Research Article Variational Histogram Equalization for...

Research ArticleVariational Histogram Equalization for Single ColorImage Defogging

Li Zhou Du Yan Bi and Lin Yuan He

Communication and Navigation Lab Aerospace Engineering College Air Force Engineering University Xirsquoan 710038 China

Correspondence should be addressed to Li Zhou zhouli 5120801163com

Received 31 March 2016 Revised 21 June 2016 Accepted 10 July 2016

Academic Editor Alberto Borboni

Copyright copy 2016 Li Zhou et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Foggy images taken in the bad weather inevitably suffer from contrast loss and color distortion Existing defoggingmethodsmerelyresort to digging out an accurate scene transmission in ignorance of their unpleasing distortion and high complexity Different fromprevious works we propose a simple but powerful method based on histogram equalization and the physical degradation modelBy revising two constraints in a variational histogram equalization framework the intensity component of a fog-free image canbe estimated in HSI color space since the airlight is inferred through a color attenuation prior in advance To cut down the timeconsumption a general variation filter is proposed to obtain a numerical solution from the revised framework After getting theestimated intensity component it is easy to infer the saturation component from the physical degradation model in saturationchannel Accordingly the fog-free image can be restored with the estimated intensity and saturation components In the end theproposed method is tested on several foggy images and assessed by two no-reference indexes Experimental results reveal that ourmethod is relatively superior to three groups of relevant and state-of-the-art defogging methods

1 Introduction

To perceive natural scenes from captured images means a lotto the computer vision applications such as image retrievalvideo analysis object recognition and car navigation In thefoggy weather the visual quality of images is impaired by thedistribution of atmospheric particles resulting in a loss ofimage contrast and color fidelity This kind of degradationundoubtedly makes a great discount on the effectivenessof those applications Therefore the technique of removingfoggy image degradation namely ldquoimage defoggingrdquo hasattracted much attention in the image processing field [1ndash3]

Since the fog degradation is largely dependent on the dis-tant from the object to the camera many existing approachesrely on the physical degradation model [4] In the model thefog-free scene can be recovered as long as the airlight andthe scene transmission (or alternatively the depth map) areestimated in advance Compared with the airlight the scenetransmission is relatively more difficult to be inferred sopeople concentrate on how to obtain an accurate scene trans-mission At the very beginning two or more images captured

in the same scene yet at different times or angles of polarizingfilters are utilized to estimate the scene transmission [5 6]Obviously it is too much for practical use to fetch multipleimages on demand Soon afterwards single-image-basedapproaches become a mainstream and are classified into twocategories one class obtains the scene transmission throughthe third party such as satellites and 3D models [7 8] theother one needs to estimate the scene transmission underpriors or assumptions that come from empirical statistics andproper analyses [9 10] Because the scene transmission staysunknown in most practical cases people lay more emphasison the second class For example He et al find a dark channelprior on purpose of obtaining a rough scene transmissionwhich is then refined by soft-matting algorithm in 2010 [9]Three years later Nishino et al try to propose a Bayesianprobabilistic approach that predicts the scene transmissionand albedo jointly under scene-specific albedo priors [10]Although these methods produce good results the color oftheir results is quite dim and crucial features are almost cov-ered up Worse still it takes so much time in achieving a pre-cisely estimated scene transmission that this kind of approach

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 9897064 17 pageshttpdxdoiorg10115520169897064

2 Mathematical Problems in Engineering

can hardly meet with the time requirement of the practicalapplications

Traditional image enhancement algorithms like intensitytransformation functions and histogram equalization whichdo not have to evaluate the scene transmission will not sufferfrom those problems mentioned above However they arelikely to produce distorted results due to their ignoranceof the physical degradation mechanism Nowadays peoplechange their minds by combining image enhancement algo-rithms with the physical degradation model In 2014 Arigelaand Asari design a sine nonlinear function to refine a roughscene transmission [11] but the method is more likely toproduce results with a dim color and a distorted sky regionThis is mainly due to the fact that it relies on preciselyestimated scene transmission information In 2015 Liu et alstop digging out an accurate scene transmission and turnto introduce a contrast-stretching transformation functionbased on rough depth layers of scenes to enhance localcontrast of foggy images [12] It is an intuitive and effectivemethod that can achieve clear visibility but the correspond-ing results are in oversaturation distortion The main reasonis that the transformation function does not contain anavailable constraint which guarantees color fidelity Differentfrom Arigelarsquos and Liursquos methods histogram equalization isable to not only enhance image contrast but also preservecolor fidelity of the input images In 2015 Ranota and Kaurdecompose a foggy image into three components in LABcolor space and apply the adaptive histogram equalizationchannel by channel to enhance contrast [13] Then a roughscene transmission is obtained by dark channel prior andrefined by an adaptive Gaussian filter The method is capableof reconstructing many fine details but the processed resultsstill suffer from heavy color distortion The reasons can beconcluded into two aspects for one thing the method turnsa blind eye to the fact that three components in HSI channelsare attenuated by fog degradation to different degrees foranother histogram equalization preserves the color fidelity ofthe foggy image instead of the fog-free one To deal with thefirst factor we use physical degradationmodel inHSI space toprocess three channels separately As to the second one it is anecessity for us to revise the color preservationmechanism ofhistogram equalization Thanks to the work of Wang and Ngin 2013 [14] we have a chance to modify the mechanism withthe aid of his proposed variational histogram equalizationframework

Here we propose an improved variational histogramequalization framework for single color image defoggingSimilar to the previously presented methods in [11ndash13]histogram equalization and the physical degradation modelare merged together into an effective and fast defoggingframework in our paper The major contributions we makecan be summarized into four aspects (1) The strategy whichtreats saturation and intensity components differently occurson the purpose of avoiding artificial color distortion (2)According to the physical degradation model we modifythe mean brightness constraint that can preserve the colorfidelity in the variational framework (3) Unlike the work ofWang and Ng we design and substitute a mixed norm fortotal variation (TV) and119867

1norms in the original variational

framework Thus there is no need to choose a proper normto be a regularization term manually Moreover a generalvariation filter as an extension of a TV filter is establishedin order to solve the framework efficiently (4) Different froma global constant airlight in many existing methods a localairlight associated with the density of fog is appropriatelyestimated under a color attenuation prior and pixel-baseddark and bright channels

The remainder of this paper is organized as follows Inthe next section a physical degradationmodel is expressed inHSI color space and our strategy for color image defogging isthen illustrated in brief in Section 3 we present the proposedvariational histogram equalization framework in detail InSection 4 our method is compared with other representativeand relevant approaches in some simulation experimentswhile this paper is summarized in the last section

2 Physical Degradation Model inHSI Color Space

Generally because of suspended particlesrsquo absorption andscattering in the foggy weather the scene-reflected light 119869(119909)goes through undesirable attenuation and dispersal Worsestill the airlight 119860 scattered by those particles pools muchlight quantity to the observers Both of these two factors areclosely associated with scene depth 119889(119909) so the observedscene appearance 119871(119909) can be illustrated in the followingexpression according to Koschmiederrsquos law [4]

119871 (119909) = 119869 (119909) 119905 (119909)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Direct attenuation+ 119860 (1 minus 119905 (119909))⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Theveiling light (1)

where 119905(119909) is the scene transmission related to 119889(119909) atposition 119909 and it can be described as 119905(119909) = 119890

minus120573119889(119909) with120573 denoting the scattering coefficient of the atmosphere Theadditive model in formula (1) is obviously comprised of twomajor parts direct attenuation and the veiling light Theformer part interprets that 119869(119909) is attenuated and dispersedwhile the latter one is the main cause of color distortionWith the physical degradation model image restoration infoggy scenes is essentially an ill-posed problem that recovers119869(119909) 119860 and 119905(119909) from the observed image 119871(119909) Notice thatHSI space is practical for human interpretation whichmakesit an ideal tool for processing images [15] The space con-tains three components (hue saturation and intensity) thatcan be transformed by RGB channels With the equivalenttransformation relationship from RGB to HSI the physicaldegradation model can be expressed in HSI color space by

119867119871= 119867119869 (2)

119878119871=119868119869

119868119871

119905119878119869 (3)

119868119871= 119868119869119905 + 119860 (1 minus 119905) (4)

where 119867119871 119878119871 and 119868

119871are the HSI components of 119871(119909) while

119867119869 119878119869 and 119868

119869represent ones of 119869(119909) respectively Formula (2)

implies that the hue component keeps constant according tothe color constancy theory while formula (3) verifies that fog

Mathematical Problems in Engineering 3

contaminates the saturation component which is easy to beoverlooked in some existingmethods Obviously formula (4)is accessible because the intensity channel can be consideredas a gray-level image Based on this model we propose acolor image defogging idea that 119868

119869is firstly inferred through

a variational framework of histogram equalization and then119878119869can be obtained provided 119860 is estimated in advance

119878119869=119868119871(119868119869minus 119860)

119868119869(119868119871minus 119860)

119878119871 (5)

where 119868119871and 119878119871are given by the decomposition of 119871 in 119868 and 119878

channels Together with estimated 119868119869and 119878119869 the fog-removal

image is recovered in the end

3 The Proposed Variational Framework forImage Defogging

Histogram equalization is one of the most representativemethods in the image enhancement field but the originalone is limited since a mean brightness constraint is notconsidered In 2007 Jafar and Ying proposed a constrainedvariational framework of histogram equalization for imagecontrast enhancement [16] With an attached constraint themean brightness of the output is approximately equal to thatof the input resulting in realistic color restoration Neverthe-less it may fail to enhance the contrast because of neglectingthe differences among the local transformations at the nearbypixel locationsOn the basis of Iyad Jafarrsquos workWang andNgmodified the framework with another constraint to a furtherstep in 2013 [14] The specific expression of the variationalframework 119864(119891) can be described as

119864TV (119891) = intΛ

1

ℎ119894(119903 119909)

119891119903(119903 119909)2119889119909 119889119903

+ 1205741intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 120583 (119909))

2

119889119909

+ 1205742intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816 119889119909 119889119903

1198641198671

(119891) = intΛ

1

ℎ119894(119903 119909)

119891119903(119903 119909)2119889119909 119889119903

+ 1205741intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 120583 (119909))

2

119889119909

+ 1205742intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816

2

119889119909 119889119903

(6)

where 119903 denotes the input of gray-level image while 119904 is theenhanced output ℎ

119894(119903) and ℎ

119900(119904) are the histograms of the

input and output respectively 119891(119903 119909) represents the localtransformation function and (119903 119909) isin Λ = (0 1) times Ω where119909 is each pixel location and Ω denotes the image domain119891119903(119903 119909) denotes the first derivative of 119891(119903 119909) with respect to

119903 nabla119891 is the gradient of 119891(119903 119909) with respect to the horizontaland vertical directions 120583(119909) is the mean brightness of theinput 120574

1and 1205742are positive constant parameters From the

framework 119864(119891) consists of three positive parts The firstpart is meant to make ℎ

119900(119904) distribute uniformly through

local transformation function resulting in enhancing localdetails of image scenes The second part is the same asthat in Iyad Jafarrsquos method aiming at preserving 120583(119909) Fortraditional image enhancement tasks this part is necessaryand helpful but may be incorrect or even harmful for foggyimage recovery The reason is that the mean brightness of afoggy image with whitening color is generally higher thanthat of a fog-free image We plan to discuss and modify 120583(119909)in Section 31 The last part in 119864(119891) is to keep structuresconsistent by narrowing down the differences among thosepairs of 119891(119903 119909) in local regions However the selection fromtwo norms in formula (6) still needs manual interventionThus a mixed norm is designed for an automatic processin Section 32 With those two improvements a modifiedvariational framework is built up in Section 33 Moreoverthe airlight is estimated by a color attenuation prior and thepixel-based dark and bright channels in Section 34 Noticethat a general variation filter is designed for calculating theproposed framework efficiently resulting in recovering theintensity of a fog-removal image finally The flowchart of ourproposed method is depicted as in Figure 1

31 Improvement for a Mean Brightness Constraint As is wellknown a foggy image possesses a high mean brightness sothe mean brightness constraint in the framework should beimproved through the physical degradation model Due to0 lt 119905 lt 1 formula (4) can be properly rearranged as

119868119869=1

119905119868119871minus(1 minus 119905)

119905119860 (7)

where 119860 is a local constant that will be estimated inSection 34 Moreover 119905 is assumed to be piecewise smoothin [17 18] which means that it can be treated as a constantin local regions Therefore after taking average of eachcomponent in formula (7) we can get

119868119869=1

119905119868119871minus(1 minus 119905)

119905119860 (8)

where 119868119869and 119868

119871represent the mean intensity values of 119869

and 119871 in a local region Λ respectively 119860 and 119905 denotethe mean values of airlight and scene transmission in Λrespectively Apparently the remaining problem is how tofetch 119905 from a foggy image Fortunately dark channel priormakes it possible to get a rough 119905 that proceeds to be refinedby soft-matting algorithm [9] as is shown in Figure 2 Fromthe figure the mean value of the rough 119905 in red or blue boxesis approximately close to that of the refined 119905 Accordinglythe mean rough 119905 may be adequate enough to equal 119905 Nowthat there is no need to calculate the refined 119905 that is the maincause for large time consumption in [9] it implies that we canobtain 119905 in patches promptly Thus when 119868

119869is substituted for


Design for a general variation filter

The intensity componentof a fog-removal image

A colorattenuation prior

Pixel-based dark and bright

channels

The estimation of airlight

Improvement fora mean brightness

constraint

Design for a spatialregularization term

The proposed variational framework

The intensity component of a foggy image

Figure 1 The flowchart of our proposed algorithm

(a) (b) (c)

Figure 2 Comparison of mean local values in both rough and refined 119905 obtained by He et al in [9] In columns a foggy image (a) rough 119905(b) and refined 119905 (c) In rows images in normal size and their zoom-in view of patches in red or blue frames

120583(119909) in formula (6) a proper mean brightness constraint canbe described as

intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 119868

119869(119909))

2

119889119909

= intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus

1

119905119868119871(119909) +

(1 minus 119905)

119905119860)

2

119889119909

(9)

when 119905 = 1 formula (9) turns to be int(int119891ℎ119894(119903 119909)119889119903 minus

119868119871(119909))2119889119909 which is only appropriate for regular image

enhancement such as Iyad Jafarrsquos or Wei Wangrsquos works Dueto 0 lt 119905 lt 1 in foggy images it is fully convinced that themodified constraint in formula (9) is more beneficial in foggyimage restoration

32 Design for a Spatial Regularization Term First emergingin image restoration field TV and 119867

1norms perform well

and have their own merits On the one hand TV normallows discontinuity of images and preserves more edges intextual regions which is proven in [19] On the other hand[20] validates that 119867

1norm is able to keep the structural

consistency in flat regions and costs fewer computer sourceswhen the minimization of its regularization is processedGiven that we seek for a spatial regularization term that canimitate both of the two norms to be specific the expectedregularization term should get close to TV norm in textualregions and behave like119867

1norm in flat areas

Here we suppose 120588(sdot) to be a function with respect tonabla119891 denoted by 119888 for the sake of clearness in the paper Ifwe have 120588(119888) = |119888| then a TV norm is formed In a similar


way a 1198671norm is established when 120588(119888) = 119888

2 In order toimitate TV and 119867

1norms it is reasonable to analyze the

diffuse behavior ofint 120588(119888)119889119909 through its corresponding Euler-Lagrange equation

div [1205881015840(|119888|)

|119888|119888] = 0 (10)

First of all we would like to decompose the divergenceterm into two orthotropic components along the level setcurve as is shown in the following expression

div [1205881015840(|119888|)

|119888|119888] =

1205881015840(|119888|)

|119888|119891120585120585

+ [1205881015840(|119888|)

|119888|+ (

1205881015840(|119888|)

|119888|)

1015840

sdot |119888|] 119891120578120578

(11)

where 119891120585120585

and 119891120578120578

represent tangential and normal com-ponents respectively Notice that it is available to controlthe diffuse speed of 119891

120585120585and 119891

120578120578 For one thing if both of

the speeds in tangential direction 120585 and normal direction120578 gradually go to zero as 119888 grows up together with thedescending rate of speed in 120585 being lower than that in theother direction it guarantees that 120588(sdot) is close to TV normin the textural areas Hence the first rule can be listed as

1205881015840(119888)

119888

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)

1205881015840 (119888) 119888

119888rarrinfin

997888997888997888997888rarr 120594 le 0

(12)

For another if the speed in 120585 keeps as fast as that indirection 120578 in the flat regions 120588(sdot) can be treated as119867

1norm

approximately Therefore the second rule is illustrated as

1205881015840(119888)

119888

119888rarr0

997888997888997888rarr 120581

(1205881015840(|119888|)

|119888|)

1015840

119888rarr0

997888997888997888rarr 0

(13)

Based on those two rules of pointed diffuse behaviormentioned above a satisfactory function is designed andturns out to be

120588 (119888) =1198882

1 + 1198882 (14)

It is easy to examine whether 120588(119888) obeys those two rulesPlugging 120588(119888) into formulas (12) and (13) we can get

1205881015840(119888)

119888=

2

(1 + 1198882)2

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888) =

2 minus 61198882

(1 + 1198882)3

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)

1205881015840 (119888) 119888=1 minus 3119888

2

1 + 1198882

119888rarrinfin

997888997888997888997888rarr minus3 le 0

1205881015840(119888)

119888=

2

(1 + 1198882)2

119888rarr0

997888997888997888rarr 2

(1205881015840(|119888|)

|119888|)

1015840

=minus8119888

(1 + 1198882)3

119888rarr0

997888997888997888rarr 0

(15)

Apparently the function 120588(119888) is availableThus the spatialregularization term in formula (6) is changed into a newversion

intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816

2

1 +1003816100381610038161003816nabla119891

1003816100381610038161003816

2119889119909 119889119903 (16)

33 Construction and Calculation of the Proposed FrameworkCombined with a mean brightness constraint in formula(9) and a spatial regularization term in formula (16) ourvariational framework of histogram equalization for imagedefogging is finally built up which is depicted as

119864 (119891) = intΛ

1

ℎ119894(119903 119909)

119891119903(119903 119909)2119889119909 119889119903

+ 1205741intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 119868

119869(119909))

2

119889119909

+ 1205742intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816

2

1 +1003816100381610038161003816nabla119891

1003816100381610038161003816

2119889119909 119889119903

(17)

From formula (17) our model is more concise in com-parison withWeiWangrsquos frameworkThe first term is utilizedto enhance contrast through local histogram equalizationwhile the second one aims at recovering the true brightnessby enforcing the output brightness being close to 119868

119869(119909) The

last one is devoted to preserving the structural consistencyby minimizing the differences among local transformationfunctions

As to the solution of the proposed framework we canlearn from Wangrsquos algorithm According to the alternatingdirection method of multipliers (ADMM) [21] formula (17)is converted into an unconstrained minimization problemthrough a pair of quadratic penalty functionsThus thewholeprocess for minimizing 119864(119891) is actually a loop iteration con-taining two corresponding Euler-Lagrange equations Rele-vant information about solving Euler-Lagrange equations canbe found in [22 23] However the time consumption is tooexpensive to be accepted A possible way to accelerate theprocess is to deal with Euler-Lagrange equation through a TVfilter [24] Nevertheless the regularization term |nabla119891|

2(1 +

|nabla119891|2) in our framework is not the same as TV norm |nabla119891|

exactly If the fitted TV energy in the filter is replaced by anew energy we have to adjust the filter coefficients especially


the weights 119908120572120573(119906) First of all we might as well define the

general form of a regularization term

119877120588(|nabla119906|) = int

Ω

120588 (|nabla119906|) 119889Ω (18)

where 120588(119888) is a monotone function Then it is easy to obtainthe energy functionrsquos Euler-Lagrange equation from an inputimage 1199060 in the discrete case [25]

sum

120572isinℓ

120597

120597ℓ(minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

+ 120582 (1199060

120572minus 119906120572) = 0 (19)

where 120572 is one node of ℓ and (120597119906120597ℓ)|120572denotes the edge

derivative Focusing on the first term of formula (19) weproceed to define the discrete versions of |nabla119906| and (120597119906120597ℓ)|

120572

as

|nabla119906| = radicsum

120572isinℓ

(120597119906

120597ℓ

10038161003816100381610038161003816100381610038161003816120572

)

2

= radicsum

120572isinℓ

(119906120573minus 119906120572)2

(20)

With formula (20) we can get

120597

120597ℓ(minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

= (minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120573

minus (minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

=

minus1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

(120597119906

120597ℓ

10038161003816100381610038161003816100381610038161003816120573

) +1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(120597119906

120597ℓ)

10038161003816100381610038161003816100381610038161003816120572

=

minus1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

(119906120572minus 119906120573) +

1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(119906120573minus 119906120572)

= (

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

) (119906120573minus 119906120572)

(21)

If formula (21) is plugged into formula (19) the discreteequation turns to be

sum

120572isinℓ

(

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

) (119906120573minus 119906120572)

+ 120582 (1199060

120572minus 119906120572) = 0

(22)

Now it is available to describe the expression of 119908120572120573(119906)

from formula (22)

119908120572120573=

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(23)

Therefore a general variation filter is formed to geta numerical solution from the energy functional frame-work precisely and promptly In particular 119908

120572120573(119906) goes to

(1|nabla120572119906| + 1|nabla

120573119906|) when 120588(119888) = 119888 which is brought into

correspondence with the weights of the TV filter Now that120588(119888) = 119888

2(1 + 119888

2) in our regularization term the newly

configured 119908120572120573(119906) should be

119908120572120573=

2

(1 +10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

2

)

2+

2

(1 +1003816100381610038161003816nabla120572119906

1003816100381610038161003816

2

)2 (24)

34 The Estimation of Airlight To recover the fog-free scenewithout yielding color shifting the airlight is another impor-tant factor which is often neglected It is simply inferredby selecting the brightest pixel of the entire image in [10]Afterwards He et al pick up a pointed pixel that correspondsto the brightest one in the dark channel as the estimatedairlight [9]Then Kim et al merge quad-tree subdivision intoa hierarchical searching strategy on purpose of obtaining aroughly inferred airlight [26] Recently people are devotedto seeking for an accurately estimated value For exampleSulami et al take a two-step estimating approach to recoverthe airlight through a geometric constraint and a global imageprior [27] Although they are remarkable in some situationsit is worth noting that these methods just provide a globalconstant airlight Unfortunately this is contrary to the factthat the airlight ought to vary with the fog density Thuswe need to estimate a local airlight associated with the fogdensity

To recover the local airlight the first step aims at measur-ing the fog density We introduce a color attenuation prior[28] to measure the density for each pixel The prior findsthat the difference between the brightness and saturation isdirectly proportional to the depth Moreover it is well knownthat when the depth increases gradually the fog density goeshigher and higher Based on these two observations we candraw a conclusion about the relationship among the fogdensity 119902(119909) the depth 119889(119909) and the difference between thebrightness V(119909) and the saturation 119904(119909)

119902 (119909) prop 119889 (119909) prop (V (119909) minus 119904 (119909)) (25)

Because 119860(119909) varies along with the changes of 119902(119909) itis reasonable to make an assumption that 119860(119909) is positivelyproportional to 119902(119909) and we can get

119860 (119909) prop (V (119909) minus 119904 (119909)) (26)

Since there is one-to-one correspondence between 119860(119909)and 119902(119909) the maximum and minimum of 119860(119909) denotedby 119860+(119909) and 119860minus(119909) correspond to the highest and lowestfog density respectively Based on pixel-based dark channel119869dark(119909) and pixel-based bright channel 119869bright(119909) [29] 119860+(119909)

and 119860minus(119909) are simply defined as the pixels with the highestand lowest values in 119869

dark(119909) and 119869

bright(119909) respectively

119869dark(119909) and 119869bright(119909) are mathematically expressed as

119869dark

(119909) = min119888isin119877119866119861

119869119888(119909) (27)

119869bright

(119909) = max119888isin119877119866119861

119869119888(119909) (28)


Figure 3 Synthetic images named as L08005 and L08010 in the columns foggy images and ground-truth images

(a) (b) (c)

Figure 4 Results of our method initialized by different 1205741 (a) 120574

1= 1 (b) 120574

1= 100 and (c) 120574

1= 10000

00592

00478004640035700346

00246002380014400137

0005

AM

BE in

dex

0001002003004005006007

L08005 L08010

110100

100010000

Parameter 1205741

(a)

00618 0060900537 00571

0048900557

0046600532

0046300525

0001002003004005006007

L08005 L08010

MSE

inde

x

110100

100010000

Parameter 1205741

(b)

Figure 5 Image quality of defogging results on L08005 and L08010 images is assessed by (a) AMBE index and (b) MSE index respectively


(a) (b) (c)


2= 1 (b) 120574

2= 100 and (c) 120574

2= 10000

79518426

66277096

5647 58084911 4851

4312 4159

0102030405060708090

EI in

dex

L08005 L08010

110100

100010000

Parameter 1205742

(a)

00532

00758

00489

00608

004880055700514 0055800554 00583

0001002003004005006007008

MSE

inde

x

L08005 L08010

110100

100010000

Parameter 1205742

(b)

Figure 7 Image quality of defogging results on L08005 and L08010 images is assessed by (a) EI index and (b) MSE index respectively

According to formula (26) with two known points(max(V(119909) minus 119904(119909)) 119860+(119909)) and (min(V(119909) minus 119904(119909)) 119860minus(119909)) alocal 119860(119909) can be estimated by

119860 (119909) =119860+(119909) minus 119860

minus(119909)

120579+ (119909) minus 120579minus(119909)

(V (119909) minus 119904 (119909))

+119860minus(119909) sdot 120579

+(119909) minus 119860

+(119909) sdot 120579

minus(119909)

120579+ (119909) minus 120579minus(119909)

(29)

where max(V(119909) minus 119904(119909)) and min(V(119909) minus 119904(119909)) denoted by120579+(119909) and 120579minus(119909) are constants With the estimated 119860(119909) we

can infer 119868119869from the variational framework and then 119878

119869is

obtained by formula (5) At last 119869(119909) can be easily recoveredwith 119868

119869and 119878119869

4 Experiments and Analysis

In order to perform a qualitative and quantitative analysis ofthe proposed method we do some simulation experimentson color foggy images in comparison with three pairs ofstate-of-the-art defogging approachesThe first pair is Ranotaand Kaurrsquos [13] and Wang and Ngrsquos [14] that are directly


(b) (c)(a)

Figure 8 Comparison of results on house image obtained by Ranotarsquos method Wangrsquos method and ours In the columns an original foggyimage fog-removal results processed by Ranotarsquos method (up) Wangrsquos method (middle) and ours (down) corresponding zoom-in view ofred or blue boxes

based on histogram equalization technique while the secondone is Arigela and Asarirsquos [11] and Liu et alrsquos [12] whichbelong to intensity transformation functions Apparentlyit is indispensable for us to choose them as comparativegroups since they are quite relevant to our method He etalrsquos [9] and Nishino et alrsquos [10] in the last pair are classicaland representative as is well known in the image defoggingfield Thus we are going to compare our method with allof them pair by pair on the foggy image set that containsbenchmarked and realistic images chosen from [9ndash14]

41 Test of Parameters Before the comparison we oughtto inform the experimental condition and the parameterselection All the mentioned approaches are carried out inthe MATLAB R2014a environment on a 35 GHz computerwith 4GB RAM On the simulation platform the parametersutilized in our method are set to be 1205820

1= 1205820

2= 1205820

3= 0 120573 = 1

1205741= 100 and 120574

2= 100 It is worth pointing out that 120574

1and

1205742are picked up from several pairs of (120574

1 1205742) where 120574

1 1205742isin

1 10 100 1000 10000 in the parameter-testing experimentSpecifically two synthetic images are chosen from Fridadatabase as shown in Figure 3 Besides three assessmentindexes are introduced to evaluate the effectiveness of ourmethod initialized by different pairs of (120574

1 1205742)The first index

is the absolute mean brightness error (AMBE) that is thedifference of mean brightness between the output and theground-truth image while the second one is edge intensity(EI) which quantifies the structural informationThe last oneis the mean square error (MSE) that measures the changeof the output in comparison with the ground-truth imageNotice that AMBE index andMSE index belong to backwardpointers Their scores range from 0 to 1 and the lower theyare the better image quality will be EI index is on theopposite side where higher scores imply better results


(b) (c)(a)

Figure 9 Comparison of results on trees image obtained by Ranotarsquos method Wangrsquos method and ours In the columns an original foggyimage fog-removal results processed by Ranotarsquos method (up) Wangrsquos method (middle) and ours (down) corresponding zoom-in view ofred or blue boxes

Firstly we might as well fix 1205742on 100 and adjust 120574

1freely

Figure 4 describes a part of results processed with different 1205741

and Figure 5 is the quantitative evaluation of performance byAMBE and MSE indexes From Figure 5(a) AMBE index isdecreasing along with the increasing 120574

1 due to the fact that 120574

1

is directly related to the second term in formula (17) It meansthat the higher 120574

1is the better the mean brightness of results

will be However it can be seen from Figure 5(b) that MSEindex keeps decreasing at a slow speed after 120574

1reaches 100

which implies that merely resorting to increasing 1205741is not

enough to remove the fog degradationWorse still a higher 1205741

is prone to induce more iterations in the solution-calculatingstepTherefore it is logical to set 120574

1to be 100 in the proposed

frameworkSecondly 120574

1is fixed on 100 and 120574

2can be adjusted from

1 to 10000 Processed results of our method initialized bydifferent 120574

2are presented partially and evaluated as shown

in Figures 6 and 7 respectively At first sight of Figure 6results suffer from a great loss of contrast when 120574

2increases

gradually This observation is strengthened by Figure 7(a)where EI index as ameasurement of image contrast becomessmaller and smaller Nevertheless it does not mean that thehigher 120574

2is the worse image quality will be From Figure 6

the structural consistency keeps better alongwith the increas-ing 1205742 This is because 120574

2affects both of image contrast and

structural consistency according to the first and third termsin formula (17) The contradictory relationship between the

contrast and consistency is validated by Figure 7(b) whereMSE index is in the shape of ldquoUrdquo with high at two ends andlow in middle given that it is a compromise that 120574

2is set to

be 100 in the proposed method

42 Qualitative Comparison

421 Qualitative Comparison with Histogram-Equalization-Based Methods Since our method is based on a variationalhistogram equalization framework it is acceptable to becompared with Ranota and Kaurrsquos [13] and Wang and Ngrsquos[14] on house and trees images From Figures 8 and 9results obtained by Ranotarsquos method are in artificial colorand dark patches In contrast results processed by Wangrsquosmethod and ours are quite superior in terms of color fidelityand local structure The reasons can be summed up intotwo aspects for one thing Wangrsquos method and ours avoiddistorted color by treating 119868 and 119878 components differentlyunlike Ranotarsquos processing all of color components in thesame way for another Gaussian filter in Ranotarsquos methodtends to smoothen the rough scene transmission as well aslocal contrast at the same time Wangrsquos method and ourswhich do not need to refine the scene transmission will notproduce the structural distortion However in comparisonwith our results Wangrsquos method seems to be brighter inthe global illumination For example the grasses in bluebox of Figure 8(c) are too bright to exhibit their real color


(b) (c)(a)

Figure 10 Comparison of results on street image obtained by Arigelarsquos method Liursquos method and ours In the columns an original foggyimage fog-removal results processed by Arigelarsquos method (up) Liursquos method (middle) and ours (down) corresponding zoom-in view of redor blue boxes

information so is the road in blue box of Figure 9(c) This islargely becauseWangrsquosmethodpreserves themeanbrightnessof an input foggy image Thanks to the revised brightnessconstraint in our framework the color is recovered properlyMoreover from the results of Wangrsquos method based on TVnorm in the house image and 119867

1norm in the trees image

the structural consistency between the wall and the branch isviolated in the middle red box of Figure 8(c) What is worsemany details of branches are lost in the middle red box ofFigure 9(c) The main reason is that TV and 119867

1norms are

unable to preserve the consistency and fine details in thediscontinuity of textural regions simultaneously With the


(b) (c)(a)

Figure 11 Comparison of results on train image obtained by Arigelarsquos method Liursquos method and ours In the columns an original foggyimage fog-removal results processed by Arigelarsquos method (up) Liursquos method (middle) and ours (down) corresponding zoom-in view of redor blue boxes

help of a designed regularization term in our method thereis no need to sacrifice the consistency for fine details and viceversa

422 Qualitative Comparison with Intensity-Transformation-Based Methods Histogram equalization and intensity-transformation-based methods are similar enhancementalgorithms so we launch a comparison among Arigelarsquosmethod [11] Liu et alrsquos method [12] and ours on street andtrain images As exhibited in Figure 10 (up) and Figure 11(up) Arigelarsquos results are still in dim color with the sky andwhite objects being accompanied with halo effect This isbecauseArigelarsquosmethod substitutes intensity transformationfunction for soft-matting algorithm used in Hersquos method [9]so as to refine a rough scene transmission It means that themethod is in close relationship with Hersquos and therefore itsresults inevitably suffer from the same unwanted distortionas Hersquos does exactly A further explanation will be found inSection 423 Liursquos method does not seek for an accuratescene transmission and its results are displayed in Figure 10(middle) and Figure 11 (middle) They show fine localdetails and abundant color information because the method

removes fog degradation with intensity transformationfunction guided by scene depth layers Nevertheless Liursquosresults present an oversaturation appearance due to a lack ofcolor fidelity constraint Compared with those two defoggingmethods ours can produce pleasing results with vivid colorand great contrast The success may owe to the constrainedvariational framework with a color fidelity term

423 Qualitative Comparison with Classical and Represen-tative Methods To make the performance of the proposedmethod more persuasive and convincing it is a must tocompare ourmethodwith several classical and representativeones such as He et alrsquos [9] and Nishino et alrsquos [10] FromFigures 12 and 13 results delivered by Hersquos and Nishinorsquosmethod are recovered up to a reasonable level so does theproposed method However the scene color of Hersquos andNishinorsquos results should have been bright but stay quite diminstead For instance the color of grasses on the rocks isnearly concealed so that we could not tell the differencebetween grasses and rocks in the blue box of Figure 12(up) and Figure 12 (middle) so is the one of buildings inthe blue box of Figure 13 (up) and Figure 13 (middle) The


(b) (c)(a)

Figure 12 Comparison of results on cliff image obtained by Hersquos method Nishinorsquos method and ours In the columns an original foggyimage fog-removal results processed by Hersquos method (up) Nishinorsquos method (middle) and ours (down) corresponding zoom-in view of redor blue boxes

reason that Hersquos method induces over-defogging results ismainly due to dark channel prior that overestimates thethickness of fog The predicted transmission value is loweror much lower sometimes than the true one when it isfiltered by ldquominrdquo filters which has been pointed out in[30] The reason for Nishinorsquos method is the same thingsince statistical distributions are considered as depth priorabsorbed in the probabilistic minimization Worse still falsecolor and blocking artifacts occur in the regions of Hersquos andNishinorsquos results as is shown in the red boxes of Figures 12and 13 It is due to the different ways of their estimating theairlight Both of them consider the roughly estimated airlightto be a global constant which goes against the basic factthat the airlight varies with the fog density Different fromtheir methods the proposed one estimates a local airlightassociatedwith the fog density under a color attenuation priorand pixel-based bright and dark channels Apparently ourmethod has the capability of removing fog effects withoutyielding unappealing distortion or information loss

43 Quantitative Comparison In order to strengthen thequalitative analyses mentioned above two no-referenceassessment indexes are introduced including EI index andno-reference image quality evaluator index (NIQE) [31] It isworth noting that among those three indexes in Section 41AMBE index and MSE index are full-reference evaluatorsThey are inappropriate for assessing image quality with theabsence of ground-truth images Accordingly only EI indexis adopted to evaluate the structural contrast again in thissection Plus NIQE index is meant to measure the distortiondegree of processed results through scene statistics of locallynormalized luminance coefficients Actually the shape ofthose coefficientsrsquo distribution in a foggy image is thinnerthan the one in a defogged image which implies NIQE indexis capable of quantifying the losses of naturalness from adistorted image Scores of the index distribute from 0 to 100and zero score represents the best result Figures 14ndash16 give aseries of quantitative assessments for images in Figures 8ndash13As displayed in the plots of Figures 14ndash16 results processed by


(b) (c)(a)

Figure 13 Comparison of results on tower image obtained by Hersquos method Nishinorsquos method and ours In the columns an original foggyimage fog-removal results processed by Hersquos method (up) Nishinorsquos method (middle) and ours (down) corresponding zoom-in view of redor blue boxes

our method gain the best scores of EI index and NIQE indexThis fully shows that the proposedmethod can producemoreplausible results compared with the other algorithms

The time consumption needs to be taken into considera-tion if the method is put into the practice Figure 17 exhibitsthe running time of several defogging methods previouslymentioned From the figure Hersquos and Nishinorsquos methods costthe most time sources above all This is because they dependmuch on the accurate scene transmission refined by the soft-matting algorithm or a jointly estimated Bayesian frameworkthat takes up plenty of time Except for Hersquos and Nishinorsquosmethods Wangrsquos method also consumes too much time upto 15 seconds which is three times as much as the cost ofthe remaining four methods The reason is that two Euler-Lagrange equations are required to be tacked in every loopiteration Compared withWangrsquos work ourmethod proposesa general variation filter to solve the Euler-Lagrange equationresulting in saving considerable time Moreover the timecost of Ranotarsquos Arigelarsquos and Liursquos methods also keep ata comfortably lower level since they are based on imageenhancement algorithms

5 Conclusion

In the paper we propose an image defogging method usinga variational histogram equalization framework A previousvariational framework on image enhancement inspires usto establish a constrained energy functional that containshistogram equalization and the physical degradation modelThemean brightness constraint in the framework is revised topreserve the brightness of a fog-free image while the regular-ization term is redesigned for avoiding manual interventionTo pursue the processing efficiency a general variation filteris proposed to solve the constrained framework promptlyAs to another important unknown quantity 119860 a colorattenuation prior and pixel-based dark and bright channelsare introduced to infer a local constant 119860 reasonably Inthe end the proposed method is tested on several bench-marked and realistic images in comparison with three groupsof representative defogging methods With qualitative andquantitative comparison it is safe to draw a conclusionthat our method performs much better in terms of coloradjustment and contrast enhancement In the future more


Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex

Ranotarsquos method Wangrsquos methodOurs

(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees

Defogging methodsRanotarsquos method Wangrsquos methodOurs

(b)

Figure 14 Image quality of defogging results on house and trees images obtained by Ranotarsquos method Wangrsquos method and ours is assessedby (a) EI index and (b) NIQE index

Defogging methodsArigelarsquos method Liursquos methodOurs

289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)

Figure 15 Image quality of defogging results on street and train images obtained by Arigelarsquos method Liursquos method and ours is assessed by(a) EI index and (b) NIQE index

Defogging methodsHersquos method Nishinorsquos methodOurs

187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)

Figure 16 Image quality of defogging results on cliff and tower images obtained by Hersquos method Nishinorsquos method and ours is assessed by(a) EI index and (b) NIQE index


371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)

Figure 17 Time consumption for three groups of defogging methods including Ranotarsquos method Wangrsquos method Arigelarsquos method Liursquosmethod Hersquos method Nishinorsquos method and ours

attention will be put on accelerating the processing speed upto a real-time level for some computer vision applications

Competing Interests

The authors declare that they have no competing interests

References

[1] N Hautiere J-P Tarel H Halmaoui R Bremond and DAubert ldquoEnhanced fog detection and free-space segmentationfor car navigationrdquoMachine Vision and Applications vol 25 no3 pp 667ndash679 2014

[2] H Liu J Yang Z Wu and Q Zhang ldquoFast single imagedehazing based on image fusionrdquo Journal of Electronic Imagingvol 24 no 1 Article ID 013020 2015

[3] J-G Wang S-C Tai and C-J Lin ldquoImage haze removal usinga hybrid of fuzzy inference system and weighted estimationrdquoJournal of Electronic Imaging vol 24 no 3 Article ID 033027pp 1ndash14 2015

[4] N S Narasimhan and S Nayar ldquoInteractive (de) weatheringof an image using physical modelsrdquo in Proceedings of the IEEEWorkshop on Color and Photometric Methods in ComputerVision pp 1ndash8 October 2003

[5] Y Y Schechner S G Narasimhan and S K Nayar ldquoInstantdehazing of images using polarizationrdquo in Proceedings of the

2001 IEEEComputer Society Conference onComputer Vision andPattern Recognition pp I325ndashI332 December 2001

[6] S G Narasimhan and S K Nayar ldquoVision and the atmosphererdquoInternational Journal of Computer Vision vol 48 no 3 pp 233ndash254 2002

[7] Y Lee K B Gibson Z Lee and T Q Nguyen ldquoStereo imagedefoggingrdquo in Proceedings of the IEEE International Conferenceon Image Processing (ICIP rsquo14) pp 5427ndash5431 Paris FranceOctober 2014

[8] J Kopf B Neubert B Chen et al ldquoDeep photo Model-basedphotograph enhancement and viewingrdquo ACM Transactions onGraphics vol 27 no 5 article 116 2008

[9] K-M He J Sun and X-O Tang ldquoSingle image haze removalusing dark channel priorrdquo IEEETransactions on PatternAnalysisand Machine Intelligence vol 33 no 12 pp 2341ndash2353 2010

[10] K Nishino L Kratz and S Lombardi ldquoBayesian defoggingrdquoInternational Journal of Computer Vision vol 98 no 3 pp 263ndash278 2012

[11] S Arigela and V K Asari ldquoEnhancement of hazy color imagesusing a self-tunable transformation functionrdquo in Advances inVisual Computing G Bebis R Boyle B Parvin et al Edsvol 8888 of Lecture Notes in Computer Science pp 578ndash587Springer New York NY USA 2014

[12] Q LiuM Y Chen andD H Zhou ldquoSingle image haze removalvia depth-based contrast stretching transformrdquo Science ChinaInformation Sciences vol 58 no 1 pp 1ndash17 2015


[13] H K Ranota and P Kaur ldquoA new single image dehazingapproach using modified dark channel priorrdquo Advances inIntelligent Systems and Computing vol 320 pp 77ndash85 2015

[14] W Wang and M K Ng ldquoA variational histogram equalizationmethod for image contrast enhancementrdquo SIAM Journal onImaging Sciences vol 6 no 3 pp 1823ndash1849 2013

[15] R C Gonzalez and R E Woods Digital Image ProcessingPublishing House of Electronics Industry Beijing China 3rdedition 2010

[16] I Jafar and H Ying ldquoImage contrast enhancement by con-strained variational histogram equalizationrdquo in Proceedingsof the IEEE International Conference on ElectroInformationTechnology (EIT rsquo07) pp 120ndash125 Chicago Ill USA May 2007

[17] R Fattal ldquoSingle image dehazingrdquoACMTransactions on Graph-ics vol 27 no 3 article 72 2008

[18] R T Tan ldquoVisibility in bad weather from a single imagerdquo inProceedings of the 26th IEEEConference onComputer Vision andPattern Recognition (CVPR rsquo08) pp 1ndash8 Anchorage AlaskaUSA June 2008

[19] L I Rudin S Osher and E Fatemi ldquoNonlinear total variationbased noise removal algorithmsrdquo Physica D Nonlinear Phenom-ena vol 60 no 1ndash4 pp 259ndash268 1992

[20] G Aubert and P Kornprobst Mathematical Problems in ImageProcessing Partial Differential Equations and the Calculus ofVariations vol 147 Springer New York NY USA 2002

[21] F Lin M Fardad and M R Jovanovic ldquoDesign of optimalsparse feedback gains via the alternating direction method ofmultipliersrdquo IEEE Transactions on Automatic Control vol 58no 9 pp 2426ndash2431 2013

[22] C R Vogel andM E Oman ldquoFast robust total variation-basedreconstruction of noisy blurred imagesrdquo IEEE Transactions onImage Processing vol 7 no 6 pp 813ndash824 1998

[23] A Marquina and S Osher ldquoExplicit algorithms for a newtime dependent model based on level set motion for nonlineardeblurring and noise removalrdquo SIAM Journal on ScientificComputing vol 22 no 2 pp 387ndash405 2000

[24] T F Chan S Osher and J Shen ldquoThe digital TV filter andnonlinear denoisingrdquo IEEE Transactions on Image Processingvol 10 no 2 pp 231ndash241 2001

[25] G Aubert and L Vese ldquoA variational method in image recov-eryrdquo SIAM Journal on Numerical Analysis vol 34 no 5 pp1948ndash1979 1997

[26] J-H Kim J-Y Sim and C-S Kim ldquoSingle image dehazingbased on contrast enhancementrdquo in Proceedings of the 36thIEEE International Conference on Acoustics Speech and SignalProcessing (ICASSP rsquo11) pp 1273ndash1276 Prague Czech RepublicMay 2011

[27] M Sulami I Glatzer R Fattal and M Werman ldquoAutomaticrecovery of the atmospheric light in hazy imagesrdquo inProceedingsof the 6th IEEE International Conference on ComputationalPhotography (ICCP rsquo14) pp 1ndash11 May 2014

[28] Q Zhu J Mai and L Shao ldquoA fast single image haze removalalgorithm using color attenuation priorrdquo IEEE Transactions onImage Processing vol 24 no 11 pp 3522ndash3533 2015

[29] C-H Yeh L-W Kang C-Y Lin and C-Y Lin ldquoEfficientimagevideo dehazing through haze density analysis basedon pixel-based dark channel priorrdquo in Proceedings of the 3rdInternational Conference on Information Security and IntelligentControl (ISIC rsquo12) pp 238ndash241 Yunlin Taiwan August 2012

[30] K Tang J Yang and J Wang ldquoInvestigating haze-relevantfeatures in a learning framework for image dehazingrdquo in

Proceedings of the 27th IEEE Conference on Computer Visionand Pattern Recognition (CVPR rsquo14) pp 2995ndash3002 ColumbusOhio USA June 2014

[31] A Mittal A K Moorthy and A Bovik ldquoNo-reference imagequality assessment in the spatial domainrdquo IEEE Transactions onImage Processing vol 21 no 12 pp 4695ndash4708 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


can hardly meet with the time requirement of the practicalapplications

Traditional image enhancement algorithms like intensitytransformation functions and histogram equalization whichdo not have to evaluate the scene transmission will not sufferfrom those problems mentioned above However they arelikely to produce distorted results due to their ignoranceof the physical degradation mechanism Nowadays peoplechange their minds by combining image enhancement algo-rithms with the physical degradation model In 2014 Arigelaand Asari design a sine nonlinear function to refine a roughscene transmission [11] but the method is more likely toproduce results with a dim color and a distorted sky regionThis is mainly due to the fact that it relies on preciselyestimated scene transmission information In 2015 Liu et alstop digging out an accurate scene transmission and turnto introduce a contrast-stretching transformation functionbased on rough depth layers of scenes to enhance localcontrast of foggy images [12] It is an intuitive and effectivemethod that can achieve clear visibility but the correspond-ing results are in oversaturation distortion The main reasonis that the transformation function does not contain anavailable constraint which guarantees color fidelity Differentfrom Arigelarsquos and Liursquos methods histogram equalization isable to not only enhance image contrast but also preservecolor fidelity of the input images In 2015 Ranota and Kaurdecompose a foggy image into three components in LABcolor space and apply the adaptive histogram equalizationchannel by channel to enhance contrast [13] Then a roughscene transmission is obtained by dark channel prior andrefined by an adaptive Gaussian filter The method is capableof reconstructing many fine details but the processed resultsstill suffer from heavy color distortion The reasons can beconcluded into two aspects for one thing the method turnsa blind eye to the fact that three components in HSI channelsare attenuated by fog degradation to different degrees foranother histogram equalization preserves the color fidelity ofthe foggy image instead of the fog-free one To deal with thefirst factor we use physical degradationmodel inHSI space toprocess three channels separately As to the second one it is anecessity for us to revise the color preservationmechanism ofhistogram equalization Thanks to the work of Wang and Ngin 2013 [14] we have a chance to modify the mechanism withthe aid of his proposed variational histogram equalizationframework

Here we propose an improved variational histogramequalization framework for single color image defoggingSimilar to the previously presented methods in [11ndash13]histogram equalization and the physical degradation modelare merged together into an effective and fast defoggingframework in our paper The major contributions we makecan be summarized into four aspects (1) The strategy whichtreats saturation and intensity components differently occurson the purpose of avoiding artificial color distortion (2)According to the physical degradation model we modifythe mean brightness constraint that can preserve the colorfidelity in the variational framework (3) Unlike the work ofWang and Ng we design and substitute a mixed norm fortotal variation (TV) and119867

1norms in the original variational

framework Thus there is no need to choose a proper normto be a regularization term manually Moreover a generalvariation filter as an extension of a TV filter is establishedin order to solve the framework efficiently (4) Different froma global constant airlight in many existing methods a localairlight associated with the density of fog is appropriatelyestimated under a color attenuation prior and pixel-baseddark and bright channels

The remainder of this paper is organized as follows Inthe next section a physical degradationmodel is expressed inHSI color space and our strategy for color image defogging isthen illustrated in brief in Section 3 we present the proposedvariational histogram equalization framework in detail InSection 4 our method is compared with other representativeand relevant approaches in some simulation experimentswhile this paper is summarized in the last section

2 Physical Degradation Model inHSI Color Space

Generally because of suspended particlesrsquo absorption andscattering in the foggy weather the scene-reflected light 119869(119909)goes through undesirable attenuation and dispersal Worsestill the airlight 119860 scattered by those particles pools muchlight quantity to the observers Both of these two factors areclosely associated with scene depth 119889(119909) so the observedscene appearance 119871(119909) can be illustrated in the followingexpression according to Koschmiederrsquos law [4]

119871 (119909) = 119869 (119909) 119905 (119909)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Direct attenuation+ 119860 (1 minus 119905 (119909))⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Theveiling light (1)

where 119905(119909) is the scene transmission related to 119889(119909) atposition 119909 and it can be described as 119905(119909) = 119890

minus120573119889(119909) with120573 denoting the scattering coefficient of the atmosphere Theadditive model in formula (1) is obviously comprised of twomajor parts direct attenuation and the veiling light Theformer part interprets that 119869(119909) is attenuated and dispersedwhile the latter one is the main cause of color distortionWith the physical degradation model image restoration infoggy scenes is essentially an ill-posed problem that recovers119869(119909) 119860 and 119905(119909) from the observed image 119871(119909) Notice thatHSI space is practical for human interpretation whichmakesit an ideal tool for processing images [15] The space con-tains three components (hue saturation and intensity) thatcan be transformed by RGB channels With the equivalenttransformation relationship from RGB to HSI the physicaldegradation model can be expressed in HSI color space by

119867119871= 119867119869 (2)

119878119871=119868119869

119868119871

119905119878119869 (3)

119868119871= 119868119869119905 + 119860 (1 minus 119905) (4)

where 119867119871 119878119871 and 119868

119871are the HSI components of 119871(119909) while

119867119869 119878119869 and 119868

119869represent ones of 119869(119909) respectively Formula (2)

implies that the hue component keeps constant according tothe color constancy theory while formula (3) verifies that fog





119878119869=119868119871(119868119869minus 119860)

119868119869(119868119871minus 119860)

119878119871 (5)






119864TV (119891) = intΛ

1

ℎ119894(119903 119909)

119891119903(119903 119909)2119889119909 119889119903

+ 1205741intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 120583 (119909))

2

119889119909

+ 1205742intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816 119889119909 119889119903

1198641198671

(119891) = intΛ

1

ℎ119894(119903 119909)

119891119903(119903 119909)2119889119909 119889119903

+ 1205741intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 120583 (119909))

2

119889119909

+ 1205742intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816

2

119889119909 119889119903

(6)


119894(119903) and ℎ









119868119869=1

119905119868119871minus(1 minus 119905)

119905119860 (7)


119868119869=1

119905119868119871minus(1 minus 119905)

119905119860 (8)

where 119868119869and 119868









channels



constraint





(a) (b) (c)



intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 119868

119869(119909))

2

119889119909

= intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus

1

119905119868119871(119909) +

(1 minus 119905)

119905119860)

2

119889119909

(9)





1norms perform well




1norm in flat areas







div [1205881015840(|119888|)

|119888|119888] = 0 (10)


div [1205881015840(|119888|)

|119888|119888] =

1205881015840(|119888|)

|119888|119891120585120585

+ [1205881015840(|119888|)

|119888|+ (

1205881015840(|119888|)

|119888|)

1015840

sdot |119888|] 119891120578120578

(11)

where 119891120585120585

and 119891120578120578


120585120585and 119891



1205881015840(119888)

119888

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)

1205881015840 (119888) 119888

119888rarrinfin

997888997888997888997888rarr 120594 le 0

(12)


1norm


1205881015840(119888)

119888

119888rarr0

997888997888997888rarr 120581

(1205881015840(|119888|)

|119888|)

1015840

119888rarr0

997888997888997888rarr 0

(13)


120588 (119888) =1198882

1 + 1198882 (14)


1205881015840(119888)

119888=

2

(1 + 1198882)2

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888) =

2 minus 61198882

(1 + 1198882)3

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)

1205881015840 (119888) 119888=1 minus 3119888

2

1 + 1198882

119888rarrinfin

997888997888997888997888rarr minus3 le 0

1205881015840(119888)

119888=

2

(1 + 1198882)2

119888rarr0

997888997888997888rarr 2

(1205881015840(|119888|)

|119888|)

1015840

=minus8119888

(1 + 1198882)3

119888rarr0

997888997888997888rarr 0

(15)


intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816

2

1 +1003816100381610038161003816nabla119891

1003816100381610038161003816

2119889119909 119889119903 (16)


119864 (119891) = intΛ

1

ℎ119894(119903 119909)

119891119903(119903 119909)2119889119909 119889119903

+ 1205741intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 119868

119869(119909))

2

119889119909

+ 1205742intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816

2

1 +1003816100381610038161003816nabla119891

1003816100381610038161003816

2119889119909 119889119903

(17)


119869(119909) The



2(1 +






119877120588(|nabla119906|) = int

Ω

120588 (|nabla119906|) 119889Ω (18)


sum

120572isinℓ

120597

120597ℓ(minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

+ 120582 (1199060

120572minus 119906120572) = 0 (19)



120572

as


120572isinℓ

(120597119906

120597ℓ

10038161003816100381610038161003816100381610038161003816120572

)

2

= radicsum

120572isinℓ

(119906120573minus 119906120572)2

(20)


120597

120597ℓ(minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

= (minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120573


|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

=

minus1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

(120597119906

120597ℓ

10038161003816100381610038161003816100381610038161003816120573

) +1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(120597119906

120597ℓ)

10038161003816100381610038161003816100381610038161003816120572

=

minus1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

(119906120572minus 119906120573) +

1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(119906120573minus 119906120572)

= (

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

) (119906120573minus 119906120572)

(21)


sum

120572isinℓ

(

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

) (119906120573minus 119906120572)

+ 120582 (1199060

120572minus 119906120572) = 0

(22)


from formula (22)

119908120572120573=

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(23)


120572120573(119906) goes to

(1|nabla120572119906| + 1|nabla



2(1 + 119888



119908120572120573=

2

(1 +10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

2

)

2+

2

(1 +1003816100381610038161003816nabla120572119906

1003816100381610038161003816

2

)2 (24)



119902 (119909) prop 119889 (119909) prop (V (119909) minus 119904 (119909)) (25)


119860 (119909) prop (V (119909) minus 119904 (119909)) (26)



dark(119909) and 119869



119869dark

(119909) = min119888isin119877119866119861

119869119888(119909) (27)

119869bright

(119909) = max119888isin119877119866119861

119869119888(119909) (28)



(a) (b) (c)


1= 1 (b) 120574

1= 100 and (c) 120574

1= 10000

00592

00478004640035700346

00246002380014400137

0005

AM

BE in

dex

0001002003004005006007

L08005 L08010

110100

100010000

Parameter 1205741

(a)

00618 0060900537 00571

0048900557

0046600532

0046300525

0001002003004005006007

L08005 L08010

MSE

inde

x

110100

100010000

Parameter 1205741

(b)



(a) (b) (c)


2= 1 (b) 120574

2= 100 and (c) 120574

2= 10000

79518426

66277096

5647 58084911 4851

4312 4159

0102030405060708090

EI in

dex

L08005 L08010

110100

100010000

Parameter 1205742

(a)

00532

00758

00489

00608

004880055700514 0055800554 00583

0001002003004005006007008

MSE

inde

x

L08005 L08010

110100

100010000

Parameter 1205742

(b)



119860 (119909) =119860+(119909) minus 119860

minus(119909)

120579+ (119909) minus 120579minus(119909)

(V (119909) minus 119904 (119909))

+119860minus(119909) sdot 120579

+(119909) minus 119860

+(119909) sdot 120579

minus(119909)

120579+ (119909) minus 120579minus(119909)

(29)



119869is


119869and 119878119869




(b) (c)(a)




1= 1205820

2= 1205820

3= 0 120573 = 1

1205741= 100 and 120574


1and


1 1205742) where 120574

1 1205742isin





(b) (c)(a)



1freely




1




1reaches 100











2increases







2is set to





(b) (c)(a)





1norms are



(b) (c)(a)







(b) (c)(a)





(b) (c)(a)




5 Conclusion



Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex


(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees


(b)



289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)



187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)



371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119878119869=119868119871(119868119869minus 119860)

119868119869(119868119871minus 119860)

119878119871 (5)






119864TV (119891) = intΛ

1

ℎ119894(119903 119909)

119891119903(119903 119909)2119889119909 119889119903

+ 1205741intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 120583 (119909))

2

119889119909

+ 1205742intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816 119889119909 119889119903

1198641198671

(119891) = intΛ

1

ℎ119894(119903 119909)

119891119903(119903 119909)2119889119909 119889119903

+ 1205741intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 120583 (119909))

2

119889119909

+ 1205742intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816

2

119889119909 119889119903

(6)


119894(119903) and ℎ









119868119869=1

119905119868119871minus(1 minus 119905)

119905119860 (7)


119868119869=1

119905119868119871minus(1 minus 119905)

119905119860 (8)

where 119868119869and 119868









channels



constraint





(a) (b) (c)



intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 119868

119869(119909))

2

119889119909

= intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus

1

119905119868119871(119909) +

(1 minus 119905)

119905119860)

2

119889119909

(9)





1norms perform well




1norm in flat areas







div [1205881015840(|119888|)

|119888|119888] = 0 (10)


div [1205881015840(|119888|)

|119888|119888] =

1205881015840(|119888|)

|119888|119891120585120585

+ [1205881015840(|119888|)

|119888|+ (

1205881015840(|119888|)

|119888|)

1015840

sdot |119888|] 119891120578120578

(11)

where 119891120585120585

and 119891120578120578


120585120585and 119891



1205881015840(119888)

119888

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)

1205881015840 (119888) 119888

119888rarrinfin

997888997888997888997888rarr 120594 le 0

(12)


1norm


1205881015840(119888)

119888

119888rarr0

997888997888997888rarr 120581

(1205881015840(|119888|)

|119888|)

1015840

119888rarr0

997888997888997888rarr 0

(13)


120588 (119888) =1198882

1 + 1198882 (14)


1205881015840(119888)

119888=

2

(1 + 1198882)2

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888) =

2 minus 61198882

(1 + 1198882)3

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)

1205881015840 (119888) 119888=1 minus 3119888

2

1 + 1198882

119888rarrinfin

997888997888997888997888rarr minus3 le 0

1205881015840(119888)

119888=

2

(1 + 1198882)2

119888rarr0

997888997888997888rarr 2

(1205881015840(|119888|)

|119888|)

1015840

=minus8119888

(1 + 1198882)3

119888rarr0

997888997888997888rarr 0

(15)


intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816

2

1 +1003816100381610038161003816nabla119891

1003816100381610038161003816

2119889119909 119889119903 (16)


119864 (119891) = intΛ

1

ℎ119894(119903 119909)

119891119903(119903 119909)2119889119909 119889119903

+ 1205741intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 119868

119869(119909))

2

119889119909

+ 1205742intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816

2

1 +1003816100381610038161003816nabla119891

1003816100381610038161003816

2119889119909 119889119903

(17)


119869(119909) The



2(1 +






119877120588(|nabla119906|) = int

Ω

120588 (|nabla119906|) 119889Ω (18)


sum

120572isinℓ

120597

120597ℓ(minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

+ 120582 (1199060

120572minus 119906120572) = 0 (19)



120572

as


120572isinℓ

(120597119906

120597ℓ

10038161003816100381610038161003816100381610038161003816120572

)

2

= radicsum

120572isinℓ

(119906120573minus 119906120572)2

(20)


120597

120597ℓ(minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

= (minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120573


|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

=

minus1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

(120597119906

120597ℓ

10038161003816100381610038161003816100381610038161003816120573

) +1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(120597119906

120597ℓ)

10038161003816100381610038161003816100381610038161003816120572

=

minus1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

(119906120572minus 119906120573) +

1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(119906120573minus 119906120572)

= (

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

) (119906120573minus 119906120572)

(21)


sum

120572isinℓ

(

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

) (119906120573minus 119906120572)

+ 120582 (1199060

120572minus 119906120572) = 0

(22)


from formula (22)

119908120572120573=

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(23)


120572120573(119906) goes to

(1|nabla120572119906| + 1|nabla



2(1 + 119888



119908120572120573=

2

(1 +10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

2

)

2+

2

(1 +1003816100381610038161003816nabla120572119906

1003816100381610038161003816

2

)2 (24)



119902 (119909) prop 119889 (119909) prop (V (119909) minus 119904 (119909)) (25)


119860 (119909) prop (V (119909) minus 119904 (119909)) (26)



dark(119909) and 119869



119869dark

(119909) = min119888isin119877119866119861

119869119888(119909) (27)

119869bright

(119909) = max119888isin119877119866119861

119869119888(119909) (28)



(a) (b) (c)


1= 1 (b) 120574

1= 100 and (c) 120574

1= 10000

00592

00478004640035700346

00246002380014400137

0005

AM

BE in

dex

0001002003004005006007

L08005 L08010

110100

100010000

Parameter 1205741

(a)

00618 0060900537 00571

0048900557

0046600532

0046300525

0001002003004005006007

L08005 L08010

MSE

inde

x

110100

100010000

Parameter 1205741

(b)



(a) (b) (c)


2= 1 (b) 120574

2= 100 and (c) 120574

2= 10000

79518426

66277096

5647 58084911 4851

4312 4159

0102030405060708090

EI in

dex

L08005 L08010

110100

100010000

Parameter 1205742

(a)

00532

00758

00489

00608

004880055700514 0055800554 00583

0001002003004005006007008

MSE

inde

x

L08005 L08010

110100

100010000

Parameter 1205742

(b)



119860 (119909) =119860+(119909) minus 119860

minus(119909)

120579+ (119909) minus 120579minus(119909)

(V (119909) minus 119904 (119909))

+119860minus(119909) sdot 120579

+(119909) minus 119860

+(119909) sdot 120579

minus(119909)

120579+ (119909) minus 120579minus(119909)

(29)



119869is


119869and 119878119869




(b) (c)(a)




1= 1205820

2= 1205820

3= 0 120573 = 1

1205741= 100 and 120574


1and


1 1205742) where 120574

1 1205742isin





(b) (c)(a)



1freely




1




1reaches 100











2increases







2is set to





(b) (c)(a)





1norms are



(b) (c)(a)







(b) (c)(a)





(b) (c)(a)




5 Conclusion



Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex


(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees


(b)



289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)



187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)



371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














channels



constraint





(a) (b) (c)



intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 119868

119869(119909))

2

119889119909

= intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus

1

119905119868119871(119909) +

(1 minus 119905)

119905119860)

2

119889119909

(9)





1norms perform well




1norm in flat areas







div [1205881015840(|119888|)

|119888|119888] = 0 (10)


div [1205881015840(|119888|)

|119888|119888] =

1205881015840(|119888|)

|119888|119891120585120585

+ [1205881015840(|119888|)

|119888|+ (

1205881015840(|119888|)

|119888|)

1015840

sdot |119888|] 119891120578120578

(11)

where 119891120585120585

and 119891120578120578


120585120585and 119891



1205881015840(119888)

119888

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)

1205881015840 (119888) 119888

119888rarrinfin

997888997888997888997888rarr 120594 le 0

(12)


1norm


1205881015840(119888)

119888

119888rarr0

997888997888997888rarr 120581

(1205881015840(|119888|)

|119888|)

1015840

119888rarr0

997888997888997888rarr 0

(13)


120588 (119888) =1198882

1 + 1198882 (14)


1205881015840(119888)

119888=

2

(1 + 1198882)2

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888) =

2 minus 61198882

(1 + 1198882)3

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)

1205881015840 (119888) 119888=1 minus 3119888

2

1 + 1198882

119888rarrinfin

997888997888997888997888rarr minus3 le 0

1205881015840(119888)

119888=

2

(1 + 1198882)2

119888rarr0

997888997888997888rarr 2

(1205881015840(|119888|)

|119888|)

1015840

=minus8119888

(1 + 1198882)3

119888rarr0

997888997888997888rarr 0

(15)


intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816

2

1 +1003816100381610038161003816nabla119891

1003816100381610038161003816

2119889119909 119889119903 (16)


119864 (119891) = intΛ

1

ℎ119894(119903 119909)

119891119903(119903 119909)2119889119909 119889119903

+ 1205741intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 119868

119869(119909))

2

119889119909

+ 1205742intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816

2

1 +1003816100381610038161003816nabla119891

1003816100381610038161003816

2119889119909 119889119903

(17)


119869(119909) The



2(1 +






119877120588(|nabla119906|) = int

Ω

120588 (|nabla119906|) 119889Ω (18)


sum

120572isinℓ

120597

120597ℓ(minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

+ 120582 (1199060

120572minus 119906120572) = 0 (19)



120572

as


120572isinℓ

(120597119906

120597ℓ

10038161003816100381610038161003816100381610038161003816120572

)

2

= radicsum

120572isinℓ

(119906120573minus 119906120572)2

(20)


120597

120597ℓ(minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

= (minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120573


|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

=

minus1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

(120597119906

120597ℓ

10038161003816100381610038161003816100381610038161003816120573

) +1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(120597119906

120597ℓ)

10038161003816100381610038161003816100381610038161003816120572

=

minus1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

(119906120572minus 119906120573) +

1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(119906120573minus 119906120572)

= (

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

) (119906120573minus 119906120572)

(21)


sum

120572isinℓ

(

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

) (119906120573minus 119906120572)

+ 120582 (1199060

120572minus 119906120572) = 0

(22)


from formula (22)

119908120572120573=

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(23)


120572120573(119906) goes to

(1|nabla120572119906| + 1|nabla



2(1 + 119888



119908120572120573=

2

(1 +10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

2

)

2+

2

(1 +1003816100381610038161003816nabla120572119906

1003816100381610038161003816

2

)2 (24)



119902 (119909) prop 119889 (119909) prop (V (119909) minus 119904 (119909)) (25)


119860 (119909) prop (V (119909) minus 119904 (119909)) (26)



dark(119909) and 119869



119869dark

(119909) = min119888isin119877119866119861

119869119888(119909) (27)

119869bright

(119909) = max119888isin119877119866119861

119869119888(119909) (28)



(a) (b) (c)


1= 1 (b) 120574

1= 100 and (c) 120574

1= 10000

00592

00478004640035700346

00246002380014400137

0005

AM

BE in

dex

0001002003004005006007

L08005 L08010

110100

100010000

Parameter 1205741

(a)

00618 0060900537 00571

0048900557

0046600532

0046300525

0001002003004005006007

L08005 L08010

MSE

inde

x

110100

100010000

Parameter 1205741

(b)



(a) (b) (c)


2= 1 (b) 120574

2= 100 and (c) 120574

2= 10000

79518426

66277096

5647 58084911 4851

4312 4159

0102030405060708090

EI in

dex

L08005 L08010

110100

100010000

Parameter 1205742

(a)

00532

00758

00489

00608

004880055700514 0055800554 00583

0001002003004005006007008

MSE

inde

x

L08005 L08010

110100

100010000

Parameter 1205742

(b)



119860 (119909) =119860+(119909) minus 119860

minus(119909)

120579+ (119909) minus 120579minus(119909)

(V (119909) minus 119904 (119909))

+119860minus(119909) sdot 120579

+(119909) minus 119860

+(119909) sdot 120579

minus(119909)

120579+ (119909) minus 120579minus(119909)

(29)



119869is


119869and 119878119869




(b) (c)(a)




1= 1205820

2= 1205820

3= 0 120573 = 1

1205741= 100 and 120574


1and


1 1205742) where 120574

1 1205742isin





(b) (c)(a)



1freely




1




1reaches 100











2increases







2is set to





(b) (c)(a)





1norms are



(b) (c)(a)







(b) (c)(a)





(b) (c)(a)




5 Conclusion



Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex


(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees


(b)



289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)



187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)



371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














div [1205881015840(|119888|)

|119888|119888] = 0 (10)


div [1205881015840(|119888|)

|119888|119888] =

1205881015840(|119888|)

|119888|119891120585120585

+ [1205881015840(|119888|)

|119888|+ (

1205881015840(|119888|)

|119888|)

1015840

sdot |119888|] 119891120578120578

(11)

where 119891120585120585

and 119891120578120578


120585120585and 119891



1205881015840(119888)

119888

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)

1205881015840 (119888) 119888

119888rarrinfin

997888997888997888997888rarr 120594 le 0

(12)


1norm


1205881015840(119888)

119888

119888rarr0

997888997888997888rarr 120581

(1205881015840(|119888|)

|119888|)

1015840

119888rarr0

997888997888997888rarr 0

(13)


120588 (119888) =1198882

1 + 1198882 (14)


1205881015840(119888)

119888=

2

(1 + 1198882)2

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888) =

2 minus 61198882

(1 + 1198882)3

119888rarrinfin

997888997888997888997888rarr 0

12058810158401015840(119888)

1205881015840 (119888) 119888=1 minus 3119888

2

1 + 1198882

119888rarrinfin

997888997888997888997888rarr minus3 le 0

1205881015840(119888)

119888=

2

(1 + 1198882)2

119888rarr0

997888997888997888rarr 2

(1205881015840(|119888|)

|119888|)

1015840

=minus8119888

(1 + 1198882)3

119888rarr0

997888997888997888rarr 0

(15)


intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816

2

1 +1003816100381610038161003816nabla119891

1003816100381610038161003816

2119889119909 119889119903 (16)


119864 (119891) = intΛ

1

ℎ119894(119903 119909)

119891119903(119903 119909)2119889119909 119889119903

+ 1205741intΩ

(int

1

0

119891ℎ119894(119903 119909) 119889119903 minus 119868

119869(119909))

2

119889119909

+ 1205742intΛ

1003816100381610038161003816nabla1198911003816100381610038161003816

2

1 +1003816100381610038161003816nabla119891

1003816100381610038161003816

2119889119909 119889119903

(17)


119869(119909) The



2(1 +






119877120588(|nabla119906|) = int

Ω

120588 (|nabla119906|) 119889Ω (18)


sum

120572isinℓ

120597

120597ℓ(minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

+ 120582 (1199060

120572minus 119906120572) = 0 (19)



120572

as


120572isinℓ

(120597119906

120597ℓ

10038161003816100381610038161003816100381610038161003816120572

)

2

= radicsum

120572isinℓ

(119906120573minus 119906120572)2

(20)


120597

120597ℓ(minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

= (minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120573


|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

=

minus1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

(120597119906

120597ℓ

10038161003816100381610038161003816100381610038161003816120573

) +1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(120597119906

120597ℓ)

10038161003816100381610038161003816100381610038161003816120572

=

minus1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

(119906120572minus 119906120573) +

1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(119906120573minus 119906120572)

= (

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

) (119906120573minus 119906120572)

(21)


sum

120572isinℓ

(

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

) (119906120573minus 119906120572)

+ 120582 (1199060

120572minus 119906120572) = 0

(22)


from formula (22)

119908120572120573=

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(23)


120572120573(119906) goes to

(1|nabla120572119906| + 1|nabla



2(1 + 119888



119908120572120573=

2

(1 +10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

2

)

2+

2

(1 +1003816100381610038161003816nabla120572119906

1003816100381610038161003816

2

)2 (24)



119902 (119909) prop 119889 (119909) prop (V (119909) minus 119904 (119909)) (25)


119860 (119909) prop (V (119909) minus 119904 (119909)) (26)



dark(119909) and 119869



119869dark

(119909) = min119888isin119877119866119861

119869119888(119909) (27)

119869bright

(119909) = max119888isin119877119866119861

119869119888(119909) (28)



(a) (b) (c)


1= 1 (b) 120574

1= 100 and (c) 120574

1= 10000

00592

00478004640035700346

00246002380014400137

0005

AM

BE in

dex

0001002003004005006007

L08005 L08010

110100

100010000

Parameter 1205741

(a)

00618 0060900537 00571

0048900557

0046600532

0046300525

0001002003004005006007

L08005 L08010

MSE

inde

x

110100

100010000

Parameter 1205741

(b)



(a) (b) (c)


2= 1 (b) 120574

2= 100 and (c) 120574

2= 10000

79518426

66277096

5647 58084911 4851

4312 4159

0102030405060708090

EI in

dex

L08005 L08010

110100

100010000

Parameter 1205742

(a)

00532

00758

00489

00608

004880055700514 0055800554 00583

0001002003004005006007008

MSE

inde

x

L08005 L08010

110100

100010000

Parameter 1205742

(b)



119860 (119909) =119860+(119909) minus 119860

minus(119909)

120579+ (119909) minus 120579minus(119909)

(V (119909) minus 119904 (119909))

+119860minus(119909) sdot 120579

+(119909) minus 119860

+(119909) sdot 120579

minus(119909)

120579+ (119909) minus 120579minus(119909)

(29)



119869is


119869and 119878119869




(b) (c)(a)




1= 1205820

2= 1205820

3= 0 120573 = 1

1205741= 100 and 120574


1and


1 1205742) where 120574

1 1205742isin





(b) (c)(a)



1freely




1




1reaches 100











2increases







2is set to





(b) (c)(a)





1norms are



(b) (c)(a)







(b) (c)(a)





(b) (c)(a)




5 Conclusion



Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex


(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees


(b)



289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)



187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)



371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119877120588(|nabla119906|) = int

Ω

120588 (|nabla119906|) 119889Ω (18)


sum

120572isinℓ

120597

120597ℓ(minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

+ 120582 (1199060

120572minus 119906120572) = 0 (19)



120572

as


120572isinℓ

(120597119906

120597ℓ

10038161003816100381610038161003816100381610038161003816120572

)

2

= radicsum

120572isinℓ

(119906120573minus 119906120572)2

(20)


120597

120597ℓ(minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

= (minus1205881015840(|nabla119906|)

|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120573


|nabla119906|

120597119906

120597ℓ)

100381610038161003816100381610038161003816100381610038161003816120572

=

minus1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

(120597119906

120597ℓ

10038161003816100381610038161003816100381610038161003816120573

) +1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(120597119906

120597ℓ)

10038161003816100381610038161003816100381610038161003816120572

=

minus1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

(119906120572minus 119906120573) +

1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(119906120573minus 119906120572)

= (

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

) (119906120573minus 119906120572)

(21)


sum

120572isinℓ

(

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

) (119906120573minus 119906120572)

+ 120582 (1199060

120572minus 119906120572) = 0

(22)


from formula (22)

119908120572120573=

1205881015840(10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816)

10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

+1205881015840(1003816100381610038161003816nabla120572119906

1003816100381610038161003816)

1003816100381610038161003816nabla1205721199061003816100381610038161003816

(23)


120572120573(119906) goes to

(1|nabla120572119906| + 1|nabla



2(1 + 119888



119908120572120573=

2

(1 +10038161003816100381610038161003816nabla12057311990610038161003816100381610038161003816

2

)

2+

2

(1 +1003816100381610038161003816nabla120572119906

1003816100381610038161003816

2

)2 (24)



119902 (119909) prop 119889 (119909) prop (V (119909) minus 119904 (119909)) (25)


119860 (119909) prop (V (119909) minus 119904 (119909)) (26)



dark(119909) and 119869



119869dark

(119909) = min119888isin119877119866119861

119869119888(119909) (27)

119869bright

(119909) = max119888isin119877119866119861

119869119888(119909) (28)



(a) (b) (c)


1= 1 (b) 120574

1= 100 and (c) 120574

1= 10000

00592

00478004640035700346

00246002380014400137

0005

AM

BE in

dex

0001002003004005006007

L08005 L08010

110100

100010000

Parameter 1205741

(a)

00618 0060900537 00571

0048900557

0046600532

0046300525

0001002003004005006007

L08005 L08010

MSE

inde

x

110100

100010000

Parameter 1205741

(b)



(a) (b) (c)


2= 1 (b) 120574

2= 100 and (c) 120574

2= 10000

79518426

66277096

5647 58084911 4851

4312 4159

0102030405060708090

EI in

dex

L08005 L08010

110100

100010000

Parameter 1205742

(a)

00532

00758

00489

00608

004880055700514 0055800554 00583

0001002003004005006007008

MSE

inde

x

L08005 L08010

110100

100010000

Parameter 1205742

(b)



119860 (119909) =119860+(119909) minus 119860

minus(119909)

120579+ (119909) minus 120579minus(119909)

(V (119909) minus 119904 (119909))

+119860minus(119909) sdot 120579

+(119909) minus 119860

+(119909) sdot 120579

minus(119909)

120579+ (119909) minus 120579minus(119909)

(29)



119869is


119869and 119878119869




(b) (c)(a)




1= 1205820

2= 1205820

3= 0 120573 = 1

1205741= 100 and 120574


1and


1 1205742) where 120574

1 1205742isin





(b) (c)(a)



1freely




1




1reaches 100











2increases







2is set to





(b) (c)(a)





1norms are



(b) (c)(a)







(b) (c)(a)





(b) (c)(a)




5 Conclusion



Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex


(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees


(b)



289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)



187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)



371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(a) (b) (c)


1= 1 (b) 120574

1= 100 and (c) 120574

1= 10000

00592

00478004640035700346

00246002380014400137

0005

AM

BE in

dex

0001002003004005006007

L08005 L08010

110100

100010000

Parameter 1205741

(a)

00618 0060900537 00571

0048900557

0046600532

0046300525

0001002003004005006007

L08005 L08010

MSE

inde

x

110100

100010000

Parameter 1205741

(b)



(a) (b) (c)


2= 1 (b) 120574

2= 100 and (c) 120574

2= 10000

79518426

66277096

5647 58084911 4851

4312 4159

0102030405060708090

EI in

dex

L08005 L08010

110100

100010000

Parameter 1205742

(a)

00532

00758

00489

00608

004880055700514 0055800554 00583

0001002003004005006007008

MSE

inde

x

L08005 L08010

110100

100010000

Parameter 1205742

(b)



119860 (119909) =119860+(119909) minus 119860

minus(119909)

120579+ (119909) minus 120579minus(119909)

(V (119909) minus 119904 (119909))

+119860minus(119909) sdot 120579

+(119909) minus 119860

+(119909) sdot 120579

minus(119909)

120579+ (119909) minus 120579minus(119909)

(29)



119869is


119869and 119878119869




(b) (c)(a)




1= 1205820

2= 1205820

3= 0 120573 = 1

1205741= 100 and 120574


1and


1 1205742) where 120574

1 1205742isin





(b) (c)(a)



1freely




1




1reaches 100











2increases







2is set to





(b) (c)(a)





1norms are



(b) (c)(a)







(b) (c)(a)





(b) (c)(a)




5 Conclusion



Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex


(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees


(b)



289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)



187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)



371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)


2= 1 (b) 120574

2= 100 and (c) 120574

2= 10000

79518426

66277096

5647 58084911 4851

4312 4159

0102030405060708090

EI in

dex

L08005 L08010

110100

100010000

Parameter 1205742

(a)

00532

00758

00489

00608

004880055700514 0055800554 00583

0001002003004005006007008

MSE

inde

x

L08005 L08010

110100

100010000

Parameter 1205742

(b)



119860 (119909) =119860+(119909) minus 119860

minus(119909)

120579+ (119909) minus 120579minus(119909)

(V (119909) minus 119904 (119909))

+119860minus(119909) sdot 120579

+(119909) minus 119860

+(119909) sdot 120579

minus(119909)

120579+ (119909) minus 120579minus(119909)

(29)



119869is


119869and 119878119869




(b) (c)(a)




1= 1205820

2= 1205820

3= 0 120573 = 1

1205741= 100 and 120574


1and


1 1205742) where 120574

1 1205742isin





(b) (c)(a)



1freely




1




1reaches 100











2increases







2is set to





(b) (c)(a)





1norms are



(b) (c)(a)







(b) (c)(a)





(b) (c)(a)




5 Conclusion



Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex


(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees


(b)



289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)



187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)



371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(b) (c)(a)




1= 1205820

2= 1205820

3= 0 120573 = 1

1205741= 100 and 120574


1and


1 1205742) where 120574

1 1205742isin





(b) (c)(a)



1freely




1




1reaches 100











2increases







2is set to





(b) (c)(a)





1norms are



(b) (c)(a)







(b) (c)(a)





(b) (c)(a)




5 Conclusion



Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex


(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees


(b)



289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)



187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)



371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(b) (c)(a)



1freely




1




1reaches 100











2increases







2is set to





(b) (c)(a)





1norms are



(b) (c)(a)







(b) (c)(a)





(b) (c)(a)




5 Conclusion



Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex


(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees


(b)



289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)



187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)



371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(b) (c)(a)





1norms are



(b) (c)(a)







(b) (c)(a)





(b) (c)(a)




5 Conclusion



Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex


(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees


(b)



289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)



187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)



371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(b) (c)(a)







(b) (c)(a)





(b) (c)(a)




5 Conclusion



Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex


(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees


(b)



289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)



187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)



371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(b) (c)(a)





(b) (c)(a)




5 Conclusion



Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex


(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees


(b)



289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)



187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)



371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(b) (c)(a)




5 Conclusion



Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex


(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees


(b)



289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)



187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)



371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Defogging methods

8851261041377

1116164

1494683

1184214

1578979

020406080

100120140160180

House Trees

EI in

dex


(a)

138762

7321557666

2657246015

18697

02468

10121416

NIQ

E in

dex

House Trees


(b)



289408 441146

685053506521

1169585

662798

020406080

100120140

Street Train

EI in

dex

(a)


98976 1006492862

70718

3350148749

0

2

4

6

8

10

12

Street Train

NIQ

E in

dex

(b)



187254 202303227732

338662

59876511282

010203040506070

Cliff Tower

EI in

dex

(a)


109864

43344

104285

364944689127977

02468

1012

Cliff Tower

NIQ

E in

dex

(b)



371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










371 399

13471536

427 463

02468

1012141618

House Trees

Tim

e cos

t (s)


(a)

447 409

296 273

538 505

0123456

Street Train

Tim

e cos

t (s)


(b)

32213515

2169 2342

524 603

05

10152025303540

Cliff Tower

Tim

e cos

t (s)


(c)



Competing Interests


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Variational Histogram Equalization for...

Documents

Transcript of Research Article Variational Histogram Equalization for...