Level Set Based Hippocampus Segmentation in MR Images with … · 2017. 2. 12. · 2 ComputationalandMathematicalMethodsinMedicine (a) (b) (c) Figure1:PositionandintensityofhippocampusinMRimage.(a)Originalimageofhippocampuswitharectangle;(b

Research ArticleLevel Set Based Hippocampus Segmentation in MR Images withImproved Initialization Using Region Growing

Xiaoliang Jiang12 Zhaozhong Zhou1 Xiaokang Ding1

Xiaolei Deng1 Ling Zou3 and Bailin Li2

1College of Mechanical Engineering Quzhou University Quzhou Zhejiang 324000 China2College of Mechanical Engineering Southwest Jiaotong University Chengdu Sichuan 610031 China3Department of Radiology West China Hospital Sichuan University Chengdu Sichuan 610031 China

Correspondence should be addressed to Xiaoliang Jiang jxl swjtu163com

Received 26 October 2016 Revised 10 December 2016 Accepted 22 December 2016 Published 15 January 2017

Academic Editor Ezequiel Lopez-Rubio

Copyright copy 2017 Xiaoliang Jiang et alThis 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

The hippocampus has been known as one of themost important structures referred to as Alzheimerrsquos disease and other neurologicaldisordersHowever segmentation of the hippocampus fromMR images is still a challenging task due to its small size complex shapelow contrast and discontinuous boundaries For the accurate and efficient detection of the hippocampus a new image segmentationmethod based on adaptive region growing and level set algorithm is proposed Firstly adaptive region growing and morphologicaloperations are performed in the target regions and its output is used for the initial contour of level set evolution method Then animproved edge-based level set method utilizing global Gaussian distributions with different means and variances is developed toimplement the accurate segmentation Finally gradient descent method is adopted to get the minimization of the energy equationAs proved by experiment results the proposed method can ideally extract the contours of the hippocampus that are very close tomanual segmentation drawn by specialists

1 Introduction

Magnetic Resonance Imaging (MRI) is a noninvasivemedicalimaging technology which provides anatomical informationon disease-related for detection of pathology and treatmentplanning Many clinical applications depend on the segmen-tation results of MRI brain structures that allow us to knowour brain anatomy changes during the deterioration of dis-ease However accurate and reliable automatic segmentationof medial temporal lobe structures such as the hippocampusis still a challenging Hippocampus being one of a small tem-poral lobe brain structure has been found to be highly relatedto many neurological pathologies and psychiatric disor-ders includingAlzheimerrsquos disease [1 2] epilepsy [3 4] schiz-ophrenia [5 6] and mild cognitive impairment [7] Unfortu-nately at the spatial resolution of MRI the structure of hip-pocampus has relatively low contrast and no distinguishableboundaries as shown in Figure 1 Traditionally clinical hip-pocampus segmentation almost relies on the manual which

is rater-dependent and highly time-consuming So how tosegment hippocampus automatically inmedical image analy-sis is critical in clinical aspects

In recent years many scholars have been engaged inhippocampus segmentation algorithm and various types ofworks also have been done For example Kim et al [8] builta surface model of hippocampus and used principal compo-nent analysis to find the shape asymmetry of hippocampusbetween schizophrenic patients and healthy controls Wanget al [9] proposed a region growth algorithm based on seedwhich is simple and effective but could fail due to the indis-tinct boundary of hippocampus Tong et al [10] proposed anewmethod for nearly automated detection of the hippocam-pus contours on MRI images based on combined sparsecoding techniques and discriminative dictionary learningHajiesmaeili et al [11] utilized contraction wave atom as anefficient method for enhancing the noisy of MR images toimprove segmentation accuracy Kwak et al [12] presentedgraph cuts algorithm to segment hippocampus Similarly

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2017 Article ID 5256346 11 pageshttpdxdoiorg10115520175256346

2 Computational and Mathematical Methods in Medicine

(a) (b) (c)

Figure 1 Position and intensity of hippocampus in MR image (a) Original image of hippocampus with a rectangle (b) zoomed view of therectangle region (c) zoomed view of the region where the hippocampus is indicated with red color

Wolz et al [13] applied a graph cuts algorithm to simultane-ously measure hippocampal atrophy in longitudinal studies

Another hotspot of segmentation approach is level setmethod It allows integration of prior information within asingle optimization framework and has proved to be powerfultools in image segmentation One popular example of edge-based methods is the Geometric Active Contour (GAC)model [14] which used the image gradient to stop evolvingcontours on object boundaries The Chan-Vese model [15]which based on theMumford-Shah segmentation framework[16] is one of themost popular region-basedmodels to detectobjects whose boundaries are not defined by strong edgesRecently Li et al [17] proposed a new edge-based variationallevel set method without reinitialization It incorporated apenalty term to preserve the level set function close to asigned distance function However these previous worksusually present some problems such as oversegmentation andlower speed

In this paper we propose a level set framework combinedwith seeded region growing algorithm for the segmentationof hippocampus The method has several advantages (1)the outcome of the region growing approach is providedautomatically as the initial contour of level set evolutionmethod (2) the global Gaussian distribution with differentmeans and variances is integrated into level set framework Byeffectively combining region growing and level set algorithmthe segmentation accuracy is improved while the processingtime is decreased The remaining part of the paper is orga-nized as follows In Section 2 the proposed method and itstheoretical background is presented In Section 3 results anddiscussion are given In Section 4 we state our conclusion

2 Methods

Level set method is defined as energy minimization processwhich can be applied for image segmentation Howevertraditional level set algorithm has some commonly known

weaknesses (1) the initial contour is difficult to extract (2) forhigh intensity inhomogeneity and discontinuous boundariesimages the edge stop function will produce high values andthe contour may be attracted to false edges In this sectionan accurate and efficient level set algorithm is presentedFirstly the evolution of the level set model is initialized usingregion growing as the prior information Besides that weintegrated the global Gaussian distributions information intoa conventional edge-based level set model proposed in [17]The role of the edge-based is to guide segmentation towardsapparent boundaries while the global Gaussian distributionsterm is to take lead on regions with weak boundaries Thedetail of the proposed segmentation approach is described inSections 21 to 22 The flowchart of the proposed method isshown in Figure 2

21 Region Growing Traditional level set algorithms canhandle the segmentation problem accurately by using somecontour or image dynamics such as curvature intensities orderivatives But all approaches need the initialization stepso that the automatic operation of the system is difficult toachieve Consequently some researchers have used shape-based methods to segment the hippocampus and proposedlocal solutions strictly depending on the shape information ofthe interest In this section the outcomeof the region growingapproach is provided automatically as the initial contour tothe distance regularized level set method The contour basedLirsquosmethod [17] is thus aided by region growing approach andimproves the segmentation performance compared to boththe region growing and Lirsquos models

The region growing [18ndash20] is a simple region-basedimage segmentationmethod that attempts to segment imagesinto many small regions based on predefined seed pointsgrow rule and stop condition As the name is implied regiongrowing starts from the seed points to the adjacent pixelsaccording to the similarity between the pixel and the markedregion and then determines whether the pixel neighbors

Computational and Mathematical Methods in Medicine 3

Prior mask construction

Brain MRI

Select initial seed points

The proposed level set model

InitializationPrior integrated

level setRegion growing

Morphological operations

Segmented hippocampus

Figure 2 Block diagram of the proposed hippocampus segmentation approach

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

Figure 3 Initialization procedure of the level set function using region growing (a) Original images of hippocampus with initial seed points(blue) (b) slice images of hippocampus (c) homogeneous region as a result of region growing (red) (d) perimeter of the homogeneous region(red) (e) convex polygon fitting of the perimeter used as zero level set (red)

should be added to the regions Obviously selecting seedpoints and the growth rules is the key of regional growthHowever this methodmay encounter difficulties with imagesin which regions both are noisy and have blurred or indistinctboundaries Therefore in this study we proposed a noveland adaptive region growing method for semiautomaticallygenerating the preliminary boundary of hippocampus inMRI Detailed process will be introduced as follows

Step 1 Select initial seed points in MRI images the hip-pocampus structure has similar gray values with its adjacentanatomical structures and it is only a small region In order toreduce the computation and improve the segment reliabilitywe only process a local area including the whole hippocam-pus and little other region Firstly an initial seed point ismanually selected on the original image of hippocampus asshown in Figure 3(a) Then we obtain slice image with thesize of 45 lowast 45 pixels of which center is the initial seed pointposition as shown in Figure 3(b) After shearing themajorityof the cerebrospinal fluid white matter and gray matter canbe filtered out so that it can reduce the interference of the

hippocampus segmentation From the gray histogram shownin Figure 4 we can see that the slice images can betterreflect the characteristics of the hippocampus and improvethe segmentation accuracy

Step 2 Region Growing given the seed regions are grownfrom it by appending those neighboring pixels whose prop-erties are most similar to the region According to thespecial position of hippocampus inMR image we choose thestandard deviation as the growth rule It can be described asfollows

10038161003816100381610038161003816119868 (119909 119910) minus 119883seed10038161003816100381610038161003816 le threshold = 120585120590Global (1)

where 119868(119909 119910) is the slice image 119883seed is the mean intensity ofcurrent seed points 120590Global is the global standard deviation ofthe slice image and 120585 is the weight of 120590Global This criterionallows including all connected pixel blocks obtained bythe quad-tree decomposition of which intensity does notdiffer from the seed point intensity more than a user-defined


0

100

200

300

400

0 100 200(a)

0

10

20

0 100 200(b)

Figure 4 Comparison of gray histogram between original image and slice image (a) The gray-level histogram of original image (b) thegray-level histogram of slice image

threshold value Based on this observation 120590Global can bedefined as

120590Global = radic 1119872 times 119873119872sum119909=1

119873sum119910=1

(119868 (119909 119910) minus 119883)2119883 = 1119872 times 119873

119872sum119909=1

119873sum119910=1

119868 (119909 119910) (2)

where119872times119873 is the size of slice image and119883 is themean inten-sity of slice image If any pixel belonging to the seed pointrsquos8-connected neighborhood satisfies the growth criterion weadd it to the seed point set The experiment results of ourregion growing algorithm are presented in Figure 3(c)

Step 3 Morphological operations morphological operationsare a collection of nonlinear operations related to the shapeor morphology of features in an image Firstly the perimeterof the homogeneous region obtaining by region growing isextracted as shown in Figure 3(d) However the result isnot the hippocampus region we desire to extract exactlyTherefore level set method is introduced to modify the resultwhich is produced by region growing In order to preventfalling into localminima a convex polygon is gained by fittingthe perimeter and used to initialize the level set algorithm asshown in Figure 3(e)

22 Level Set Algorithm

221 Traditional Level Set Algorithm Level set methods havebeen widely used for image segmentation [21ndash26] In generalthese methods can be broadly classified as either edge-basedor region-based In implementing the traditional level setmethods it is numerically essential to keep the evolving levelset function close to a signed distance function This processis called reinitialization and it can be quite complicated andtime-consuming In [17] Li et al proposed a new variational

formulation of level set method without reinitialization Theenergy functional is defined as

120576 (120601) = 120583119875 (120601) + 120582119871119892 (120601) + ]119860119892 (120601)= 12 120583 int

Ω

(1003816100381610038161003816nabla1206011003816100381610038161003816 minus 1)2 119889119909 119889119910+ 120582 intΩ

119892120575120576 (120601) 1003816100381610038161003816nabla1206011003816100381610038161003816 119889119909 119889119910+ ]intΩ

119892119867120576 (minus120601) 119889119909 119889119910(3)

where 120583 120582 gt 0 ] are vary constants and 120601 is a level setfunctionThe 119875(120601) is the penalizing term 119871119892(120601) is the lengthof the interface 119860119892(120601) is the area of the subregion enclosedby the zero level curve the function 119867120576(sdot) and 120575120576(sdot) are theHeaviside and Dirac function and 119892 is an edge indicatorfunction defined by

119892 = 11 + 1003816100381610038161003816nabla119866120590 lowast 11986810038161003816100381610038162 (4)

where 119866120590 is the Gaussian kernel having the standard devia-tion 120590

Due to the penalizing term 119875(120601) the involving level setfunction driven by (3) can automatically close to a signed dis-tance function thus reinitialization procedure is eliminatedcompletely However this method above has at least threeintrinsic limitations in applications First it is highly sensitiveto the contour initializations therefore we have to choosesuitable initial contours to correctly detect object of interestin given images Second it is prone to getting ldquotrappedrdquo byextraneous edges caused by image noise or concavity causedby high intensity variations near these points Third it onlycan detect objects with edges defined by gradient

222 The Proposed Level Set Algorithm As discussed in theabove section Lirsquos method is highly sensitive to strong noise


and cannot extract edge without gradient or cognitive edgesFor the accurate and efficient detection of the hippocampuswe use the region growing as the prior information andintegrate the global Gaussian distributions information into aconventional edge-based level set model In image segmenta-tion the role of global information plays a leading on regionswith weak boundaries Particularly it is crucial to reduce thesensitivity to the initialization of the contour From statisticalpoint of view in the proposed model the fitting energy isdescribed by a combination of Lirsquos model and global Gaussiandistributionswith differentmeans and variances respectivelyThe global Gaussian distributions fitting energy is defined asfollows

119864GDF = 2sum119894=1

intΩ119894

minus log119901119894 (119868 (119909)) 119889119909 119894 = 1 2 (5)

where Ω1 = in(119862) Ω2 = out(119862) and 119901119894(119868(119909)) is119901119894 (119868 (119909)) = 1radic2120587120590119894 exp(minus (119868 (119909) minus 119906119894)221205902119894 ) (6)

where 119906119894 and 120590119894 are global intensity means and standarddeviation respectively

Let Ω sub 1198772 be a two-dimensional image space and let119868 Ω rarr 119877 be a given gray image We assume that theimage domain can be partitioned into two regionsThese tworegions can be represented as the regions outside and insidethe zero level 120601 that is Ω1 = 120601 gt 0 and Ω2 = 120601 lt 0 Usingthe Heavside function 119867120576 the global Gaussian distributionfitting energy function can be rewritten as follows

119864GDF (120601 (119909))= intΩ

minus log1199011 (119868 (119909)) 119867120576 (120601 (119909)) 119889119909+ intΩ

minus log1199012 (119868 (119909)) (1 minus 119867120576 (120601 (119909))) 119889119909(7)

In this study we define the energy function as follows

119865 (120601 (119909)) = 120583119875 (120601 (119909)) + 120582119871119892 (120601 (119909)) + ]119860119892 (120601 (119909))+ 120591119864GDF (120601 (119909)) = 12 120583 int

Ω

(1003816100381610038161003816nabla120601 (119909)1003816100381610038161003816 minus 1)2 119889119909+ 120582 intΩ

119892120575120576 (120601 (119909)) 1003816100381610038161003816nabla120601 (119909)1003816100381610038161003816 119889119909+ ]intΩ

119892119867120576 (minus120601 (119909)) 119889119909+ 120591 [int

Ω

minus log1199011 (119868 (119909)) 119867120576 (120601 (119909)) 119889119909+ intΩ

minus log1199012 (119868 (119909)) (1 minus 119867120576 (120601 (119909))) 119889119909]

(8)

where 120591 is weighting constants of 119864GDF

By calculus of variations it can be shown that the parame-ters 119906119894 and 1205902119894 that minimize the energy functional in (8)satisfy the following Euler-Lagrange equations

int (119868 (119909) minus 119906119894) 119872119894120576 (120601 (119909)) 119889119909 = 0int (1205901198942 minus (119868 (119909) minus 119906119894)2) 119872119894120576 (120601 (119909)) 119889119909 = 0 (9)

where 1198721120576(120601(119909)) = 119867120576(120601(119909)) and 1198722120576(120601(119909)) = 1minus119867120576(120601(119909))From (9) we obtain

119906119894 = int 119868 (119909) 119872119894120576 (120601 (119909)) 119889119909int 119872119894120576 (120601 (119909)) 119889119909 (10)

1205901198942 = int (119868 (119909) minus 119906119894)2119872119894120576 (120601 (119909)) 119889119909int 119872119894120576 (120601 (119909)) 119889119909 (11)

which minimize the energy functional 119865(120601) for fixed 120601Minimization of the energy functional (8) with respect

to 120601 can be achieved by solving the gradient descent flowequation

120597120601120597119905 = 120583 [Δ120601 minus div ( Δ1206011003816100381610038161003816Δ1206011003816100381610038161003816 )] + 120582120575120576 (120601) div(119892 Δ1206011003816100381610038161003816Δ1206011003816100381610038161003816 )+ ]119892120575120576 (120601) minus 120591120575120576 (120601) (1198901 minus 1198902) (12)

where

1198901 = log (radic21205871205901) + (119868 (119909) minus 1199061)2212059012 (13)

1198902 = log (radic21205871205902) + (119868 (119909) minus 1199062)2212059022 (14)

119867120576 (119909) = 12 (1 + 2120587 arctan( 119909120576 )) (15)

120575120576 (119909) = 1120587 1205761205762 + 1199092 (16)

223 Algorithms and Numerical Approximation In this sub-section we briefly present the numerical algorithms and pro-cedures to solve the evolution (12) Due to the regularizationterm 119875(120601) (12) in the continuous domain can be discretizedby means of simple finite difference rather than complexupwind difference scheme We recall first the usual notationLet ℎ be the space step letΔ119905 be the time step and let (119909119894 119910119894) =(119894ℎ 119895ℎ) be the grid points Let 120601119899119894119895 = 120601(119899Δ119905 119909119894 119910119895) be anapproximation of the level set function 120601(119905 119909 119910) with 119899 ge 01206010 = 1206010 where 1206010 is the initial level set function The centraldifferences of spatial partial derivatives are expressed in thefollowing notations

Δ1199090120601119894119895 = 120601119894+1119895 minus 120601119894minus11198952ℎ Δ1199100120601119894119895 = 120601119894119895+1 minus 120601119894119895minus12ℎ

(17)


seed seed seed seed seed

(a)

(b)

(c)

(d)

Figure 5 Segmentation of hippocampus with different algorithms (a) Five original images of hippocampus with initial seed points (blue)(b) Results of Lirsquos model (red) (c) Results of our method (red) (d) Results of artificial segmentation by the specialist (red)

All the spatial partial derivatives 120597120601120597119909 and 120597120601120597119910 areapproximated by the central difference and the temporal par-tial derivative 120597120601120597119905 is discretized as the forward differenceSet 119899 = 0 and start with initial level set function 1206010119894119895 knowing120601119899119894119895 we first compute 1198901198991 and 1198901198992 according to (13) and (14)Then the numerical approximation to (12) is given by thefollowing discretization

120601119899+1119894119895 minus 120601119899119894119895Δ119905 = 120583 (Δ120601119899119894119895 minus 119896119899119894119895) + 120582120575120576 (120601119899119894119895)sdot (119892119896119899119894119895 + Δ1199090119892 ( Δ11990901206011198991198941198951003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895) + Δ1199100119892 ( Δ11991001206011198991198941198951003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895))+ ]119892120575120576 (120601119899119894119895) minus 120591120575120576 (120601119899119894119895) (1198901198991 minus 1198901198992)

(18)

where the corresponding curvature 119896119899119894119895 can be discretizedusing a second-order central difference scheme

119896119899119894119895 = Δ1199090( Δ11990901206011198991198941198951003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895) + Δ1199100( Δ11991001206011198991198941198951003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895) (19)

with

1003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895 = radic(Δ1199090120601119899119894119895)2 + (Δ1199100120601119899119894119895)2 (20)

Next a direct implementation of the proposed level setalgorithm is presented in Algorithm 1


Input 119868Initialize 1206010 = 1206010(119909) 120583 120582 ] 120591 120576 120590Choose Δ119905119899 = 1while the solution is not converged doCompute 119892 1199061 1199062 12059021 12059022 by using Eq (4) (10) (11) sequentiallyUpdate level set function 120601119899+1 by solving level set formulation (12)119899 = 119899 + 1

end while

Algorithm 1 The implementation of the proposed level set algorithm

Our methodLirsquo method

05

06

07

08

09

1

DSC

5 10 15 200


05

06

07

08

09

1

DSC

5 10 15 200


05

06

07

08

09

1

DSC

5 10 15 200

Figure 6 Quantitative comparison of our method with Lirsquos model using the DSC standards from three different group MRI images

3 Results and Discussion

To assess the performance of the proposed method quanti-tative and visual experiments have been carried out on threegroups of the whole brainMRI images (the image size is 157lowast189 and 2D axial brain images with T1-weighted sequenceof size 078 lowast 078 lowast 200mm) For each image set 17 to 19frames which correctly resemble the shape of hippocampuswere used to make the model These images are from West

ChinaHospital SichuanUniversity China Lirsquosmodel and themethod of this paper are applied to the hippocampus brainsegmentation while the experiment results were comparedTwo methods set the same parameters and initial conditionsas follows the level set function 120601 can be simply initialized asa binary valueminus1198880 inside the region and a positive value 1198880 out-side the region Unless otherwise specified we use the follow-ing default setting of the parameters in the experiments timestep Δ119905 = 4 1198880 = 2 120583 = 005 120582 = 10 ] = 2 120591 = 001 120576 = 2


(a)

(b)

(c)

Figure 7 Segmentation results of our method on other brain tissue images

and 120590 = 1 The results are all carried out by Matlab2011a onthe PC with Pentium CPU 250 and 4GB of RAM Windows7 64 bits

31 Segmentation of Hippocampus Images Figure 5 offers thesegmentation results of five hippocampus images where thered curve surrounded is the segmentation area Our methodrequires a seed point located inside the targeted structure bythe user as shown in Figure 5(a) The segmentation resultsby Lirsquos model and our method are shown in the second rowand the third row respectively The last row is the results ofartificial segmentation by specialist The experiment resultswith the same seed points show that when the images arebadly under the conditions of intensity inhomogeneity lowcontrast and discontinuous boundaries only using the edgeinformationmay fail to discriminate the intensity between anobject and background leading to inaccurate segmentationHowever our method both uses the edge information andregion information of images making our method have avery strong discriminative capability for the object and back-ground So the segmentation results of this paper are moreaccurate which can distinguish regions similar intensitymeans but different variances

Furthermore in order to show the superiority of ourmodel we utilize the dice similarity coefficient (DSC) [27] toevaluate the performances on three different group images If

1198781 and 1198782 stand for the areas enclosed by contours obtained bythemodel and themanualmethod respective then themetricis defined as follows

DSC = 2119873 (1198781 cap 1198782)119873 (1198781) + 119873 (1198782) (21)

where 119873(sdot) indicates the number of pixels in the enclosedregion and Ω is the image domain The value of DSC rangesfrom 0 to 1 with a higher value representing a more accuratesegmentation result A comparison plot of DSC for the threegroups of thewhole brainMRI imageswhich correctly resem-ble the shape of hippocampus is provided in Figure 6 Byquantitative comparison we can see from the overlap ratiosthat there is a very good correspondence between ourmethodandmanually extracted boundaries and this demonstrates thesignificant advantage of our model in terms of segmentationaccuracy

In addition Figure 7 shows the results of our method onother brain tissue imagesThe first to last rows show temporallobe amygdale andmidbrain respectively It can be observedthat the good result can be obtained by our method

32 Robustness to the Initial Seed Point In order to evaluatethe robustness of ourmethod we apply it to the hippocampusimages in Figure 5 with different initial seed points as shownin Figure 8 In Figure 8 the original images and different ini-


(a)

(b)

(c)

(d)

(e)

Figure 8 Segmentation of hippocampus with different initial seed points (a) Five original images of hippocampus (bndashe) Segmentationresults of our method with different initial seed points


0

02

04

06

08

1

DSC

2 3 4 51Image with different initial seed points

Figure 9 Segmentation accuracy of our method for images withdifferent initial seed points

tial seed points and the corresponding segmentation resultsare shown In spite of huge difference of these seed pointsthe corresponding results are nearly the same and the objectboundaries are accurately obtained To further illustrate therobustness to initialization of our method the segmentationaccuracy is quantitatively verified with DSC values in Fig-ure 9 It can be seen that the 119909-axis represents five originalimages of hippocampus with four different initial seed pointsand the 119910-axis represents the DSC value of each initializationFrom the results we can see that the segmentation resultsobtained by each initial seed point are almost the sameThus experiment proved that the proposedmodel has a goodrobustness to the initial seed point

4 Conclusion

In summary we have introduced an approach to extractthe pathway of hippocampus based on level set algorithmwith an improved initialization using region growing whichallows us to optimize the location of hippocampus in a globalsenseThismodel is derived from Lirsquos model by incorporatingthe global Gaussian distributions with different means andvariances into level set framework avoiding the problems ofboundary leakage low contrast and discontinuous boundariesof hippocampus images Meanwhile to solve the problemof initialization sensitivity we have discussed and analyzedthe adaptive region growing and morphological operationswhich are used to gain a rough segmentation result ofhippocampus to initialize the level set function Comparedwith Lirsquos model our method provides more accurate andefficient segmentation results that are closer to the manualsegmentation results obtained by a specialist In future workwe will further improve the segmentation efficiency by GPUacceleration and study the adaptive change of the shape con-straintrsquos effect according to the quality of the hippocampusimages to acquire better segmentation results

Competing Interests

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

Acknowledgments

The authors would like to thank Dr Ling Zou of the WestChina Hospital of Sichuan University for kindly providingthem with the hippocampus images as well as manuallysegmenting of the target regions of hippocampus imagesBesides this work is supported by the National NaturalScience Foundation of China (nos 51275272 51605253 and30800263) and Zhejiang Provincial Natural Science Founda-tion of China (nos LY16E050011 and LQ17C160001)

References

[1] J Guo X Feng S Zhou W Yan and D Meng ldquoPotentialanti-Alzheimerrsquos disease activities of the roots of Desmodiumcaudatumrdquo Industrial Crops and Products vol 90 no 15 pp 94ndash99 2016

[2] A Chaalal R Poirier D Blum et al ldquoPTU-induced hypothy-roidism in rats leads to several early neuropathological signsof Alzheimerrsquos disease in the hippocampus and spatial memoryimpairmentsrdquoHippocampus vol 24 no 11 pp 1381ndash1393 2014

[3] F Imparato ldquoCeliac disease could have been the cause ofCaesarrsquos epilepsyrdquo Journal of Clinical Gastroenterology vol 50no 9 p 797 2016

[4] B Wang and L Meng ldquoFunctional brain network alterations inepilepsy a magnetoencephalography studyrdquo Epilepsy Researchvol 126 pp 62ndash69 2016

[5] S Troube ldquoPhilosophy of psychiatry and phenomenologyof everyday life the disruptions of ordinary experience inschizophreniardquo Revue de Synthese vol 137 no 1-2 pp 61ndash862016

[6] R Pinacho E M Valdizan F Pilar-Cuellar et al ldquoIncreasedSP4 and SP1 transcription factor expression in the postmortemhippocampus of chronic schizophreniardquo Journal of PsychiatricResearch vol 58 pp 189ndash196 2014

[7] C R Jack Jr M M Shiung J L Gunter et al ldquoComparison ofdifferentMRI brain athrophy ratemeasures with clinical diseaseprogression in ADrdquoNeurology vol 62 no 4 pp 591ndash600 2004

[8] S H Kim J-M Lee H-P Kim et al ldquoAsymmetry analysis ofdeformable hippocampal model using the principal componentin schizophreniardquoHumanBrainMapping vol 25 no 4 pp 361ndash369 2005

[9] Y Wang Y Zhu W Gao Y Fu Y Wang and Z Lin ldquoThemeasurement of the volume of stereotaxtic MRI hippocampalformation applying the Region growth algorithm based onseedsrdquo in Proceedings of the International Conference on Com-plex Medical Engineering (CME rsquo11) pp 489ndash492 May 2011

[10] T Tong R Wolz P Coupe J V Hajnal and D RueckertldquoSegmentation of MR images via discriminative dictionarylearning and sparse coding application to hippocampus label-ingrdquo NeuroImage vol 76 pp 11ndash23 2013

[11] M Hajiesmaeili B Bagherinakhjavanlo J Dehmeshki andT Ellis ldquoSegmentation of the hippocampus for detection ofAlzheimerrsquos diseaserdquo in Proceedings of the 8th InternationalSymposium on Visual Computing (ISVC rsquo12) pp 42ndash50 CreteGreece 2012

[12] K Kwak U Yoon D-K Lee et al ldquoFully-automated approachto hippocampus segmentation using a graph-cuts algorithmcombined with atlas-based segmentation and morphologicalopeningrdquo Magnetic Resonance Imaging vol 31 no 7 pp 1190ndash1196 2013


[13] R Wolz R A Heckemann P Aljabar et al ldquoMeasurementof hippocampal atrophy using 4D graph-cut segmentationapplication to ADNIrdquo NeuroImage vol 52 no 1 pp 109ndash1182010

[14] V Caselles R Kimmel and G Sapiro ldquoGeodesic active con-toursrdquo International Journal of Computer Vision vol 22 no 1pp 61ndash79 1997

[15] T F Chan and L A Vese ldquoActive contours without edgesrdquo IEEETransactions on Image Processing vol 10 no 2 pp 266ndash2772001

[16] DMumford and J Shah ldquoOptimal approximations by piecewisesmooth functions and associated variational problemsrdquo Com-munications on Pure and Applied Mathematics vol 42 no 5pp 577ndash685 1989

[17] C Li C Xu C Gui andM D Fox ldquoLevel set evolution withoutre-initialization a new variational formulationrdquo in Proceedingsof the IEEE Computer Society Conference on Computer Visionand Pattern Recognition (CVPR rsquo05) pp 430ndash436 San DiegoCalif USA June 2005

[18] S A Hojjatoleslami and J Kittler ldquoRegion growing a newapproachrdquo IEEE Transactions on Image Processing vol 7 no 7pp 1079ndash1084 1998

[19] D Holz and S Behnke ldquoApproximate triangulation and regiongrowing for efficient segmentation and smoothing of rangeimagesrdquo Robotics and Autonomous Systems vol 62 no 9 pp1282ndash1293 2014

[20] R Rouhi M Jafari S Kasaei and P Keshavarzian ldquoBenign andmalignant breast tumors classification based on region growingand CNN segmentationrdquo Expert Systems with Applications vol42 no 3 pp 990ndash1002 2015

[21] L D Cohen ldquoOn active contour models and balloonsrdquo CVGIPImage Understanding vol 53 no 2 pp 211ndash218 1991

[22] C Xu and J L Prince ldquoGeneralized gradient vector flowexternal forces for active contoursrdquo Signal Processing vol 71 no2 pp 131ndash139 1998

[23] V Jaouen P Gonzalez S Stute et al ldquoVariational segmentationof vector-valued images with gradient vector flowrdquo IEEE Trans-actions on Image Processing vol 23 no 11 pp 4773ndash4785 2014

[24] C Feng J Zhang and R Liang ldquoA method for lung boundarycorrection using split Bregman method and geometric activecontour modelrdquo Computational and Mathematical Methods inMedicine vol 2015 Article ID 789485 14 pages 2015

[25] Y N Law H K Lee and A M Yip ldquoSemi-supervised subspacelearning for Mumford-Shah model based texture segmenta-tionrdquo Optics Express vol 18 no 5 pp 4434ndash4448 2010

[26] J Wang S Zhao Z Liu Y Tian F Duan and Y Pan ldquoAn activecontour model based on adaptive threshold for extraction ofcerebral vascular structuresrdquo Computational and MathematicalMethods in Medicine vol 2016 Article ID 6472397 9 pages2016

[27] B Liu H D Cheng J Huang J Tian X Tang and J Liu ldquoProb-ability density difference-based active contour for ultrasoundimage segmentationrdquo Pattern Recognition vol 43 no 6 pp2028ndash2042 2010


(a) (b) (c)

Figure 1 Position and intensity of hippocampus in MR image (a) Original image of hippocampus with a rectangle (b) zoomed view of therectangle region (c) zoomed view of the region where the hippocampus is indicated with red color

Wolz et al [13] applied a graph cuts algorithm to simultane-ously measure hippocampal atrophy in longitudinal studies

Another hotspot of segmentation approach is level setmethod It allows integration of prior information within asingle optimization framework and has proved to be powerfultools in image segmentation One popular example of edge-based methods is the Geometric Active Contour (GAC)model [14] which used the image gradient to stop evolvingcontours on object boundaries The Chan-Vese model [15]which based on theMumford-Shah segmentation framework[16] is one of themost popular region-basedmodels to detectobjects whose boundaries are not defined by strong edgesRecently Li et al [17] proposed a new edge-based variationallevel set method without reinitialization It incorporated apenalty term to preserve the level set function close to asigned distance function However these previous worksusually present some problems such as oversegmentation andlower speed

In this paper we propose a level set framework combinedwith seeded region growing algorithm for the segmentationof hippocampus The method has several advantages (1)the outcome of the region growing approach is providedautomatically as the initial contour of level set evolutionmethod (2) the global Gaussian distribution with differentmeans and variances is integrated into level set framework Byeffectively combining region growing and level set algorithmthe segmentation accuracy is improved while the processingtime is decreased The remaining part of the paper is orga-nized as follows In Section 2 the proposed method and itstheoretical background is presented In Section 3 results anddiscussion are given In Section 4 we state our conclusion

2 Methods

Level set method is defined as energy minimization processwhich can be applied for image segmentation Howevertraditional level set algorithm has some commonly known

weaknesses (1) the initial contour is difficult to extract (2) forhigh intensity inhomogeneity and discontinuous boundariesimages the edge stop function will produce high values andthe contour may be attracted to false edges In this sectionan accurate and efficient level set algorithm is presentedFirstly the evolution of the level set model is initialized usingregion growing as the prior information Besides that weintegrated the global Gaussian distributions information intoa conventional edge-based level set model proposed in [17]The role of the edge-based is to guide segmentation towardsapparent boundaries while the global Gaussian distributionsterm is to take lead on regions with weak boundaries Thedetail of the proposed segmentation approach is described inSections 21 to 22 The flowchart of the proposed method isshown in Figure 2

21 Region Growing Traditional level set algorithms canhandle the segmentation problem accurately by using somecontour or image dynamics such as curvature intensities orderivatives But all approaches need the initialization stepso that the automatic operation of the system is difficult toachieve Consequently some researchers have used shape-based methods to segment the hippocampus and proposedlocal solutions strictly depending on the shape information ofthe interest In this section the outcomeof the region growingapproach is provided automatically as the initial contour tothe distance regularized level set method The contour basedLirsquosmethod [17] is thus aided by region growing approach andimproves the segmentation performance compared to boththe region growing and Lirsquos models

The region growing [18ndash20] is a simple region-basedimage segmentationmethod that attempts to segment imagesinto many small regions based on predefined seed pointsgrow rule and stop condition As the name is implied regiongrowing starts from the seed points to the adjacent pixelsaccording to the similarity between the pixel and the markedregion and then determines whether the pixel neighbors



Brain MRI








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









0

100

200

300

400

0 100 200(a)

0

10

20

0 100 200(b)




119873sum119910=1

(119868 (119909 119910) minus 119883)2119883 = 1119872 times 119873

119872sum119909=1

119873sum119910=1

119868 (119909 119910) (2)






120576 (120601) = 120583119875 (120601) + 120582119871119892 (120601) + ]119860119892 (120601)= 12 120583 int

Ω

(1003816100381610038161003816nabla1206011003816100381610038161003816 minus 1)2 119889119909 119889119910+ 120582 intΩ

119892120575120576 (120601) 1003816100381610038161003816nabla1206011003816100381610038161003816 119889119909 119889119910+ ]intΩ

119892119867120576 (minus120601) 119889119909 119889119910(3)


119892 = 11 + 1003816100381610038161003816nabla119866120590 lowast 11986810038161003816100381610038162 (4)






119864GDF = 2sum119894=1

intΩ119894

minus log119901119894 (119868 (119909)) 119889119909 119894 = 1 2 (5)




119864GDF (120601 (119909))= intΩ

minus log1199011 (119868 (119909)) 119867120576 (120601 (119909)) 119889119909+ intΩ

minus log1199012 (119868 (119909)) (1 minus 119867120576 (120601 (119909))) 119889119909(7)


119865 (120601 (119909)) = 120583119875 (120601 (119909)) + 120582119871119892 (120601 (119909)) + ]119860119892 (120601 (119909))+ 120591119864GDF (120601 (119909)) = 12 120583 int

Ω

(1003816100381610038161003816nabla120601 (119909)1003816100381610038161003816 minus 1)2 119889119909+ 120582 intΩ

119892120575120576 (120601 (119909)) 1003816100381610038161003816nabla120601 (119909)1003816100381610038161003816 119889119909+ ]intΩ

119892119867120576 (minus120601 (119909)) 119889119909+ 120591 [int

Ω

minus log1199011 (119868 (119909)) 119867120576 (120601 (119909)) 119889119909+ intΩ

minus log1199012 (119868 (119909)) (1 minus 119867120576 (120601 (119909))) 119889119909]

(8)





119906119894 = int 119868 (119909) 119872119894120576 (120601 (119909)) 119889119909int 119872119894120576 (120601 (119909)) 119889119909 (10)

1205901198942 = int (119868 (119909) minus 119906119894)2119872119894120576 (120601 (119909)) 119889119909int 119872119894120576 (120601 (119909)) 119889119909 (11)




where

1198901 = log (radic21205871205901) + (119868 (119909) minus 1199061)2212059012 (13)

1198902 = log (radic21205871205902) + (119868 (119909) minus 1199062)2212059022 (14)

119867120576 (119909) = 12 (1 + 2120587 arctan( 119909120576 )) (15)

120575120576 (119909) = 1120587 1205761205762 + 1199092 (16)



(17)



(a)

(b)

(c)

(d)




(18)


119896119899119894119895 = Δ1199090( Δ11990901206011198991198941198951003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895) + Δ1199100( Δ11991001206011198991198941198951003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895) (19)

with

1003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895 = radic(Δ1199090120601119899119894119895)2 + (Δ1199100120601119899119894119895)2 (20)




end while



05

06

07

08

09

1

DSC

5 10 15 200


05

06

07

08

09

1

DSC

5 10 15 200


05

06

07

08

09

1

DSC

5 10 15 200






(a)

(b)

(c)






DSC = 2119873 (1198781 cap 1198782)119873 (1198781) + 119873 (1198782) (21)





(a)

(b)

(c)

(d)

(e)



0

02

04

06

08

1

DSC




4 Conclusion


Competing Interests


Acknowledgments


References































Brain MRI








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









0

100

200

300

400

0 100 200(a)

0

10

20

0 100 200(b)




119873sum119910=1

(119868 (119909 119910) minus 119883)2119883 = 1119872 times 119873

119872sum119909=1

119873sum119910=1

119868 (119909 119910) (2)






120576 (120601) = 120583119875 (120601) + 120582119871119892 (120601) + ]119860119892 (120601)= 12 120583 int

Ω

(1003816100381610038161003816nabla1206011003816100381610038161003816 minus 1)2 119889119909 119889119910+ 120582 intΩ

119892120575120576 (120601) 1003816100381610038161003816nabla1206011003816100381610038161003816 119889119909 119889119910+ ]intΩ

119892119867120576 (minus120601) 119889119909 119889119910(3)


119892 = 11 + 1003816100381610038161003816nabla119866120590 lowast 11986810038161003816100381610038162 (4)






119864GDF = 2sum119894=1

intΩ119894

minus log119901119894 (119868 (119909)) 119889119909 119894 = 1 2 (5)




119864GDF (120601 (119909))= intΩ

minus log1199011 (119868 (119909)) 119867120576 (120601 (119909)) 119889119909+ intΩ

minus log1199012 (119868 (119909)) (1 minus 119867120576 (120601 (119909))) 119889119909(7)


119865 (120601 (119909)) = 120583119875 (120601 (119909)) + 120582119871119892 (120601 (119909)) + ]119860119892 (120601 (119909))+ 120591119864GDF (120601 (119909)) = 12 120583 int

Ω

(1003816100381610038161003816nabla120601 (119909)1003816100381610038161003816 minus 1)2 119889119909+ 120582 intΩ

119892120575120576 (120601 (119909)) 1003816100381610038161003816nabla120601 (119909)1003816100381610038161003816 119889119909+ ]intΩ

119892119867120576 (minus120601 (119909)) 119889119909+ 120591 [int

Ω

minus log1199011 (119868 (119909)) 119867120576 (120601 (119909)) 119889119909+ intΩ

minus log1199012 (119868 (119909)) (1 minus 119867120576 (120601 (119909))) 119889119909]

(8)





119906119894 = int 119868 (119909) 119872119894120576 (120601 (119909)) 119889119909int 119872119894120576 (120601 (119909)) 119889119909 (10)

1205901198942 = int (119868 (119909) minus 119906119894)2119872119894120576 (120601 (119909)) 119889119909int 119872119894120576 (120601 (119909)) 119889119909 (11)




where

1198901 = log (radic21205871205901) + (119868 (119909) minus 1199061)2212059012 (13)

1198902 = log (radic21205871205902) + (119868 (119909) minus 1199062)2212059022 (14)

119867120576 (119909) = 12 (1 + 2120587 arctan( 119909120576 )) (15)

120575120576 (119909) = 1120587 1205761205762 + 1199092 (16)



(17)



(a)

(b)

(c)

(d)




(18)


119896119899119894119895 = Δ1199090( Δ11990901206011198991198941198951003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895) + Δ1199100( Δ11991001206011198991198941198951003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895) (19)

with

1003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895 = radic(Δ1199090120601119899119894119895)2 + (Δ1199100120601119899119894119895)2 (20)




end while



05

06

07

08

09

1

DSC

5 10 15 200


05

06

07

08

09

1

DSC

5 10 15 200


05

06

07

08

09

1

DSC

5 10 15 200






(a)

(b)

(c)






DSC = 2119873 (1198781 cap 1198782)119873 (1198781) + 119873 (1198782) (21)





(a)

(b)

(c)

(d)

(e)



0

02

04

06

08

1

DSC




4 Conclusion


Competing Interests


Acknowledgments


References






























0

100

200

300

400

0 100 200(a)

0

10

20

0 100 200(b)




119873sum119910=1

(119868 (119909 119910) minus 119883)2119883 = 1119872 times 119873

119872sum119909=1

119873sum119910=1

119868 (119909 119910) (2)






120576 (120601) = 120583119875 (120601) + 120582119871119892 (120601) + ]119860119892 (120601)= 12 120583 int

Ω

(1003816100381610038161003816nabla1206011003816100381610038161003816 minus 1)2 119889119909 119889119910+ 120582 intΩ

119892120575120576 (120601) 1003816100381610038161003816nabla1206011003816100381610038161003816 119889119909 119889119910+ ]intΩ

119892119867120576 (minus120601) 119889119909 119889119910(3)


119892 = 11 + 1003816100381610038161003816nabla119866120590 lowast 11986810038161003816100381610038162 (4)






119864GDF = 2sum119894=1

intΩ119894

minus log119901119894 (119868 (119909)) 119889119909 119894 = 1 2 (5)




119864GDF (120601 (119909))= intΩ

minus log1199011 (119868 (119909)) 119867120576 (120601 (119909)) 119889119909+ intΩ

minus log1199012 (119868 (119909)) (1 minus 119867120576 (120601 (119909))) 119889119909(7)


119865 (120601 (119909)) = 120583119875 (120601 (119909)) + 120582119871119892 (120601 (119909)) + ]119860119892 (120601 (119909))+ 120591119864GDF (120601 (119909)) = 12 120583 int

Ω

(1003816100381610038161003816nabla120601 (119909)1003816100381610038161003816 minus 1)2 119889119909+ 120582 intΩ

119892120575120576 (120601 (119909)) 1003816100381610038161003816nabla120601 (119909)1003816100381610038161003816 119889119909+ ]intΩ

119892119867120576 (minus120601 (119909)) 119889119909+ 120591 [int

Ω

minus log1199011 (119868 (119909)) 119867120576 (120601 (119909)) 119889119909+ intΩ

minus log1199012 (119868 (119909)) (1 minus 119867120576 (120601 (119909))) 119889119909]

(8)





119906119894 = int 119868 (119909) 119872119894120576 (120601 (119909)) 119889119909int 119872119894120576 (120601 (119909)) 119889119909 (10)

1205901198942 = int (119868 (119909) minus 119906119894)2119872119894120576 (120601 (119909)) 119889119909int 119872119894120576 (120601 (119909)) 119889119909 (11)




where

1198901 = log (radic21205871205901) + (119868 (119909) minus 1199061)2212059012 (13)

1198902 = log (radic21205871205902) + (119868 (119909) minus 1199062)2212059022 (14)

119867120576 (119909) = 12 (1 + 2120587 arctan( 119909120576 )) (15)

120575120576 (119909) = 1120587 1205761205762 + 1199092 (16)



(17)



(a)

(b)

(c)

(d)




(18)


119896119899119894119895 = Δ1199090( Δ11990901206011198991198941198951003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895) + Δ1199100( Δ11991001206011198991198941198951003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895) (19)

with

1003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895 = radic(Δ1199090120601119899119894119895)2 + (Δ1199100120601119899119894119895)2 (20)




end while



05

06

07

08

09

1

DSC

5 10 15 200


05

06

07

08

09

1

DSC

5 10 15 200


05

06

07

08

09

1

DSC

5 10 15 200






(a)

(b)

(c)






DSC = 2119873 (1198781 cap 1198782)119873 (1198781) + 119873 (1198782) (21)





(a)

(b)

(c)

(d)

(e)



0

02

04

06

08

1

DSC




4 Conclusion


Competing Interests


Acknowledgments


References































119864GDF = 2sum119894=1

intΩ119894

minus log119901119894 (119868 (119909)) 119889119909 119894 = 1 2 (5)




119864GDF (120601 (119909))= intΩ

minus log1199011 (119868 (119909)) 119867120576 (120601 (119909)) 119889119909+ intΩ

minus log1199012 (119868 (119909)) (1 minus 119867120576 (120601 (119909))) 119889119909(7)


119865 (120601 (119909)) = 120583119875 (120601 (119909)) + 120582119871119892 (120601 (119909)) + ]119860119892 (120601 (119909))+ 120591119864GDF (120601 (119909)) = 12 120583 int

Ω

(1003816100381610038161003816nabla120601 (119909)1003816100381610038161003816 minus 1)2 119889119909+ 120582 intΩ

119892120575120576 (120601 (119909)) 1003816100381610038161003816nabla120601 (119909)1003816100381610038161003816 119889119909+ ]intΩ

119892119867120576 (minus120601 (119909)) 119889119909+ 120591 [int

Ω

minus log1199011 (119868 (119909)) 119867120576 (120601 (119909)) 119889119909+ intΩ

minus log1199012 (119868 (119909)) (1 minus 119867120576 (120601 (119909))) 119889119909]

(8)





119906119894 = int 119868 (119909) 119872119894120576 (120601 (119909)) 119889119909int 119872119894120576 (120601 (119909)) 119889119909 (10)

1205901198942 = int (119868 (119909) minus 119906119894)2119872119894120576 (120601 (119909)) 119889119909int 119872119894120576 (120601 (119909)) 119889119909 (11)




where

1198901 = log (radic21205871205901) + (119868 (119909) minus 1199061)2212059012 (13)

1198902 = log (radic21205871205902) + (119868 (119909) minus 1199062)2212059022 (14)

119867120576 (119909) = 12 (1 + 2120587 arctan( 119909120576 )) (15)

120575120576 (119909) = 1120587 1205761205762 + 1199092 (16)



(17)



(a)

(b)

(c)

(d)




(18)


119896119899119894119895 = Δ1199090( Δ11990901206011198991198941198951003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895) + Δ1199100( Δ11991001206011198991198941198951003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895) (19)

with

1003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895 = radic(Δ1199090120601119899119894119895)2 + (Δ1199100120601119899119894119895)2 (20)




end while



05

06

07

08

09

1

DSC

5 10 15 200


05

06

07

08

09

1

DSC

5 10 15 200


05

06

07

08

09

1

DSC

5 10 15 200






(a)

(b)

(c)






DSC = 2119873 (1198781 cap 1198782)119873 (1198781) + 119873 (1198782) (21)





(a)

(b)

(c)

(d)

(e)



0

02

04

06

08

1

DSC




4 Conclusion


Competing Interests


Acknowledgments


References































(a)

(b)

(c)

(d)




(18)


119896119899119894119895 = Δ1199090( Δ11990901206011198991198941198951003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895) + Δ1199100( Δ11991001206011198991198941198951003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895) (19)

with

1003816100381610038161003816nabla1206011003816100381610038161003816119899119894119895 = radic(Δ1199090120601119899119894119895)2 + (Δ1199100120601119899119894119895)2 (20)




end while



05

06

07

08

09

1

DSC

5 10 15 200


05

06

07

08

09

1

DSC

5 10 15 200


05

06

07

08

09

1

DSC

5 10 15 200






(a)

(b)

(c)






DSC = 2119873 (1198781 cap 1198782)119873 (1198781) + 119873 (1198782) (21)





(a)

(b)

(c)

(d)

(e)



0

02

04

06

08

1

DSC




4 Conclusion


Competing Interests


Acknowledgments


References































end while



05

06

07

08

09

1

DSC

5 10 15 200


05

06

07

08

09

1

DSC

5 10 15 200


05

06

07

08

09

1

DSC

5 10 15 200






(a)

(b)

(c)






DSC = 2119873 (1198781 cap 1198782)119873 (1198781) + 119873 (1198782) (21)





(a)

(b)

(c)

(d)

(e)



0

02

04

06

08

1

DSC




4 Conclusion


Competing Interests


Acknowledgments


References






























(a)

(b)

(c)






DSC = 2119873 (1198781 cap 1198782)119873 (1198781) + 119873 (1198782) (21)





(a)

(b)

(c)

(d)

(e)



0

02

04

06

08

1

DSC




4 Conclusion


Competing Interests


Acknowledgments


References






























(a)

(b)

(c)

(d)

(e)



0

02

04

06

08

1

DSC




4 Conclusion


Competing Interests


Acknowledgments


References






























0

02

04

06

08

1

DSC




4 Conclusion


Competing Interests


Acknowledgments


References













































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