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Galley Proof 21/12/2007; 13:59 File: ica280.tex; BOKCTP/ljl p. 1 Integrated Computer-Aided Engineering 15 (2008) 1–17 1 IOS Press New algorithms for contrast enhancement in grayscale images based on the variational definition of histogram equalization Iyad Jafar and Hao Ying Department of Electrical and Computer Engineering, Wayne State University, Detroit, MI 48202, USA Abstract. Images captured in dark or bright environments are usually characterized of low contrast. It is important to preprocess these images to make them suitable for other image processing applications. The histogram equalization (HE) algorithm is widely used for this purpose due to its simplicity and effectiveness. However, it can result in a significant change in the mean brightness and produce undesirable visual artifacts. This paper introduces the Constrained Variational Histogram Equalization (CVHE) algorithm which basically extends the variational definition of the HE algorithm by adding a mean brightness constraint to formulate a functional optimization problem, the solution of which defines a new graylevel transformation function for contrast enhancement. Preserving the mean brightness is expected to add more control on histogram stretching, thus reducing the artifacts and change in brightness. We also develop two variants of the CVHEalgorithm. The first variant is the Constrained Variational Local Histogram Equalization (CVLHE) algorithm which works in a similar manner to the popular local histogram equalization (LHE) algorithm; however it uses the CVHE transformation function. This variant achieves better performance than the CVHE algorithm but with higher computational requirements. The second variant is the Accelerated CVLHE (ACVLHE) algorithm which uses a modified nonoverlapped block processing approach to reduce the CVLHE computations. The ACVLHE strikes a balance between the speed of the CVHE and the performance of the CVLHE. The choice between the CVHE algorithm and its two local variants is a tradeoff between speed and desired enhancement levels. Visual and quantitative evaluation involving benchmark images show our algorithms to be better than their HE counterparts. Keywords: Blocking effects, contrast, graylevel transformation, histogram equalization, image enhancement 1. Introduction Capturing images in dark or bright environments makes them less contrasted. It is important that these images are processed in order to improve their appear- ance to the human viewers or to make them more suit- able for other image processing applications such as segmentation and pattern recognition. The problem of image contrast enhancement is subjective and applica- tion dependent, and this justifies the presence of numer- ous contrast enhancement and modification algorithms in the literature. Typically,these algorithms can be clas- Corresponding author. Tel.: +1 313 467 6127; Fax: +1 313 577 1101; E-mail: [email protected]. sified into two main categories: the direct techniques and indirect techniques. In the direct algorithms, the basic idea is to emphasize some features in the images such as edges and local graylevel statistics [3]. The feature(s) to be enhanced is (are) selected based on pri- or knowledge of the application. For example, unsharp masks can be convoluted with the original image to enhance the edges [15]. Local means, variances, and other statistical moments can also be calculated and then manipulated in a specific manner to achieve bet- ter image quality [9,24]. Conversely, the indirect con- trast enhancement algorithms are usually applied to the whole image without the computation and utilization of a quantitative measure of contrast. Instead, these algorithms attempt to modify the contrast by defining a transformation function T (r) that manipulates each ISSN 1069-2509/08/$17.00 © 2008 – IOS Press and the author(s). All rights reserved

Transcript of New algorithms for contrast enhancement in grayscale images based …hying/images/ICAE 2008.pdf ·...

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Galley Proof 21/12/2007; 13:59 File: ica280.tex; BOKCTP/ljl p. 1

Integrated Computer-Aided Engineering 15 (2008) 1–17 1IOS Press

New algorithms for contrast enhancement ingrayscale images based on the variationaldefinition of histogram equalization

Iyad Jafar and Hao Ying∗Department of Electrical and Computer Engineering, Wayne State University, Detroit, MI 48202, USA

Abstract. Images captured in dark or bright environments are usually characterized of low contrast. It is important to preprocessthese images to make them suitable for other image processing applications. The histogram equalization (HE) algorithm iswidely used for this purpose due to its simplicity and effectiveness. However, it can result in a significant change in the meanbrightness and produce undesirable visual artifacts. This paper introduces the Constrained Variational Histogram Equalization(CVHE) algorithm which basically extends the variational definition of the HE algorithm by adding a mean brightness constraintto formulate a functional optimization problem, the solution of which defines a new graylevel transformation function for contrastenhancement. Preserving the mean brightness is expected to add more control on histogram stretching, thus reducing the artifactsand change in brightness. We also develop two variants of the CVHE algorithm. The first variant is the Constrained VariationalLocal Histogram Equalization (CVLHE) algorithm which works in a similar manner to the popular local histogram equalization(LHE) algorithm; however it uses the CVHE transformation function. This variant achieves better performance than the CVHEalgorithm but with higher computational requirements. The second variant is the Accelerated CVLHE (ACVLHE) algorithmwhich uses a modified nonoverlapped block processing approach to reduce the CVLHE computations. The ACVLHE strikesa balance between the speed of the CVHE and the performance of the CVLHE. The choice between the CVHE algorithm andits two local variants is a tradeoff between speed and desired enhancement levels. Visual and quantitative evaluation involvingbenchmark images show our algorithms to be better than their HE counterparts.

Keywords: Blocking effects, contrast, graylevel transformation, histogram equalization, image enhancement

1. Introduction

Capturing images in dark or bright environmentsmakes them less contrasted. It is important that theseimages are processed in order to improve their appear-ance to the human viewers or to make them more suit-able for other image processing applications such assegmentation and pattern recognition. The problem ofimage contrast enhancement is subjective and applica-tion dependent, and this justifies the presence of numer-ous contrast enhancement and modification algorithmsin the literature. Typically, these algorithms can be clas-

∗Corresponding author. Tel.: +1 313 467 6127; Fax: +1 313 5771101; E-mail: [email protected].

sified into two main categories: the direct techniquesand indirect techniques. In the direct algorithms, thebasic idea is to emphasize some features in the imagessuch as edges and local graylevel statistics [3]. Thefeature(s) to be enhanced is (are) selected based on pri-or knowledge of the application. For example, unsharpmasks can be convoluted with the original image toenhance the edges [15]. Local means, variances, andother statistical moments can also be calculated andthen manipulated in a specific manner to achieve bet-ter image quality [9,24]. Conversely, the indirect con-trast enhancement algorithms are usually applied to thewhole image without the computation and utilizationof a quantitative measure of contrast. Instead, thesealgorithms attempt to modify the contrast by defininga transformation function T (r) that manipulates each

ISSN 1069-2509/08/$17.00 © 2008 – IOS Press and the author(s). All rights reserved

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2 I. Jafar and H. Ying / New algorithms for contrast enhancement in grayscale images based on the variational definition

graylevel r in the image such that the dynamic rangeof the display device is fully utilized and the imagedetails are more distinct. Contrast stretching using lin-ear and nonlinear functions [9,21,24], histogram pro-cessing [8–11,13,24], iterative histogram thinning [19,23], and fuzzy contrast intensification [18] are exam-ples of the indirect enhancement techniques. In par-ticular, histogram equalization (HE) is one of the mostcommonly known indirect techniques. The underlyingprinciple in this algorithm is that for maximum transferof information, the perceived distribution (histogram)of graylevels in an image should be uniform. It can beeasily shown that histogram equalization achieves thisby using the cumulative distribution function of the nor-malized histogram of the image graylevels as the trans-formation function [9]. Nonetheless, histogram equal-ization suffers from some problems. First, histogramequalization transforms the histogram of the originalimage into a flat uniform histogram that spans to theentire graylevel range. Accordingly, the mean bright-ness of the output image is always at the middle of thegraylevel range regardless of the mean of the originalimage. For images with high and low mean brightnessvalues, this means a significant change in the imageoutlook for the price of enhancing the contrast. Sec-ond, the stretching of the histogram’s graylevels in thehistogram equalization algorithm is controlled by theshape of the histogram only. This may result in overen-hancement artifacts due to excessive lousy stretchingof graylevels. Additionally, this excessive stretchingmay increase the mergence between neighboring dis-crete histogram bars. Thus, for regions containing suchneighboring graylevels, they will contain one level inthe output image, which can be referred to as saturationartifacts. Finally, histogram equalization performs theenhancement based on the global content of the image,thus it may not increase, or even may decrease, thelocal details, especially when the image contains morethan one object.

Several algorithms addressed the problem of bright-ness change in HE [5,6,12,26]. Although they areproven to preserve the mean brightness to some extent,they are still based on HE, which means they are notvery efficient in local enhancement and still produceoverenhancement and saturation artifacts. Adaptive orlocal histogram equalization (LHE) [17,20] extends HEto allow for local enhancements. In LHE, a rectangularblock of the input image is defined and the transforma-tion function of that region is computed by histogramequalization. Afterwards, the center pixel of that win-dow is modified using this function. This process is

repeated for all the pixels in the image by moving thecenter of the block. This extension of histogram equal-ization allows each pixel to adapt to its neighborhood,so that high contrast can be obtained for all locationsin the image. However, the LHE algorithm usuallyresults in an unnatural modification in the processedimage due to excessive noise amplification, especial-ly in smooth regions. Also, Rehm and Dallas pointedout that LHE produces edge artifacts at sharp boundarypoints where the local transformation changes abruptlydue to rapid change of the local histogram [22]. This isbecause LHE is only the local extension of HE, thus itinherits its noise amplification and saturation problemsthat mainly result from the absence of a limit on theamount of stretching of the graylevel values.

In this paper we develop three new algorithms forimage contrast enhancement. The main algorithm iscalled the Constrained Variational Histogram Equal-ization (CVHE). This algorithm combines the powerof the histogram equalization algorithm in contrast en-hancement for graylevel images with the target of pre-serving the global outlook of the image. The CVHE al-gorithm is based on extension of the variational defini-tion of the histogram equalization algorithm by addinga constraint that would make the mean brightness of theprocessed image as close to that of the original imageas possible. The rationale behind this new approachis that maintaining the original mean brightness wouldreduce the amount of stretch caused by the histogramequalization algorithm, thus preserving the appearanceof the image and reducing saturation and overenhance-ment artifacts. The other two algorithms are the basi-cally the local extensions of the CVHE algorithm. Thefirst extension is called the Constrained Variational Lo-cal Histogram Equalization (CVLHE) and its operationis similar to the LHE algorithm; however it uses theCVHE transformation function of blocks instead of theHE transformation functions. The CVLHE algorithmincreases the enhancement capabilities of the CVHEalgorithm because it operates locally. However, it isassociated with excessive computational requirements.For this reason we introduce the third algorithm whichwe call Accelerated Constrained Variational Local His-togram Equalization (ACVLHE) to reduce the process-ing time of the CVLHE algorithm and obtain com-parable performance. Thus the ACVLHE algorithmstrikes a balance between the CVHE and CVLHE algo-rithms in terms of the level of enhancement and speed.The ACVLHE is basically based on the nonoverlappedblock processing where the image is partitioned intoa set of nonoverlapping blocks, then all the pixels in-

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I. Jafar and H. Ying / New algorithms for contrast enhancement in grayscale images based on the variational definition 3

(a) CVHE with blocking (b) HE with blocking

Fig. 1. An example for the blocking effect when the non-overlapped block approach is used with the (a) CVHE (b) HE algorithms.

side each block is modified using the CVHE transfor-mation function of that block. To reduce the problemof blocking effects that arises due to the shape differ-ences between the transformation functions of adjacentnonoverlapping blocks in the image (Fig. 1), especiallyfor pixels near the borders of the blocks, the ACVL-HE algorithm uses a weighted sum of the neighboringblocks’ transformation functions to modify each pixelin the image. Choosing between the CVHE algorithmand its two local variants is application-dependent asthere is always a tradeoff between performance andspeed. We have conducted an extensive experimentalevaluation on these three new algorithms and comparedthem to the HE, LHE, and the accelerated version ofthe LHE algorithm. Our comparison included visualevaluation and the use of three measures to quantifythe increase in image contrast and distortion introducedinto the image. On overall, our new algorithms out-performed the HE, LHE, and an accelerated version ofthe LHE algorithm using the speed up approach usedwith the ACVLHE algorithm. Also, the experimentalevaluation proved the ACVLHE algorithm capabilityin reducing the processing time required in the CVL-HE algorithm dramatically with acceptable decrease inperformance when compared to the CVLHE algorithmand with negligible blocking effects.

The rest of this paper is organized as follows. In Sec-tion 2, we provide an overview of the classical and vari-ational formulations of histogram equalization. Sec-

tion 3 discusses the formulation and the derivation ofthe CVHE algorithm. The CVLHE and ACVLHE al-gorithms are detailed in Section 4. Experimental resultsand discussion are provided in Section 5, and Section 6concludes our paper.

2. Mathematical overview of histogramequalization

As mentioned in the previous section, the CVHE al-gorithm extends the histogram equalization algorithmby adding a mean brightness constraint to its varia-tional formulation. Accordingly, we find it importantto review the mathematical details of the classical andvariational formulations of the histogram equalizationalgorithm before proceeding to the derivation of theCVHE algorithm

2.1. Classical formulation

In image processing context, the normalized his-togram of an image can be used as the probability den-sity function for the graylevel distribution in that im-age. So, let’s assume that hi(r) and ho(s) representthe normalized histograms for the input and output im-ages, respectively, where r and s stand for the continu-ous normalized random variables for the graylevels, i.e.0 � r, s � 1. In transform-based image enhancement

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4 I. Jafar and H. Ying / New algorithms for contrast enhancement in grayscale images based on the variational definition

techniques, a transformation function

s = T (r) (1)

is used to map each input graylevel rk to a new level sk

to achieve the required enhancement objective. It is im-portant to mention here that any graylevel transforma-tion function for image contrast enhancement shouldhave the following properties:

(i) T (r) values are bounded between 0 and 1, i.e.

0 � T (r) � 1, ∀r ∈ [0, 1] (2)

(ii) T (r) should be monotonically increasing in orderto preserve the graylevel order from black to white. Inother words,

T ′(r) � 0, ∀r ∈ [0, 1] (3)

where T ′(r) is the first derivative of T (r). If suchtransformation is used, it follows from basic probabilitytheory that the output histogram ho(s) is related to theinput histogram hi(r) by

ho(s) = hi(r)dr

ds(4)

For the case of histogram equalization, we know thatthe output histogram is a uniform distribution function,i.e. ho(s) = 1, thus Eq. (4) becomes

ds = hi(r)dr (5)

Integrating both sides and assuming T (0) = 0, wecan find the transformation function T (r) in histogramequalization to be

s = T (r) =∫ r

0

hi(w)dw (6)

This implies that the transformation function T (r)is the cumulative distribution function (CDF) of theimage histogram.

2.2. Variational formulation

The variational formulation of the histogram equal-ization algorithm was proposed by Altas et al. [1].Their formulation is based on the analysis of the his-togram equalization algorithm which can be understoodas minimizing the total (cumulative) distances (spac-ing) between histogram bars ((r)) in output histogramho(s) subject to a weighting by the input histogramhi(r). Accordingly, they proposed the following func-tional

J(T (r)) =∫ 1

0

F (r, T ′(r), T (r))(7)

dr =∫ 1

0

1hi(r)

(T ′(r))2dr

and showed that minimizing it produced the same trans-formation function in the classical formulation of thehistogram equalization algorithm. The derivation de-tails are briefly explained as follows. In the calculus ofvariations, the functional in Eq. (7) can be minimizedby applying the Euler equation

∂F

∂T− d

dr

(∂F

∂T ′

)= 0 (8)

to define the differential equation

− d

dr

(2T ′

hi(r)

)= 0 (9)

that can be easily solved to find the relation betweenT (r) with hi(r). Integrating both sides with respect tor and solving for T ′(r) gives

T ′(r) = Ahi(r) (10)

where A is an integration constant. If we integrate oncemore with respect to r, we find that the transformationT (r) is given by

T (r) = T (0) + A

∫ r

0

hi(w)dw (11)

Since T (0) and T (1) are assumed to be 0 and 1,respectively; in order to make use of the full graylevelrange, it can be found that A = 1. Thus, the transfor-mation function in the variational version of histogramequalization is given by

T (r) =∫ r

0

hi(w)dw (12)

which is the same as the one obtained using the classicalformulation in Eq. (6).

3. The constrained variational histogramequalization algorithm

3.1. Formulation and general solution

Unlike the classical formulation of the histogramequalization algorithm, the variational approach viewsthe equalization process as a minimization problem.This directly implies that constraints can be added tothe objective function in Eq. (7) to achieve specifictasks. This is the basis of our approach Constrained

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I. Jafar and H. Ying / New algorithms for contrast enhancement in grayscale images based on the variational definition 5

Variational Histogram Equalization (CVHE), where weuse the variational formulation of histogram equaliza-tion given in Eq. (7) and add the constraint that wouldmake the mean brightness of the output image similaror closer to that of the input image. The idea behindadding such constraint is that conserving the originalmean brightness would limit the amount of stretch inhistogram equalization and thus preserve the global ap-pearance of the image while enhancing the contrast.Mathematically, this can be written as

J(T (r)) =∫ 1

0

1hi(r)

(T ′(r))2dr

(13)

[∫ 1

0

s ho(s)ds − µi

]where µi is the mean brightness of the original image,λ is the Lagrange multiplier, and the integral term in theconstraint is the mean brightness of the output image.Using Eqs (1) and (4), the new objective function canbe rewritten as

J(T (r)) =∫ 1

0

1hi(r)

(T ′(r))2dr

(14)

[∫ 1

0

T (r)hi(r)dr − µi

]Next, applying the Euler equation in Eq. (8) to the

functional in Eq. (14) gives the differential equationthat relates T (r) with the input histogram hi(r) as

λhi(r) − d

dr

(2T ′(r)hi(r)

)= 0 (15)

This is a first order separable differential equation andcan be solved using the same approach discussed inSection 2. In view of that, the solution for T (r) interms of the input histogram hi(r) is

T (r)=λ

(∫ r

0

hi(w)dw

)2

∫ r

0

hi(w)dw (16)

where β is an integration constant. We know thatT (1) = 1, thus we can eliminate β and find the generalsolution for the transformation function to be

T (r) =∫ r

0

hi(w)dw (17)

[(∫ r

0

hi(w)dw

)2

−∫ r

0

hi(w)dw

]

This new transformation function can be viewed es-sentially as the sum of the transformation function ofhistogram equalization in Eq. (6) and a conditioningterm that controls the amount of enhancement. For dig-

ital images, the discrete form for the general solutiongiven in Eq. (17) can be written as

s = T (r) =r∑

w=0

hi(w)

(18)

⎡⎣( r∑

w=0

hi(w)

)2

−r∑

w=0

hi(w)

⎤⎦

3.2. Computing the conditioning scale factor λ

The conditioning scale factor λ is the Lagrange mul-tiplier associated with the added constraint that wouldpreserve the input mean brightness. Thus, to find λ,we should calculate the mean brightness of the outputimage µo and equate it to the input mean µi. FromEq. (14), the discrete form of the constraint will be

µo =L∑

r=0

T (r)hi(r) = µi (19)

where L is the maximum available graylevel (255 for 8-bit grayscale images). Substituting T (r) from Eq. (18)and solving for λ gives

λ=µi −

L∑r=0

r∑w=0

hi(r)hi(w)

L∑r=0

h(r)(

r∑w=0

hi(w))2

−L∑

r=0

r∑w=0

hr(r)hi(w)

(20)

Let’s call this value the exact conditioning scale fac-tor λexact. If this value is substituted in Eq. (18), itmakes the transformation function T (r) able to pro-duce an image with the same mean brightness as theinput image. However, all the discussion till this pointdid not take into account the two basic properties ofgraylevel transformation functions given in Eqs (2) and(3). So, for some images, the transformation functionwith the exact conditioning scale factor λexact will beable to preserve the mean, though it may not be positiveor/and monotonically increasing The reason why wedid not include these constraints in the objective func-tion in Eq. (14) is to simplify the solution as these con-straints will impose global point-wise inequality con-straints which require a functional expression of h i(r)in order to find the solution [25]; something that cannotbe obtained in this case. Accordingly, to account forthese two constraints, we should check if λexact makesthe transformation satisfy Eqs (2) and (3). Actually,these conditions will impose lower and upper boundsfor the possible value of conditioning scale factor. Thepermissible range of the conditioning scale factor λ can

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6 I. Jafar and H. Ying / New algorithms for contrast enhancement in grayscale images based on the variational definition

be found as follows. In the first property, if we usethe expression of T (r) given in Eq. (18) to solve eachinequality separately in terms of λ, we find that thisproperty enforces a lower bound Lower1 and an upperbound Upper1 such that

Lower1 � λ � Upper1 (21)

where

Lower1 = maxr∈[0,L]

⎛⎜⎜⎜⎝

r∑w=0

hi(w) − 1

r∑w=0

hi(w)−(

r∑w0

hi(w))2

⎞⎟⎟⎟⎠ (22)

and

Upper1 = minr∈[0,L]

⎛⎜⎜⎜⎝

r∑w=0

hi(w)

r∑w=0

hi(w)−(

r∑w0

hi(w))2

⎞⎟⎟⎟⎠ (23)

For the second property, the discrete first derivativecan be approximated by

T ′(r) ≈ T (r + 1) − T (r) (24)

and the property can be rewritten as

T (r + 1) � T (r), ∀r ∈ [0, L − 1] (25)

By mathematical manipulation,Eq. (25) requires that

λ

[h2

i (r+1)+2hi(r+1)r∑

w=0

hi(w)−hi(r+1)

]

� −hi(r + 1), ∀r ∈ [0, L − 1] (26)

Since the sign of the term in brackets (TIB) on theleft side of the inequality depends on the values of theinput histogram, the condition in Eq. (26) may enforcea lower bound or an upper bound, or both. If the TIBis positive, then there exists a lower bound Lower2 thatis defined as

Lower2 = maxr∈[0,L−1]|TIB >0

(f1(r)) (27)

where

f1(r) =

−h1(r + 1)

|h2i (r+1)+2hi(r+1)

r∑w=0

hi(w)−hi(r+1)|, (28)

∀r ∈ [0, L − 1]|TIB > 0

When the TIB is negative, then there exists an upperbound Upper2 given by

Upper2 = minr∈[0,L−1]|TIB <0

(f2(r)) (29)

such that

f2(r) = −f1(r), ∀r ∈ [0, L − 1]|TIB < 0 (30)

Now, using Eqs (21), (27), and (29) we can findthe smallest permissible range R∗ for the conditioningscale factor λ by

Lower = max(Lower1, Lower2)

Upper = min(Upper1, Upper2) (31)

R∗ = [Lower, Upper]For the case when the condition in Eq. (26) does not

enforce a lower bound, then Lower is set to Lower1.Also, if no upper bound is available from Eq. (26),Upper is set to Upper1.

Having defined the permissible range R∗ for the con-ditioning scale factor λ for a specific image, the ques-tion that remains is: How would CVHE process thatimage if the exact conditioning scale factor λexact com-puted using Eq. (20) is out of the permissible rangeR∗? When λexact is out of range, this implies that theoriginal mean cannot be preserved in the output imagewithout violating one/all of the basic properties for thetransformation functions. So, in order to make the newalgorithm capable of processing all images, we shouldtolerate some deviation in the output mean from theoriginal mean by using a new value for the conditioningscale factor other than λexact. We will refer to this val-ue as the alternative conditioning scale actor λalt and itshould be a permissible value that is as close as possibleto λexact and would produce the minimum deviation inthe mean brightness. Since the relation between λ andµi given in Eq. (20) (effectively it is the desired mean inthe output image) is linear, it can be easily deduced thatthe using the bound that is closer to λexact as the valuefor the alternative conditioning scale factor λalt willgive the optimal deviation in the mean brightness. Inother words, for the cases when the exact conditioningscale factor cannot be used, we can use the alternativeconditioning scale factor λalt given by

λalt ={

Lower, λexact < LowerUpper, λexact < Upper

(32)

to define the transformation function in Eq. (17).

4. Variations of the CVHE algorithm

4.1. The constrained variational local histogramequalization algorithm

As we discussed in the introduction section, the glob-al histogram equalization algorithm may not increase

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I. Jafar and H. Ying / New algorithms for contrast enhancement in grayscale images based on the variational definition 7

or it may even decrease the local contrast and details.This is because the graylevel transformation functionis based on the global histogram of the image. Localhistogram equalization was proposed to address thisproblem. It is simply the application of the histogramequalization algorithm in a small neighborhood aroundeach pixel in the image. Although the LHE algorithmcan significantly increase the details in the image, it isusually associated with noise amplification in smoothregions and near the edges. This reason and fact thatthe CVHE algorithm outperforms the global HE algo-rithm (as explained in Section 5), inspired us to ex-tend the CVHE algorithm by applying it locally at eachpixel in the image. We refer to this extension as theConstrained Variational Local Histogram Equalization(CVLHE) and it is summarized as follows. At each pix-el in the image, a block of size MxM pixels is specifiedsuch that the pixel is at the center of the block. Thenthe histogram of the graylevel of the pixels inside theblock is constructed. This histogram is used to com-pute the permissible range R∗ and the exact condition-ing scale factor λexact as discussed on Section 3. If theexact conditioning scale factor λexact falls in the per-missible range, then it is used to define the transformedgraylevel value for the pixel under consideration usingEq. (18). Otherwise, the alternative conditioning scalefactor λalt defined in Eq. (32) is used. This process isrepeated over and over until all image pixels are pro-cessed. Modifying the pixel values with transforma-tion functions that preserve the mean brightness of theblocks is expected to alleviate the problems of localhistogram equalization (LHE) in terms of brightnesschange, artifacts, and noise amplification.

4.2. The accelerated constrained variationalhistogram equalization algorithm

Despite the ability of local enhancement algorithmsto improve image details, they are associated with anincreased processing overhead as they are applied ona pixel-by-pixel basis. For example, for a 512 × 512image, the use of LHE algorithm requires the evalua-tion of histogram equalization for 262,144 times. Theextension of the CVHE algorithm, the CVLHE algo-rithm, is not an exception for this problem. In fact,the CVLHE algorithm adds additional processing over-head when compared to the LHE algorithm since it re-quires the calculation of the conditioning scale factorλ and the permissible range R∗ at each pixel. One ap-proach for reducing such overhead in local algorithmsis to use nonoverlapped block processing, where the

Fig. 2. Partitioning the image into blocks based on the shift valuessx and sy .

image is partitioned into nonoverlapped blocks and allthe pixels inside a single block are processed with thesame transformation function of that block. Althoughthis approach reduces the processing time dramatically,it cannot avoid the appearance of the blocking effects,especially for pixels near the boundaries of the blocks.Figures 1-a and 1-b show the blocking effect when theCVHE and HE algorithms, respectively, are used toprocess all pixels inside each individual nonoverlap-ping block. We can see from these figures that theblocking effects are not avoidable even when using theCVHE algorithm. However, the level of blocking ef-fects seems to be less when the CVHE algorithm isused because it modifies each block such that its meanbrightness is the closest to the original value.

Accordingly, we introduce the following multistagespeed-up approach that takes advantage of the CVLHEalgorithm capabilities and reduces its computation re-quirements to produces images with enhanced contrastand with negligible blocking effects. Let’s assume thatthe input image have Nx × Ny pixels. Then, this im-age is partitioned into nonoverlapped blocks of size Bx

× By such that the coordinates of the point at whichpartitioning starts is given by

(xp, yp) = (sxBx, syBy), sxsy ∈ [0, 1] (33)

where sx and sy are user defined values to controlthe point location. Figure 2 shows two examples for

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partitioning the image with (sx = sy = 1.0) and(sx = sy = 0.5). After partitioning, the CVLHEtransformation function for each of these blocks andthe coordinates of their center pixels are computed andstored. Next, instead of using each of these transfor-mation functions to process the pixels inside each cor-responding block, the pixels’ values are modified asfollows. For any arbitrary pixel with graylevel r in theimage, the block to which it belongs B(m,n) is identi-fied and the distance D(m,n) to its center is found. Al-so, the distance between the pixel under considerationand the center of each of the eight neighboring blocksis also calculated (see Fig. 3). The computed distancesare used then to find the new graylevel s value for thatpixel by

s = TACVLHE(r) =

l∑i=−1

l∑j=−1

T(m+i,n+j)(r)

D(m+i,n+j)

l∑i=−1

l∑j=−1

1D(m+i,n+j)

(34)

where T(m+i,n+j)(r) represents the transformationfunction of block B(m+i,n+j) with reference to blockB(m,n). Effectively, Eq. (34) represents the transforma-tion function in the ACVLHE algorithm TACVLHE(r)that is equal to the weighted sum for the transformationfunctions of the eight neighboring blocks and the blockthat contains the pixel with the weights being inverselyproportional to distance. In essence, this transforma-tion satisfies the conditions in Eqs (2) and (3) since thetransformation functions T(m+i,n+j)(r) are governedto be bounded and monotonic by the CVHE algorithm.

The function in Eq. (34) can be applied similarly toall pixels in image except for the pixels at the center ofthe blocks since the distance D(m,n) is zero, and pixelsinside the outlier blocks (blocks at image borders) be-cause some of the neighboring blocks are not available.For the center pixels, the new value is found using thetransformation function of its block only. In case of out-lier blocks, the averaging of transformation functionsis performed on the available neighboring blocks only.We argue that the computation of the new pixel valueusing Eq. (34) helps reducing the blocking for two ba-sic reasons: 1) it utilizes the transformation functionsof neighboring blocks, and 2) it allows the transforma-tion functions for adjacent pixels to change smoothlybased on the pixel location with respect its neighboringblocks. Once all pixels in the image are processed, thisprocedure may be repeated many times but with differ-ent values for sx and sy . We call each of these repeti-tions a stage. The output at each stage is accumulated

Fig. 3. Calculation of transformation value for an arbitrary pixelusing neighboring blocks.

with those of previous stages and once the final stage isfinished, the accumulated result is averaged. As shownin Fig. 1, this step improves the reduction of blockingeffects by averaging regions with blocking effects inone stage with others without blocking effects in otherstages. The more the number of stages, the better thereduction of blocking effects. However, this impliesincreased processing overhead and more decrease inthe contrast obtained in the CVLHE algorithm due tothe averaging (smoothing) operation. So, for a specificapplication, the ACVLHE parameters; block size, shiftvalues sx and sy , and the number of stages, can betuned to meet the desired performance. The use of thisspeed-up approach is not restricted to the CVLHE algo-rithm. It can be applied to any local algorithm that usestransformation functions for enhancement. However,the performance is limited by the capabilities of thetransformation functions of these algorithms. We referto the use of this speed-up approach with the CVL-HE algorithm as the Accelerated CVLHE (ACVLHE)algorithm.

5. Performance evaluation

5.1. Performance measures and evaluation setup

In this paper, the evaluation of different algorithmsis carried out using visual inspection and three quanti-

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tative measures. We used the quantitative measures toassist the discussion and as a guideline for the compari-son. In addition to these measures, we also included theaverage time required for each algorithm to process thetest images. The first quantitative measure is adoptedfrom [3,16] and is used to compute the average localcontrast for an image of M × N pixels by

Contrast =1

MN

M−1∑i=0

N−1∑j=0

|rij − Eij

rij + Eij(35)

where rij is the graylevel value at pixel (i, j), andEij is the mean edge graylevel that is computed in aneighborhood N W of size w × w pixels and centeredat pixel (i, j) by

Eij =

∑(k,l)∈Nw

∆klrkl∑(k,l)∈Nw

∆kl(36)

with ∆kl being the edge value computed by any edgeoperator, e.g. Sobel or Robert operators [9,24]. Essen-tially, this contrast measure is based on the detection ofobject edges. This is based on the fact that the percep-tion mechanisms are very sensitive to edges present inthe image, in addition to the graylevel variation. Thatexplains using the edge information Eij along with thegray value rij in computing the contrast at pixel (i, j).It is apparent from Eq. (35) that the average contrastvalue is always less than 1, with higher values indicat-ing better contrast around that pixel.

The second measure is used to quantify thechange/distortion introduced into the in the image afterprocessing. Several distortion measures have been pro-posed in the literature [2,4]. For simplicity, we used thewell known Root Mean Square Error (RMSE) which isdefined

RMSE=

⎛⎝ 1

MN

M−1∑i=0

N−1∑j=0

(I(i, j) − I∗(i, j))2

⎞⎠

1/2

(37)

where I and I∗ are the original and enhanced images,respectively. Higher RMSE values are usually relat-ed to higher change/distortion in the processed image.However, it is important to mention here that the en-hancement operation causes the image to change andthus distortion cannot be avoided. Essentially, the en-hancement process can be viewed as distortion. Ac-cordingly, we are generally looking for algorithms thatare capable of increasing the image contrast with lowerdistortion values.

The last measure is the Absolute Mean BrightnessError (AMBE) [5,6] which is defined as

AMBE = |µo − µi| (38)

where µi and µo are mean brightness of the input andoutput images, respectively. The AMBE measures thechange in the global image mean brightness before andafter processing. Lower AMBE values are preferredas they indirectly reflect how the global outlook of theimage changes after processing.

In our performance evaluation we included the fol-lowing algorithms: the histogram equalization (HE),the Constrained Variational Histogram Equalization(CVHE), the Local Histogram Equalization (LHE), theConstrained Variational Local Histogram Equalization(CVLHE), the Accelerated LHE (ALHE), and Accel-erated CVLHE (ACVLHE). The ALHE algorithm issimply the use of the speed up approach discussed insection 4.2 with the block transformation functions thatare computed by applying the histogram equalizationalgorithm to the blocks. These six algorithms wereused to process a large number of commonly used im-ages (512 × 512 pixels) that were obtained from [7] ona PC with a Pentium 4 processor and 1 GB of RAM.The block size in both the LHE and CVLHE algorithmswas set to 64 × 64 pixels. The number of blocks in theACVLHE and ALHE algorithms was set to 64 blocks,i.e. eight blocks in each direction, each with the sizeof 64 × 64 pixels. In the accelerated versions of LHEand CVLHE, we used three stages each with equal sx

and sy values. The values were 1.0, 0.5, and 0.25. TheSobel edge detector was used in computing the con-trast values, thus the neighborhood N W size was 3 ×3 pixels.

5.2. Performance of the CVHE algorithm

In this subsection we present some of the processingresults that we obtained in our evaluation of the globalalgorithms: CVHE and HE. Fig. 4 shows the originalimages Airport, Crowd, Bottle, and Toys and their pro-cessed versions using the CVHE and HE algorithms.For the images Airport and Crowd the exact condition-ing scale factor value λexact was used. This was not thecase for the Bottle and Toys images, where we used thealternative conditioning scale factor value λalt as thevalue for λ. Visual inspection reveals that the CVHEalgorithm is able to enhance the contrast with reducedproduction of saturation artifacts and less change in theimage appearance. On the other hand, the uncontrolledhistogram stretching in the HE algorithm caused satu-

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10 I. Jafar and H. Ying / New algorithms for contrast enhancement in grayscale images based on the variational definition

Airport

Crowd

Bottle

Toys (a) Original (b) CVHE (c) HE

Fig. 4. (a) Original images (b) CVHE results (c) HE results.

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Table 1The contrast values for original and processed images

Original CVHE HE CVLHE LHE ACVLHE ALHE

Airport 0.071 0.254 0.149 0.296 0.186 0.268 0.184Crowd 0.037 0.113 0.058 0.127 0.069 0.105 0.067Bottle 0.039 0.084 0.046 0.103 0.062 0.095 0.063Toys 0.043 0.182 0.111 0.227 0.133 0.208 0.134

ration and overenhancement artifacts to appear in theprocessed images. This caused the output images to beless contrasted than their corresponding CVHE images.Also, the HE algorithm forced dark (bright) images tolook aggressively brighter (darker) as it always pushesthe image mean brightness toward the middle of thegraylevel range regardless of the original mean value.

Quantitatively, the numerical values for the threemeasures support the visual inspection. In Table 1,we see that the CVHE algorithm increased the averagecontrast of the images more than the HE algorithm.Higher contrast values are justified by the ability of theCVHE algorithm to change the graylevel values withreduced saturation and overenhancement artifacts byincorporating the mean brightness preservation condi-tion in the derivation of the transformation function.Such control mechanism is not available in the HE al-gorithm and this explains why these artifacts are moreperceptible in the images processed by the HE algo-rithm. The computed RMSE and AMBE measures aregiven in Tables 2 and 3, respectively. For images pro-cessed by the CVHE algorithm, the values for thesetwo measures were always less than those for imagesprocessed by the HE algorithm. This is reflected onthe CVHE images by reduced change in the image ap-pearance and content. Although the CVHE algorithmhas lower RMSE values which might be interpreted aslower change/enhancement, this is negated by its capa-bility of producing images with higher contrast values.Examining the AMBE values, we see that the CVHEalgorithm outperforms the HE algorithm in preservingthe mean brightness of the image after processing, evenwhen we used the alternative conditioning scale factorinstead of the exact one for the images Toys and Bot-tle. Again, this explains why the outlook of the CVHEimages is closer to the original images. Theoretically,the output image in CVHE should have the same meanbrightness as the original image if the actual condition-ing scale factor is used. However, this is correct if weapply the CVHE algorithm in the continuous domain,not in the discrete domain where we have discretizationerror due to the possibility of mapping more than oneneighboring levels to the same level. This explains whythe AMBE values for the images Crowd and Airportare nonzero.

The ability of CVHE to limit the change in the im-age appearance during enhancement can be further ex-plained by inspecting the corresponding transformationfunctions of the test images for the CVHE and HE algo-rithms. Figure 5 shows these transformation functionsalong with the identity transformation function

s= T (r)= r (39)

which basically maps the image to itself. Compar-ing the transformation functions, we see that those forthe CVHE algorithm are closer to the identity trans-formation over most of the graylevel range, and thisjustifies the ability of the CVHE algorithm to modifythe graylevels in the image with lower and controlledhistogram stretching.

Comparing the CVHE algorithm to the LHE and AL-HE algorithms visually and quantitatively shows thatthe CVHE algorithm is better, despite the fact it is ap-plied globally. This is because the LHE and ALHEalgorithms inherit the same problems associated by theHE algorithm in terms of the change in mean imagebrightness and production of saturation and overen-hancement artifacts in addition to noise amplificationin smooth regions and around the edges.

In terms of processing time, Table 4 shows that theCVHE algorithm is slower than the HE algorithm byorder of magnitudes. This is because the CVHE algo-rithm involves extra overhead in computing the permis-sible range and the conditioning scale factor. Howev-er, the performance achieved by the CVHE when com-pared to the HE algorithm makes it the algorithm ofchoice in applications that can tolerate this increasedprocessing time for the price of better enhancement.

5.3. Performance of the CVLHE and ACVLHE

As we mentioned earlier, local enhancement algo-rithms are supposed to produce images with higher con-trast than global algorithms because they allow eachpixel to adapt with its neighborhoodinstead of the glob-al image content. Accordingly, if we compare the re-sults of the CVLHE and ACVLHE algorithms in Figs 6,7, 8, and 9 to the corresponding results for the CVHEalgorithm in Fig. 4, we see that images processed by the

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12 I. Jafar and H. Ying / New algorithms for contrast enhancement in grayscale images based on the variational definition

Table 2The RMSE values for images processed by different algorithms

CVHE HE CVLHE LHE ACVLHE ALHE

Airport 42.37 62.71 47.34 66.38 42.43 46.03Crowd 31.57 57.73 37.73 61.18 33.81 43.57Bottle 28.35 58.32 41.47 67.41 36.79 45.72Toys 48.76 84.46 65.34 95.38 57.23 73.87

Table 3The AMBE values for images processed by different algorithms

CVHE HE CVLHE LHE ACVLHE ALHE

Airport 3.77 46.07 6.20 45.97 7.01 25.25Crowd 1.10 43.68 5.89 45.19 5.49 21.73Bottle 6.95 49.52 13.36 51.05 15.11 28.85Toys 27.64 72.12 34.95 77.69 31.07 55.15

0 260

260

Tran

sfor

mat

ion

Func

tion

T(r)

r

CVHEHEIdentity

(a)

0 260

260Tr

ansf

orm

atio

n Fu

nctio

n T(

r)

r

CVHEHEIdentity

(b)

0 260

260

Tran

sfor

mat

ion

Func

tion

T(r)

r

CVHEHEIdentity

(c)

0 260

260

Tran

sfor

mat

ion

Func

tion

T(r)

r

CVHEHEIdentity

(d)

Fig. 5. The Identity, CVHE, and HE transformation functions for images (a) Airport (b) Crowd (c) Bottle (d) Toys.

two local algorithms have more details than the CVHEimages. This is supported numerically by the increasein the value of the average contrast listed in Table 1.Although one may say that this increase is small, wehave to keep in mind that the local algorithms increase

both the global and local details. So, the increase inthe contrast values for the CVLHE and ACVLHE al-gorithms in Table 1 corresponds to the local details ig-nored by the global algorithms. Examining the RMSEvalues in Table 2 shows that the local versions of the

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I. Jafar and H. Ying / New algorithms for contrast enhancement in grayscale images based on the variational definition 13

(a) CVLHE (b) LHE

(c) ACVLHE (d) ALHE

Fig. 6. Processing results for the image Airport using: (a) CVLHE, (b) LHE, (c) ACVLHE (d) ALHE algorithms.

CVHE algorithm have relatively higher values than theglobal CVHE algorithm. This can be justified by theincrease in the local details that can be viewed as dis-tortion. As we mentioned earlier this increase in theRMSE values is acceptable with the achieved increasein the contrast values. The AMBE values for the threenew algorithms were comparable and this can be per-ceived by the reduced change in the overall outlook andcontent of the original images. Accordingly, we cansay that the CVLHE and ACVLHE algorithms performbetter than the global CVHE. The only concern whenusing the CVLHE and ACVLHE is the increased com-putation time (see Table 4). Based on the discussion inthe previous section we saw that the CVHE algorithmoutperforms the three algorithms: HE, LHE, and AL-

HE. Accordingly, it is obvious that the local variantsof the CVHE algorithm; the CVLHE and ACVLHE,outperform those algorithms. This is supported by theprocessing results of the images in the figures and thenumerical values for the quantitative measures whichindicate higher contrast and lower distortion for the newalgorithms.

Let’s now study the effect of the speed approach thatis used in the ACVLHE algorithm on the performanceof the CVLHE algorithm. From the processed images,we can see that the ACVLHE images have compara-ble contrast to that of the CVLHE images. Also, it isnoticeable that the ACVLHE images have almost neg-ligible blocking effects. The only difference betweenthe results of the CVLHE algorithm when compared

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14 I. Jafar and H. Ying / New algorithms for contrast enhancement in grayscale images based on the variational definition

(a) CVLHE (b) LHE

(c) ACVLHE (d) ALHE

Fig. 7. Processing results for the image Crowd using: (a) CVLHE, (b) LHE, (c) ACVLHE (d) ALHE algorithms.

to the results of its accelerated version is that some ofthe details look blurred in the images of the acceleratedalgorithm. This is due to the averaging of the resultsof different stages in the accelerated versions. Howev-er, the performance of the ACVLHE algorithm is stillbetter than all of the other global and local algorithms.In terms of computation time, Table 4 shows that thespeed-up approach discussed in Section 4.2 is able toreduce the original computation time for the LHE andCVLHE algorithms by about 84% and 95%, respec-tively. However, the processing time for the ACVLHEalgorithm is still higher than the ALHE algorithm dueto the time needed to compute the conditioning scalefactor λ and the permissible range R∗ for each block.Also, the processing time gap between the accelerated

CVLHE and the accelerated LHE (around three sec-onds) is much less than the gap between the CVLHEand the LHE (around 634 seconds). This is because theadditional overhead in the ACVLHE algorithm is pro-portional to the number of nonoverlapped blocks andnot to the number of pixels (effectively the number ofoverlapped blocks) as in the CVLHE algorithm.

Although the new algorithms were tested on gray-scale images, they may be extended to color imagesby either operating on the luminance component or oneach color channel separately. However, special modi-fications and restrictions might be needed to avoid dis-tortion of the colors since there is a strong correlationbetween the color components [9]. This is an interest-ing topic for future research.

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I. Jafar and H. Ying / New algorithms for contrast enhancement in grayscale images based on the variational definition 15

(a) CVLHE (b) LHE

(c) ACVLHE (d) ALHE

Fig. 8. Processing results for the image Bottle using: (a) CVLHE, (b) LHE, (c) ACVLHE (d) ALHE algorithms.

Table 4The average processing time re-quired for different algorithms

Method Time

CVHE 0.731HE 0.036CVLHE 862.1LHE 228.3ACVLHE 39.6ALHE 36.5

6. Conclusion

This paper discussed the CVHE algorithm and itstwo variants; the CVLHE and ACVLHE algorithms forcontrast enhancement in grayscale images. The pur-

pose of these algorithms was to reduce the problems as-sociated with the traditional histogram equalization al-gorithm and its local version in terms of the significantchange in the image outlook and the production of un-desired visual artifacts. The CVLHE algorithm is oneextension of the CVHE algorithm that operates in a lo-cal fashion and was able to produce images with higherdetails when compared to the global CVHE algorithm.However this was at the expense of dramatic increasein the processing time. The ACVLHE is another ex-tension for the CVHE algorithm and was introduced toreduce the processing time of the CVLHE algorithmwith minimal decrease in performance. Although itis based on the nonoverlapped block processing, the

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16 I. Jafar and H. Ying / New algorithms for contrast enhancement in grayscale images based on the variational definition

(a) CVLHE (b) LHE

(c) ACVLHE (d) ALHE

Fig. 9. Processing results for the image Toys using: (a) CVLHE, (b) LHE, (c) ACVLHE (d) ALHE algorithms.

design of the ACVLHE succeeded in significantly re-ducing the blocking effects. The three new algorithmsshowed different levels of contrast enhancement withdifferent processing requirements. However, experi-mental evaluation proved these algorithms to be betterthan the HE, LHE and ALHE algorithms but with ad-ditional computational overheads. Generally, we cansay that the new algorithms are preferable over the HE,LHE, and ALHE algorithms in applications that cantolerate the increase the processing time as the price formore enhancement. This argument is also applicablewhen choosing between the three new algorithms sincethere is always a tradeoff between enhancement leveland speed.

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