Brightness Preserving Histogram Equalization with Maximum Entropy: A Variational Perspective Chao Wang and Zhongfu Ye

Transcription

1 326 IEEE Tansactions on Consume Electonics, Vol. 5, No. 4, NOVEMBER 25 Bightness Peseving Histogam Equalization with Maximum Entopy: A Vaiational Pespective Chao Wang and Zhongfu Ye Abstact Histogam equalization (HE) is a simple and effective image enhancing technique, howeve, it tends to change the mean bightness of the image to the middle level of the pemitted ange, and hence is not vey suitable fo consume electonic poducts, whee peseving the oiginal bightness is essential to avoid annoying atifacts. This pape poposes a novel extension of histogam equalization, actually histogam specification, to ovecome such dawback as HE. To maximize the entopy is the essential idea of HE to make the histogam as flat as possible. Following that, the essence of the poposed algoithm, named Bightness Peseving Histogam Equalization with Maximum Entopy (BPHEME), ties to find, by the vaiational appoach, the taget histogam that maximizes the entopy, unde the constaints that the mean bightness is fixed, then tansfoms the oiginal histogam to that taget one using histogam specification. Compaing to the existing methods including HE, Bightness peseving Bi-Histogam Equalization (BBHE), equal aea Dualistic Sub-Image Histogam Equalization (DSIHE), and Minimum Mean Bightness Eo Bi-Histogam Equalization (MMBEBHE), expeimental esults show that BPHEME can not only enhance the image effectively, but also peseve the oiginal bightness quite well, so that it is possible to be utilized in consume electonic poducts. Index Tems Image enhancement, histogam equalization, histogam specification, maximum entopy, vaiational appoach, mean bightness peseving. I. INTRODUCTION Histogam is defined as the statistic pobability distibution of each gay level in a digital image []. Histogam equalization (HE) is one of the well-known methods fo enhancing the contast of given images, making the esult image have a unifom distibution of the gay levels []. It flattens and stetches the dynamic ange of the image s histogam and esults in oveall contast impovement. HE has been widely applied when the image needs enhancement, such as medical images enhancement [2]. Howeve, in consume electonics such as TV, HE is aely employed because it may significantly change the bightness of an input image and cause C. Wang is with the Institute of Statistical Signal Pocessing, Depatment of Electonic Engineeing and Infomation Science, Univesity of Science and Technology of China (USTC), Hefei, Anhui, 2327, P.R.China ( chaowang@mail.ustc.edu.cn). Z. Ye is the coesponding autho with the Institute of Statistical Signal Pocessing, Depatment of Electonic Engineeing and Infomation Science, Univesity of Science and Technology of China (USTC), Hefei, Anhui, 2327, P.R.China, and he is also with the National Laboatoy of Patten Recognition (NLPR), Institute of Automation, Chinese Academy of Sciences, Beijing, 8, P.R.China. ( yezf@ustc.edu.cn). Contibuted Pape Manuscipt eceived Octobe 9, /5/$2. 25 IEEE undesiable atifacts. In theoy, the mean bightness of its output image is always the middle gay level egadless of the input mean, because the desied histogam is flat. This is not a desiable popety in some applications whee bightness pesevation is necessay. Bightness peseving Bi-Histogam Equalization (BBHE) has been poposed to ovecome that poblem [3]. BBHE fist sepaates the input image s histogam into two by its mean, and thus two non-ovelapped anges of the histogam ae obtained. Next, it equalizes the two sub-histogams independently. It has been analyzed that BBHE can peseve the oiginal bightness to a cetain extent when the input histogam has a quasi-symmetical distibution aound its mean. Late, equal aea Dualistic Sub-Image Histogam Equalization (DSIHE) has been poposed, it claims that if the sepaating level of histogam is the median of the input image s bightness, it will yield the maximum entopy afte two independent sub-equalizations [4]. DSIHE will change the bightness to the middle level between the median level and the middle one of the input image. Nevetheless, neithe BBHE no DSIHE could peseve the mean bightness. Then Minimum Mean Bightness Eo Bi- Histogam Equalization (MMBEBHE) is poposed to peseve the bightness optimally [5]. MMBEBHE is to pefom the sepaation based on the theshold level, which would yield minimum diffeence between input and output mean. This theshold level is essentially chosen by enumeation. Anothe scheme, named Recusive Mean-Sepaate Histogam Equalization (RMSHE), has been poposed to peseve the bightness [6]. RMSHE uses the BBHE iteatively. Fist RMSHE sepaates the input histogam into two pieces, by the mean. Then, to each piece, it uses this opeation many times to geneate 2 n -pieces histogams. Finally, it equalizes each histogam piece independently. It is claimed theoetically that when the iteation level n gows lage, the output mean conveges to the input mean, and thus yields good bightness pesevation. Actually, when n gows to infinite, the output histogam is exactly the input histogam, and thus the input image will be output without any enhancement at all. In the consume electonics such as TV, the pesevation of bightness is highly demanded. The afoementioned algoithms (HE, BBHE, DSIHE, MMBEBHE and RMSHE) peseve the bightness to some extent, howeve, they do not meet that desiable popety quite well. In this pape, a novel enhancement method is poposed which can yield the optimal equalization in the sense of entopy maximization, unde the constaint of the mean

2 C. Wang and Z. Ye: Bightness Peseving Histogam Equalization with Maximum Entopy: A Vaiational Pespective 327 bightness, called Bightness Peseving Histogam Equalization with Maximum Entopy (BPHEME). BPHEME, togethe with the afoementioned algoithms, is essentially a kind of histogam specification [] in geneal, except that diffeent ideal histogams ae employed in diffeent algoithms. In the next section, histogam specification will be eviewed, and HE, BBHE, DSIHE, MMBEBHE will be quickly intoduced as special cases of histogam specification. In Section III, the poposed algoithm, BPHEME, will be pesented, which is ideal in the sense of maximum entopy with an invaiant mean bightness. Section IV will list goups of expeimental esults to claim the pefomance of BPHEME, compaing with the ones of HE, BBHE, DSIHE and MMBEBHE. Section V makes some concluding emaks. whee S is the suppot set of input gay level. (4) is called SML (Single Mapping Law) in [7], and a geneal mapping law (GML) is poposed in [7], to impove the accuacy of HS. In this pape, we use SML, i.e., eq.(4), fo its simplicity to ealize. B. Special Cases of Histogam Specification HS can appoximately yield a desiable histogam, howeve what is a desiable one? It is a poblem deseving discussion. The afoementioned algoithms such as HE, BBHE, DSIHE and MMBEBHE, ae all histogam specification essentially, see Fig. fo the geneal shape of thei desiable histogams. Without loss of geneality, we assume that has been nomalized to the inteval S=[,], with = epesenting black and = epesenting white. II. HISTOGRAM SPECIFICATION When histogam tansfomation method is consideed, many applications equie a desiable shape of histogam. We want to geneate a pocessed image that has the specified desiable histogam, which is called Histogam Specification (HS) o histogam matching. Since a histogam can be viewed as the pobability density function of the vaiable fo gay levels, we intoduce the histogam pocessing methods fom a continuous view in the following pats. Fist this section coves the details of HS fom a geneal pespective, actually a epint of coesponding pat in []. Then we egad HE, BBHE, DSIHE and MMBEBHE as the special cases of HS, give thei coesponding desiable histogams. A. Geneal Histogam Specification Let the gay levels and z be continuous andom vaiables with coesponding continuous pobability density functions (PDFs) denoted by p () and p z (z). In this notation, and z denote the gay levels of the input and output (pocessed) images, espectively. We can estimate p () fom the given input image, while p z (z) is the specified PDF that we hope the output image to have. Let s be a andom vaiable with the popety s = T ( ) = p ( w) dw () whee T is essentially the cumulative opeato and s is the cumulative histogam, i.e., the distibution function of the vaiable. Suppose next that we define a andom vaiable z with the popety G( z) = z p z ( t) dt = s (2) It then follows fom these two equations that G(z)=T() and, theefoe, that z must satisfy the condition z = G ( s) = G [ T ( )] (3) That is the theoetical case. In application, we would like to map each given input gay level to an output one, z, which yields the closest distance between the input and output cumulative histogam, i.e., z( ) = ag min z T ( ) G( z), S (4) Fig. Desied histogam (PDF) of HE, BBHE, DSIHE and MMBEBHE. x is the sepaating gay level fo BBHE, DSIHE and MMBEBHE. ) Histogam Equalization We focus ou attention on the following tansfomation z = T(), (5) that maps to a level z fo evey pixel value in the oiginal image. T() satisfies the following fundamental conditions: T() is single-valued and monotonically inceasing in the inteval ; T() fo. If we choose T ( ) = p ( w) dw like (), it is not difficult to find that the PDF of the output gay level z follows a unifom distibution anging fom to. O, we may say that the detemination fo the vaiable z at a given pixel will povide us the maximum infomation, because z equals to any gay level with equal pobability, thus, it contains the most uncetainty. HE can be egaded as a special case of HS when we choose the taget output histogam as a unifom one, p z (z)=, z. 2) Bightness peseving Bi-Histogam Equalization As mentioned befoe, the desied histogam of HE is unifom, and thus its desied mean bightness is.5, the middle level of S. So the mean bightness can not be peseved by HE. BBHE [3] fist sepaates the input histogam into two pats based on its mean bightness, and then equalizes the two subhistogams independently. Thus thee is a step in the histogam

3 328 IEEE Tansactions on Consume Electonics, Vol. 5, No. 4, NOVEMBER 25 at the gay level x B = μ = p ( ) d. The desiable histogam of BBHE is a piecewise constant one as xb p ( ) d, z < xb x B p z ( z) = (6) p ( ) d, xb z x x B B It is epoted that BBHE can peseve the mean bightness, if the input histogam has a symmetical distibution aound its mean. Howeve, that assumption is not the fact in many cases, and thus BBHE can not always peseve the bightness well. 3) Dualistic Sub-Image Histogam Equalization DSIHE [4] is vey simila to BBHE, except that the sepaating point x D is selected as the median gay level of the input image, i.e., x D satisfies xd p ( ) d =.5 (7) Fo the applicable case, it may be modified as x x D = ag min x p ( ) d.5 (8) The pupose of DSIHE is to find a sepaating point, based on which the desied histogam (PDF) can obtain a maximum entopy. And it has been poved that the bightness of the output image is z =.5 (x D +.5) (9) It is clea that DSIHE always pulls the output bightness towad the input middle level fom the input median level. 4) RMSHE and MMBEBHE RMSHE [6] is to ecusively implement BBHE to the histogam. Obviously, ecusive sepaation on the histogam will divide the histogam into vey small pieces. If the ecusive level tends to infinite, equalization to each small piece will not change the whole histogam, and thus can peseve the mean bightness. So an infinite ecusive level esults in the same output image as the input one. BBHE and DSIHE belong to two-piece-sepaating histogam equalization algoithm, and so does HE in a boad sense. MMBEBHE [5] is anothe modification to two-piecesepaating histogam equalization algoithm. MMBEBHE diectly consides the gay level, based on which the input histogam is sepaated. It chooses the sepaating level that poduces the minimum absolute mean bightness eo (AMBE) to the oiginal image. It eally can peseve the bightness quite well. It is essentially an enumeation method though the authos of [5] povided a ecusive method to compute AMBE. Futhe moe, this method fixed the type of desiable histogam to a piecewise constant function with a step, this choice does not seem to have a convincing theoetical stand. III. BRIGHTNESS PRESERVING HISTOGRAM EQUALIZATION WITH MAXIMUM ENTROPY Let us e-focus ou attention on HE. As mentioned in section II-B., HE is to make the output histogam as flat as possible. A supeficial eason fo HE elies on that a flat histogam makes all the gay levels unifom, and thus will cause a moe comfotable peception. A futhe compehension is that a unifom distibution limited to a given ange gives the maximum infomation, measued by entopy! So we would like to find an ideal histogam (PDF) as a taget one to pefom the histogam specification. That ideal histogam peseves the mean bightness of the input image, and has the maximum entopy. That is the basic thought of ou poposed method. In this section we quickly intoduce the entopy of a continuous vaiable fist, then we give the coesponding functional extemum poblem and solve it. A. Peliminay Infomation Theoy [8] Definition : Let X be a andom vaiable with cumulative distibution function F(x)=P(Xx). If F(x) is continuous, the andom vaiable is said to be continuous. Let f(x)=f (x) when the deivative is defined. If + f ( x) dx =, then f(x) is called the pobability density function (PDF) fo X. The set whee f(x)> is called the suppot set of X. Definition 2: The diffeential entopy h(x) of a continuous andom vaiable X with a density f(x) is defined as h ( X ) = f ( x) log f ( x) dx () S whee S is the suppot set of the andom vaiable. The diffeential entopy is a convex function ove a convex set, and depends only on the pobability density of the andom vaiable, hence the diffeential entopy is sometimes witten as h(f) athe than h(x). h(x) chaacteizes the andomness of the vaiable X, the moe andom (chaotic) the vaiable is, the moe infomation it may povide, the moe the diffeential entopy is. Since we will not compae the andomness between continuous and discete vaiables, we will not diffeentiate between a continuous vaiable s diffeential entopy and a discete vaiable s entopy. So in the following pats, h(x) is also named as the continuous vaiable s entopy. B. BPHEME: Bightness Peseving Histogam Equalization with Maximum Entopy Afte the entopy of a continuous vaiable has been defined, we can etun to what we ae discussing. Since the pesevation of the mean bightness is of high demands in consume electonics such as TV, we may find an enhancement method with the mean bightness being constained. As has been said in the peface of this section, let an image be enhanced optimally means that the histogam (PDF) may have the most entopy. Thus an optimal bightness peseving enhancement method using histogam tansfomation may be to maximize the taget histogam s entopy unde the constaints of bightness. Mathematically speaking, we want to maximize h(f) ove all pobability densities f subject to some constaints:

4 C. Wang and Z. Ye: Bightness Peseving Histogam Equalization with Maximum Entopy: A Vaiational Pespective 329, s S max f { log ds},s.t. ds = () S S = s ds μ S whee S=[,] is the nomalized suppot set of gay level, and μ = p ( ) d is the mean bightness of the input image. Because a whole white o a whole black image is not the input image in geneal, we assume μ (,). Since h(f) has the convexity, we fom the functional J ( f ) = ln ds + λ [ ds ] + λ [ s ds μ ] 2 (2) whee, 2 ae the undetemined paametes, and hee we specify the logaithmical function as the natue one. Using the vaiational appoach, we can diffeentiate with espect to f(s), the s th component of f, to obtain J = ln + λ + λ 2 s =, s S (3) f ( s) thus λ s = e e, s S (4) Using the second and thid constaints in (), we may find,if μ =.5 s = λ e s S (5) 2,if μ (,.5) (.5,) e In (5), 2 can be detemined. When.5, 2 is the solution of equation e e + μ = (6) ( e ) and 2 = when =.5. It is a single-valued function of with espect to 2, as shown in Fig.2. So given a μ (,), thee exists the only 2 that geneates the mean bightness as. s c( s) = f ( t) dt s,if μ =.5 (7) s = e s S,if μ (,.5) (.5,) e Had f(s) o c(s) been given, we can specify the input image s histogam unde the instuction of f(s) o c(s), using histogam specification intoduced in section II-A. In applications, eal-time pocessing is desied. It had bette not cost too much time in solving (6). Fotunately, ( 2 ) is monotonically inceasing, and we can pe-list a table that designates a 2 to each. It is not necessay to discetize too delicately, since the following opeation HS is implemented to a discete case, and is an appoximation in essence. What s moe, Fig.2 shows the elation ( 2 ) is antisymmetical about (, 2 )=(,.5), so the memoy cost can be futhe educed by half. The pe-listing table will not cost too much computing time and memoy at all. C. Discussions on Discete Vesion of BPHEME The BPHEME scheme intoduced in section III-B can be easily implemented to the discete image. Howeve, in this section we must claify something elated. The expession with maximum entopy is only suitable fo the continuous case. The histogam mapping methods, including all the afoementioned methods, ae likely to mege some neighboing gay levels into a single one in the output image, and thus the pocessing of histogam mapping will decease (o at most peseve) the discete entopy of the histogam undoubtedly. So in a discete case of BPHEME, theoetically speaking, BPHEME may not maximize the discete entopy, the mean bightness peseving output image with maximum discete entopy is cetainly the input one, and thus the enhancement makes no sense. Though BPHEME does not maximize the discete entopy, the continuous vaiable s diffeential entopy defined by () can still be maximized by BPHEME. Fig.2 Relation between the mean bightness and the paamete 2. Since ( 2 ) is monotonically inceasing, it is easy to use a Newton iteation [9] to solve the equation fo a numeical solution. Thus we have the cumulative histogam, o cumulative distibution function, c(s) as following: Fig.3 Equal discete entopy may cause diffeent image quality. The fomulations in section III-B, togethe with all HS methods, ae all based on the assumption that the gay level be continuous and the histogam line can split into moe than one, thus we can obtain exactly the same histogam as the desiable

5 33 IEEE Tansactions on Consume Electonics, Vol. 5, No. 4, NOVEMBER 25 oigin BPHEME HE Fig.4 Enhancement fo bottle based on HE, BBHE, DSIHE, MMBEBHE and BPHEME. Fig.5 Histogams of each enhanced image bottle based on afoementioned algoithms in Fig.4. The line plot in each sub-figue is the desiable histogam of BPHEME. one. The whole contast is much moe intepetable using continuous vaiable s diffeential entopy than using the discete vaiable entopy, as shown in Fig.3. The input image has the gay levels in a small inteval, and the output image is to stetch the gay levels in a lage inteval. Thus they have the same discete entopy. Howeve, the output is obviously enhanced due to its lage dynamic ange than the input. In a continuous view, see the dotted line in Fig.3, the histogam of the output image is moe flat and the gay level of a pixel is moe uncetain than the input image, so the diffeential entopy of the output is lage than the input, and thus it may be an enhanced vesion of the input image. IV. EXPERIMENTAL RESULTS A. Expeiments of HE, BBHE, DSIHE MMBEBHE and BPHEME In this section, we pesent some expeimental esults of ou

6 C. Wang and Z. Ye: Bightness Peseving Histogam Equalization with Maximum Entopy: A Vaiational Pespective 33 poposed method, togethe with HE, BBHE, DSIHE and MMBEBHE fo compaison. The souce image fo the fist expeiment is bottle. The souce image, togethe with the esults based on HE, BBHE, DSIHE, MMBEBHE and BPHEME, is shown in Fig.4. Fom Fig.4, we can see that the esults based on HE, BBHE and DSIHE seem to have lage contast than the ones based on MMBEBHE and BPHEME. Howeve, the mean bightness of HE, BBHE and DSIHE deviate vey much fom the oiginal image s. The mean bightness of bottle, BPHEME, HE, BBHE, DSIHE and MMBEBHE ae 77.27, 77.93, 28.37, 93.23, 96.7 and 8.4 espectively, as listed in Table I. BPHEME peseves the mean bightness vey well. The histogam of the six images ae shown in Fig.5, and the desiable histogam of BPHEME is supeimposed on each one denoted by the line plot. Fig.5 shows clealy that BPHEME is in well accodance with the desiable one, and MMBEBHE can also peseve the mean bightness well to some extent. Diffeent output histogams match the desied one of BPHEME diffeently, and esult in diffeent bightness pesevation in diffeent algoithms. Fo the afoementioned five algoithms, we cut the same pat of the esults, togethe with the oiginal image, as shown in Fig.6, which is focusing on the the text of the bottle mak. We can see that the last two chaactes NE in HE, BBHE and DSIHE ae not vey clea to ecognize, howeve the same chaactes can be easily ecognized in the esults based on MMBEBHE and the poposed BPHEME. oigin BPHEME HE Fig.6 Coesponding piece of the images infig.4, showing the text on the bottle. Moe examples ae pesented. The souce images ae F6, Einstein, house and gil. These fou goups of expeiments ae shown as Fig.7-Fig.. The five algoithms ae implemented to these souces espectively. The mean bightness of all these esults ae computed and listed in Table I. As to each souce image, it can be found that BPHEME eaches the minimum absolute mean bightness eo (AMBE), and MMBEBHE is the unneup. AMBE is defined as AMBE(X,R)= E(X)-E(R) (8) whee E(.) denotes the expectation (i.e., mean), R is the efeenced image, and hee R epesents the oiginal image. We define the mean AMBE (MAMBE) as N MAMBE = AMBE( X i, R ) (9) i N = Since we have five souce images hee, N=5. MAMBE geneally epesents the bightness peseving ability of an algoithm. Using this measuement, the MAMBE of HE, BBHE, DSIHE, MMBEBHE and BPHEME ae espectively 39.75, 4.95, 7.24,.6 and.33. That is to say, BPHEME i oigin BPHEME HE Fig.7 Enhancement fo F6 based on HE, BBHE, DSIHE, MMBEBHE and BPHEME.

7 332 IEEE Tansactions on Consume Electonics, Vol. 5, No. 4, NOVEMBER 25 oigin BPHEME HE Fig.8 Enhancement fo Einstein based on HE, BBHE, DSIHE, MMBEBHE and BPHEME. oigin BPHEME HE Fig.9 Enhancement fo house based on HE, BBHE, DSIHE, MMBEBHE and BPHEME. outpefoms the existing histogam pocessing methods in the sense that BPHEME can peseve the mean bightness quite well, which is vey suitable fo consume electonics such as TV. In the view of visual quality, fom Fig.4, Fig.7-Fig., it is shown that BPHEME does enhance the oiginal image, and it is at least not wose than the existing methods.

8 C. Wang and Z. Ye: Bightness Peseving Histogam Equalization with Maximum Entopy: A Vaiational Pespective 333 oigin BPHEME HE Fig. Enhancement fo gil based on HE, BBHE, DSIHE, MMBEBHE and BPHEME. TABLE I MEAN BRIGHTNESS OF THE EXPERIMENTAL RESULTS Image Oigin BPHEME HE Bottle F Einstein house gil Though the discete entopy does not necessaily epesent the enhancement quality, it can depict the ichness of details to some extent. In Table II, we pesents the discete entopy of each esults, which is defined as 255 ENT( q ) = q( i) log q( i) (bits) (2) i= whee q(i) is the nomalized pobability of the gay level i. In Table II, we can see that fo each souce image, the esultant discete entopy based on BPHEME is equal to, o even slightly highe than MMBEBHE. When the souce images Einstein and house ae consideed, the discete entopy of BPHEME falls behind HE, BBHE and DSIHE, howeve, 2 TABLE II DISCRETE ENTROPY OF THE EXPERIMENTAL RESULTS (:BITS) Image Oigin BPHEME HE Bottle F Einstein house gil fo these two test images, HE, BBHE and DSIHE lost the mean bightness seiously. B. Potential Application of BPHEME The theoy and expeimental esults show that BPHEME peseves the mean bightness exactly, the mino eo comes fom the discetizaion eo and the non-cleavability of the histogam line. In applications, maybe thee is some theshold of mean bightness change fo people s peception to bea. That is, when enhancement is implemented, the mean bightness is pemitted to vay in a small inteval [ -, +]. This poblem can also be solved using BPHEME with mino modification. We modify the taget mean bightness as

9 334 IEEE Tansactions on Consume Electonics, Vol. 5, No. 4, NOVEMBER 25 μ = ag min μ.5 = μ + δ μ δ { μ.5 : μ δ μ μ + δ},if,if,if.5 [ μ δ, μ + δ ] μ + δ <.5 μ δ >.5 Fig. Relation between the entopy and the paamete 2. (2) To choose the taget mean bightness like (2) is easonable. Because the mean bightness is monotonically inceasing with 2 (see Fig.2), and the elation between entopy and 2 is shown as Fig., the taget bightness chosen as eq.(2) can ensue the maximum output entopy when mean bightness is constained within a small inteval. V. CONCLUSION In this pape, we pesent a novel case of histogam specification, which can peseve the mean bightness with maximum entopy (BPHEME), in a continuous view. BPHEME ties, using vaiational appoach, to find the optimal histogam, which has the maximum diffeential entopy unde the mean bightness constaint, and then implements the histogam specification unde the instuction of that desied histogam. Expeimental esults show that BPHEME can enhance the image quite well when peseving the mean bightness, which is vey suitable fo consume electonics such as TV. BPHEME may find its potential applications consideing the toleant theshold fo the human visual systems. ACKNOWLEDGEMENT The authos would like to thank M. Liangliang Cao (Chinese Univesity of Hong Kong) fo occasional and useful discussion. REFERENCES [] R.C.Gonzalez and R.E.Woods, Digital Image Pocessing, 2 nd Edition, Pentice Hall, 22. [2] J.B.Zimmeman, S.M.Pize, E.V.Staab, J.R.Pey, W.McCatney and B.C.Benton, An evaluation of the effectiveness of adaptive histogam equalization fo contast enhancement, IEEE Tansactions on Medical Imaging, Vol.7, No.4, pp:34-32, 988. [3] Y.-T Kim, Contast Enhancement Using Bightness Peseving Bi- Histogam Equalization, IEEE Tansactions on Consume Electonics, Vol.43, No., pp:-8, 997. [4] Y.Wang, Q.Chen and B.M.Zhang, Image Enhancement based on Equal Aea Dualistic Sub-image Histogam Equalization Method, IEEE Tansactions on Consume Electonics, Vol.45, No., pp:68-75, 999. [5] S.-D.Chen and A.R.Ramli, Minimum Mean Bightness Eo Bi- Histogam Equalization in Contast Enhancement, IEEE Tansactions on Consume Electonics, Vol.49, No.4, pp:3-39, 23. [6] S.-D.Chen and A.R.Ramli, Contast Enhancement using Recusive Mean-Sepaate Histogam Equalization fo Scalable Bightness Pesevation, IEEE Tansactions on Consume Electonics, Vol.49, No.4, pp:3-39, 23. [7] Y.J.Zhang, Impoving the Accuacy of Diect Histogam Specification, Electonics Lettes, Vol.28, No.3, pp:23-24, 992. [8] T.M.Cove and J.A.Thomas, Elements of Infomation Theoy, John Wiley & Sons, Inc., 99, and Tsinghua Univesity Pess, 23. [9] W.H.Pess, S.A.Teukolsky and W.T.Vetteling. Numeical Recipes in C++ --The At of Scientific Computing, 2 nd Edition, Publishing House of Electonics Industy, 23 Chao Wang was bon in May, 98 in the city of Maanshan, Anhui Povince, P.R.China. In 22, he eceived the B.E. degee in electonics and infomation engineeing fom Univesity of Science and Technology of China (USTC). He is now a Ph.D candidate of USTC. His cuent eseach inteests include image pocessing and coding, especially the vaiational image pocessing. Zhongfu Ye was bon in Decembe 24, 959. He has eceived the B.E., M.Sc and Ph.D degee in electonics and infomation engineeing in 982, 986 and 995 espectively. He is now a pofesso of USTC. He has been to the Chinese Univesity of Hong Kong and the Univesity of Hong Kong, as a visiting schola in 997, 998 and 2 espectively. His cuent eseach inteests include image pocessing, statistical and aay signal pocessing, ada and communication signal pocessing.