research article multifocus image fusion using...

Research ArticleMultifocus Image Fusion UsingBiogeography-Based Optimization

Ping Zhang1 Chun Fei2 Zhenming Peng1 Jianping Li2 and Hongyi Fan3

1School of Optoelectronic Information University of Electronic Science and Technology of China Chengdu 611731 China2School of Computer Science amp Engineering University of Electronic Science and Technology of China Chengdu 611731 China3School of Engineering Brown University Providence RI 02912 USA

Correspondence should be addressed to Ping Zhang pingzhuestceducn

Received 11 October 2014 Revised 4 February 2015 Accepted 7 February 2015

Academic Editor George S Dulikravich

Copyright copy 2015 Ping Zhang 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

For multifocus image fusion in spatial domain sharper blocks from different source images are selected to fuse a new image Blocksize significantly affects the fusion results and a fixed block size is not applicable in various multifocus images In this paper anovel multifocus image fusion algorithm using biogeography-based optimization is proposed to obtain the optimal block size Thesharper blocks of each source image are first selected by sum modified Laplacian and morphological filter to contain an initialfused image Then the proposed algorithm uses the migration and mutation operation of biogeography-based optimization tosearch the optimal block size according to the fitness function in respect of spatial frequency The chaotic search is adopted duringiteration to improve optimization precision The final fused image is constructed based on the optimal block size Experimentalresults demonstrate that the proposed algorithm has good quantitative and visual evaluations

1 Introduction

Optical lenses with long focal lengths often suffer from theproblem of limited depth of field It is impossible to getan image that contains all relevant objects in focus Theobjects only on the focus plane are sharpness and otherobjects in front of or behind the focus plane are blurred [1]Multifocus fusionmethod which synthesizes multiple imagesof the same view point under different focal settings canbe used to extend depth of field and obtain an all-in focusimage The fused image has more useful information of theview point and is more suitable for many applications thanany individual images Multifocus image fusion technologyhas played important roles in many fields such as targetrecognition remote sensing medical diagnosis and militaryapplication [2]

Many multifocus fusion algorithms have been proposedin recent years Basically these fusion algorithms can becategorized into two groups spatial domain fusion andtransform domain fusion [3] For spatial domain fusion anew image is fused by directly selecting different regions

from source images Firstly source images are divided intononoverlapping blocks Then the sharpness values of blocksare calculated based on different sharpnessmeasuremethodsFinally the sharper blocks from different source images areselected to fuse a new image The common algorithms inspatial domain include average variance energy of imagegradient (EOG) sum modified Laplacian (SML) and spatialfrequency (SF) [4] Recently many new algorithms in spatialdomain have been proposed to improve efficiency such asartificial neural network (ANN) [5] pulse coupled neu-ral network (PCNN) [6] independent component analysis(ICA) [7] robust principal component analysis (RPCA) [8]and neighbor distance [9] For transform domain fusion anew image is generated with certain frequency transformsFirstly source images are converted into a transform domainto obtain the corresponding transform coefficients Thenthe transform coefficients are integrated together based ondifferent fusion rules Finally the fused image is constructedby applying the inverse transform The common algorithmsin transformdomain are based onpyramid transform such asLaplacian pyramid (LAP) gradient pyramid (GRP) ratio of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 340675 14 pageshttpdxdoiorg1011552015340675

2 Mathematical Problems in Engineering

low-pass pyramid (RAP) andmorphological pyramid (ROP)[10] Some fusion algorithms based on wavelet transformare proposed which are generally superior to the fusionalgorithms based on pyramid transform such as discretewavelet transform (DWT) [11] stationary wavelet trans-form (SWT) [12] and dual-tree complex wavelet transform(DTCWT) [13] Recently many new multiscale multireso-lution transform algorithms have been widely proposed inimage fusion such as curvelet transform [14] contourlettransform [15] nonsubsampled contourlet transform (NSCT)[16] and nonsubsampled shearlet transform (NSST) [17]

The transform domain algorithms have no block artifactbut they usually are complicated and time-consuming toimplement And the multiscale multiresolution transformalgorithms are generally shift-variant and sensitive to noiseThe spatial domain algorithms are simple to implement andhave low complexity [18] However block artifacts inevitablyexist because the block size is fixed in these algorithms If theblock size is too large there are both in-focus part and out-of-focus part in the same block The blockrsquos sharpness maybe incorrect and the out-of-focus part may be selected as thepart of the fused image when considering the integrity of thesegmented part If the block size is too small the fused resultis sensitive to noise and the computational complexity is toohigh Obviously a fixed block size may not be applicable invarious multifocus images

To solve the fixed block size problem some image fusionalgorithms based on optimization are proposed such asgenetic algorithm (GA) [19 20] particle swarm optimization(PSO) [21] and differential evolution (DE) [22]These fusionalgorithms utilize the global optimization characteristics toobtain the optimal block size which effectively suppress theblock artifact and increase the performance of the fusedimages However the efficiency of these fusion algorithmsdepends largely on the performance of the optimizationalgorithms GA PSO and DE are not so satisfactory becauseof low convergence rate and low optimization accuracy

A novel multifocus image fusion algorithm using bio-geography-based optimization is proposed in this paperBiogeography-based optimization (BBO) [23] is a new swarmintelligence optimization which is proposed by Dr Simonin 2008 BBO obtains the global optimum through itsmigration mechanisms and mutation operation BBO hasfast convergence rate and high search precision comparedwith GA PSO and DE [23] These advantages make BBOsolve more effectively complex optimization problem It hasbeen applied to image and video processing such as imageclassification [24] image matching [25] image segmentation[26] image enhancement [27] and motion estimation forvideo coding [28] In this paper the proposed algorithmutilizes the biogeography-based optimization technique tofind the best block sizeMoreover chaotic search is embeddedto improve optimization accuracy Experiments on variousmultifocus images demonstrate that the proposed algorithmhas good performance in terms of quantitative and visualevaluations

The rest of this paper is organized as follows In Section 2the biogeography-based optimization is briefly reviewedSection 3 describes the proposed image fusion algorithm in

Migration rate

120583

120582

Immigration rate

Emigration rate

HSI

E = I

Figure 1 Island migration rates versus HSI

detail The experiment results and discussions are presentedin Section 4 The conclusions are given in Section 5

2 Biogeography-Based Optimization

Dr Simon proposed biogeography-based optimization(BBO) in 2008 [23] It is based on the mathematics ofbiogeography which describes how species migrate from oneisland to another how new species arise and how speciesbecome extinct BBO which is similar to GA PSO and DEis a stochastic algorithm to solve optimization problem Ithas two important operations one is migration and the otherone is mutation Species among neighboring islands sharetheir information through migration Individual speciesimprove their diversity through mutation Global optimumcan be effectively obtained with migration and mutationBBO has good performance because of its fast convergencerate and high search precision [23]

21 Migration In BBO each solution of optimization prob-lem is regarded as an island The feature of each solution isregarded as a suitability index variable (SIV) and the fitnessvalue of each solution is regarded as its habitat suitabilityindex (HSI) The higher the HIS of an island the betterthe performance of the optimization problem High HSIand low HSI use the emigration and immigration rates ofeach solution to probabilistically share information betweenislands as in Figure 1

The immigration rate 120582 and the emigration rate 120583 are thefunctions of the number of species in the island

120582119904= 119868(1 minus

119878

119878max)

120583119904=119864119878

119878max

(1)

where 119864 is the maximum emigration rate 119868 is the maximumimmigration rate 119878 is the number of species of the 119878th

Mathematical Problems in Engineering 3

individual and 119878max is the maximum number of species Forsimplicity here assume 119864 = 119868 = 1 that is 120582

119896+ 120583

119896= 119864

The basic step of migration is as follows Firstly calculatethe HSI values of all islands and sort them in descendingorder Second select one island that is needed to immigratebased on immigration rate and choose its adjacent islandsbased on emigration rate Then randomly select SIV valuefrom the adjacent islands to replace the SIV value of thatisland Recalculate the HSI values of all islands The islandwith highest HSI value is the optimal solution

22 Mutation An islandrsquos HSI may change suddenly dueto apparently random events such as disease and naturalcatastrophes In BBO this phenomenon is called mutationThe mutation rates119898

119904is determined as follows

119898119904= 119898max (1 minus

119875119904

119875max) (2)

where 119898max is a mutation parameter and 119875max = argmax119875119894

119894 = 1 119873 119875119904is probability of species The relationship

between 119875119904and migration rate is shown as follows

119875119904=

minus (120582119904+ 120583

119904) 119875

119904+ 120583

119904+1119875119904+1

119878 = 0

minus (120582119904+ 120583

119904) 119875

119904+ 120582

119904minus1119875119904minus1+ 120583

119904+1119875119904+1

1 le 119878 le 119878max minus 1

minus (120582119904+ 120583

119904) 119875

119904+ 120582

119904minus1119875119904minus1

119878 = 119878max

(3)

This mutation approach makes low HSI solutions likelyto mutate which gives them a chance of improving It alsomakes high HSI solutions likely to mutate which gives thema chance of improving more than they already had Thus theBBO mutation strategy tends to increase diversity of islands

23 Elitism Strategy BBO incorporate elitism strategy inorder to save the features of the habitat that has the bestsolution in the iterative process as with GA PSO and DEThis prevents the best solutions from being corrupted byimmigration or being ruined by mutation So even if thehabitat with highest HSI is destroyed BBO has saved it andcan revert back to it if neededThe proposed algorithm in thispaper retains two higher islands as elitesThese elites are keptfrom this generation to the next generation

3 Multifocus Image Fusion UsingBiogeography-Based Optimization

In this paper a novelmultifocus image fusion algorithmusingbiogeography-based optimization is proposed Consideringthe characteristic of image fusion the island SIV and HSI ofBBO is regarded as block size width and height of block sizeand image quality assessment respectively So the dimensionof optimization problem is only two the width and height of

block size Here are the key steps of the proposed algorithmFirstly it selects the random block size as initial islands forutilizing the global random characteristics of BBO Then itcalculates the HSI value of each block size During iterationprocess the width and height of the block size are updatedthrough themigration andmutation operationsThe iterationstops if the termination condition is met Finally the blocksize with the highest HSI is the optimal block sizeThe detailsof the multifocus image fusion using BBO are as follows

31 HSI Function Selection Spatial frequency [4] can mea-sure the overall activity level in images and reflect the abilityto express small details in images The larger the value ofspatial frequency in fused image the better the performanceof fusion In the image fusion algorithms based on GA PSOand DE spatial frequency is chosen as the fitness function[20ndash22] It is a good fitness function to assess the fused imagequality In this paper we also choose the spatial frequency asthe HSI function of BBO

HSI (119909 119910) = SF (119909 119910) (4)

Spatial frequency SF(119909 119910) of119872times119873 size image is definedas follows

SF (119909 119910) = radic1198652119877(119909 119910) + 1198652

119862(119909 119910) (5)

where

119865119877(119909 119910) = radic

1

119872 times119873

119872

sum

119909=1

119873

sum

119910=1

[119868 (119909 119910 + 1) minus 119868 (119909 119910)]2

119865119862(119909 119910) = radic

1

119872 times119873

119872

sum

119909=1

119873

sum

119910=1

[119868 (119909 + 1 119910) minus 119868 (119909 119910)]2

(6)

are the row and column gradients respectively 119868(119909 119910) is thepixel of the image

32 Sharpness Measurement For multifocus image thesharpness areas of different source images are selected toconstruct a fused image How to define the sharpness areas isvery important There are many typical sharpness measure-ments such as EOG SF and SML [18] They measure thevariation of pixels of blocks in source images to determinethe sharpness The blocks with greater values are consideredas in-focus blocks Experimental results of Huang and Jing[18] show that SMLmethodprovides better performance thanEOG and SF It differs from the usual Laplacian operator inthat the absolute values of the partial second derivatives aresummed instead of their actual values The SML is defined asfollows

SML (119868 (119909 119910)) =119882

sum

119908=minus119882

119867

sum

ℎ=minus119867

[119871 (119868 (119909 + 119908 119910 + ℎ))]2

(7)

where 119868(119909 119910) is the pixel of the image119882 and119867 is thewindowsize 119871(119868(119909 119910)) is the modified Laplacian given as follows


Here step is a variable that is used to set the distance betweenthe central pixel and the pixels are used to compute the secondorder derivative In this paper a value of step = 1 is found toproduce the best results

119871 (119868 (119909 119910)) =10038161003816100381610038162119868 (119909 119910) minus 119868 (119909 minus step 119910) minus 119868 (119909 + step 119910)

1003816100381610038161003816

+10038161003816100381610038162119868 (119909 119910) minus 119868 (119909 119910 minus step) minus 119868 (119909 119910 + step)

1003816100381610038161003816

(8)

Source images 119860 and 119861 are divided into nonoverlappedblocks with the size of119898 times 119899 Refer to the 119894th block of sourceimages119860 and119861 images by 119868119860

119894and 119868119861

119894 respectivelyThe focus

region 119877sml119860119894(119909 119910) according to SML is defined as follows

119877sml119860119894(119909 119910) =

1 SML119860119894(119909 119910) gt SML119861

119894(119909 119910)

0 otherwise(9)

However there are always some small holes in focusregion in practice Here the morphological filter is used tosmooth blocks and remove the holes and gulfs defects whichis shown as follows

119874119862 (119868 (119909 119910)) = max 119874 (119868 (119909 119910)) 119862 (119868 (119909 119910))

119874 (119868 (119909 119910)) = 119868 (119909 119910) minus 119868 (119909 119910) ∘ 119863

119862 (119868 (119909 119910)) = 119868 (119909 119910) sdot 119863 minus 119868 (119909 119910)

(10)

The structure element 119863 is 5 times 5 ldquodiamondrdquo matrix119874(119868(119909 119910)) and 119862(119868(119909 119910)) is the opening and closing opera-tionThe focus region119877119900119888119860

119894(119909 119910) according tomorphological

operation is defined as follows

119877119900119888119860

119894(119909 119910) =

1 119874119862119860

119894(119909 119910) gt 119874119862

119861

119894(119909 119910)

0 otherwise(11)

So the final focus region 119877119860119894(119909 119910) is defined as

119877119860

119894(119909 119910) =

1 119877sml119860119894(119909 119910) = 1 119877119900119888

119860

119894(119909 119910) = 1

0 otherwise(12)

The focus region119877119861119894(119909 119910) is defined same as119877119860

119894(119909 119910)The

blocks of fusion image 119868119865119894are combined with 119868119860

119894and 119868119861

119894in

focused regions

119868119865119894(119909 119910) =

119868119860119894(119909 119910) if 119877119860

119894(119909 119910) = 1

119868119861119894(119909 119910) if 119877119861

119894(119909 119910) = 1

(13)

33 Chaotic Search To further enhance the optimizationaccuracy of BBO the modified chaotic search algorithm isadopted during each late period of iteration Chaotic searchalgorithm makes full use of the ergodicity randomness andregularity of chaotic motion It obtains the optimal solutionby traversing every state in a small area without repetition[29] Therefore further chaotic search can generate several

neighborhood islands based on the island with the highestHSI value to improve the performance of the proposedalgorithm

The basic process is that the chaotic variables are firstlygenerated based on the current optimal solution Then thefitness value of each variable is calculated and the optimalsolution of these chaotic variables is compared with theprevious global optimal solution Better solution is chosen asthe current global optimal solutionThis paper adopts typicallogistic mapping as shown in (14) where 119905 = 1 2 119879and 119879 is the maximum of iteration which is set as 5 120578 is thecontrol parameter The system of function in (14) with 120578 = 41199101isin (0 1) and 119910

1= 025 05 075 is a chaotic system

119910119905+1= 120578119910

119905(1 minus 119910

119905) (14)

The chaotic search is as follows

Step 1 The current optimal solution 119875119887119890119904119905 (the highest HSIpoint of current 119896th iteration) is mapped to domain [0 1]of logistic formula (15) The search area is confined in[119871min 119871max] = [minus4 4]

1199101=119875119887119890119904119905 minus 119871min119871max minus 119871min

(15)

Step 2 The chaotic variable 119910119905is generated by logistic

function in (14) and then returned by inverse mapping interm of the following equations

119901119887119890119904119905119905= 119871min + (119871max minus 119871min) lowast 119910119905 (16)

Each fitness value of each solution of 119901119887119890119904119905119905=

(1199011198871198901199041199051 119901119887119890119904119905

2 119901119887119890119904119905

119879) is calculated and the fitness val-

ues of the optimal solution of them with 119875119887119890119904119905 are comparedIf the former is better than 119875119887119890119904119905 it replaces 119875119887119890119904119905 as globaloptimal solution otherwise return

119875119887119890119904119905 =

119901119887119890119904119905119905

HSI (119901119887119890119904119905119905) lt HSI (119875119887119890119904119905)

119875119887119890119904119905 otherwise(17)

34 Fusion Algorithm Description The multifocus imagefusion algorithm using biogeography-based optimization issummarized as follows

Step 1 Define some initial parameters such as the maximumnumber of species 119875119873 and maximum iteration number 119879max

Step 2 Randomly generate initial islands Island = Island1

Island2 Island

119875119873 Each island contains two parameters

Island119894= 119898

119894 119899

119894119898

119894and 119899

119894are width and height of 119894th block

size respectively Divide source images into nonoverlappingblocks according these initial islands

Step 3 Calculate the sharpness value of blocks to definefocus regions and use these focus regions to construct afused image Compute the HSI functions set HSI

119894(119909 119910) =

HSI1(119909 119910)HSI

2(119909 119910) HSI

119875119873(119909 119910) of the fused image

set 119868119865119894(119909 119910) = 119868119865

1(119909 119910) 119868119865

2(119909 119910) 119868119865

119875119873(119909 119910)


Step 4 BBO and chaotic searches are executed to update thestate of islands as follows

(1) Calculate the immigration rate 120582 the emigration rate120583 and mutation rates119898

119904of each island

(2) Migration operation is as follows the parameters119898

119894 119899

119894 of Island

119894will be changed to generate new

parameters based on immigration rate and emigra-tion rate

(3) Mutation operation is as follows if the mutation rateis not zero mutate the island with the highest HSIvalue

(4) Chaotic search is as follows generate chaotic variables119910119905based on current optimal solution and obtain new

solutions of 119901119887119890119904119905119905= (119901119887119890119904119905

1 119901119887119890119904119905

2 119901119887119890119904119905

119879)

Update the global optimal solution

Step 5 Repeat Steps 3 and 4 until maximum iteration num-ber is reached Finally the optimal block size is obtainedbased on the highest HSI to construct the optimal fusedimage

4 Experiments

The proposed algorithm is called CSBBO In order to verifythe performance of CSBBO algorithm such classical algo-rithms include the pulse coupled neural network (PCNN)[6] in spatial domain nonsubsampled contourlet transform(NSCT) [16] and nonsubsampled shearlet transform (NSST)[17] in frequency domain genetic algorithm (GA) [20]particle swarm optimization (PSO) [21] and differentialevolution (DE) [22] based on intelligence optimization

Two kinds of experiment results with a reference imageand without a reference image are shown in this paperDifferent objective evaluation is used in these two kindsof experiments A reference image is regarded as a groundtruth image and various algorithms are assessed using thedifference or structure similarity between the fused imageand reference image Without reference image various algo-rithms are assessed using the information which transfersfrom the source images to the fused image In the imple-mentation it is assumed that two multifocus source imagesare registered before the image fusion process All multifocussource images and reference images are downloaded fromhttpwwwimagefusionorg

41 Fusion with Reference Image The parameters of variousalgorithms are set as follows For PCNN 120572

119871= 006931

120572120579= 02 and 120573 = 02 The maximal iterative number is

120578 = 200 119881119871= 10 and 119881

120579= 20 [6] For NSTC and

NSST the decomposition level is four and the directionsfrom coarse scale to finer scale are (4 8 8 16) [15 17] Thelow-pass subband coefficients and the band-pass subbandcoefficients are simplymerged by ldquoaveragingrdquo scheme and theldquoabsolute maximum choosingrdquo scheme respectively For GACrossoverRate = 001 and MutationRate = 001 [20] ForPSO 119888

1= 119888

2= 2 and weight = 09 [21] For DE the version

is rand1bin CrossoverRate = 06 and ScalingFactor = 09[22]

With the reference image root mean squared error(RMSE) peak signal-to-noise ratio (PSNR) and structuralsimilarity metric (SSIM) [30] are used as quantitative assess-ment metrics to compare various fusion algorithms RMSEand PSNR are commonly used to evaluate the differencebetween the fused image and the reference image SSIMcommonly evaluates the structure similarity between thefused image and reference imageThe higher PSNR and SSIMvalues and the less RMSE value indicate better fusion resultsThey are defined as

RMSE = radic 1

119872 times119873

119872

sum

119894=1

119873

sum

119895=1

(119877 (119894 119895) minus 119865 (119894 119895))2

PSNR = 10log10

(119872 times 119873)2

sum119872

119894=1sum

119873

119895=1[119877 (119894 119895) minus 119865 (119894 119895)]

2

SSIM (119877 119865) = (2120583

119903120583119891+ 119888

1

1205832119903+ 1205832

119891+ 119888

1

)

120572

sdot (2120590

119903120590119891+ 119888

2

1205902119903+ 1205902

119891+ 119888

2

)

120573

sdot (120590119903119891+ 119888

3

120590119903120590119891+ 119888

3

)

120574

(18)

where 119877(119894 119895) and 119865(119894 119895) are the pixel values of the referenceimage and the fused image respectively The image size is119872 times 119873 120583

119903 120583

119891 120590

2

119903 120590

2

119891 120590

119903119891is mean variance and covariance

respectively 1198881 1198882 1198883are constants closing to zero

It is impossible to guarantee that an optimal solutioncould be reached after a number of iterations because ofthe stochastic behavior of GA PSO DE and BBO [20ndash22]Moreover the optimal solution almost could not be improvedin late of iterations because of the premature behavior of GAPSO DE and BBO Too much iteration will result in veryhigh computational complexity The experimental result in[22] has shown that the performances of image fusion basedon GA and DE cannot be significantly improved after 20iterations There is the same trend of the image fusion basedon PSO BBO and CSBBO in our experimentsTherefore themaximum iteration number of all intelligence optimizationalgorithms is set to 20 in order to reduce complexity

In order to analyze the parameters of CSBBO populationnumber and maximummutation parameter the experimentof ldquoPepperrdquo image fusion is given in Table 1 Three images ofldquoPepperrdquo are shown as in Figure 2 Figure 2(a) is the referenceimage in focus everywhere Figures 2(b) and 2(c) focus oncenter and surround respectively All images have 512 times 512pixels with 256 level gray scales Considering the randomnessof four intelligence optimization algorithms experiments arerepeated 30 times and the average results are saved FromTable 1 we can see that the difference between RMSE valuesobtained for different population number (PN) is not veryhigh Then PN is set to 10 taking into account the efficiencyand complexity And the most suitable mutation parameter119898max that gives the best RMSE value is 02 when PN is 10


(a) (b) (c)

Figure 2 Pepper images (a) reference image (b) focus on center (c) focus on surround

(a) PCNN (b) NSCT (c) NSST (d) GA (13 times 8)

(e) PSO (10 times 15) (f) DE (9 times 14) (g) BBO(8 times 9) (h) CSBBO (10 times 10)

Figure 3 Fusion results for Pepper images (a) PCNN (b) NSCT (c) NSST (d) GA (e) PSO (f) DE (g) BBO (h) CSBBO

Table 1 Average RMSE results obtained by CSBBO for Pepperimages fusion

119898max = 01 119898max = 02 119898max = 03 119898max = 04

PN = 5 13576 13440 13763 13863PN = 10 13019 12922 13528 13577PN = 15 12854 12718 13403 13325

The fused results of various algorithms are shown inFigure 3 The value inside the parentheses represents theoptimal block size Each algorithm can obtain a good fusedimage which is almost the same as the reference image Forclearer comparison the difference results between the fusedimage and the reference image are shown in Figure 4 Ideallythe difference should be zero So less residual features inthe difference results means better performance of the fusion

algorithm It can be observed that the difference imagesobtained by PCNN NSCT and NSST have more residualfeatures than GA PSO DE BBO and CSBBOThis indicatesthat these fusion algorithms based on block optimizationof intelligent techniques can improve the performance ofimage fusion Moreover we can see that the difference imageobtained by CSBBO has the least residue among them

Table 2 shows the quantitative evaluation results of vari-ous image fusion algorithms with the reference image As wecan see GA PSO DE BBO and CSBBO based on intelligentoptimization are able to obtain higher PSNR SSIMvalues andless RMSE value Their performances are superior to PCNNNSCT and NSST Among them the performance of BBO isslightly better than GA PSO and DE And CSBBO has thehighest PSNR and SSIM values and the least RMSE valuewhich can obtainmore useful features from the source imagesand has better performance than other fusion algorithms


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

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

Figure 4 Difference results for Pepper images (a)sim(h) is the difference between Figure 2(a) and Figures 3(a)sim3(h) respectively

Table 2 Evaluation of various image fusion algorithms with the reference image

PCNN NSCT NSST GA PSO DE BBO CSBBORMSE 99285 13422 13396 13013 12942 13010 12953 12922PSNR 38195 46886 46894 47020 47036 47021 47044 47051SSIM 09442 09986 09986 09989 09990 09989 09990 09991

42 Fusion without Reference Image Natural digital cameraimages have no reference images which contain multipleobjects in focus and defocus parts because of their locationat different distance from the camera We choose two grayimages Plane (160 times 160) Clock (256 times 256) and two colorimages Flower (480 times 480) Book (512 times 512) They all havetwo source images with different focus parts as shown inFigure 5 respectively

For these four image pairs the fused results of variousalgorithms are presented in Figures 6 8 10 and 12 respec-tively To make better comparisons the difference results ofvarious algorithms are presented in Figures 7 9 11 and 13respectively

From Figures 6 and 8 for gray image Plane and Clockwe can see that each algorithm can get a clear fused imageexcept for PCNN The fused image obtained by PCNN hasno good visual effect because there are some blurred blocksin focus area such as the front plane and the clock surfaceMoreover the fused image obtained by PCNN does notmaintain continuous edge features such as the book edgeand the clock edge in Figure 8 The fused images obtained byNSCT NSST GA PSO DE BBO and CSBBO have almostno difference for visual effects

However we can distinguish the fusion performance ofthese algorithms from the difference results from Figures 7and 9 It can be observed that the difference images obtained

by NSCT and NSST have more residual features than GAPSO DE BBO and CSBBO There is large residual informa-tion of front plane obtained by NSCT and NSST in Figures7(b) and 7(c) The numbers of clock surface are obviouslyremained in Figures 9(b) and 9(c) From Figures 7(d)sim7(g)and Figures 9(d)sim9(g) it can be found that the differenceresults obtained by GA PSO DE BBO and CSBBO arebetter than other algorithms Among them it is obvious thatCSBBO has the least residual features The difference imagesobtained by GA PSO DE and BBO have more residualinformation especially in the edge parts of the focus areasuch as Figures 9(d)sim9(f) and they also have more blockartifacts in Figures 7(d)sim7(f) Figures 6(g) 8(g) 7(g) and 9(g)show that CSBBO can detect the focus region accurately andtransfer more useful features from source images to the fusedimage

From Figures 10 and 12 for color image Flower and Bookwe can see the performance of each algorithm in vision Thefused image obtained by PCNNhas distortion brightness andblurred edges such as the switch and bricks in Figure 10The fused images obtained by NSCT and NSST preserve agood brightness and edge features but they still have someobvious blurred area such as the words at the top right cornerin Figure 12 The fused images obtained by GA PSO DEBBO and CSBBO improve the visual effects because thesealgorithms depress efficiently the blurred vision such as the


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

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

Figure 5 Source images (a) Plane left focus (b) Plane right focus (c) Clock left focus (d) Clock right focus (e) Flower left focus (f) Flowerright focus (g) Book left focus (h) Book right focus


(e) PSO (9 times 6) (f) DE (14 times 4) (g) BBO (11 times 5) (h) CSBBO (9 times 9)

Figure 6 Fusion results for Plane images (a) PCNN (b) NSCT (c) NSST (d) GA (e) PSO (f) DE (g) BBO (h) CSBBO

words at the top right corner in these images which areclearer in Figure 12 Moreover we can distinguish the fusionperformance of these algorithms from the difference resultsfrom Figures 11 and 13 The difference image obtained byPCNN has the most residual features in Figures 11(a) and13(a) And the difference images obtained byNSCT andNSSThave more residual features than GA PSO DE BBO andCSBBO in Figures 11(b) 11(c) 13(b) and 13(c) Compared

with GA PSO DE and BBO CSBBO has the least residualfeatures From the above-mentioned results it can be seenthat CSBBO can obtain clearer and more nature fused imagethan other algorithms

For objective assessment without the reference imagenatural multifocus images should be evaluated by nonref-erence fusion metrics In this paper the spatial frequency(SF) [4] feature mutual information (FMI) [31] the edge


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

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

Figure 7 Difference results for Plane images (a)sim(h) is the difference between Figure 5(b) and Figures 6(a)sim6(h) respectively

information based metric 119876119860119861

119865[32] and similarity based on

block metric 119876119887(119860 119861 119865) [33] are used to evaluate the fusion

performance of various fusion algorithms SF evaluates theability to express small details in the fused images definedas formula (5) FMI calculates the amount of informationconducted from the source images to the fused image definedas

FMI119860119861119865= sum

119891119886

119901119865119860(119909 119910 119911 119908) log

119901119865119860(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119860(119911 119908)

+sum

119891119887

119901119865119861(119909 119910 119911 119908) log

119901119865119861(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119861(119911 119908)

(19)where119860 and119861 are the two source images119865 is the fused image119901119865119860(119909 119910 119911 119908) and 119901

119865119861(119909 119910 119911 119908) are joint probability distri-

bution 119901119860(119911 119908) and 119901

119861(119911 119908) are probability distribution

119876119860119861

119865measures the relative amount of edge information

that is transferred from the source images into the fusedimage It is defined as

119876119860119861

119865= (

119872

sum

119898=1

119873

sum

119899=1

119876119860119865

(119898 119899)119908119860119865

(119898 119899)

+ 119876119861119865

(119898 119899) 119908119861119865

(119898 119899))

sdot(

119872

sum

119898=1

119873

sum

119899=1

119908119860119865

(119898 119899) + 119908119861119865

(119898 119899))

minus1

(20)

where 119876119860119865

(119898 119899) and 119876119861119865

(119898 119899) are the edge preserva-tion values for two 119872 times 119873 source images 119876119894119865

(119898 119899) =

119876119894119865

119892(119898 119899)119876

119894119865

119900(119898 119899) (119894 = 119860 119861) 119876119894119865

119892(119898 119899) and 119876119894119865

119900(119898 119899) are

the Sobel edge strength and orientation preservation valueat location (119898 119899) respectively 119908119860119865

(119898 119899) and 119908119861119865

(119898 119899) areweighted coefficients119876119887(119860 119861 119865) is based on the universal image quality index

and uses the similarity between blocks of pixels in sourceimages and the fused image as the weighting factors for themetrics It is defined as

119876119887(119860 119861 119865)

= sum

119908isin119882

sim (119860 119861 119865 | 119908)

sdot (119876 (119860 119865 | 119908) minus 119876 (119861 119865 | 119908)) + 119876 (119861 119865 | 119908)

(21)

where 119908 is the analysis window and 119882 is the family of allwindows 119876(119894 119865 | 119908) (119894 = 119860 119861) is the image quality indexsim(119860 119861 119865 | 119908) is the similarity in spatial domain betweenthe input image and the fused image

For all the nonreference fusion metrics the larger thevalue is the better the fusion performance is Table 3 showsthe objective assessments of the fused images obtained byvarious fusion algorithms without the reference image Fromthis table it can be observed that CSBBO outperforms otheralgorithms in terms of SF FMI and 119876119860119861

119865for all images

For 119876119887(119860 119861 119865) PCNN performs a little bit better than

CSBBO for Plane image but there are obvious defects in thefused image obtained by PCNN in Figure 6(a) Based on the




Figure 8 Fusion results for Clock images (a) PCNN (b) NSCT (c) NSST (d) GA (e) PSO (f) DE (g) BBO (h) CSBBO

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

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

Figure 9 Difference results for Clock images (a)sim(h) is the difference between Figure 5(d) and Figures 8(a)sim8(h) respectively

analysis above it can be concluded that CSBBO has goodperformance in terms of quantitative and visual evaluations

5 Conclusion

For spatial domain fusion the fixed block size of sourceimages will result in block artifacts in multifocus image

fusion To solve this problem a novel multifocus imagefusion algorithm using biogeography-based optimization isproposed in this paper An optimized block size of eachsource image is obtained by BBOand chaotic searchThefinalfused image consisted of the optimal block size not a fixedblock sizeThe experiment results with or without a referenceimage show that CSBBO outperforms the traditional PCNNNSCT and NSST algorithms Furthermore among image




Figure 10 Fusion results for Flower images (a) PCNN (b) NSCT (c) NSST (d) GA (e) PSO (f) DE (g) BBO (h) CSBBO

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

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

Figure 11 Difference results for Flower images (a)sim(h) is the difference between Figure 5(f) and Figures 10(a)sim10(h) respectively




Figure 12 Fusion results for Book images (a) PCNN (b) NSCT (c) NSST (d) GA (e) PSO (f) DE (g) BBO (h) CSBBO

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

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

Figure 13 Difference results for Book images (a)sim(h) is the difference between Figure 5(h) and Figures 12(a)sim12(h) respectively


Table 3 Evaluation of various image fusion algorithms without the reference image

PCNN NSCT NSST GA PSO DE BBO CSBBO

Plane

SF 107549 124720 125931 126338 126364 125951 126426 126454FMI 09258 09273 09310 09362 09375 09370 09372 09378119876

119860119861

11986507195 07298 07321 07432 07524 07535 07530 07546

119876119887(119860 119861 119865) 08944 08645 08634 08566 08668 08769 08762 08812

Clock

SF 215565 227596 230138 229630 230049 230087 229725 230105FMI 08828 08805 08829 08834 08840 08842 08842 08843119876

119860119861

11986506869 06830 06889 07165 07180 07186 07213 07219

119876119887(119860 119861 119865) 08739 08633 08624 08751 08745 08785 08795 08803

Flower

SF 147468 163808 166530 166400 166500 166129 166396 166539FMI 08479 08755 08774 08808 08812 08813 08813 08815119876

119860119861

11986506506 06980 06994 07161 07163 07166 07172 07173

119876119887(119860 119861 119865) 08660 08974 08959 08972 08968 08978 08990 08995

Book

SF 279575 286606 289587 289555 289917 289727 289969 289991FMI 08678 08631 08653 08672 08671 08670 08670 08673119876

119860119861

11986506961 06975 07014 07206 07211 07199 07205 07214

119876119887(119860 119861 119865) 08848 08546 08563 08930 08930 08939 08933 08950

fusion algorithms based on intelligence optimization CSBBOis superior to GA PSO DE and BBO in terms of visualanalysis and quantitative evaluation In the future it is worthyto further investigate other types of fitness functions thatmayaffect fusion performance And it is possible to apply theproposed algorithm to other image fusions of different typesof images

Conflict of Interests

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

Acknowledgments

The authors would like to thank the editor and anonymousreviewersThis research is supported by the National NaturalScience Foundation of China (no 61308102) SpecializedResearch Fund for the Doctoral Program of Higher Edu-cation (no 20130185120038) China Postdoctoral ScienceFoundation (no 2013M531946) Science and TechnologyPlanning Project of Sichuan Province (nos 2014GZ0005and 2014JY0228) and Fundamental Research Funds for theCentral Universities China (no ZYGX2013J059)

References

[1] A Ardeshir Goshtasby and S Nikolov ldquoImage fusion advancesin the state of the artrdquo Information Fusion vol 8 no 2 pp 114ndash118 2007

[2] A P James and B V Dasarathy ldquoMedical image fusion a surveyof the state of the artrdquo Information Fusion vol 19 no 1 pp 4ndash192014

[3] A Anish and T J Jebaseeli ldquoA survey on multi-focus imagefusion methodsrdquo International Journal of Advanced Research inComputer Engineering amp Technology vol 1 no 8 pp 319ndash3242012

[4] S T Li J T Kwok and Y N Wang ldquoCombination of imageswith diverse focuses using the spatial frequencyrdquo InformationFusion vol 2 no 3 pp 169ndash176 2001

[5] S Li J T Kwok and Y Wang ldquoMultifocus image fusion usingartificial neural networksrdquo Pattern Recognition Letters vol 23no 8 pp 985ndash997 2002

[6] Z Wang Y Ma and J Gu ldquoMulti-focus image fusion usingPCNNrdquo Pattern Recognition vol 43 no 6 pp 2003ndash2016 2010

[7] N Cvejic D Bull and N Canagarajah ldquoRegion-based multi-modal image fusion using ICA basesrdquo IEEE Sensors Journal vol7 no 5 pp 743ndash750 2007

[8] T Wan C C Zhu and Z C Qin ldquoMultifocus image fusionbased on robust principal component analysisrdquo Pattern Recog-nition Letters vol 34 no 9 pp 1001ndash1008 2013

[9] H J Zhao Z W Shang Y Y Tang and B Fang ldquoMulti-focus image fusion based on the neighbor distancerdquo PatternRecognition vol 46 no 3 pp 1002ndash1011 2013

[10] A Toet L J van Ruyven and J M Valeton ldquoMerging thermaland visual images by a contrast pyramidrdquo Optical Engineeringvol 28 no 7 pp 789ndash792 1989

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

[12] M Beaulieu S Faucher and L Gagnon ldquoMulti-spectral imageresolution refinement using stationary wavelet transformrdquo inProceedings of the IEEE International Conference on Geoscienceand Remote Sensing pp 4032ndash4034 July 2003

[13] J J Lewis R J OrsquoCallaghan S G Nikolov D R Bull andN Canagarajah ldquoPixel- and region-based image fusion withcomplex waveletsrdquo Information Fusion vol 8 no 2 pp 119ndash1302007

[14] F Nencini A Garzelli S Baronti and L Alparone ldquoRemotesensing image fusion using the curvelet transformrdquo InformationFusion vol 8 no 2 pp 143ndash156 2007

[15] S Yang M Wang L Jiao R Wu and Z Wang ldquoImage fusionbased on a new contourlet packetrdquo Information Fusion vol 11no 2 pp 78ndash84 2010


[16] C Fei and J-P Li ldquoMulti-focus image fusion based on non-subsampled contourlet transform and multi-objective opti-mizationrdquo in Proceedings of the International Conference onWavelet Active Media Technology and Information Processing(ICWAMTIP rsquo12) pp 189ndash192 chn December 2012

[17] Q-G Miao C Shi P-F Xu M Yang and Y-B Shi ldquoA novelalgorithm of image fusion using shearletsrdquo Optics Communica-tions vol 284 no 6 pp 1540ndash1547 2011

[18] W Huang and Z Jing ldquoEvaluation of focus measures in multi-focus image fusionrdquo Pattern Recognition Letters vol 28 no 4pp 493ndash500 2007

[19] X M Zhang J Q Han and P Liu ldquoRestoration and fusionoptimization scheme of multifocus image using genetic searchstrategiesrdquo Optica Applicata vol 35 no 4 pp 927ndash942 2005

[20] J Kong K Zheng J Zhang and X Feng ldquoMulti-focus imagefusion using spatial frequency and genetic algorithmrdquo Interna-tional Journal of Computer Science and Network Security vol 8no 2 pp 220ndash224 2008

[21] X M Zhang L B Sun J Han and G Chen ldquoAn applicationof swarm intelligence binary particle swarm optimization algo-rithm to multi-focus image fusionrdquo Optica Applicata vol 40no 4 pp 949ndash964 2010

[22] V Aslantas and R Kurban ldquoFusion of multi-focus imagesusing differential evolution algorithmrdquo Expert Systems withApplications vol 37 no 12 pp 8861ndash8870 2010

[23] D Simon ldquoBiogeography-based optimizationrdquo IEEE Transac-tions on Evolutionary Computation vol 12 no 6 pp 702ndash7132008

[24] V K Panchal P Singh N Kaur and H Kundra ldquoBiogeographybased satellite image classificationrdquo International Journal ofComputer Science and Information Security vol 6 no 2 pp269ndash274 2009

[25] X HWang H B Duan and D L Luo ldquoCauchy biogeography-based optimization based on lateral inhibition for imagematch-ingrdquo Optik vol 124 no 22 pp 5447ndash5453 2013

[26] A Chatterjee P Siarry A Nakib and R Blanc ldquoAn improvedbiogeography based optimization approach for segmentationof human head CT-scan images employing fuzzy entropyrdquoEngineering Applications of Artificial Intelligence vol 25 no 8pp 1698ndash1709 2012

[27] J Jasper S B Shaheema and S B Shiny ldquoNatural imageenhancement using a biogeography based optimizationenhanced with blended migration operatorrdquo MathematicalProblems in Engineering vol 2014 Article ID 232796 11 pages2014

[28] P Zhang P Wei and H-Y Yu ldquoBiogeography-based optimisa-tion search algorithm for block matching motion estimationrdquoIET Image Processing vol 6 no 7 pp 1014ndash1023 2012

[29] C-G Fei and Z-Z Han ldquoImproved chaotic optimizationalgorithmrdquo Control Theory and Applications vol 23 no 3 pp471ndash474 2006

[30] 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

[31] M B A Haghighat A Aghagolzadeh and H Seyedarabi ldquoAnon-reference image fusion metric based on mutual informa-tion of image featuresrdquo Computers and Electrical Engineeringvol 37 no 5 pp 744ndash756 2011

[32] C S Xydeas and V Petrovic ldquoObjective image fusion perfor-mance measurerdquo Electronics Letters vol 36 no 4 pp 308ndash3092000

[33] N Cvejic and A Łoza ldquoA novel metric for performance evalu-ation of image fusion algorithmsrdquo Transactions on EngineeringComputing and Technology vol 7 pp 80ndash85 2005

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


low-pass pyramid (RAP) andmorphological pyramid (ROP)[10] Some fusion algorithms based on wavelet transformare proposed which are generally superior to the fusionalgorithms based on pyramid transform such as discretewavelet transform (DWT) [11] stationary wavelet trans-form (SWT) [12] and dual-tree complex wavelet transform(DTCWT) [13] Recently many new multiscale multireso-lution transform algorithms have been widely proposed inimage fusion such as curvelet transform [14] contourlettransform [15] nonsubsampled contourlet transform (NSCT)[16] and nonsubsampled shearlet transform (NSST) [17]

The transform domain algorithms have no block artifactbut they usually are complicated and time-consuming toimplement And the multiscale multiresolution transformalgorithms are generally shift-variant and sensitive to noiseThe spatial domain algorithms are simple to implement andhave low complexity [18] However block artifacts inevitablyexist because the block size is fixed in these algorithms If theblock size is too large there are both in-focus part and out-of-focus part in the same block The blockrsquos sharpness maybe incorrect and the out-of-focus part may be selected as thepart of the fused image when considering the integrity of thesegmented part If the block size is too small the fused resultis sensitive to noise and the computational complexity is toohigh Obviously a fixed block size may not be applicable invarious multifocus images

To solve the fixed block size problem some image fusionalgorithms based on optimization are proposed such asgenetic algorithm (GA) [19 20] particle swarm optimization(PSO) [21] and differential evolution (DE) [22]These fusionalgorithms utilize the global optimization characteristics toobtain the optimal block size which effectively suppress theblock artifact and increase the performance of the fusedimages However the efficiency of these fusion algorithmsdepends largely on the performance of the optimizationalgorithms GA PSO and DE are not so satisfactory becauseof low convergence rate and low optimization accuracy

A novel multifocus image fusion algorithm using bio-geography-based optimization is proposed in this paperBiogeography-based optimization (BBO) [23] is a new swarmintelligence optimization which is proposed by Dr Simonin 2008 BBO obtains the global optimum through itsmigration mechanisms and mutation operation BBO hasfast convergence rate and high search precision comparedwith GA PSO and DE [23] These advantages make BBOsolve more effectively complex optimization problem It hasbeen applied to image and video processing such as imageclassification [24] image matching [25] image segmentation[26] image enhancement [27] and motion estimation forvideo coding [28] In this paper the proposed algorithmutilizes the biogeography-based optimization technique tofind the best block sizeMoreover chaotic search is embeddedto improve optimization accuracy Experiments on variousmultifocus images demonstrate that the proposed algorithmhas good performance in terms of quantitative and visualevaluations

The rest of this paper is organized as follows In Section 2the biogeography-based optimization is briefly reviewedSection 3 describes the proposed image fusion algorithm in

Migration rate

120583

120582

Immigration rate

Emigration rate

HSI

E = I

Figure 1 Island migration rates versus HSI

detail The experiment results and discussions are presentedin Section 4 The conclusions are given in Section 5

2 Biogeography-Based Optimization

Dr Simon proposed biogeography-based optimization(BBO) in 2008 [23] It is based on the mathematics ofbiogeography which describes how species migrate from oneisland to another how new species arise and how speciesbecome extinct BBO which is similar to GA PSO and DEis a stochastic algorithm to solve optimization problem Ithas two important operations one is migration and the otherone is mutation Species among neighboring islands sharetheir information through migration Individual speciesimprove their diversity through mutation Global optimumcan be effectively obtained with migration and mutationBBO has good performance because of its fast convergencerate and high search precision [23]

21 Migration In BBO each solution of optimization prob-lem is regarded as an island The feature of each solution isregarded as a suitability index variable (SIV) and the fitnessvalue of each solution is regarded as its habitat suitabilityindex (HSI) The higher the HIS of an island the betterthe performance of the optimization problem High HSIand low HSI use the emigration and immigration rates ofeach solution to probabilistically share information betweenislands as in Figure 1

The immigration rate 120582 and the emigration rate 120583 are thefunctions of the number of species in the island

120582119904= 119868(1 minus

119878

119878max)

120583119904=119864119878

119878max

(1)

where 119864 is the maximum emigration rate 119868 is the maximumimmigration rate 119878 is the number of species of the 119878th



119896+ 120583

119896= 119864




119898119904= 119898max (1 minus

119875119904

119875max) (2)




119875119904=

minus (120582119904+ 120583

119904) 119875

119904+ 120583

119904+1119875119904+1

119878 = 0

minus (120582119904+ 120583

119904) 119875

119904+ 120582

119904minus1119875119904minus1+ 120583

119904+1119875119904+1

1 le 119878 le 119878max minus 1

minus (120582119904+ 120583

119904) 119875

119904+ 120582

119904minus1119875119904minus1

119878 = 119878max

(3)







HSI (119909 119910) = SF (119909 119910) (4)


SF (119909 119910) = radic1198652119877(119909 119910) + 1198652

119862(119909 119910) (5)

where

119865119877(119909 119910) = radic

1

119872 times119873

119872

sum

119909=1

119873

sum

119910=1

[119868 (119909 119910 + 1) minus 119868 (119909 119910)]2

119865119862(119909 119910) = radic

1

119872 times119873

119872

sum

119909=1

119873

sum

119910=1

[119868 (119909 + 1 119910) minus 119868 (119909 119910)]2

(6)



SML (119868 (119909 119910)) =119882

sum

119908=minus119882

119867

sum

ℎ=minus119867

[119871 (119868 (119909 + 119908 119910 + ℎ))]2

(7)





1003816100381610038161003816


1003816100381610038161003816

(8)


119894and 119868119861



119877sml119860119894(119909 119910) =

1 SML119860119894(119909 119910) gt SML119861

119894(119909 119910)

0 otherwise(9)


119874119862 (119868 (119909 119910)) = max 119874 (119868 (119909 119910)) 119862 (119868 (119909 119910))

119874 (119868 (119909 119910)) = 119868 (119909 119910) minus 119868 (119909 119910) ∘ 119863

119862 (119868 (119909 119910)) = 119868 (119909 119910) sdot 119863 minus 119868 (119909 119910)

(10)




119877119900119888119860

119894(119909 119910) =

1 119874119862119860

119894(119909 119910) gt 119874119862

119861

119894(119909 119910)

0 otherwise(11)


119877119860

119894(119909 119910) =

1 119877sml119860119894(119909 119910) = 1 119877119900119888

119860

119894(119909 119910) = 1

0 otherwise(12)


119894(119909 119910)The


119894and 119868119861

119894in

focused regions

119868119865119894(119909 119910) =

119868119860119894(119909 119910) if 119877119860

119894(119909 119910) = 1

119868119861119894(119909 119910) if 119877119861

119894(119909 119910) = 1

(13)





119910119905+1= 120578119910

119905(1 minus 119910

119905) (14)




(15)





(1199011198871198901199041199051 119901119887119890119904119905

2 119901119887119890119904119905



119875119887119890119904119905 =

119901119887119890119904119905119905

HSI (119901119887119890119904119905119905) lt HSI (119875119887119890119904119905)

119875119887119890119904119905 otherwise(17)




Island2 Island


Island119894= 119898

119894 119899

119894119898

119894and 119899




119894(119909 119910) =

HSI1(119909 119910)HSI

2(119909 119910) HSI


set 119868119865119894(119909 119910) = 119868119865

1(119909 119910) 119868119865

2(119909 119910) 119868119865

119875119873(119909 119910)






119894 119899

119894 of Island





solutions of 119901119887119890119904119905119905= (119901119887119890119904119905

1 119901119887119890119904119905

2 119901119887119890119904119905

119879)



4 Experiments




119871= 006931


120578 = 200 119881119871= 10 and 119881

120579= 20 [6] For NSTC and


1= 119888




RMSE = radic 1

119872 times119873

119872

sum

119894=1

119873

sum

119895=1

(119877 (119894 119895) minus 119865 (119894 119895))2

PSNR = 10log10

(119872 times 119873)2

sum119872

119894=1sum

119873

119895=1[119877 (119894 119895) minus 119865 (119894 119895)]

2

SSIM (119877 119865) = (2120583

119903120583119891+ 119888

1

1205832119903+ 1205832

119891+ 119888

1

)

120572

sdot (2120590

119903120590119891+ 119888

2

1205902119903+ 1205902

119891+ 119888

2

)

120573

sdot (120590119903119891+ 119888

3

120590119903120590119891+ 119888

3

)

120574

(18)


119903 120583

119891 120590

2

119903 120590

2

119891 120590






(a) (b) (c)






119898max = 01 119898max = 02 119898max = 03 119898max = 04

PN = 5 13576 13440 13763 13863PN = 10 13019 12922 13528 13577PN = 15 12854 12718 13403 13325





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

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











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

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









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

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






FMI119860119861119865= sum

119891119886

119901119865119860(119909 119910 119911 119908) log

119901119865119860(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119860(119911 119908)

+sum

119891119887

119901119865119861(119909 119910 119911 119908) log

119901119865119861(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119861(119911 119908)



bution 119901119860(119911 119908) and 119901


119876119860119861



119876119860119861

119865= (

119872

sum

119898=1

119873

sum

119899=1

119876119860119865

(119898 119899)119908119860119865

(119898 119899)

+ 119876119861119865

(119898 119899) 119908119861119865

(119898 119899))

sdot(

119872

sum

119898=1

119873

sum

119899=1

119908119860119865

(119898 119899) + 119908119861119865

(119898 119899))

minus1

(20)

where 119876119860119865

(119898 119899) and 119876119861119865


(119898 119899) =

119876119894119865

119892(119898 119899)119876

119894119865

119900(119898 119899) (119894 = 119860 119861) 119876119894119865

119892(119898 119899) and 119876119894119865

119900(119898 119899) are


(119898 119899) and 119908119861119865



119876119887(119860 119861 119865)

= sum

119908isin119882

sim (119860 119861 119865 | 119908)

sdot (119876 (119860 119865 | 119908) minus 119876 (119861 119865 | 119908)) + 119876 (119861 119865 | 119908)

(21)










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

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



5 Conclusion







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

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






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

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





Plane

SF 107549 124720 125931 126338 126364 125951 126426 126454FMI 09258 09273 09310 09362 09375 09370 09372 09378119876

119860119861

11986507195 07298 07321 07432 07524 07535 07530 07546

119876119887(119860 119861 119865) 08944 08645 08634 08566 08668 08769 08762 08812

Clock

SF 215565 227596 230138 229630 230049 230087 229725 230105FMI 08828 08805 08829 08834 08840 08842 08842 08843119876

119860119861

11986506869 06830 06889 07165 07180 07186 07213 07219

119876119887(119860 119861 119865) 08739 08633 08624 08751 08745 08785 08795 08803

Flower

SF 147468 163808 166530 166400 166500 166129 166396 166539FMI 08479 08755 08774 08808 08812 08813 08813 08815119876

119860119861

11986506506 06980 06994 07161 07163 07166 07172 07173

119876119887(119860 119861 119865) 08660 08974 08959 08972 08968 08978 08990 08995

Book

SF 279575 286606 289587 289555 289917 289727 289969 289991FMI 08678 08631 08653 08672 08671 08670 08670 08673119876

119860119861

11986506961 06975 07014 07206 07211 07199 07205 07214

119876119887(119860 119861 119865) 08848 08546 08563 08930 08930 08939 08933 08950




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896+ 120583

119896= 119864




119898119904= 119898max (1 minus

119875119904

119875max) (2)




119875119904=

minus (120582119904+ 120583

119904) 119875

119904+ 120583

119904+1119875119904+1

119878 = 0

minus (120582119904+ 120583

119904) 119875

119904+ 120582

119904minus1119875119904minus1+ 120583

119904+1119875119904+1

1 le 119878 le 119878max minus 1

minus (120582119904+ 120583

119904) 119875

119904+ 120582

119904minus1119875119904minus1

119878 = 119878max

(3)







HSI (119909 119910) = SF (119909 119910) (4)


SF (119909 119910) = radic1198652119877(119909 119910) + 1198652

119862(119909 119910) (5)

where

119865119877(119909 119910) = radic

1

119872 times119873

119872

sum

119909=1

119873

sum

119910=1

[119868 (119909 119910 + 1) minus 119868 (119909 119910)]2

119865119862(119909 119910) = radic

1

119872 times119873

119872

sum

119909=1

119873

sum

119910=1

[119868 (119909 + 1 119910) minus 119868 (119909 119910)]2

(6)



SML (119868 (119909 119910)) =119882

sum

119908=minus119882

119867

sum

ℎ=minus119867

[119871 (119868 (119909 + 119908 119910 + ℎ))]2

(7)





1003816100381610038161003816


1003816100381610038161003816

(8)


119894and 119868119861



119877sml119860119894(119909 119910) =

1 SML119860119894(119909 119910) gt SML119861

119894(119909 119910)

0 otherwise(9)


119874119862 (119868 (119909 119910)) = max 119874 (119868 (119909 119910)) 119862 (119868 (119909 119910))

119874 (119868 (119909 119910)) = 119868 (119909 119910) minus 119868 (119909 119910) ∘ 119863

119862 (119868 (119909 119910)) = 119868 (119909 119910) sdot 119863 minus 119868 (119909 119910)

(10)




119877119900119888119860

119894(119909 119910) =

1 119874119862119860

119894(119909 119910) gt 119874119862

119861

119894(119909 119910)

0 otherwise(11)


119877119860

119894(119909 119910) =

1 119877sml119860119894(119909 119910) = 1 119877119900119888

119860

119894(119909 119910) = 1

0 otherwise(12)


119894(119909 119910)The


119894and 119868119861

119894in

focused regions

119868119865119894(119909 119910) =

119868119860119894(119909 119910) if 119877119860

119894(119909 119910) = 1

119868119861119894(119909 119910) if 119877119861

119894(119909 119910) = 1

(13)





119910119905+1= 120578119910

119905(1 minus 119910

119905) (14)




(15)





(1199011198871198901199041199051 119901119887119890119904119905

2 119901119887119890119904119905



119875119887119890119904119905 =

119901119887119890119904119905119905

HSI (119901119887119890119904119905119905) lt HSI (119875119887119890119904119905)

119875119887119890119904119905 otherwise(17)




Island2 Island


Island119894= 119898

119894 119899

119894119898

119894and 119899




119894(119909 119910) =

HSI1(119909 119910)HSI

2(119909 119910) HSI


set 119868119865119894(119909 119910) = 119868119865

1(119909 119910) 119868119865

2(119909 119910) 119868119865

119875119873(119909 119910)






119894 119899

119894 of Island





solutions of 119901119887119890119904119905119905= (119901119887119890119904119905

1 119901119887119890119904119905

2 119901119887119890119904119905

119879)



4 Experiments




119871= 006931


120578 = 200 119881119871= 10 and 119881

120579= 20 [6] For NSTC and


1= 119888




RMSE = radic 1

119872 times119873

119872

sum

119894=1

119873

sum

119895=1

(119877 (119894 119895) minus 119865 (119894 119895))2

PSNR = 10log10

(119872 times 119873)2

sum119872

119894=1sum

119873

119895=1[119877 (119894 119895) minus 119865 (119894 119895)]

2

SSIM (119877 119865) = (2120583

119903120583119891+ 119888

1

1205832119903+ 1205832

119891+ 119888

1

)

120572

sdot (2120590

119903120590119891+ 119888

2

1205902119903+ 1205902

119891+ 119888

2

)

120573

sdot (120590119903119891+ 119888

3

120590119903120590119891+ 119888

3

)

120574

(18)


119903 120583

119891 120590

2

119903 120590

2

119891 120590






(a) (b) (c)






119898max = 01 119898max = 02 119898max = 03 119898max = 04

PN = 5 13576 13440 13763 13863PN = 10 13019 12922 13528 13577PN = 15 12854 12718 13403 13325





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

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











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

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









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

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






FMI119860119861119865= sum

119891119886

119901119865119860(119909 119910 119911 119908) log

119901119865119860(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119860(119911 119908)

+sum

119891119887

119901119865119861(119909 119910 119911 119908) log

119901119865119861(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119861(119911 119908)



bution 119901119860(119911 119908) and 119901


119876119860119861



119876119860119861

119865= (

119872

sum

119898=1

119873

sum

119899=1

119876119860119865

(119898 119899)119908119860119865

(119898 119899)

+ 119876119861119865

(119898 119899) 119908119861119865

(119898 119899))

sdot(

119872

sum

119898=1

119873

sum

119899=1

119908119860119865

(119898 119899) + 119908119861119865

(119898 119899))

minus1

(20)

where 119876119860119865

(119898 119899) and 119876119861119865


(119898 119899) =

119876119894119865

119892(119898 119899)119876

119894119865

119900(119898 119899) (119894 = 119860 119861) 119876119894119865

119892(119898 119899) and 119876119894119865

119900(119898 119899) are


(119898 119899) and 119908119861119865



119876119887(119860 119861 119865)

= sum

119908isin119882

sim (119860 119861 119865 | 119908)

sdot (119876 (119860 119865 | 119908) minus 119876 (119861 119865 | 119908)) + 119876 (119861 119865 | 119908)

(21)










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

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



5 Conclusion







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

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






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

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





Plane

SF 107549 124720 125931 126338 126364 125951 126426 126454FMI 09258 09273 09310 09362 09375 09370 09372 09378119876

119860119861

11986507195 07298 07321 07432 07524 07535 07530 07546

119876119887(119860 119861 119865) 08944 08645 08634 08566 08668 08769 08762 08812

Clock

SF 215565 227596 230138 229630 230049 230087 229725 230105FMI 08828 08805 08829 08834 08840 08842 08842 08843119876

119860119861

11986506869 06830 06889 07165 07180 07186 07213 07219

119876119887(119860 119861 119865) 08739 08633 08624 08751 08745 08785 08795 08803

Flower

SF 147468 163808 166530 166400 166500 166129 166396 166539FMI 08479 08755 08774 08808 08812 08813 08813 08815119876

119860119861

11986506506 06980 06994 07161 07163 07166 07172 07173

119876119887(119860 119861 119865) 08660 08974 08959 08972 08968 08978 08990 08995

Book

SF 279575 286606 289587 289555 289917 289727 289969 289991FMI 08678 08631 08653 08672 08671 08670 08670 08673119876

119860119861

11986506961 06975 07014 07206 07211 07199 07205 07214

119876119887(119860 119861 119865) 08848 08546 08563 08930 08930 08939 08933 08950




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1003816100381610038161003816


1003816100381610038161003816

(8)


119894and 119868119861



119877sml119860119894(119909 119910) =

1 SML119860119894(119909 119910) gt SML119861

119894(119909 119910)

0 otherwise(9)


119874119862 (119868 (119909 119910)) = max 119874 (119868 (119909 119910)) 119862 (119868 (119909 119910))

119874 (119868 (119909 119910)) = 119868 (119909 119910) minus 119868 (119909 119910) ∘ 119863

119862 (119868 (119909 119910)) = 119868 (119909 119910) sdot 119863 minus 119868 (119909 119910)

(10)




119877119900119888119860

119894(119909 119910) =

1 119874119862119860

119894(119909 119910) gt 119874119862

119861

119894(119909 119910)

0 otherwise(11)


119877119860

119894(119909 119910) =

1 119877sml119860119894(119909 119910) = 1 119877119900119888

119860

119894(119909 119910) = 1

0 otherwise(12)


119894(119909 119910)The


119894and 119868119861

119894in

focused regions

119868119865119894(119909 119910) =

119868119860119894(119909 119910) if 119877119860

119894(119909 119910) = 1

119868119861119894(119909 119910) if 119877119861

119894(119909 119910) = 1

(13)





119910119905+1= 120578119910

119905(1 minus 119910

119905) (14)




(15)





(1199011198871198901199041199051 119901119887119890119904119905

2 119901119887119890119904119905



119875119887119890119904119905 =

119901119887119890119904119905119905

HSI (119901119887119890119904119905119905) lt HSI (119875119887119890119904119905)

119875119887119890119904119905 otherwise(17)




Island2 Island


Island119894= 119898

119894 119899

119894119898

119894and 119899




119894(119909 119910) =

HSI1(119909 119910)HSI

2(119909 119910) HSI


set 119868119865119894(119909 119910) = 119868119865

1(119909 119910) 119868119865

2(119909 119910) 119868119865

119875119873(119909 119910)






119894 119899

119894 of Island





solutions of 119901119887119890119904119905119905= (119901119887119890119904119905

1 119901119887119890119904119905

2 119901119887119890119904119905

119879)



4 Experiments




119871= 006931


120578 = 200 119881119871= 10 and 119881

120579= 20 [6] For NSTC and


1= 119888




RMSE = radic 1

119872 times119873

119872

sum

119894=1

119873

sum

119895=1

(119877 (119894 119895) minus 119865 (119894 119895))2

PSNR = 10log10

(119872 times 119873)2

sum119872

119894=1sum

119873

119895=1[119877 (119894 119895) minus 119865 (119894 119895)]

2

SSIM (119877 119865) = (2120583

119903120583119891+ 119888

1

1205832119903+ 1205832

119891+ 119888

1

)

120572

sdot (2120590

119903120590119891+ 119888

2

1205902119903+ 1205902

119891+ 119888

2

)

120573

sdot (120590119903119891+ 119888

3

120590119903120590119891+ 119888

3

)

120574

(18)


119903 120583

119891 120590

2

119903 120590

2

119891 120590






(a) (b) (c)






119898max = 01 119898max = 02 119898max = 03 119898max = 04

PN = 5 13576 13440 13763 13863PN = 10 13019 12922 13528 13577PN = 15 12854 12718 13403 13325





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

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











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

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









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

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






FMI119860119861119865= sum

119891119886

119901119865119860(119909 119910 119911 119908) log

119901119865119860(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119860(119911 119908)

+sum

119891119887

119901119865119861(119909 119910 119911 119908) log

119901119865119861(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119861(119911 119908)



bution 119901119860(119911 119908) and 119901


119876119860119861



119876119860119861

119865= (

119872

sum

119898=1

119873

sum

119899=1

119876119860119865

(119898 119899)119908119860119865

(119898 119899)

+ 119876119861119865

(119898 119899) 119908119861119865

(119898 119899))

sdot(

119872

sum

119898=1

119873

sum

119899=1

119908119860119865

(119898 119899) + 119908119861119865

(119898 119899))

minus1

(20)

where 119876119860119865

(119898 119899) and 119876119861119865


(119898 119899) =

119876119894119865

119892(119898 119899)119876

119894119865

119900(119898 119899) (119894 = 119860 119861) 119876119894119865

119892(119898 119899) and 119876119894119865

119900(119898 119899) are


(119898 119899) and 119908119861119865



119876119887(119860 119861 119865)

= sum

119908isin119882

sim (119860 119861 119865 | 119908)

sdot (119876 (119860 119865 | 119908) minus 119876 (119861 119865 | 119908)) + 119876 (119861 119865 | 119908)

(21)










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

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



5 Conclusion







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

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






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

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





Plane

SF 107549 124720 125931 126338 126364 125951 126426 126454FMI 09258 09273 09310 09362 09375 09370 09372 09378119876

119860119861

11986507195 07298 07321 07432 07524 07535 07530 07546

119876119887(119860 119861 119865) 08944 08645 08634 08566 08668 08769 08762 08812

Clock

SF 215565 227596 230138 229630 230049 230087 229725 230105FMI 08828 08805 08829 08834 08840 08842 08842 08843119876

119860119861

11986506869 06830 06889 07165 07180 07186 07213 07219

119876119887(119860 119861 119865) 08739 08633 08624 08751 08745 08785 08795 08803

Flower

SF 147468 163808 166530 166400 166500 166129 166396 166539FMI 08479 08755 08774 08808 08812 08813 08813 08815119876

119860119861

11986506506 06980 06994 07161 07163 07166 07172 07173

119876119887(119860 119861 119865) 08660 08974 08959 08972 08968 08978 08990 08995

Book

SF 279575 286606 289587 289555 289917 289727 289969 289991FMI 08678 08631 08653 08672 08671 08670 08670 08673119876

119860119861

11986506961 06975 07014 07206 07211 07199 07205 07214

119876119887(119860 119861 119865) 08848 08546 08563 08930 08930 08939 08933 08950




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119894 119899

119894 of Island





solutions of 119901119887119890119904119905119905= (119901119887119890119904119905

1 119901119887119890119904119905

2 119901119887119890119904119905

119879)



4 Experiments




119871= 006931


120578 = 200 119881119871= 10 and 119881

120579= 20 [6] For NSTC and


1= 119888




RMSE = radic 1

119872 times119873

119872

sum

119894=1

119873

sum

119895=1

(119877 (119894 119895) minus 119865 (119894 119895))2

PSNR = 10log10

(119872 times 119873)2

sum119872

119894=1sum

119873

119895=1[119877 (119894 119895) minus 119865 (119894 119895)]

2

SSIM (119877 119865) = (2120583

119903120583119891+ 119888

1

1205832119903+ 1205832

119891+ 119888

1

)

120572

sdot (2120590

119903120590119891+ 119888

2

1205902119903+ 1205902

119891+ 119888

2

)

120573

sdot (120590119903119891+ 119888

3

120590119903120590119891+ 119888

3

)

120574

(18)


119903 120583

119891 120590

2

119903 120590

2

119891 120590






(a) (b) (c)






119898max = 01 119898max = 02 119898max = 03 119898max = 04

PN = 5 13576 13440 13763 13863PN = 10 13019 12922 13528 13577PN = 15 12854 12718 13403 13325





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

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











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

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









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

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






FMI119860119861119865= sum

119891119886

119901119865119860(119909 119910 119911 119908) log

119901119865119860(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119860(119911 119908)

+sum

119891119887

119901119865119861(119909 119910 119911 119908) log

119901119865119861(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119861(119911 119908)



bution 119901119860(119911 119908) and 119901


119876119860119861



119876119860119861

119865= (

119872

sum

119898=1

119873

sum

119899=1

119876119860119865

(119898 119899)119908119860119865

(119898 119899)

+ 119876119861119865

(119898 119899) 119908119861119865

(119898 119899))

sdot(

119872

sum

119898=1

119873

sum

119899=1

119908119860119865

(119898 119899) + 119908119861119865

(119898 119899))

minus1

(20)

where 119876119860119865

(119898 119899) and 119876119861119865


(119898 119899) =

119876119894119865

119892(119898 119899)119876

119894119865

119900(119898 119899) (119894 = 119860 119861) 119876119894119865

119892(119898 119899) and 119876119894119865

119900(119898 119899) are


(119898 119899) and 119908119861119865



119876119887(119860 119861 119865)

= sum

119908isin119882

sim (119860 119861 119865 | 119908)

sdot (119876 (119860 119865 | 119908) minus 119876 (119861 119865 | 119908)) + 119876 (119861 119865 | 119908)

(21)










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

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



5 Conclusion







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

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






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

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





Plane

SF 107549 124720 125931 126338 126364 125951 126426 126454FMI 09258 09273 09310 09362 09375 09370 09372 09378119876

119860119861

11986507195 07298 07321 07432 07524 07535 07530 07546

119876119887(119860 119861 119865) 08944 08645 08634 08566 08668 08769 08762 08812

Clock

SF 215565 227596 230138 229630 230049 230087 229725 230105FMI 08828 08805 08829 08834 08840 08842 08842 08843119876

119860119861

11986506869 06830 06889 07165 07180 07186 07213 07219

119876119887(119860 119861 119865) 08739 08633 08624 08751 08745 08785 08795 08803

Flower

SF 147468 163808 166530 166400 166500 166129 166396 166539FMI 08479 08755 08774 08808 08812 08813 08813 08815119876

119860119861

11986506506 06980 06994 07161 07163 07166 07172 07173

119876119887(119860 119861 119865) 08660 08974 08959 08972 08968 08978 08990 08995

Book

SF 279575 286606 289587 289555 289917 289727 289969 289991FMI 08678 08631 08653 08672 08671 08670 08670 08673119876

119860119861

11986506961 06975 07014 07206 07211 07199 07205 07214

119876119887(119860 119861 119865) 08848 08546 08563 08930 08930 08939 08933 08950




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)






119898max = 01 119898max = 02 119898max = 03 119898max = 04

PN = 5 13576 13440 13763 13863PN = 10 13019 12922 13528 13577PN = 15 12854 12718 13403 13325





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

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











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

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









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

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






FMI119860119861119865= sum

119891119886

119901119865119860(119909 119910 119911 119908) log

119901119865119860(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119860(119911 119908)

+sum

119891119887

119901119865119861(119909 119910 119911 119908) log

119901119865119861(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119861(119911 119908)



bution 119901119860(119911 119908) and 119901


119876119860119861



119876119860119861

119865= (

119872

sum

119898=1

119873

sum

119899=1

119876119860119865

(119898 119899)119908119860119865

(119898 119899)

+ 119876119861119865

(119898 119899) 119908119861119865

(119898 119899))

sdot(

119872

sum

119898=1

119873

sum

119899=1

119908119860119865

(119898 119899) + 119908119861119865

(119898 119899))

minus1

(20)

where 119876119860119865

(119898 119899) and 119876119861119865


(119898 119899) =

119876119894119865

119892(119898 119899)119876

119894119865

119900(119898 119899) (119894 = 119860 119861) 119876119894119865

119892(119898 119899) and 119876119894119865

119900(119898 119899) are


(119898 119899) and 119908119861119865



119876119887(119860 119861 119865)

= sum

119908isin119882

sim (119860 119861 119865 | 119908)

sdot (119876 (119860 119865 | 119908) minus 119876 (119861 119865 | 119908)) + 119876 (119861 119865 | 119908)

(21)










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

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



5 Conclusion







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

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






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

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





Plane

SF 107549 124720 125931 126338 126364 125951 126426 126454FMI 09258 09273 09310 09362 09375 09370 09372 09378119876

119860119861

11986507195 07298 07321 07432 07524 07535 07530 07546

119876119887(119860 119861 119865) 08944 08645 08634 08566 08668 08769 08762 08812

Clock

SF 215565 227596 230138 229630 230049 230087 229725 230105FMI 08828 08805 08829 08834 08840 08842 08842 08843119876

119860119861

11986506869 06830 06889 07165 07180 07186 07213 07219

119876119887(119860 119861 119865) 08739 08633 08624 08751 08745 08785 08795 08803

Flower

SF 147468 163808 166530 166400 166500 166129 166396 166539FMI 08479 08755 08774 08808 08812 08813 08813 08815119876

119860119861

11986506506 06980 06994 07161 07163 07166 07172 07173

119876119887(119860 119861 119865) 08660 08974 08959 08972 08968 08978 08990 08995

Book

SF 279575 286606 289587 289555 289917 289727 289969 289991FMI 08678 08631 08653 08672 08671 08670 08670 08673119876

119860119861

11986506961 06975 07014 07206 07211 07199 07205 07214

119876119887(119860 119861 119865) 08848 08546 08563 08930 08930 08939 08933 08950




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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

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











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

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









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

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






FMI119860119861119865= sum

119891119886

119901119865119860(119909 119910 119911 119908) log

119901119865119860(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119860(119911 119908)

+sum

119891119887

119901119865119861(119909 119910 119911 119908) log

119901119865119861(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119861(119911 119908)



bution 119901119860(119911 119908) and 119901


119876119860119861



119876119860119861

119865= (

119872

sum

119898=1

119873

sum

119899=1

119876119860119865

(119898 119899)119908119860119865

(119898 119899)

+ 119876119861119865

(119898 119899) 119908119861119865

(119898 119899))

sdot(

119872

sum

119898=1

119873

sum

119899=1

119908119860119865

(119898 119899) + 119908119861119865

(119898 119899))

minus1

(20)

where 119876119860119865

(119898 119899) and 119876119861119865


(119898 119899) =

119876119894119865

119892(119898 119899)119876

119894119865

119900(119898 119899) (119894 = 119860 119861) 119876119894119865

119892(119898 119899) and 119876119894119865

119900(119898 119899) are


(119898 119899) and 119908119861119865



119876119887(119860 119861 119865)

= sum

119908isin119882

sim (119860 119861 119865 | 119908)

sdot (119876 (119860 119865 | 119908) minus 119876 (119861 119865 | 119908)) + 119876 (119861 119865 | 119908)

(21)










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

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



5 Conclusion







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

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






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

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





Plane

SF 107549 124720 125931 126338 126364 125951 126426 126454FMI 09258 09273 09310 09362 09375 09370 09372 09378119876

119860119861

11986507195 07298 07321 07432 07524 07535 07530 07546

119876119887(119860 119861 119865) 08944 08645 08634 08566 08668 08769 08762 08812

Clock

SF 215565 227596 230138 229630 230049 230087 229725 230105FMI 08828 08805 08829 08834 08840 08842 08842 08843119876

119860119861

11986506869 06830 06889 07165 07180 07186 07213 07219

119876119887(119860 119861 119865) 08739 08633 08624 08751 08745 08785 08795 08803

Flower

SF 147468 163808 166530 166400 166500 166129 166396 166539FMI 08479 08755 08774 08808 08812 08813 08813 08815119876

119860119861

11986506506 06980 06994 07161 07163 07166 07172 07173

119876119887(119860 119861 119865) 08660 08974 08959 08972 08968 08978 08990 08995

Book

SF 279575 286606 289587 289555 289917 289727 289969 289991FMI 08678 08631 08653 08672 08671 08670 08670 08673119876

119860119861

11986506961 06975 07014 07206 07211 07199 07205 07214

119876119887(119860 119861 119865) 08848 08546 08563 08930 08930 08939 08933 08950




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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

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









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

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






FMI119860119861119865= sum

119891119886

119901119865119860(119909 119910 119911 119908) log

119901119865119860(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119860(119911 119908)

+sum

119891119887

119901119865119861(119909 119910 119911 119908) log

119901119865119861(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119861(119911 119908)



bution 119901119860(119911 119908) and 119901


119876119860119861



119876119860119861

119865= (

119872

sum

119898=1

119873

sum

119899=1

119876119860119865

(119898 119899)119908119860119865

(119898 119899)

+ 119876119861119865

(119898 119899) 119908119861119865

(119898 119899))

sdot(

119872

sum

119898=1

119873

sum

119899=1

119908119860119865

(119898 119899) + 119908119861119865

(119898 119899))

minus1

(20)

where 119876119860119865

(119898 119899) and 119876119861119865


(119898 119899) =

119876119894119865

119892(119898 119899)119876

119894119865

119900(119898 119899) (119894 = 119860 119861) 119876119894119865

119892(119898 119899) and 119876119894119865

119900(119898 119899) are


(119898 119899) and 119908119861119865



119876119887(119860 119861 119865)

= sum

119908isin119882

sim (119860 119861 119865 | 119908)

sdot (119876 (119860 119865 | 119908) minus 119876 (119861 119865 | 119908)) + 119876 (119861 119865 | 119908)

(21)










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

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



5 Conclusion







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

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






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

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





Plane

SF 107549 124720 125931 126338 126364 125951 126426 126454FMI 09258 09273 09310 09362 09375 09370 09372 09378119876

119860119861

11986507195 07298 07321 07432 07524 07535 07530 07546

119876119887(119860 119861 119865) 08944 08645 08634 08566 08668 08769 08762 08812

Clock

SF 215565 227596 230138 229630 230049 230087 229725 230105FMI 08828 08805 08829 08834 08840 08842 08842 08843119876

119860119861

11986506869 06830 06889 07165 07180 07186 07213 07219

119876119887(119860 119861 119865) 08739 08633 08624 08751 08745 08785 08795 08803

Flower

SF 147468 163808 166530 166400 166500 166129 166396 166539FMI 08479 08755 08774 08808 08812 08813 08813 08815119876

119860119861

11986506506 06980 06994 07161 07163 07166 07172 07173

119876119887(119860 119861 119865) 08660 08974 08959 08972 08968 08978 08990 08995

Book

SF 279575 286606 289587 289555 289917 289727 289969 289991FMI 08678 08631 08653 08672 08671 08670 08670 08673119876

119860119861

11986506961 06975 07014 07206 07211 07199 07205 07214

119876119887(119860 119861 119865) 08848 08546 08563 08930 08930 08939 08933 08950




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










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

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






FMI119860119861119865= sum

119891119886

119901119865119860(119909 119910 119911 119908) log

119901119865119860(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119860(119911 119908)

+sum

119891119887

119901119865119861(119909 119910 119911 119908) log

119901119865119861(119909 119910 119911 119908)

119901119865(119909 119910) sdot 119901

119861(119911 119908)



bution 119901119860(119911 119908) and 119901


119876119860119861



119876119860119861

119865= (

119872

sum

119898=1

119873

sum

119899=1

119876119860119865

(119898 119899)119908119860119865

(119898 119899)

+ 119876119861119865

(119898 119899) 119908119861119865

(119898 119899))

sdot(

119872

sum

119898=1

119873

sum

119899=1

119908119860119865

(119898 119899) + 119908119861119865

(119898 119899))

minus1

(20)

where 119876119860119865

(119898 119899) and 119876119861119865


(119898 119899) =

119876119894119865

119892(119898 119899)119876

119894119865

119900(119898 119899) (119894 = 119860 119861) 119876119894119865

119892(119898 119899) and 119876119894119865

119900(119898 119899) are


(119898 119899) and 119908119861119865



119876119887(119860 119861 119865)

= sum

119908isin119882

sim (119860 119861 119865 | 119908)

sdot (119876 (119860 119865 | 119908) minus 119876 (119861 119865 | 119908)) + 119876 (119861 119865 | 119908)

(21)










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

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



5 Conclusion







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

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






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

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





Plane

SF 107549 124720 125931 126338 126364 125951 126426 126454FMI 09258 09273 09310 09362 09375 09370 09372 09378119876

119860119861

11986507195 07298 07321 07432 07524 07535 07530 07546

119876119887(119860 119861 119865) 08944 08645 08634 08566 08668 08769 08762 08812

Clock

SF 215565 227596 230138 229630 230049 230087 229725 230105FMI 08828 08805 08829 08834 08840 08842 08842 08843119876

119860119861

11986506869 06830 06889 07165 07180 07186 07213 07219

119876119887(119860 119861 119865) 08739 08633 08624 08751 08745 08785 08795 08803

Flower

SF 147468 163808 166530 166400 166500 166129 166396 166539FMI 08479 08755 08774 08808 08812 08813 08813 08815119876

119860119861

11986506506 06980 06994 07161 07163 07166 07172 07173

119876119887(119860 119861 119865) 08660 08974 08959 08972 08968 08978 08990 08995

Book

SF 279575 286606 289587 289555 289917 289727 289969 289991FMI 08678 08631 08653 08672 08671 08670 08670 08673119876

119860119861

11986506961 06975 07014 07206 07211 07199 07205 07214

119876119887(119860 119861 119865) 08848 08546 08563 08930 08930 08939 08933 08950




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













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

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



5 Conclusion







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

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






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

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





Plane

SF 107549 124720 125931 126338 126364 125951 126426 126454FMI 09258 09273 09310 09362 09375 09370 09372 09378119876

119860119861

11986507195 07298 07321 07432 07524 07535 07530 07546

119876119887(119860 119861 119865) 08944 08645 08634 08566 08668 08769 08762 08812

Clock

SF 215565 227596 230138 229630 230049 230087 229725 230105FMI 08828 08805 08829 08834 08840 08842 08842 08843119876

119860119861

11986506869 06830 06889 07165 07180 07186 07213 07219

119876119887(119860 119861 119865) 08739 08633 08624 08751 08745 08785 08795 08803

Flower

SF 147468 163808 166530 166400 166500 166129 166396 166539FMI 08479 08755 08774 08808 08812 08813 08813 08815119876

119860119861

11986506506 06980 06994 07161 07163 07166 07172 07173

119876119887(119860 119861 119865) 08660 08974 08959 08972 08968 08978 08990 08995

Book

SF 279575 286606 289587 289555 289917 289727 289969 289991FMI 08678 08631 08653 08672 08671 08670 08670 08673119876

119860119861

11986506961 06975 07014 07206 07211 07199 07205 07214

119876119887(119860 119861 119865) 08848 08546 08563 08930 08930 08939 08933 08950




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













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

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






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

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





Plane

SF 107549 124720 125931 126338 126364 125951 126426 126454FMI 09258 09273 09310 09362 09375 09370 09372 09378119876

119860119861

11986507195 07298 07321 07432 07524 07535 07530 07546

119876119887(119860 119861 119865) 08944 08645 08634 08566 08668 08769 08762 08812

Clock

SF 215565 227596 230138 229630 230049 230087 229725 230105FMI 08828 08805 08829 08834 08840 08842 08842 08843119876

119860119861

11986506869 06830 06889 07165 07180 07186 07213 07219

119876119887(119860 119861 119865) 08739 08633 08624 08751 08745 08785 08795 08803

Flower

SF 147468 163808 166530 166400 166500 166129 166396 166539FMI 08479 08755 08774 08808 08812 08813 08813 08815119876

119860119861

11986506506 06980 06994 07161 07163 07166 07172 07173

119876119887(119860 119861 119865) 08660 08974 08959 08972 08968 08978 08990 08995

Book

SF 279575 286606 289587 289555 289917 289727 289969 289991FMI 08678 08631 08653 08672 08671 08670 08670 08673119876

119860119861

11986506961 06975 07014 07206 07211 07199 07205 07214

119876119887(119860 119861 119865) 08848 08546 08563 08930 08930 08939 08933 08950




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













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

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





Plane

SF 107549 124720 125931 126338 126364 125951 126426 126454FMI 09258 09273 09310 09362 09375 09370 09372 09378119876

119860119861

11986507195 07298 07321 07432 07524 07535 07530 07546

119876119887(119860 119861 119865) 08944 08645 08634 08566 08668 08769 08762 08812

Clock

SF 215565 227596 230138 229630 230049 230087 229725 230105FMI 08828 08805 08829 08834 08840 08842 08842 08843119876

119860119861

11986506869 06830 06889 07165 07180 07186 07213 07219

119876119887(119860 119861 119865) 08739 08633 08624 08751 08745 08785 08795 08803

Flower

SF 147468 163808 166530 166400 166500 166129 166396 166539FMI 08479 08755 08774 08808 08812 08813 08813 08815119876

119860119861

11986506506 06980 06994 07161 07163 07166 07172 07173

119876119887(119860 119861 119865) 08660 08974 08959 08972 08968 08978 08990 08995

Book

SF 279575 286606 289587 289555 289917 289727 289969 289991FMI 08678 08631 08653 08672 08671 08670 08670 08673119876

119860119861

11986506961 06975 07014 07206 07211 07199 07205 07214

119876119887(119860 119861 119865) 08848 08546 08563 08930 08930 08939 08933 08950




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Plane

SF 107549 124720 125931 126338 126364 125951 126426 126454FMI 09258 09273 09310 09362 09375 09370 09372 09378119876

119860119861

11986507195 07298 07321 07432 07524 07535 07530 07546

119876119887(119860 119861 119865) 08944 08645 08634 08566 08668 08769 08762 08812

Clock

SF 215565 227596 230138 229630 230049 230087 229725 230105FMI 08828 08805 08829 08834 08840 08842 08842 08843119876

119860119861

11986506869 06830 06889 07165 07180 07186 07213 07219

119876119887(119860 119861 119865) 08739 08633 08624 08751 08745 08785 08795 08803

Flower

SF 147468 163808 166530 166400 166500 166129 166396 166539FMI 08479 08755 08774 08808 08812 08813 08813 08815119876

119860119861

11986506506 06980 06994 07161 07163 07166 07172 07173

119876119887(119860 119861 119865) 08660 08974 08959 08972 08968 08978 08990 08995

Book

SF 279575 286606 289587 289555 289917 289727 289969 289991FMI 08678 08631 08653 08672 08671 08670 08670 08673119876

119860119861

11986506961 06975 07014 07206 07211 07199 07205 07214

119876119887(119860 119861 119865) 08848 08546 08563 08930 08930 08939 08933 08950




Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article multifocus image fusion using...

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