c d e f g Fig. 4. Contrat enhancement reult for image Iland: a original image; b GHE; c BPHEME; d FHSABP; e HMF; f CEBGA; and g propoed. entropy. Thu, the mapping function of BPHEME achieve almot one-to-one mapping between input and output to guarantee the maximum entropy. The reult of HMF i viually pleaing, providing high contrat a well a a high dynamic range. However, there are two different park cluter due to the firework and the moke between park. HMF over enhance the brighter pixel of the park and the urrounding moke, o that the moke pixel are alo identified a park pixel. Thi over enhancement i repreented a a harp change in the mapping function. Furthermore, due to over enhancement of the brighter pixel, the park due to the firework cannot be clearly differentiated. The over enhancement i due to forming a hitogram from pixel with ignificant grey-level difference with their neighbour. The moke around the park ha imilar but lower grey-level value. Thu, mot of the moke pixel cannot be differentiated from the park pixel which reult in mapping them to the ame output grey-level a that of the park pixel. Due to the not harp image detail caued by the moke from firework, CEBGA can only improve the overall brightne of the image. Thi i verified by the mapping function which

14 13 a b c d e f g Fig. 5. Contrat enhancement reult for image City: a original image; b GHE; c BPHEME; d FHSABP; e HMF; f CEBGA; and g propoed. a b c d e f g Fig. 6. Contrat enhancement reult for image Girl: a original image; b GHE; c BPHEME; d FHSABP; e HMF; f CEBGA; and g propoed.

15 y grey level 15 1 y grey level 15 1 y grey level 15 1 y grey level x grey level x grey level x grey level x grey level a b c d Fig. 7. Mapping function of enhanced image: a Fig. 3; b Fig. 4; c Fig. 5; and d Fig. 6. Key: Green olid line - no-change mapping; black dah-dotted line - GHE; red olid line - BPHEME; red dah-dotted line - FHSABP; blue olid line - HMF; blue dah-dotted line - CEBGA; and black olid line - propoed algorithm normalized frequency normalized frequency normalized frequency normalized frequency grey level grey level grey level grey level a b c d normalized frequency normalized frequency.15.1 normalized frequency grey level grey level grey level e f g Fig. 8. propoed. Hitogram of original and enhanced image hown Fig. 6: a original image; b GHE; c BPHEME; d FHSABP; e HMF; f CEBGA; and g i almot parallel to the no-change mapping. However, in the propoed algorithm, the dynamic range of the input image i modelled with GMM, which make it poible to model the intenity value of park and moke differently. Input grey-level value are aigned to output grey-level value according to their repreentative Gauian component. The non-linear mapping i deigned to utilie the full dynamic range of the output image. Thu, the propoed algorithm improve the overall contrat while preerving the detail of the image. The input image of an iland in Fig. 4 [22] ha average brightne value of 125. The reult obtained by the different algorithm are imilar a verified by the imilar mapping function in Fig. 7b. BPHEME and FHSABP behave exactly the ame way a GHE when the average brightne value i [13], [14]. Since the average brightne value of the input image i very cloe to 127.5, BPHEME and FHSABP algorithm obtain the imilar target hitogram. The light difference of FHSABP from BPHEME i due to exact hitogram pecification ued in FHSABP. The reult of HMF and CEBGA are alo a match becaue both algorithm employ imilar edge information. Where the ky and ea converge, GHE, BPHEME and FHSABP provide a higher contrat than HMF and CEBGA. The propoed algorithm provide a contrat which i neither too high nor too low.

16 15 a b c d e f g Fig. 9. Contrat enhancement reult for image Plane: a original image; b GHE; c BPHEME; d FHSABP; e HMF; f CEBGA; and g propoed. The input image in Fig. 5 how an aerial view of a junction in a city [23] with an average brightne value of 181 which i too high for recognizing the different object. GHE increae the overall contrat of the image ignificantly, but the image look darker a i verified by it mapping function in Fig. 7c. The contrat improvement obtained by uing BPHEME, FHSABP, HMF and CEBGA are very light. HMF fail to provide an improvement due to weak edge information. The propoed algorithm, on the other hand, doe not darken the image and produce ufficient contrat for the different object to be recognized. The image Girl hown in Fig. 6 conit of challenging condition for an enhancement algorithm: very bright and dark object, and average brightne value of 139 a can be verified from the hitogram of the Girl image hown in Fig. 8a. The hitogram reveal that mot of the grey level of the input image are accumulated around grey level 144. Meanwhile, there are uniform grey level ditribution on the left and right ide of the harp peak in the middle. Becaue average brightne value of the input image i near to 127.5, GHE, BPHEME, and FHSABP perform very imilar. Thi can be verified from the viual reult, mapping function hown Fig. 7d, and hitogram demontrated in Fig. 8b-d. A can be een from the hitogram, the output hitogram fail to achieve mooth ditribution between high and low value of grey level. Thu, the enhancement reult of GHE, BPHEME, and FHSABP are viually unpleaing. Meanwhile, the output hitogram of HMF achieve moother ditribution in between low and high grey value, thu HMF achieve more natural looking output a hown in Fig. 8e when it i compared with that of GHE, BPHEME, and FHSABP. CEBGA produce natural looking output image, however the overall enhancement i not ignificant. A can be een from mapping function and hitogram in Fig. 7d and Fig. 8f, CEBGA produce minor alteration on the input image. The reult of the propoed algorithm i hown Fig. 6g. The propoed algorithm preerve the overall ditribution hape, meanwhile it achieve to reditribute the grey level of the input image within the dynamic range without detroying natural look of the enhanced image. Although it lightly darken the hair of woman, till the overall natural look i not detroyed while the perceived contrat i improved ignificantly. 2 Colour Image: In Fig. 9, the over enhancement provided by GHE, BPHEME and FHSABP whiten ome area of the concrete ground. HMF and CEBGA provide imilar reult where the light contrat enhancement with repect to the input image i apparent, wherea the propoed algorithm enhance the contrat and the average brightne to improve the overall image quality. In Fig. 1, GHE, BPHEME, FHSABP and HMF caue part of the ky to be too bright. CEBGA and the

17 16 a b c d e f g Fig. 1. Contrat enhancement reult for image Ruin: a original image; b GHE; c BPHEME; d FHSABP; e HMF; f CEBGA; and g propoed. a b c d e f g Fig. 11. Contrat enhancement reult for image Hat: a original image; b GHE; c BPHEME; d FHSABP; e HMF; f CEBGA; and g propoed. propoed algorithm improve the overall contrat coniderably and maintain high viual quality. In Fig. 11, GHE, BPHEME and FHSABP reult in lo of detail in the cloud and on the top of the yellow hat, wherea HMF and CEBGA retain the detail while increaing the contrat. However, the contrat between the right ide of the wall and the ky i not ufficiently high. The propoed algorithm keep the detail while improving the overall contrat. Finally, in Fig. 12, GHE make the tone around the window and the pink flower very bright; hence, the enhanced image ha an unnatural look. Although BPHEME and FHSABP perform better than GHE, it till doe not remove thi effect completely. Thi effect i reduced by HMF, CEBGA and the propoed algorithm. Alo, the colour of window and wall are better differentiated in the reult of the propoed algorithm. Turgay: Pleae kip thi paragraph a it need new evaluation core. In order to aign a viual aement core to each algorithm for each enhanced image, ubjective perceived quality tet are performed by a group of ten ubject on the reult of the five algorithm for the eight tet image. For each tet on a tet image, a ubject i hown two image: the tet image and the image proceed by one of the five algorithm. The ubject i then aked to core the quality of the proceed image

18 17 a b c d e f g Fig. 12. Contrat enhancement reult for image Window: a original image; b GHE; c BPHEME; d FHSABP; e HMF; f CEBGA; and g propoed. TABLE II AVERAGE OF SUBJECTIVE QUALITY TEST SCORES. Image GHE FHSABP HMF CEBGA Prop. Firework Bad Bad Good Good Good Iland Bad Bad Good Good Good City Good Bad Bad Bad Good Wolf Bad Bad Bad Good Good Plane Bad Bad Good Good Good Ruin Bad Bad Bad Good Good Hat Good Good Bad Good Good Window Good Good Bad Bad Good by aigning a fuzzy core of Bad for weak enhancement, and Good for viually pleaing enhancement. The tet on the ame input image i repeated for the proceed image generated by the other four algorithm. For each proceed image of an algorithm, we average the core from the ten ubject. In the averaging operation, when the number of Good core are higher than Bad core, then the proceed image i deemed a Good, and vice vera. The viual aement core a hown in Table II validate our ubjective evaluation that the propoed algorithm provide good viual quality enhancement. B. Quantitative Aement The quantitative meaure AMBE, DE, and EBCM may fail to provide enhancement meaure which are parallel with perceived image quality. For intance for Girl image, GHE, BPHEME, FHSABP, HMF, CEBGA, and propoed method produce AMBE value of 5.6, 5.5, 5.5, 1.3, 1.6, and 12.9, repectively. CEBGA achieve the bet in term of brightne preervation where the propoed method perform the wort. The Girl image ha DE value of 3.87, meanwhile GHE, BPHEME, FHSABP, HMF, CEBGA, and propoed method produce DE value of 3.65, 3.65, 3.65, 3.7, 3.45, and 3.81, repectively. The propoed method achieve the bet DE value. The Girl image ha very low contrat meaure EBCM value of.8. GHE, BPHEME, FHSABP, HMF, CEBGA, and propoed method reult in EBCM value of.23,.22,.22,.17,.11, and.12. All method achieve to produce higher value of EBCM with repect to the EBCM value of original Girl image. GHE, BPHEME, and FHSABP produce the highet value of EBCM, however perceived viual quality i not natural. Although there i a

19 18 2 GHE 2 BPHEME 2 FHSABP 5.5 GHE 5.5 BPHEME 5.5 FHSABP.5 GHE.5 BPHEME.5 FHSABP MB 15 1 MB 15 1 MB 15 1 DE DE DE EBCM.3.2 EBCM.3.2 EBCM index index index HMF 2 CEBGA 2 Propoed 5.5 HMF 5.5 CEBGA 5.5 Propoed.5 HMF.5 CEBGA.5 Propoed MB 15 1 MB 15 1 MB 15 1 DE DE DE EBCM.3.2 EBCM.3.2 EBCM index index index a b c Fig. 13. Quantitative performance reult on 3 image from Berkeley dataet [24]: a reult for MB; b reult for DE; and c reult for EBCM. The reference meaurement from the original image i hown in red colour, meanwhile the meaurement from the proceed image reulted from different algorithm are hown in black colour. correlation between between EBCM and perceived contrat enhancement, it doe not alway mean that the higher value of EBCM mean better perceived contrat enhancement. In order to tet algorithm performance quantitatively in term of brightne and entropy preervation a well a contrat improvement, they are applied on 3 tet image of Berkeley image dataet [24]. MB, DE, and EBCM value are computed from original and proceed image. In reported reult, the meaurement value from the original image are orted in acending order and the image are indexed accordingly. The quantitative reult for MB, DE, and EBCM are hown in Fig. 13a, b, and c, repectively. The MB value in Fig. 13a how that except GHE all algorithm follow general trend in the mean brightne value, i.e.when the mean brightne value of the original image i low o do the output image, and vice vera. GHE conitently map the mean brightne value of the output image very cloe to which i the mid value of the 8-bit grey-level dynamic range. The average of AMBE reulted from GHE over dataet i Meanwhile, BPHEME and FHSABP achieve the bet brightne preervation a can be een from the plot. Both algorithm produce very imilar reult for the whole dataet. Thi i mainly due to the target hitogram of BPHEME and FHSABP are imilar to each other. The average of AMBE reulted from BPHEME and FHSABP over dataet are 1.3 and 1.28, repectively. On the average, the propoed method perform better than HMF and HMF perform better than CEBGA in term of brightne preervation. The average of AMBE reulted from HMF, CEBGA, and the propoed method are 1.7, 12.23, and 8.8, repectively. GHE, BPHEME, and FHSABP perform very imilar reult in term of DE on Berkeley dataet a hown Fig. 13b. The average abolute dicrete entropy difference between the input and out image over dataet for GHE, BPHEME, and FHSABP are.12,.12, and.11, repectively. Fig. 13b alo how that CEBGA perform the wort with the average abolute dicrete entropy difference of.38. Meanwhile, HMF and the propoed method achieve good performance in term of entropy. The average abolute dicrete entropy difference between the input and out image over dataet for HMF and the propoed method are.5, and.4, repectively. Since the entropy i related with the overall image content, one can ay that for Berkeley dataet the propoed method can preerve the overall content of the image while improving it contrat. The EBCM meaure are hown in Fig. 13c. Although high EBCM doe not alway mean a good and natural image enhancement, however it i, at leat, expected that the output image EBCM value i higher than that of the input image. Out of 3 image in the dataet, GHE, BPHEME, FHSABP, HMF, CEBGA, and propoed method produce 294, 3, 296, 293,

20 19 a b c Fig. 14. High dynamic range compreion reult. a Original image. Proceed image obtained uing: b [27]; and c propoed algorithm. 286, and 3 output image, repectively, which have higher than or equal to EBCM value with that of the input image. Meanwhile, the average abolute EBCM difference between the input and out image over dataet for GHE, BPHEME, FHSABP, HMF, CEBGA, and propoed method are.652,.63,.573,.361,.278, and.366, repectively. A expected, GHE provide the highet contrat improvement in term of EBCM. Meanwhile, BPHEME and FHSABP perform very imilar and lightly wore than GHE. BPHEME perform better than FHSABP, becaue of lowpa filtering employed in exact hitogram pecification ued in FHSABP. Meanwhile, the propoed method and HMF perform imilar and CEBGA provide the wort performance. It i worth to note that only two algorithm, BPHEME and the propoed method, achieve to make EBCM improvement with repect to the input image. C. Application to High Dynamic Range Compreion The propoed algorithm can be applied for rendering high dynamic range HDR image on conventional diplay. Thu, we compare ome of our reult with thoe of the tate-of-the-art method propoed by Fattal et al. [27]. In the Fattal et al. method, the gradient field of the luminance image i manipulated by attenuating the magnitude of large gradient. A low dynamic range image i then obtained by olving a Poion equation on the modified gradient field. The reult in [27], a few of which are in Fig. 14, how that the method i capable of dratic dynamic range compreion, while preerving fine detail and avoiding common artefact uch a halo, gradient reveral, or lo of local contrat. Fig. 14 alo how that the propoed algorithm produce comparable reult. It i worth noting that our reult are obtained without any parameter tuning. IV. CONCLUSIONS In thi paper, we propoed an automatic image enhancement algorithm which employ Gauian mixture modelling of an input image to perform non-linear data mapping for generating viually pleaing enhancement on different type of image. Performance comparion with tate-of-the-art technique how that the propoed algorithm can achieve good enough image equalization even under divere illumination condition. The propoed algorithm can be applied to both grey-level and colour

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