research article multiscale single image dehazing...

Research ArticleMultiscale Single Image Dehazing Based onAdaptive Wavelet Fusion

Wei Wang12 Wenhui Li1 Qingji Guan3 and Miao Qi4

1College of Computer Science and Technology Jilin University Changchun 130023 China2Computer School Dalian Nationalities University Dalian 116600 China3Beijing Key Lab of Traffic Data Analysis and Mining Beijing Jiaotong University Beijing 100044 China4School of Computer Science and Information Technology Northeast Normal University Changchun 130117 China

Correspondence should be addressed to Miao Qi qim801nenueducn

Received 26 May 2015 Revised 28 August 2015 Accepted 30 August 2015

Academic Editor Chih-Cheng Hung

Copyright copy 2015 Wei Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Removing the haze effects on images or videos is a challenging and meaningful task for image processing and computer visionapplications In this paper we propose a multiscale fusion method to remove the haze from a single image Based on the existingdark channel prior and optics theory two atmospheric veils with different scales are first derived from the hazy image Then anovel and adaptive local similarity-basedwavelet fusionmethod is proposed for preserving the significant scene depth property andavoiding blocky artifacts Finally the clear haze-free image is restored by solving the atmospheric scattering model Experimentalresults demonstrate that the proposed method can yield comparative or even better results than several state-of-the-art methodsby subjective and objective evaluations

1 Introduction

Outdoor scene images captured in the bad weather are oftendegraded due to the existence of the haze fog mist or othermedia The main reason is that light rays reflected by theobject scene are absorbed by the medium aerosols resultingin the direct attenuation At the same time the airlight is alsoscattered by these aerosols during the propagation from thescene points to the observer Consequently captured imageseasily display poor quality such as low contrast distortedcolor and inferior visibility

Moreover most of the vision systems such as outdoorsurveillance remote sensing object detection and trackingsystem would not work effectively due to the influenceof haze Accordingly attention has been paid widely toimage dehazing techniques in computer vision and imageprocessing applications in recent years However recoveringthe original clear scene is a challenging task especially whenonly one hazy image is given At present some dehazingmethods have been proposed They are mainly divided intotwo categories multiple images dehazing and single image

dehazing Narasimhan and Nayar [1 2] presented a physics-based model that described the appearances of scene infog or haze weather conditions Based on the observationthat the scene depth was invariant in different bad weatherconditions while the camera was static they estimated thedepth map and restored the scene contrast using two ormore captured images with the same scene Although thesemethods can produce good results in certain circumstancesit is hard to obtain such multiple images from the same scenein different bad weathers Assuming the airlight was partiallypolarized or unpolarized whereas the scene was polarizedpolarization based methods [3ndash7] were proposed to removethe haze effects by using multi-images with different anglesof polarized filter However polarization filtering might loseefficacy to dense fog situations due to the scattering Oneof the key points for removing the unwanted influence ofthe hazy image is to get the accurate transmission map andthe atmosphere light Kopf et al [8] obtained the scenedepth map and corrected the color bias with the aid ofexisting approximate 3D model or georeferenced digitalterrain and urban models Nevertheless it is limited in case

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 131082 14 pageshttpdxdoiorg1011552015131082

2 Mathematical Problems in Engineering

no accurate or approximate models are obtained for someexisting images

Recently a number of single image dehazing methodshave been developed for overcoming the weakness of multi-images or getting 3D model in advance Fattal [9] estimatedthe scene albedo and depth with the hypothesis that thesurface shading was locally uncorrelated with the transmis-sionThis method is effective but fails when the hypothesis isbroken Tan [10] observed that the images in clear days hadstronger contrast than those in bad weather Simultaneouslythe variation of airlight mainly depended on the scene depthand tended to be smooth Consequently they developed amaximization local contrast method for restoring the sceneAlthough this method can enhance the visibility it yieldsserious halos and oversaturated colors Based on the darkchannel prior of the haze-free natural images He et al [11]estimated the thickness of the haze and recovered a highquality haze-free image Although this method can achieveideal dehazing results the cost of softmatting for refining thetransmission map is very large and unsuitable for real-timeapplications Tarel and Hautiere [12] proposed a linear com-plexity visibility restoration method based on median filterWhile this method can meet the real-time requirements themedian filter is not conformal and edge-preserving To getaround this problem Xiao and Gan [13] proposed a guidedjoint bilateral filter for haze removal By using bilateral filterZhang et al [14] introduced a new algorithm to removehaze from a single image based on the assumption thatonly the variation of transmission could result in large-scale luminance variation in the fog-free image Howeverthis method will fail in case the assumption does not holdNishino et al [15] introduced a Bayesian probabilisticmethodthat jointly estimated the scene albedo and depth from asingle hazy image However since it is hard to estimatethe multiple statistical prior models accurately the dehazingresults obtained by this method may contain unnaturallybright colors More recently C O Ancuti and C Ancuti [16]presented a fusion-based strategy that was able to accuratelyrestore images using only the original hazy or foggy imageThis method has high computing speed and yields similarresults with [8 9 11] Kim et al [17] restored the hazy image byan optimized contrast enhancement procedure that balancedthe cost of contrast and information loss Besides theyproposed a quad-tree subdivision based searching methodfor estimating the atmospheric light Nevertheless the quad-tree searching method will not be robust if the maximumvalue of the final subblock is noise Zhang et al [18] describeda hazy layer estimationmethod using low-rank technique andoverlap averaging schemeThe disadvantage is that it may notwork well for far scenes with heavy fog and great depth jump

In this paper we present a novel single image dehazingmethod based on atmospheric scattering model In thebasis of existing dark channel prior and optics theory twoatmospheric veils with different scales are first derived fromthe hazy imageThen a local similarity-basedwaveletmethodis proposed to fuse the two atmospheric veils adaptivelywhich can reflect the scene depth information well At lastthe scene is restored by solving the atmospheric scatteringmodel

Comparedwith previous single image dehazingmethodsour proposed method exhibits the following advantagesFirstly the proposed adaptive local similarity-based waveletfusion method shows better performance in preserving themost discriminant scene depth Secondly our dehazingmethodperforms per-pixel computation efficiently and yieldsvery little artifacts Moreover the proposed method does notneed the postprocessing like He et alrsquos exposure [11] andTarel and Hautierersquos tone mapping [12] Finally we test ourmethod on a number of natural and synthetic hazy imagesThe comparative results display comparative or even betterdehazing results by subjective and objective evaluations

The rest of this paper is organized as follows In Section 2we review the atmospheric scattering model widely usedin dehazing task Section 3 describes the proposed dehaz-ing method in detail mainly including atmospheric lightand atmospheric veil estimations image fusion and scenerestoration In Section 4 we report and discuss the experi-ment results Finally conclusions are provided in Section 5

2 Atmospheric Scattering Model

In computer vision and image processing the formationof hazy image is commonly described by the atmosphericscattering model [9ndash12]

119868 (119909) = 119869 (119909) 119905 (119909) + 119860 (1 minus 119905 (119909)) (1)

where 119868 is the observed hazy image at location 119909 119869 is thescene radiance (the clear haze-free image) 119860 is the globalatmospheric light and 119905 is the medium transmission whichis depended on the distance 119889(119909) between the object sceneand the observer while the atmosphere is homogenous Thetransmission is described as follows

119905 (119909) = 119890minus120573119889(119909)

(2)

where 120573 is the scattering coefficient of atmosphereThe first term 119869(119909)119905(119909) describes that the reflected light

by the scene points is partially attenuated during the propa-gationThe second term119860(1minus119905(119909)) is called atmospheric veil[12] which is generated by atmosphere scattering and leadsto color shift The task of dehazing is to recover 119869 from 119868

3 Proposed Dehazing Method

The proposed method can be mainly divided into four stepsFirstly the atmospheric light is estimated by the improvedquad-tree subdivision method Then atmospheric veils withdifferent scales are derived from the hazy image Next theaccurate atmospheric veil is obtained by fusing the twoveils Finally the scene radiance is restored by solving theatmospheric scatteringmodelThe flow chart of the proposeddehazing method is shown as in Figure 1

31 Estimation of Atmospheric Light Generally the atmo-spheric light 119860 is always considered as the brightest intensityin the entire image because a large amount of haze makesthe object scene brighter than itself However this will be notreliable when a white object is present in the scene Kim et

Mathematical Problems in Engineering 3

Haze-free image

Coarse-scale veil

Fusion

Transmission map

Fine-scale veil Smooth version Atmospheric light

Hazy image

Restoration

Figure 1 The flow chart of the proposed dehazing method

al [17] proposed a quad-tree subdivision method to estimateatmospheric light aiming at avoiding choosing the whiteobjectThe basic idea of this method is that atmospheric lightshould be estimated from the region with brighter intensityand less texture It searches the region with the largest scoreobtained by the difference of its mean and standard deviationiterativelyThe atmospheric light is estimated as the value thathas the least distance to the pure white

In this work we opt for this method since it is simplebut efficient However it is not robust because the estimatedatmospheric light may be the noise So we regard the meanvalue of the top 5 brightest values in the selected regionas the atmospheric light for rejecting the noise influenceeffectively

32 Dark Channel Prior He et al [11] proposed a dark chan-nel prior to the haze-free outdoor images which supports thatmost of local regions excluding the sky have very low intensityin at least one color (119903119892119887) channel The intensities of thesedark pixels are mainly caused by (1) shadows (2) colorfulobjects or surfaces (3) dark objects or surfaces Similarly ithas been observed that the pixels in sky or hazy regions havehigh values in all color channels The dark channel is definedas

119869dark

(119909) = min119888isin119903119892119887

( min119910isinΩ(119909)

(119869119888

(119910))) (3)

where 119869119888 is a channel of 119869 and Ω(119909) is a local neighborhood

centered at 119909 Based on the dark channel prior 119905(119909) is aconstant in a local image patch Thus we define the localtransmission as (119909) and take the min operation with a localpatch on (1)

min119888isin119903119892119887

( min119910isinΩ(119909)

(119868119888

(119910)))

= min119888isin119903119892119887

( min119910isinΩ(119909)

(119869119888

(119910))) (119909) + 119860 (1 minus (119909))

(4)

Since the 119869dark is close to zero we can estimate the

atmospheric veil term as

119881 = 119860 (1 minus (119909)) = min119888isin119903119892119887

( min119910isinΩ(119909)

(119868119888

(119910))) (5)

It is coarsely good but contains some block effects sincethe atmospheric veil is not always constant in a patch That isto say a local patch may contain the region where the scenedepth is discontinuous

For the sake of avoiding this problem Tarel and Hautiere[12] estimated the atmospheric veil using median filter onthe minimal component of the hazy image Although thismethod is pixel-based and can get neutral atmospheric veilthe scene depth cannot be preserved well due to two medianfilter operations Inspired by the above two methods [11 12]we will present an adaptive wavelet fusion method to obtainaccurate atmospheric veil in the next subsection

33 Estimation of Atmospheric Veil Now we describe anadaptive wavelet image fusion method to estimate the atmo-spheric veil accurately Firstly we acquire two atmosphericveils with different scales Then a local similarity-basedwavelet fusion method is introduced to preserve the mostscene depth features

331 Coarse Scale Atmospheric Veil The image pyramid [19]is a highly efficient and remarkable tool for image processingsuch as enhancement resolution and image compression Itextracts the image structure to represent image informationin different scales Therefore we reduce the resolution levelsof the pyramid representation for getting a coarse scaleversion of the original image A simple iterative imagepyramid algorithm is used for obtaining a relative smoothimage The zero level of the pyramid 119866

0

is equal to theoriginal image The next pyramid level 119866

119894

is subsample bya factor 120578 to the last level 119866

119894minus1

Figure 2 shows the levels ofimage pyramid expanded to the size of the original image Bythis procedure the image is smoother and fewer details areleft in a larger scale At last the coarse scale version of originalimage is obtained by

1198680

(119909) =1

119873

119873

sum

119894=1

119866119894

(119909) (6)

where 119873 is the number of pyramid level and 119866119894

is each levelof the image pyramid

The coarse scale of the original image is defined as theminimum operation

119881119862

(119909) = min119888isin119903119892119887

(119868119888

0

(119909)) (7)

where 1198680

(119909) is the coarse scale version of the originalhazy image Figure 3(a) shows an example of hazy imageFigure 3(b) is the coarse scale atmospheric veil estimated byour method

332 Fine Scale Atmospheric Veil During the propagationof the visible spectrum from the light source to the sensor


(a) G0 (b) G1

(c) G3 (d) G5

(e) G7

Figure 2 The levels of image pyramid of ldquohouserdquo image

the light with different wavelengths is scattered in differentdegrees Rayleigh scattering [20] describes a physical phe-nomenon that light is scattered in propagation directionsby very small particles According to Rayleigh law thescattered intensity is inversely proportional to biquadrate ofthewavelengthThe blue and purple lights of visible spectrum

are scattered strongly at the very distant sky while a little lightis scattered in a close shot However the human visual systemis not sensitive to the purple light This is the reason whythe distant sky seems blue in sunny day (grey in cloudy orhazy day) In the condition of haze weather the blue light isscattered severely in long shots while it is scattered weakly


(a) (b)

(c) (d)

Figure 3 Atmospheric veil estimation (a) input hazy image (b) coarse scale atmospheric veil (c) fine scale atmospheric veil and (d) fusedatmospheric veil

in close shots which can reflect the change of scene depthindirectly Hence the blue channel of original hazy imageis chosen as the fine scale atmospheric veil 119881

119865

Figure 3(c)shows an example of the fine scale atmospheric veil

333 Local Similarity-Based Wavelet Image Fusion Overpast several decades wavelet transform has been a pop-ular tool for image fusion which is a frequency domainprocessing technique in multiscale levels [21ndash23] Moreoverthe results are considered more suitable for human andmachine perception or further image processing tasks such assegmentation feature extraction and object tracking In thissection we describe a local similarity-based wavelet imagefusion method for preserving the most important features ofthe coarse and fine scale atmospheric veils

(A) Local Similarity Definition For each pixel 119909 we define thelocal similarity as

LS (119909) =1

1 + sum119910isinΩ(119909)

10038171003817100381710038171198681 (119910) minus 1198682

(119910)1003817100381710038171003817

2

2

(8)

where 1198681

and 1198682

are two corresponding local image patcheswith same position Ω(sdot) is a neighborhood of 119909 and sdot

2

2

represents the Euclidean distance between 1198681

(119910) and 1198682

(119910)Alternatively the 2-norm can be replaced by other normsSince 2-norm is a common measurement and without lossof generality we adopt 2-norm in our experimentsThe value

range of LS is [0 1] which ensures that the neighborhoods oftwo patches get high similarity if they are close to each otherand vice versa This local similarity is more robust comparedto computing similarity of the two corresponding pixels sincesome pixels in the coarse scale veil might not hold true valuesafter the image pyramid operation

(B) Adaptive Wavelet-Based Fusion Algorithm As analyzed inSections 331 and 332 the two atmospheric veils reflect thescene depth information with different scales Fusing themeffectively is very crucial for getting an accurate atmosphericveil Since the wavelet-based fusion strategies can fuse thetwo images naturally and without artifacts they have beenwidely explored and applied in the fields of image or videorestoration [24 25] In this study we propose an adaptivewavelet-based fusion method Given the coarse scale atmo-spheric veil 119881

119862

and fine scale atmospheric veil 119881119865

multilevelwavelet decomposition is performed on the two imagesrespectively For the two approximation subbands the meanof corresponding coefficients is used as the fused resultFor one pair of corresponding high frequency subbands thelocal similarity of each pixel is calculated using (8) Sincethe edges in coarse scale can reflect the scene depth jumpwith large probability while only a part of edges related toouter contour in fine scale can reflect the scene depth jumpwe use the local similarity to measure the edge consistencybetween the two atmospheric veils Due to the side effectof pyramid decomposition the edges in coarse scale might


not be the original ones Hence the edges in fine scaleatmospheric veil should be preserved as much as possiblein case a large local similarity is obtained between the twoscale atmospheric veils The small similarity illustrates thatit might be texture in fine scale atmospheric veil and thevalue in coarse scale atmospheric veil should be maintainedas much as possible Therefore the fused coefficient at 119897thlevel of wavelet decomposition can be computed as

119862119897

(119909) = (1 minus LS (119909)) sdot 119881119897

119862

(119909) + LS (119909) sdot 119881119897

119865

(119909) (9)

After fusing each pair of corresponding high frequencysubbands the inverse wavelet transform is carried out Thefusion procedure is described in Algorithm 1

Algorithm 1 (the adaptive wavelet-based fusion algorithm)Consider the followingInput Two images 119881

119862

and 119881119865

the number of decompositionlevels in wavelet transformation 119871 and the size of neighbor-hood in computing local similarity 119903Output The fused image 119881

(1) Decompose119881119862

and119881119865

into approximation sub-bands119881119886

119862

and 119881119886

119865

and high frequency sub-bands 119881ℎ119897

119862

and119881ℎ119897

119865

using wavelet transformation with 119871 levels

[119881119886

119862

119881ℎ119897

119862

] = wavedec2 (119881119862

119871)

[119881119886

119865

119881ℎ119897

119865

] = wavedec2 (119881119865

119871)

(10)

where wavedec2(sdot) is a function of two dimensionwavelet decomposition

(2) Calculate the approximation coefficient Coef119886 =

(12)(119881119886

119862

+ 119881119886

119865

)(3) Initialize 119897 larr 0(4) Do 119897 larr 119897 + 1(5) Compute the local similarity LS119897

119903

between119881ℎ119897

119862

and119881ℎ119897

119865

with 119903 neighborhood

LS119897119903

(119909) =1

1 + sum119910isinΩ(119909)

10038171003817100381710038171003817119881ℎ119897

119862

(119910) minus 119881ℎ119897

119865

(119910)10038171003817100381710038171003817

2

2

(11)

(6) Compute the high frequency coefficients at 119897th levelwavelet decomposition

Coefℎ119897 (119909) = (1 minus LS119897119903

(119909)) sdot 119881ℎ119897

119862

(119909) + LS119897119903

(119909)

sdot 119881ℎ119897

119865

(119909)

(12)

(7) Until 119897 = 119871(8) Reconstruct the fused image 119881 using inverse wavelet

transform on [Coef119886Coefℎ119897]

119881 = waverec2 (Coef119886Coefℎl) (13)

where waverec2(sdot) is a function of two dimensionwavelet reconstruction

(9) Return 119881

As we can see the coarse scale atmospheric veil(Figure 3(b)) preserves rough image details and ensuresthe scene depth and the boundaries are not destroyed InFigure 3(c) the close shots are not influenced However thecloud of the fine scale veil in the white rectangle characterizesno visibility due to the scattering of the blue and the purplelights Fortunately the coarse scale atmospheric veil can bea relative complement for the lost information Figure 3(d)shows the fused atmospheric veil by applying Algorithm 1 on119881119862

and 119881119865

It exhibits complementary results and the cloudsin the white rectangle show better visibility than the onesin the fine scale veil while the scene depth information ispreserved better than the coarse scale veil

34 The Scene Restoration Once atmospheric veil 119881 andatmospheric light119860 are obtained the transmissionmap 119905 canbe estimated simply by

119905 (119909) = 1 minus119881 (119909)

119860 (14)

As we know the observed scene is shrouded by theatmosphere more or less even in sunny days If we removethe haze directly using (14) the restored image may seemunnatural Therefore a reference parameter 120596 (0 le 120596 le 1)

[11 12] is introduced

119905 (119909) = 1 minus 120596 sdot119881 (119909)

119860 (15)

According to the transmissionmap 119905 and the atmosphericlight 119860 we can recover the scene radiance according to(1) However the direct attenuation 119869(119909)119905(119909) will be zerowhen 119905(119909) is close to zero and the sky or infinite distantregions of the directly recovered 119869(119909) tend to be incorrectConsequently the scene radiance 119869(119909) is recovered by

119869 (119909) =119868 (119909) minus 119860

max (119905 1199050

)+ 119860 (16)

where 1199050

is a lower bound used for restricting the transmis-sion map Algorithm 2 shows our dehazing procedure

Algorithm 2 (the proposed single image dehazing algorithm)Consider the followingInputs Hazy image 119868 the number of pyramid 119873 wavelettransformation levels 119871 the size of neighborhood in comput-ing local similarity 119903 and parameter 120596 119905

0

Outputs Haze-free image 119869

(1) Calculate the atmospheric light 119860 as described inSection 31

(2) Generate the coarse scale atmospheric veil 119881119862

usingimage pyramid with (6) and min operation with (7)

(3) Yield the fine scale atmospheric veil 119881119865

by extractingthe blue channel of hazy image


(a) Hazy image (b) 120596 = 06 (c) 120596 = 07 (d) 120596 = 08 (e) 120596 = 09

Figure 4 Dehazing results with different 120596

(4) Get the final atmospheric veil 119881 by fusing 119881119862

and 119881119865

using Algorithm 1(5) Compute the transmission map 119905 with (15)(6) Recover the scene radiance 119869 with (16)

4 Results and Discussion

In this section the performance of proposed dehazingmethod is evaluated on a series of hazy images and comparedwith several well-known single image dehazing methods

In the experiments we found that the parameters of 119873and 119871 have little effect on the results For example as seenfrom the subjective effect the results with 119871 = 2 and 119871 =

3 have no obvious difference Therefore we set these twoparameters with fixed values as 119873 = 3 and 119871 = 2 Incontrast 120596 has a greater impact on the results Generally thevalue of 120596 is proportional to the density and set from 05 to10 That is to say small 120596 is assigned for removing the thinhaze Figure 4 shows the dehazing results with changed 120596The size of neighborhood 119903 in computing local similarity localsimilarity should be small Since large 119903 will generate smallsimilarity according to Algorithm 1 the edges in coarse scalewill be preserved with small similarity Thus edges related toouter contour in fine scale will be neglected which will makescene depth not accurate In addition large 119903 will generatelarge computational quantity Therefore 119903 is set to 3 times 3 inexperiments Typically 119905

0

is equal to 01 as mentioned in [11]

41 Subjective Observation Figure 5 shows the dehazingresults of several well-known methods and the proposed

method in terms of subjective visibility on the so-calledldquohouserdquo image From this figure we can see that there aresome color biases and halos in the dehazing results obtainedby Fattal [9] and Tarel and Hautiere [12] as shown in Figures5(b) and 5(c) For other methods Ancuti et al [26] andWangand Feng [27] recover the original scenes better relativelyas shown in Figures 5(f) and 5(g) However Ancuti et alrsquosmethod introduces the oversaturated color in the close treepointsWang and Fengrsquosmethod [27] cannot remove the hazecompletely Kimet alrsquosmethod [17] yields serious artifacts andthe color of the close tree is also oversaturated (Figure 5(d))Similarly the haze is not dehazed completely in Zhang et alrsquosresult (Figure 5(e)) More unfortunately the color is shiftedseverely on some boundaries with abrupt scene depth jumpsDespite the dark color Meng et alrsquos [28] and Lan et alrsquos [29]results exhibit similar effect and look very real and natural asa whole as shown in Figures 5(h) and 5(i) Nevertheless thecolor of white rectangular regions above the windows has notbeen restored well By contrast our method can removemostof the haze without introducing any artifacts (Figure 5(j))The restored scene images show better visibility and highercontrast and preserve the real color The contents in thered rectangles indicate the obvious differences of variousmethods

Figure 6 shows the comparative results of the proposedmethod with other four methods on ldquomountainrdquo image Seenfrom the region of mountain the color is obviously distortedinYeh et alrsquos [30] and oversaturated in Shiau et alrsquos [31] resultsas shown in Figures 6(d) and 6(e) On the whole the otherthreemethods produce similar results except for the region ofsky (shown in Figures 6(b) 6(c) and 6(f)) However it seems


(a) (b) (c) (d)

(e) (f) (g) (h)

(i) (j)Figure 5 Comparative results of the popular methods and the proposed method on ldquohouserdquo image (a) input hazy image (b) Fattalrsquos result[9] (c) Tarel and Hautierersquos result [12] (d) Kim et alrsquos result [17] (e) Zhang et alrsquos result [18] (f) Ancuti et alrsquos result [26] (g) Wang and Fengrsquosresult [27] (h) Meng et alrsquos result [28] (i) Lan et alrsquos result [29] and (j) our result with 120596 = 08

that there is still a veil of mist over the scene in Figure 6(b)And Lan et alrsquos [29] result is a little oversaturated in the sky(Figure 6(c)) Taken as a whole our method gains relativelysatisfactory result (Figure 6(f)) It is worth noting that thevalue of 119905

0

is set to 03 for ldquomountainrdquo imageNext we compare the proposed method with Tanrsquos [10]

and Nishino et alrsquos [15] methods on four representativehazy images Clearly the color is distorted in Tanrsquos results(Figure 7(b)) which might be caused by the maximumlocal contrast operation Nishino et alrsquos results seem tobe more natural But Nishino et alrsquos method produces thehalos in the sky (shown in the third row of Figure 7(c))Relatively our method again exhibits better dehazing results(Figure 7(d))

Figure 8(a) shows a nonhomogeneous hazy image andFigures 8(b)ndash8(e) are the results of different methods Com-pared with others the haze is removed in a more completeway using Fattalrsquos method [9] However the phenomenon

of color shift appears to be visible in Figure 8(b) Tarel andHautierersquos [12] and Zhang et alrsquos [14] results yield unpleasingartifacts C O Ancuti and C Ancuti [16] are not helpful todeal with this situation Although our method also cannotremove the haze nicely the color and depth are kept morereal and natural

42 Objective Evaluation Except for the natural outdoorscenes we also test our method on synthetic hazy imagesand evaluate it with objective criteria The synthetic imagesare from the project website provided by Tang et al [32]Figures 9(a) and 9(b) show the synthetic hazy images andtheir corresponding haze-free images Figures 9(c)ndash9(e) givethe comparative results of He et alrsquos Tang et alrsquos and oursWecan see that our results aremore close to the haze-free imageswhich is especially clear in the red rectangles

Images captured in haze weather often suffer fromdegrading contrast distorting color and missing image


(a) (b)

(c) (d)

(e) (f)

Figure 6 Comparative results of the popular methods and the proposed method on ldquomountainrdquo image (a) input hazy image (b) He et alrsquosresult [11] (c) Lan et alrsquos result [29] (d) Yeh et alrsquos result [30] (e) Shiau et alrsquos result [31] and (f) our results with 120596 = 085

contents Focusing on these issues three criteria are appliedfor objective evaluation as follows

(i) Average Gradient [33] For the sake of testing imagecontrast between the recovered haze-free image andthe original haze-free image the image average gra-dient is calculated

AG =1

119872119873sum

(119909119910)

[(120597119868 (119909 119910)

120597119909)

2

+ (120597119868 (119909 119910)

120597119910)

2

]

12

(17)

where (119909 119910) is the location of a pixel 119868(119909 119910) is theintensity of (119909 119910) 119872119873 is the number of pixels of animage

(ii) Color Consistency [34] For measuring the colorconsistency between the restored haze-free image andthe original haze-free image the color descriptor on

HSV color space and the color histogram intersectionare used to calculate the consistency

CC = 1 minus

119873minus1

sum

119899=0

min (ℎ119901

119899

ℎ119902

119899

) (18)

where 119899 is the color level and ℎ119901

119899

ℎ119902

119899

are the numbersof the pixels in 119901 and 119902 with level 119899 respectively

(iii) Structure Similarity [35] Generally human visualperception is highly adapted for extracting structuralinformation from a scene Exactly the dehazing taskis closely connected with the human visual systemSo SSIM index is employed to measure the restoredimage quality

SSIM (119901 119902) =

(2120583119901

120583119902

+ 1198881

) (2120590119901119902

+ 1198882

)

(1205832119901

+ 1205832119902

+ 1198881

) (1205902119901

+ 1205902119902

+ 1198882

)

(19)


(a) (b) (c) (d)

Figure 7 Comparative results of the popular methods and the proposed method on ldquony12rdquo ldquony17rdquo ldquoy01rdquo and ldquoy16rdquo (a) input hazy images(b) Tanrsquos results [10] (c) Nishino et alrsquos results [15] and (d) our results with 120596 = 05 065 065 and 06 respectively


(a) (b)

(c) (d)

(e) (f)

Figure 8 Fog comparative results of the proposed algorithm and the conventional algorithms on ldquoforestrdquo image (a) input hazy image (b)Fattalrsquos result [9] (c) Tarel and Hautierersquos result [12] (d) C O Ancuti and C Ancutirsquos result [16] (e) Zhang et alrsquos result [14] and (f) our resultwith 120596 = 09

where 119901 119902 are restored haze-free image and theoriginal haze-free image respectively 120583

119901

120583119902

are themean intensities of 119901 119902 1205902

119901

1205902

119902

are the variances of119901 119902 120590

119901119902

is the covariance of 119901 and 119902 1198881

= (1198961

119871)2

and 1198882

= (1198962

119871)2 are two variables for stabilizing

the division with weak denominator where 119871 is thedynamic range of the pixel-values 119896

1

= 001 and1198962

= 003 by default

Table 1 lists the objective comparisons of different meth-ods on ldquodollsrdquo ldquoteddyrdquo and ldquovenusrdquo For ldquodollsrdquo image our

results are slightly inferior to the other methods on AGCC and SSIM For ldquoteddyrdquo image our results yield the bestperformances in terms of AG and CC which means that ourrestored images hold higher contrast and less color distortionFor the criterion of structure similarity all these methodsdisplay very similar values which are close to 1This indicatesthat the structure information of restored images is wellmaintained Visually the three methods yield very similarresults of the ldquoteddyrdquo image (the second row) For ldquovenusrdquoimage our method yields much better results in terms ofAG and SSIM The value of CC is lower than He et alrsquos by023 which demonstrates that the restored image of He et alrsquos


(a) (b) (c) (d) (e)

Figure 9 Synthetic hazy images and the comparative dehazing results (a) Hazy images (called ldquodollsrdquo ldquoteddyrdquo and ldquovenusrdquo) (b) haze-freeimages (c) He et alrsquos [11] results (d) Tang et alrsquos [32] results and (e) ours results with 120596 = 09 09 and 099 respectively

Table 1 Objective evaluation on synthetic hazy images

Hazy images Methods AG CC SSIM

ldquoDollsrdquoHe et al [11] 461 021 08860

Tang et al [32] 475 010 09037Our 433 020 08255

ldquoTeddyrdquoHe et al [11] 654 024 09987

Tang et al [32] 679 017 09989Our 776 025 09983

ldquoVenusrdquoHe et al [11] 577 070 09911Du et al [24] 563 054 09944

Our 590 057 09972

method can better maintain the color information Howeveras seen from Figure 9(c) the color is obviously distortedespecially in the upper left part This phenomenon illustratesthat some objective criteria are not consistent with subjectivevisualization That is to say these criteria can only revealpartially the level of dehazing Therefore it is necessary toexplore some integrated criteria for evaluating the restoredresult

43 Running Time Furthermore the running time is con-sidered in this paper For fair comparison we downloadthe code from the original authorrsquos project website and runthe procedures on the same platform (Intel core i7 CPU16GB RAM) Table 2 lists the running time of four methodsBecause our proposed method is a linear function of the

number of the input image pixels it is obviously faster thanothers For an image of size 119882 times 119867 the complexity of theproposed dehazing algorithm is 119874 (119873 times 119882 times 119867 + 119882 times 119867 +

119871 times 119882 times 119867 + 119882 times 119867) That is the complexity of our methodis linear and comparative or even lower than conventionalimage dehazing methods Although the complexity of Tarelet alrsquos method is also linear two median filter operationsare very time-consuming Obviously Zhang et alrsquos methodcosts the longest time which might be caused by the utilityof overlap scheme and median filter Instead Meng et alrsquosmethod is very more time-saving compared with Tarel et alrsquosand Zhang et alrsquos methods In particular our method runsfive times as fast as Meng et alrsquos method for 1024 times 768

image

5 Conclusions

In this paper we address a novel single image dehazingmethod based on wavelet fusion Two atmospheric veilswith different scales are derived from the hazy image oneof which is minimum channel after image pyramid whilethe other one is the blue component In order to get anaccurate atmospheric veil we proposed an effectively local-similarity based adaptive wavelet fusionmethod for reducingblocky artifacts which is the major novelty of this workThe experimental results indicate that our method achievescomparative or even better result by visual effect and objectiveassessments

However ourmethod cannot remove the haze completelyfor nonhomogeneous hazy and heavy foggy image In addi-tion there are still unnecessary textures in the image ofscene depth To overcome these limitations a more robust


Table 2 The comparisons of running time (s) with Zhang et al [18] Tarel and Hautiere [12] and Meng et al [28]

Methods 465 times 384 440 times 450 600 times 400 600 times 525 768 times 576 1024 times 768Zhang 1367 1506 1893 2527 3814 7635Tarel 332 373 683 943 1929 6036Meng 185 205 235 314 425 773Our 052 058 063 071 091 141

and practical dehazing method will be studied in the futurework

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Science and Technology ResearchProject of Liaoning Province Education Department (noL2014450) National Natural Science Foundation of China(no 61403078) Science and Technology Development Planof Jilin Province (no 20140101181JC) and Science and Tech-nology Project of Jinzhou New District of China (no KJCX-ZTPY-2014-0004)

References

[1] S G Narasimhan and S K Nayar ldquoContrast restoration ofweather degraded imagesrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 25 no 6 pp 713ndash724 2003

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

[3] L J Denes M S Gottlieb B Kaminsky P Metes and R JMericsko ldquoAOTF polarization difference imagingrdquo in Proceed-ings of the 27th AIPRWorkshop Advances in Computer-AssistedRecognition Washington DC USA 1998

[4] Y Y Schechner S G Narasimhan and S K Nayar ldquoInstantdehazing of images using polarizationrdquo in Proceedings of theIEEE Computer Society Conference on Computer Vision andPattern Recognition (CVPR rsquo01) vol 1 pp 325ndash332 IEEE KauaiHawaii USA 2001

[5] J G Walker P C Y Chang and K I Hopcraft ldquoVisibilitydepth improvement in active polarization imaging in scatteringmediardquo Applied Optics vol 39 no 27 pp 4933ndash4941 2000

[6] D B Chenault and J L Pezzaniti ldquoPolarization imagingthrough scattering mediardquo in Polarization Analysis Measure-ment and Remote Sensing III vol 4133 of Proceedings of SPIEp 124 San Diego Calif USA July 2000

[7] M P Rowe E N Pugh Jr J S Tyo and N EnghetaldquoPolarization-difference imaging a biologically inspired tech-nique for observation through scattering mediardquoOptics Lettersvol 20 no 6 pp 608ndash610 1995

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

[10] R T Tan ldquoVisibility in bad weather from a single imagerdquo inProceedings of the 26th IEEE Conference on Computer Visionand Pattern Recognition (CVPR rsquo08) pp 1ndash8 IEEE AnchorageAlaska USA June 2008

[11] K He J Sun and X Tang ldquoSingle image haze removal usingdark channel priorrdquo in Proceedings of the IEEE Conference onComputer Vision and Pattern Recognition (CVPR rsquo09) pp 1956ndash1963 IEEE San Francisco Calif USA June 2009

[12] J Tarel andN Hautiere ldquoFast visibility restoration from a singlecolor or gray level imagerdquo inProceedings of the 12th InternationalConference on Computer Vision (ICCV rsquo09) pp 2201ndash2208IEEE Kyoto Japan September 2009

[13] C X Xiao and J J Gan ldquoFast image dehazing using guided jointbilateral filterrdquo Visual Computer vol 28 no 6ndash8 pp 713ndash7212012

[14] J W Zhang L Li G Q Yang Y Zhang and J Z Sun ldquoLocalalbedo-insensitive single image dehazingrdquo Visual Computervol 26 no 6-8 pp 761ndash768 2010

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

[16] C O Ancuti and C Ancuti ldquoSingle image dehazing by multi-scale fusionrdquo IEEE Transactions on Image Processing vol 22 no8 pp 3271ndash3282 2013

[17] J-H Kim W-D Jang J-Y Sim and C-S Kim ldquoOptimizedcontrast enhancement for real-time image and video dehazingrdquoJournal of Visual Communication and Image Representation vol24 no 3 pp 410ndash425 2013

[18] Y-Q Zhang Y Ding J-S Xiao J Liu and Z Guo ldquoVisibilityenhancement using an image filtering approachrdquo EURASIPJournal on Advances in Signal Processing vol 2012 article 2202012

[19] B K Choudhary N K Sinhaand and P Shanker ldquoPyramidmethod in image processingrdquo Journal of Information Systemsand Communication vol 3 no 1 pp 269ndash273 2012

[20] A Bucholtz ldquoRayleigh-scattering calculations for the terrestrialatmosphererdquoApplied Optics vol 34 no 15 pp 2765ndash2773 1995

[21] L J Chipman C Intergraph A L Huntsville T M Orr andL N Graham ldquoWavelets and image fusionrdquo in Proceedings ofthe IEEE International Conference on Image Processing (ICIPrsquo95) vol 3 pp 248ndash251 International Society for Optics andPhotonics Washington DC USA 1995

[22] K Amolins Y Zhang and P Dare ldquoWavelet based image fusiontechniquesmdashan introduction review and comparisonrdquo ISPRSJournal of Photogrammetry and Remote Sensing vol 62 no 4pp 249ndash263 2007

[23] G Pajares and J M de la Cruz ldquoA wavelet-based image fusiontutorialrdquo Pattern Recognition vol 37 no 9 pp 1855ndash1872 2004

[24] Y Du B Guindon and J Cihlar ldquoHaze detection and removalin high resolution satellite image with wavelet analysisrdquo IEEETransactions on Geoscience and Remote Sensing vol 40 no 1pp 210ndash216 2002


[25] N Anantrasirichai A Achim N G Kingsbury and D R BullldquoAtmospheric turbulence mitigation using complex wavelet-based fusionrdquo IEEE Transactions on Image Processing vol 22no 6 pp 2398ndash2408 2013

[26] C O Ancuti C Ancuti C Hermans and P Bekaert ldquoA fastsemi-inverse approach to detect and remove the haze froma single imagerdquo in Proceedings of the 10th Asian Conferenceon Computer Vision (ACCV rsquo10) November 2010 pp 501ndash514Springer Berlin Germany 2010

[27] Z Wang and Y Feng ldquoFast single haze image enhancementrdquoComputers and Electrical Engineering vol 40 no 3 pp 785ndash795 2014

[28] G Meng Y Wang J Duan S Xiang and C Pan ldquoEfficientimage dehazing with boundary constraint and contextual reg-ularizationrdquo in Proceedings of the 14th IEEE InternationalConference on Computer Vision (ICCV rsquo13) pp 617ndash624 IEEESydney Australia December 2013

[29] X Lan L Zhang H Shen Q Yuan and H Li ldquoSingle imagehaze removal considering sensor blur and noiserdquo EURASIPJournal on Advances in Signal Processing vol 2013 article 862013

[30] C-H Yeh L-W Kang M-S Lee and C-Y Lin ldquoHaze effectremoval from image via haze density estimation in opticalmodelrdquo Optics Express vol 21 no 22 pp 27127ndash27141 2013

[31] Y-H Shiau P-YChenH-Y YangC-HChen and S-SWangldquoWeighted haze removal method with halo preventionrdquo Journalof Visual Communication and Image Representation vol 25 no2 pp 445ndash453 2014

[32] K Tang J Yang and J Wang ldquoInvestigating haze-relevantfeatures in a learning framework for image dehazingrdquo inProceedings of the 27th IEEE Conference on Computer Visionand Pattern Recognition (CVPR rsquo14) pp 2995ndash3002 IEEEColumbus Ohio USA June 2014

[33] R C Gonzalez and R E Woods Digital Image ProcessingPearson Education Upper Saddle River NJ USA 2nd edition2002

[34] B S Manjunath J-R Ohm V V Vasudevan and A YamadaldquoColor and texture descriptorsrdquo IEEE Transactions on Circuitsand Systems for Video Technology vol 11 no 6 pp 703ndash7152001

[35] ZWang A C Bovik H R Sheikh and E P Simoncelli ldquoImagequality assessment from error visibility to structural similarityrdquoIEEE Transactions on Image Processing vol 13 no 4 pp 600ndash612 2004

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


no accurate or approximate models are obtained for someexisting images

Recently a number of single image dehazing methodshave been developed for overcoming the weakness of multi-images or getting 3D model in advance Fattal [9] estimatedthe scene albedo and depth with the hypothesis that thesurface shading was locally uncorrelated with the transmis-sionThis method is effective but fails when the hypothesis isbroken Tan [10] observed that the images in clear days hadstronger contrast than those in bad weather Simultaneouslythe variation of airlight mainly depended on the scene depthand tended to be smooth Consequently they developed amaximization local contrast method for restoring the sceneAlthough this method can enhance the visibility it yieldsserious halos and oversaturated colors Based on the darkchannel prior of the haze-free natural images He et al [11]estimated the thickness of the haze and recovered a highquality haze-free image Although this method can achieveideal dehazing results the cost of softmatting for refining thetransmission map is very large and unsuitable for real-timeapplications Tarel and Hautiere [12] proposed a linear com-plexity visibility restoration method based on median filterWhile this method can meet the real-time requirements themedian filter is not conformal and edge-preserving To getaround this problem Xiao and Gan [13] proposed a guidedjoint bilateral filter for haze removal By using bilateral filterZhang et al [14] introduced a new algorithm to removehaze from a single image based on the assumption thatonly the variation of transmission could result in large-scale luminance variation in the fog-free image Howeverthis method will fail in case the assumption does not holdNishino et al [15] introduced a Bayesian probabilisticmethodthat jointly estimated the scene albedo and depth from asingle hazy image However since it is hard to estimatethe multiple statistical prior models accurately the dehazingresults obtained by this method may contain unnaturallybright colors More recently C O Ancuti and C Ancuti [16]presented a fusion-based strategy that was able to accuratelyrestore images using only the original hazy or foggy imageThis method has high computing speed and yields similarresults with [8 9 11] Kim et al [17] restored the hazy image byan optimized contrast enhancement procedure that balancedthe cost of contrast and information loss Besides theyproposed a quad-tree subdivision based searching methodfor estimating the atmospheric light Nevertheless the quad-tree searching method will not be robust if the maximumvalue of the final subblock is noise Zhang et al [18] describeda hazy layer estimationmethod using low-rank technique andoverlap averaging schemeThe disadvantage is that it may notwork well for far scenes with heavy fog and great depth jump

In this paper we present a novel single image dehazingmethod based on atmospheric scattering model In thebasis of existing dark channel prior and optics theory twoatmospheric veils with different scales are first derived fromthe hazy imageThen a local similarity-basedwaveletmethodis proposed to fuse the two atmospheric veils adaptivelywhich can reflect the scene depth information well At lastthe scene is restored by solving the atmospheric scatteringmodel

Comparedwith previous single image dehazingmethodsour proposed method exhibits the following advantagesFirstly the proposed adaptive local similarity-based waveletfusion method shows better performance in preserving themost discriminant scene depth Secondly our dehazingmethodperforms per-pixel computation efficiently and yieldsvery little artifacts Moreover the proposed method does notneed the postprocessing like He et alrsquos exposure [11] andTarel and Hautierersquos tone mapping [12] Finally we test ourmethod on a number of natural and synthetic hazy imagesThe comparative results display comparative or even betterdehazing results by subjective and objective evaluations

The rest of this paper is organized as follows In Section 2we review the atmospheric scattering model widely usedin dehazing task Section 3 describes the proposed dehaz-ing method in detail mainly including atmospheric lightand atmospheric veil estimations image fusion and scenerestoration In Section 4 we report and discuss the experi-ment results Finally conclusions are provided in Section 5

2 Atmospheric Scattering Model

In computer vision and image processing the formationof hazy image is commonly described by the atmosphericscattering model [9ndash12]

119868 (119909) = 119869 (119909) 119905 (119909) + 119860 (1 minus 119905 (119909)) (1)

where 119868 is the observed hazy image at location 119909 119869 is thescene radiance (the clear haze-free image) 119860 is the globalatmospheric light and 119905 is the medium transmission whichis depended on the distance 119889(119909) between the object sceneand the observer while the atmosphere is homogenous Thetransmission is described as follows

119905 (119909) = 119890minus120573119889(119909)

(2)

where 120573 is the scattering coefficient of atmosphereThe first term 119869(119909)119905(119909) describes that the reflected light

by the scene points is partially attenuated during the propa-gationThe second term119860(1minus119905(119909)) is called atmospheric veil[12] which is generated by atmosphere scattering and leadsto color shift The task of dehazing is to recover 119869 from 119868

3 Proposed Dehazing Method

The proposed method can be mainly divided into four stepsFirstly the atmospheric light is estimated by the improvedquad-tree subdivision method Then atmospheric veils withdifferent scales are derived from the hazy image Next theaccurate atmospheric veil is obtained by fusing the twoveils Finally the scene radiance is restored by solving theatmospheric scatteringmodelThe flow chart of the proposeddehazing method is shown as in Figure 1

31 Estimation of Atmospheric Light Generally the atmo-spheric light 119860 is always considered as the brightest intensityin the entire image because a large amount of haze makesthe object scene brighter than itself However this will be notreliable when a white object is present in the scene Kim et


Haze-free image

Coarse-scale veil

Fusion

Transmission map


Hazy image

Restoration





119869dark

(119909) = min119888isin119903119892119887

( min119910isinΩ(119909)

(119869119888

(119910))) (3)



min119888isin119903119892119887

( min119910isinΩ(119909)

(119868119888

(119910)))

= min119888isin119903119892119887

( min119910isinΩ(119909)

(119869119888

(119910))) (119909) + 119860 (1 minus (119909))

(4)



119881 = 119860 (1 minus (119909)) = min119888isin119903119892119887

( min119910isinΩ(119909)

(119868119888

(119910))) (5)





0


119894


119894minus1


1198680

(119909) =1

119873

119873

sum

119894=1

119866119894

(119909) (6)




119881119862

(119909) = min119888isin119903119892119887

(119868119888

0

(119909)) (7)

where 1198680




(a) G0 (b) G1

(c) G3 (d) G5

(e) G7





(a) (b)

(c) (d)



119865




LS (119909) =1

1 + sum119910isinΩ(119909)

10038171003817100381710038171198681 (119910) minus 1198682

(119910)1003817100381710038171003817

2

2

(8)

where 1198681

and 1198682


2

2


(119910) and 1198682




119862





119862119897

(119909) = (1 minus LS (119909)) sdot 119881119897

119862

(119909) + LS (119909) sdot 119881119897

119865

(119909) (9)



119862

and 119881119865


(1) Decompose119881119862

and119881119865


119862

and 119881119886

119865


119862

and119881ℎ119897

119865


[119881119886

119862

119881ℎ119897

119862

] = wavedec2 (119881119862

119871)

[119881119886

119865

119881ℎ119897

119865

] = wavedec2 (119881119865

119871)

(10)



(12)(119881119886

119862

+ 119881119886

119865


119903


119862

and119881ℎ119897

119865


LS119897119903

(119909) =1

1 + sum119910isinΩ(119909)

10038171003817100381710038171003817119881ℎ119897

119862

(119910) minus 119881ℎ119897

119865

(119910)10038171003817100381710038171003817

2

2

(11)


Coefℎ119897 (119909) = (1 minus LS119897119903

(119909)) sdot 119881ℎ119897

119862

(119909) + LS119897119903

(119909)

sdot 119881ℎ119897

119865

(119909)

(12)





(9) Return 119881


and 119881119865



119905 (119909) = 1 minus119881 (119909)

119860 (14)



119905 (119909) = 1 minus 120596 sdot119881 (119909)

119860 (15)


119869 (119909) =119868 (119909) minus 119860

max (119905 1199050

)+ 119860 (16)

where 1199050



0








(a) Hazy image (b) 120596 = 06 (c) 120596 = 07 (d) 120596 = 08 (e) 120596 = 09



and 119881119865






0






(a) (b) (c) (d)

(e) (f) (g) (h)



0








(a) (b)

(c) (d)

(e) (f)




AG =1

119872119873sum

(119909119910)

[(120597119868 (119909 119910)

120597119909)

2

+ (120597119868 (119909 119910)

120597119910)

2

]

12

(17)




CC = 1 minus

119873minus1

sum

119899=0

min (ℎ119901

119899

ℎ119902

119899

) (18)


119899

ℎ119902

119899



SSIM (119901 119902) =

(2120583119901

120583119902

+ 1198881

) (2120590119901119902

+ 1198882

)

(1205832119901

+ 1205832119902

+ 1198881

) (1205902119901

+ 1205902119902

+ 1198882

)

(19)


(a) (b) (c) (d)



(a) (b)

(c) (d)

(e) (f)



119901

120583119902


119901

1205902

119902


119901119902


= (1198961

119871)2

and 1198882

= (1198962



1

= 001 and1198962

= 003 by default




(a) (b) (c) (d) (e)





Tang et al [32] 475 010 09037Our 433 020 08255


Tang et al [32] 679 017 09989Our 776 025 09983


Our 590 057 09972





image

5 Conclusions









Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Haze-free image

Coarse-scale veil

Fusion

Transmission map


Hazy image

Restoration





119869dark

(119909) = min119888isin119903119892119887

( min119910isinΩ(119909)

(119869119888

(119910))) (3)



min119888isin119903119892119887

( min119910isinΩ(119909)

(119868119888

(119910)))

= min119888isin119903119892119887

( min119910isinΩ(119909)

(119869119888

(119910))) (119909) + 119860 (1 minus (119909))

(4)



119881 = 119860 (1 minus (119909)) = min119888isin119903119892119887

( min119910isinΩ(119909)

(119868119888

(119910))) (5)





0


119894


119894minus1


1198680

(119909) =1

119873

119873

sum

119894=1

119866119894

(119909) (6)




119881119862

(119909) = min119888isin119903119892119887

(119868119888

0

(119909)) (7)

where 1198680




(a) G0 (b) G1

(c) G3 (d) G5

(e) G7





(a) (b)

(c) (d)



119865




LS (119909) =1

1 + sum119910isinΩ(119909)

10038171003817100381710038171198681 (119910) minus 1198682

(119910)1003817100381710038171003817

2

2

(8)

where 1198681

and 1198682


2

2


(119910) and 1198682




119862





119862119897

(119909) = (1 minus LS (119909)) sdot 119881119897

119862

(119909) + LS (119909) sdot 119881119897

119865

(119909) (9)



119862

and 119881119865


(1) Decompose119881119862

and119881119865


119862

and 119881119886

119865


119862

and119881ℎ119897

119865


[119881119886

119862

119881ℎ119897

119862

] = wavedec2 (119881119862

119871)

[119881119886

119865

119881ℎ119897

119865

] = wavedec2 (119881119865

119871)

(10)



(12)(119881119886

119862

+ 119881119886

119865


119903


119862

and119881ℎ119897

119865


LS119897119903

(119909) =1

1 + sum119910isinΩ(119909)

10038171003817100381710038171003817119881ℎ119897

119862

(119910) minus 119881ℎ119897

119865

(119910)10038171003817100381710038171003817

2

2

(11)


Coefℎ119897 (119909) = (1 minus LS119897119903

(119909)) sdot 119881ℎ119897

119862

(119909) + LS119897119903

(119909)

sdot 119881ℎ119897

119865

(119909)

(12)





(9) Return 119881


and 119881119865



119905 (119909) = 1 minus119881 (119909)

119860 (14)



119905 (119909) = 1 minus 120596 sdot119881 (119909)

119860 (15)


119869 (119909) =119868 (119909) minus 119860

max (119905 1199050

)+ 119860 (16)

where 1199050



0








(a) Hazy image (b) 120596 = 06 (c) 120596 = 07 (d) 120596 = 08 (e) 120596 = 09



and 119881119865






0






(a) (b) (c) (d)

(e) (f) (g) (h)



0








(a) (b)

(c) (d)

(e) (f)




AG =1

119872119873sum

(119909119910)

[(120597119868 (119909 119910)

120597119909)

2

+ (120597119868 (119909 119910)

120597119910)

2

]

12

(17)




CC = 1 minus

119873minus1

sum

119899=0

min (ℎ119901

119899

ℎ119902

119899

) (18)


119899

ℎ119902

119899



SSIM (119901 119902) =

(2120583119901

120583119902

+ 1198881

) (2120590119901119902

+ 1198882

)

(1205832119901

+ 1205832119902

+ 1198881

) (1205902119901

+ 1205902119902

+ 1198882

)

(19)


(a) (b) (c) (d)



(a) (b)

(c) (d)

(e) (f)



119901

120583119902


119901

1205902

119902


119901119902


= (1198961

119871)2

and 1198882

= (1198962



1

= 001 and1198962

= 003 by default




(a) (b) (c) (d) (e)





Tang et al [32] 475 010 09037Our 433 020 08255


Tang et al [32] 679 017 09989Our 776 025 09983


Our 590 057 09972





image

5 Conclusions









Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) G0 (b) G1

(c) G3 (d) G5

(e) G7





(a) (b)

(c) (d)



119865




LS (119909) =1

1 + sum119910isinΩ(119909)

10038171003817100381710038171198681 (119910) minus 1198682

(119910)1003817100381710038171003817

2

2

(8)

where 1198681

and 1198682


2

2


(119910) and 1198682




119862





119862119897

(119909) = (1 minus LS (119909)) sdot 119881119897

119862

(119909) + LS (119909) sdot 119881119897

119865

(119909) (9)



119862

and 119881119865


(1) Decompose119881119862

and119881119865


119862

and 119881119886

119865


119862

and119881ℎ119897

119865


[119881119886

119862

119881ℎ119897

119862

] = wavedec2 (119881119862

119871)

[119881119886

119865

119881ℎ119897

119865

] = wavedec2 (119881119865

119871)

(10)



(12)(119881119886

119862

+ 119881119886

119865


119903


119862

and119881ℎ119897

119865


LS119897119903

(119909) =1

1 + sum119910isinΩ(119909)

10038171003817100381710038171003817119881ℎ119897

119862

(119910) minus 119881ℎ119897

119865

(119910)10038171003817100381710038171003817

2

2

(11)


Coefℎ119897 (119909) = (1 minus LS119897119903

(119909)) sdot 119881ℎ119897

119862

(119909) + LS119897119903

(119909)

sdot 119881ℎ119897

119865

(119909)

(12)





(9) Return 119881


and 119881119865



119905 (119909) = 1 minus119881 (119909)

119860 (14)



119905 (119909) = 1 minus 120596 sdot119881 (119909)

119860 (15)


119869 (119909) =119868 (119909) minus 119860

max (119905 1199050

)+ 119860 (16)

where 1199050



0








(a) Hazy image (b) 120596 = 06 (c) 120596 = 07 (d) 120596 = 08 (e) 120596 = 09



and 119881119865






0






(a) (b) (c) (d)

(e) (f) (g) (h)



0








(a) (b)

(c) (d)

(e) (f)




AG =1

119872119873sum

(119909119910)

[(120597119868 (119909 119910)

120597119909)

2

+ (120597119868 (119909 119910)

120597119910)

2

]

12

(17)




CC = 1 minus

119873minus1

sum

119899=0

min (ℎ119901

119899

ℎ119902

119899

) (18)


119899

ℎ119902

119899



SSIM (119901 119902) =

(2120583119901

120583119902

+ 1198881

) (2120590119901119902

+ 1198882

)

(1205832119901

+ 1205832119902

+ 1198881

) (1205902119901

+ 1205902119902

+ 1198882

)

(19)


(a) (b) (c) (d)



(a) (b)

(c) (d)

(e) (f)



119901

120583119902


119901

1205902

119902


119901119902


= (1198961

119871)2

and 1198882

= (1198962



1

= 001 and1198962

= 003 by default




(a) (b) (c) (d) (e)





Tang et al [32] 475 010 09037Our 433 020 08255


Tang et al [32] 679 017 09989Our 776 025 09983


Our 590 057 09972





image

5 Conclusions









Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)



119865




LS (119909) =1

1 + sum119910isinΩ(119909)

10038171003817100381710038171198681 (119910) minus 1198682

(119910)1003817100381710038171003817

2

2

(8)

where 1198681

and 1198682


2

2


(119910) and 1198682




119862





119862119897

(119909) = (1 minus LS (119909)) sdot 119881119897

119862

(119909) + LS (119909) sdot 119881119897

119865

(119909) (9)



119862

and 119881119865


(1) Decompose119881119862

and119881119865


119862

and 119881119886

119865


119862

and119881ℎ119897

119865


[119881119886

119862

119881ℎ119897

119862

] = wavedec2 (119881119862

119871)

[119881119886

119865

119881ℎ119897

119865

] = wavedec2 (119881119865

119871)

(10)



(12)(119881119886

119862

+ 119881119886

119865


119903


119862

and119881ℎ119897

119865


LS119897119903

(119909) =1

1 + sum119910isinΩ(119909)

10038171003817100381710038171003817119881ℎ119897

119862

(119910) minus 119881ℎ119897

119865

(119910)10038171003817100381710038171003817

2

2

(11)


Coefℎ119897 (119909) = (1 minus LS119897119903

(119909)) sdot 119881ℎ119897

119862

(119909) + LS119897119903

(119909)

sdot 119881ℎ119897

119865

(119909)

(12)





(9) Return 119881


and 119881119865



119905 (119909) = 1 minus119881 (119909)

119860 (14)



119905 (119909) = 1 minus 120596 sdot119881 (119909)

119860 (15)


119869 (119909) =119868 (119909) minus 119860

max (119905 1199050

)+ 119860 (16)

where 1199050



0








(a) Hazy image (b) 120596 = 06 (c) 120596 = 07 (d) 120596 = 08 (e) 120596 = 09



and 119881119865






0






(a) (b) (c) (d)

(e) (f) (g) (h)



0








(a) (b)

(c) (d)

(e) (f)




AG =1

119872119873sum

(119909119910)

[(120597119868 (119909 119910)

120597119909)

2

+ (120597119868 (119909 119910)

120597119910)

2

]

12

(17)




CC = 1 minus

119873minus1

sum

119899=0

min (ℎ119901

119899

ℎ119902

119899

) (18)


119899

ℎ119902

119899



SSIM (119901 119902) =

(2120583119901

120583119902

+ 1198881

) (2120590119901119902

+ 1198882

)

(1205832119901

+ 1205832119902

+ 1198881

) (1205902119901

+ 1205902119902

+ 1198882

)

(19)


(a) (b) (c) (d)



(a) (b)

(c) (d)

(e) (f)



119901

120583119902


119901

1205902

119902


119901119902


= (1198961

119871)2

and 1198882

= (1198962



1

= 001 and1198962

= 003 by default




(a) (b) (c) (d) (e)





Tang et al [32] 475 010 09037Our 433 020 08255


Tang et al [32] 679 017 09989Our 776 025 09983


Our 590 057 09972





image

5 Conclusions









Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119862119897

(119909) = (1 minus LS (119909)) sdot 119881119897

119862

(119909) + LS (119909) sdot 119881119897

119865

(119909) (9)



119862

and 119881119865


(1) Decompose119881119862

and119881119865


119862

and 119881119886

119865


119862

and119881ℎ119897

119865


[119881119886

119862

119881ℎ119897

119862

] = wavedec2 (119881119862

119871)

[119881119886

119865

119881ℎ119897

119865

] = wavedec2 (119881119865

119871)

(10)



(12)(119881119886

119862

+ 119881119886

119865


119903


119862

and119881ℎ119897

119865


LS119897119903

(119909) =1

1 + sum119910isinΩ(119909)

10038171003817100381710038171003817119881ℎ119897

119862

(119910) minus 119881ℎ119897

119865

(119910)10038171003817100381710038171003817

2

2

(11)


Coefℎ119897 (119909) = (1 minus LS119897119903

(119909)) sdot 119881ℎ119897

119862

(119909) + LS119897119903

(119909)

sdot 119881ℎ119897

119865

(119909)

(12)





(9) Return 119881


and 119881119865



119905 (119909) = 1 minus119881 (119909)

119860 (14)



119905 (119909) = 1 minus 120596 sdot119881 (119909)

119860 (15)


119869 (119909) =119868 (119909) minus 119860

max (119905 1199050

)+ 119860 (16)

where 1199050



0








(a) Hazy image (b) 120596 = 06 (c) 120596 = 07 (d) 120596 = 08 (e) 120596 = 09



and 119881119865






0






(a) (b) (c) (d)

(e) (f) (g) (h)



0








(a) (b)

(c) (d)

(e) (f)




AG =1

119872119873sum

(119909119910)

[(120597119868 (119909 119910)

120597119909)

2

+ (120597119868 (119909 119910)

120597119910)

2

]

12

(17)




CC = 1 minus

119873minus1

sum

119899=0

min (ℎ119901

119899

ℎ119902

119899

) (18)


119899

ℎ119902

119899



SSIM (119901 119902) =

(2120583119901

120583119902

+ 1198881

) (2120590119901119902

+ 1198882

)

(1205832119901

+ 1205832119902

+ 1198881

) (1205902119901

+ 1205902119902

+ 1198882

)

(19)


(a) (b) (c) (d)



(a) (b)

(c) (d)

(e) (f)



119901

120583119902


119901

1205902

119902


119901119902


= (1198961

119871)2

and 1198882

= (1198962



1

= 001 and1198962

= 003 by default




(a) (b) (c) (d) (e)





Tang et al [32] 475 010 09037Our 433 020 08255


Tang et al [32] 679 017 09989Our 776 025 09983


Our 590 057 09972





image

5 Conclusions









Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) Hazy image (b) 120596 = 06 (c) 120596 = 07 (d) 120596 = 08 (e) 120596 = 09



and 119881119865






0






(a) (b) (c) (d)

(e) (f) (g) (h)



0








(a) (b)

(c) (d)

(e) (f)




AG =1

119872119873sum

(119909119910)

[(120597119868 (119909 119910)

120597119909)

2

+ (120597119868 (119909 119910)

120597119910)

2

]

12

(17)




CC = 1 minus

119873minus1

sum

119899=0

min (ℎ119901

119899

ℎ119902

119899

) (18)


119899

ℎ119902

119899



SSIM (119901 119902) =

(2120583119901

120583119902

+ 1198881

) (2120590119901119902

+ 1198882

)

(1205832119901

+ 1205832119902

+ 1198881

) (1205902119901

+ 1205902119902

+ 1198882

)

(19)


(a) (b) (c) (d)



(a) (b)

(c) (d)

(e) (f)



119901

120583119902


119901

1205902

119902


119901119902


= (1198961

119871)2

and 1198882

= (1198962



1

= 001 and1198962

= 003 by default




(a) (b) (c) (d) (e)





Tang et al [32] 475 010 09037Our 433 020 08255


Tang et al [32] 679 017 09989Our 776 025 09983


Our 590 057 09972





image

5 Conclusions









Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c) (d)

(e) (f) (g) (h)



0








(a) (b)

(c) (d)

(e) (f)




AG =1

119872119873sum

(119909119910)

[(120597119868 (119909 119910)

120597119909)

2

+ (120597119868 (119909 119910)

120597119910)

2

]

12

(17)




CC = 1 minus

119873minus1

sum

119899=0

min (ℎ119901

119899

ℎ119902

119899

) (18)


119899

ℎ119902

119899



SSIM (119901 119902) =

(2120583119901

120583119902

+ 1198881

) (2120590119901119902

+ 1198882

)

(1205832119901

+ 1205832119902

+ 1198881

) (1205902119901

+ 1205902119902

+ 1198882

)

(19)


(a) (b) (c) (d)



(a) (b)

(c) (d)

(e) (f)



119901

120583119902


119901

1205902

119902


119901119902


= (1198961

119871)2

and 1198882

= (1198962



1

= 001 and1198962

= 003 by default




(a) (b) (c) (d) (e)





Tang et al [32] 475 010 09037Our 433 020 08255


Tang et al [32] 679 017 09989Our 776 025 09983


Our 590 057 09972





image

5 Conclusions









Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)

(e) (f)




AG =1

119872119873sum

(119909119910)

[(120597119868 (119909 119910)

120597119909)

2

+ (120597119868 (119909 119910)

120597119910)

2

]

12

(17)




CC = 1 minus

119873minus1

sum

119899=0

min (ℎ119901

119899

ℎ119902

119899

) (18)


119899

ℎ119902

119899



SSIM (119901 119902) =

(2120583119901

120583119902

+ 1198881

) (2120590119901119902

+ 1198882

)

(1205832119901

+ 1205832119902

+ 1198881

) (1205902119901

+ 1205902119902

+ 1198882

)

(19)


(a) (b) (c) (d)



(a) (b)

(c) (d)

(e) (f)



119901

120583119902


119901

1205902

119902


119901119902


= (1198961

119871)2

and 1198882

= (1198962



1

= 001 and1198962

= 003 by default




(a) (b) (c) (d) (e)





Tang et al [32] 475 010 09037Our 433 020 08255


Tang et al [32] 679 017 09989Our 776 025 09983


Our 590 057 09972





image

5 Conclusions









Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c) (d)



(a) (b)

(c) (d)

(e) (f)



119901

120583119902


119901

1205902

119902


119901119902


= (1198961

119871)2

and 1198882

= (1198962



1

= 001 and1198962

= 003 by default




(a) (b) (c) (d) (e)





Tang et al [32] 475 010 09037Our 433 020 08255


Tang et al [32] 679 017 09989Our 776 025 09983


Our 590 057 09972





image

5 Conclusions









Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)

(e) (f)



119901

120583119902


119901

1205902

119902


119901119902


= (1198961

119871)2

and 1198882

= (1198962



1

= 001 and1198962

= 003 by default




(a) (b) (c) (d) (e)





Tang et al [32] 475 010 09037Our 433 020 08255


Tang et al [32] 679 017 09989Our 776 025 09983


Our 590 057 09972





image

5 Conclusions









Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c) (d) (e)





Tang et al [32] 475 010 09037Our 433 020 08255


Tang et al [32] 679 017 09989Our 776 025 09983


Our 590 057 09972





image

5 Conclusions









Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Acknowledgments


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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