research article a novel fuzzy level set approach for...

Research ArticleA Novel Fuzzy Level Set Approach for Image Contour Detection

Yingjie Zhang1 Jianxing Xu1 and H D Cheng2

1College of Electronic Information and Automation Civil Aviation University of China Tianjin 300300 China2Department of Computer Science Utah State University Logan UT 84322 USA

Correspondence should be addressed to Yingjie Zhang jill1124163com

Received 24 January 2016 Revised 1 May 2016 Accepted 10 May 2016

Academic Editor Erik Cuevas

Copyright copy 2016 Yingjie 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

The level set methods have provided powerful frameworks for image segmentation However to obtain accurate boundaries of theobjects especiallywhen they haveweak edges or inhomogeneous intensities is still a very challenging task Actually we have studiedthe popular existing level set approaches and discovered that they failed to segment the images with weak edges or inhomogeneousintensities in many casesThe weakblurry edges and inhomogeneous intensities cause uncertainty and fuzziness for segmentationIn this paper a novel fuzzy level set approach is proposed At first 119878-function based on themaximum fuzzy entropy principle (MEP)is used to map the image from space domain to fuzzy domain Then an energy function is formulated according to the differencesbetween the actual and estimated probability densities of the intensities in different regions A partial differential equation is derivedfor finding the minimum of the energy functionThe proposed approach has been tested on both synthetic images and real imagesand evaluated by several popular metrics The experimental results demonstrate that the proposed approach can locate the trueobject boundaries even for objects with blurry boundaries low contrast and inhomogeneous intensities

1 Introduction

Image segmentation is an important component of imageanalysis and computer vision The results of segmentationare not always satisfactory because of low contrast blurryboundaries noise and inhomogeneous intensities Henceimage segmentation is still a quite difficult task [1]

Recently active contour models have attracted greatattention from researchers [2] There are two kinds of activecontour models One is the snake model which defines aparametric curve All snake properties and its behavior arespecified through an energy function A partial differentialequation controlling the snake makes it evolve to reduce theenergy [3] The physical analog can be extended and themotion of the snake can be viewed as the simulated forceacts on it The other one is the geometric model [4 5] Themain difference between these two kinds of approaches is thatthe geometric active contour introduces the level set functionrepresenting the evolving curve into the energy functionTheimplicit boundary representation does not depend on a spe-cific parameterization During the propagating no controlpoint mechanisms need to be employed Level set-based

active contour models have many advantages and amongthem the most important one is the ability to track thetopological variations of the boundary curves [6 7]

Level set-based segmentation models can be divided intofour categories ldquoregion-basedrdquo [5 8ndash11] ldquoedge-basedrdquo [412ndash14] ldquoshape-priorrdquo [15 16] and ldquomultifeature-basedrdquo [17]Region-based approach utilizes global information such asregion statistics region mean and weighted mean of theconstrained or scalable neighboring regions and can producemore semantically meaningful results [18] The major limi-tation of region-based approach is that it is very difficult todefine a suitable region descriptor for image with inhomoge-neous intensity [19] Some methods are based on a generalpiecewise matter [20 21] These methods do not assume thehomogeneity of the intensities therefore they are able to seg-ment images with inhomogeneous intensities and they havebeen actively studied recently [5 8] However these methodsare quite sensitive to the noise and cannot work well withblurry and weak edges It will be discussed in detail in theExperimental Results

Shape-prior approach incorporates the shape informa-tionThe method introduces a representation for deformable

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 2602647 12 pageshttpdxdoiorg10115520162602647

2 Mathematical Problems in Engineering

shapes and defines a probability distribution over the vari-ances of a set of training shapes Obviously such methodcannot work well if the prior shapes of the objects areunknown

Edge-based approaches do not need to assume the homo-geneity of image intensities hence they can even be appliedto images with inhomogeneous intensities [12] and has beenapplied to many segmentation tasks An edge representationis used to find the boundary curve with strong edge response[13]However these approaches suffer from serious boundaryleakage problems especially when the objects have inhomo-geneous intensities and they cannot converge on the realboundaries of the objects in many cases either [22] This willbe discussed in detail in the Experimental Results as well

Multifeature-based approaches use the surround inhi-bition weights of individual features including orientationluminance and luminance contrast Features are combinedaccording to a scale-guided strategy and the combinedweights are then used to modulate the final surround inhi-bition of the objects [17]

The blurry and weak edges the noise and the inhomoge-neous intensities can cause uncertainty and fuzziness whichwill result in poor outcomes of segmentation In fact imageshave uncertainty and fuzziness due to the following (1)Whenperforming 3D to 2D projection some information was lost(2) Some definitions of images such as edges and contrast areuncertain and fuzzy For instance an edge is defined as thereis intensity difference between a pixel and its neighbor pixelshowever how large the difference should be is not preciselyspecified and defined and eventually it is task-dependentTherefore we should use fuzzy logic to handle the uncertaintyand fuzziness of the images [23 24]

Many methods try to detect contour by calculating highgradients of color or gray levels As a result they are verysensitive to noise and textures Entropies can be used as ameasure of dissimilarity or inverse cohesion between two (ormore) probability distributions [25] For example in [26]a thresholding scheme is proposed to minimize the Tsalliscross-entropy between the original image and the thresh-olded image Then the contours of the objects are obtainedAverage Entropy (AE) is defined as a new informationmeasurement of regions in image [27]

In this paper a novel fuzzy level set active contour modelis proposed It will combine the advantages of fuzzy logicand level set theory and can generate much better resultsin segmentation In experiments a series of images areemployed for evaluating the proposed method and compar-ing with existing segmentation algorithmsThe experimentalresults demonstrate that the proposed approach can segmentboth synthetic and real images satisfactorily Furthermore itmakes the evolving function converge to the real boundarieseven with low contrast inhomogeneous intensities andblurry edges

The rest of the paper is organized as follows In Section 2the proposed fuzzy level set approach for segmentation isdescribed A variety of images have been tested and validatedthe proposed approach (however due to page limit only a fewof them are shown here) and the performance is evaluated

in Section 3 Finally the conclusions are summarized inSection 4

2 Proposed Fuzzy Level Set Approach

The proposed fuzzy edge-based level set approach consists ofthe following major components fuzzification fuzzy energyfunction and evolution equation

21 Image Fuzzification Assume that the size of image 119868 is119872times119873 and 119868(119909 119910) is the gray level of the pixel at coordinates(119909 119910) In order to apply fuzzy logic to deal with the fuzzinessand uncertainty of the image a suitablemembership functionis necessaryThemost commonly usedmembership functionis the standard 119878-function [24]

120583 (119868 (119909 119910)) = 119878 (119868 (119909 119910) 119886 119887 119888)

=

0 119868 (119909 119910) le 119886

(119868 (119909 119910) minus 119886)2

(119887 minus 119886) (119888 minus 119886)119886 le 119868 (119909 119910) le 119887

1 minus(119868 (119909 119910) minus 119888)

2

(119888 minus 119887) (119888 minus 119886)119887 le 119868 (119909 119910) le 119888

1 119868 (119909 119910) ge 119888

(1)

The value of 120583(119868(119909 119910)) represents the membership of119868(119909 119910) which is simplified as 120583(119909 119910) and the fuzzified imageis denoted by 120583 Parameters 119886 119887 and 119888 determine the shapeof the 119878-function Based on information theory the maximumentropy corresponds to the maximum information We use themaximum fuzzy entropy principle to determine parameters119886 119887 and 119888 We will find the combination of the parameterscorresponding to the maximum entropy

119867(119868) =1

119872119873

119872minus1

sum

119909=0

119873minus1

sum

119910=0

119878119899(120583 (119909 119910)) (2)

where 119867(119868) is the entropy of the image and 119878119899(sdot) is the

Shannon function

119878119899(120583 (119909 119910)) = minus120583 (119909 119910) log

2120583 (119909 119910)

minus (1 minus 120583 (119909 119910)) log2(1 minus 120583 (119909 119910))

(3)

There aremanyways to find themaximumvalue of (2) such assimulated annealing neural networks and genetic algorithmIn this paper we use simulated annealing algorithm [24]to find the optimum values of (119886opt 119887opt 119888opt) and to avoidsticking at local minimum

119867max (119868 119886opt 119887opt 119888opt)

= max 119867 (119868 119886 119887 119888) | 119868min le 119886 lt 119887 lt 119888 le 119868max (4)

where 119868min and 119868max are the minimum and maximum inten-sity values of the image respectively

Using above fuzzification process we can have the max-imum information when transforming image from space

Mathematical Problems in Engineering 3

Begin

Image fuzzification using (1)

Get fuzzy edge indicator function g

Initialize fuzzy level set function 1206010

120583

Get 120601n+1120583

from 120601n

120583by (12)

Yesn lt nmax

No

End

Figure 1 The flowchart of the proposed method bases on the fuzzy sets

(a) (b)

(c) (d)

Figure 2 Results of Leaf (a) Original image (b) Result by LIF method (c) Result by IGAC model (d) Result by fuzzy based approach


(a) (b)

(c) (d)

Figure 3 Results of Rabbit (a) Original image (b) Result by LIF method (c) Result by the IGACmodel (d) Result by fuzzy based approach

domain to fuzzy domain according to information theory Inaddition the 119878-function can enhance the images [23] that isit can improve the weak edges and prevent leakage further

Then the original image 119868 is transformed to a fuzzifiedimage 120583 according to (1)

22 Fuzzy Energy Function and Evolution Equation Consid-ering a fuzzified image 120583 as a real positive function defined indomain Ω the boundaries are defined as the fuzzy zero levelset of 120601

120583(119909 119910) which will be simplified as 120601

120583 Given the fuzzy

level set function 120601120583 the basic form of the energy function in

the ordinary space [10] can be adapted and transformed intothe fuzzy domain

119864 (120601120583) = 120572119875 (120601

120583) + 120573119871

119892(120601120583) + 120574119860

119892(120601120583) (5)

where 120572 gt 0 is a parameter controlling the effect of penalizingthe deviation of 120601

120583from a signed distance function 120573 and 120574

are positive constants and 119875(120601120583) is a penalizing term defined

as the integral below

119875 (120601120583) = intΩ

1

2(10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816minus 1)2

119889119909 119889119910 (6)

119875(120601120583) characterizes how close function 120601

120583is to a signed

distance function in domainΩAn external energy for function 120601

120583is defined as

119871119892(120601120583) = intΩ

119892120575 (120601120583)10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816119889119909 119889119910

119860119892(120601120583) = intΩ

119892119867(minus120601120583) 119889119909 119889119910

(7)

where 120575 is the univariate Dirac function 119867 is the Heavisidefunction 119871

119892(120601120583) is the length of the zero level curve 119860

119892(120601120583)

is used to speed up curve evolutionwhich is theweighted areaof the subregion and 119892 is the fuzzy edge indicator function

119892 =1

1 +10038161003816100381610038161003816nabla119866120590lowast 120601120583

10038161003816100381610038161003816

2 (8)

where 119866120590is the Gaussian kernel with standard deviation 120590

and lowast is the convolution operatorThe energy function drivesthe fuzzy zero level curve towards the boundaries and stopsevolving with the strongest boundary response


(a) (b)

(c) (d)

Figure 4 Results of Box (a) Original image (b) Result by LIF method (c) Result by the IGAC model (d) Result by fuzzy based approach

(a) (b)

(c) (d)

Figure 5 Results of Diabolo (a) Original image (b) Result by LIFmethod (c) Result by the IGACmodel (d) Result by fuzzy based approach


(a) (b)

(c) (d)

Figure 6 Results of Coffee can (a) Original image (b) Result by LIF method (c) Result by the IGAC model (d) Result by fuzzy basedapproach

The fuzzy zero level curve evolves to the gradient flowcorrespondingly anddrives the evolution equation for findingthe minimum of the energy function With total variationmethod the associated gradient flow is derived

120597120601120583

120597119905= 120572[Δ120601

120583minus div(

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)]

+ 120573120575 (120601120583) div(119892 sdot

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

+ 120574119892120575 (120601120583)

120601120583(119909 119910 0) = 120601

0

120583(119909 119910) in Ω

120575 (120601120583)

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

sdot120597120601120583

120597= 0 on 120597Ω

(9)

where 1206010120583is the initial condition defined in fuzzy domain and

the last equation in (9) is the boundary condition The initialcondition can be formulated as

1206010

120583(119909119910)=

minus1198890 (119909 119910) isin Ω

0minus 120597Ω0

0 (119909 119910) isin 120597Ω0

1198890 (119909 119910) isin Ω minus Ω

0

(10)

where 1198890is a predetermined constant larger than 2120576 which is

set to 4 for all experiments and 120597Ω0is the initial boundary

For numerical calculation the Dirac function 120575(sdot) issmoothed as

120575120576(119911) =

0 |119911| gt 120576

1

2120576[1 + cos(120587119911

120576)] |119911| le 120576

(11)

where 120576 = 1 is used in all experimentsEquation (9) is discretized by the central difference and

the approximation is

120601119899+1

120583(119894119895)minus 120601119899

120583(119894119895)

Δ119905= 120572 (119875

119894119895minus 119870119894119895) + 120573120575

120576(120601120583) sdot 119876119894119895+ 120574119892

sdot 120575120576(120601120583)

119875119894119895= Δ120601120583= 1198630119909119909

119894119895120601120583+ 1198630119910119910

119894119895120601120583

119870119894119895= div(

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

= 1198630119909

119894119895

1198630119909

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2


(a) (b)

(c) (d)

Figure 7 Results of Swan (a) Original image (b) Result by LIF method (c) Result by the IGAC model (d) Result by fuzzy based approach

+ 1198630119910

119894119895

1198630119910

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

119876119894119895= div(119892 sdot

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

= 119892 sdot 119870119894119895+

1198630119909

119894119895119892 sdot 1198630119909

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

+

1198630119910

119894119895119892 sdot 1198630119910

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

(12)

where1198630 is the central difference operatorThe flowchart of the proposed method is described in

Figure 1 The steps of the proposed method are summarizedas follows

(1) Initialize the fuzzy level set function using (1)

(2) Calculate fuzzy edge indicator function 119892 using (8)

(3) Initialize the fuzzy level set function 1206010120583

(4) Calculate 120601120583

119899+1 from 120601120583

119899 by (12)

(5) Check whether the convergence of 120601120583is satisfied if

it is not steady or has not reached the predeterminednumber of iterations go to step 4

3 Experimental Results

We conduct five groups of experiments using synthetic andreal images The same images were also tested by IGAC(improved geometric active contours) model [4] and LIF(local image fitting) model [5] Due to page limit we only usea few of the images to demonstrate the effectiveness and use-fulness of the proposed approach here The parameters usedhere are as followsThe step timeΔ119905 can be chosen from 01 to100 and here it is set to 10The time stepΔ119905 and the coefficient120572 must satisfy 120572 sdot Δ119905 lt 025 The coefficient 120573 determines


(a) (b)

(c) (d)

Figure 8 Results of Boat (a) Original image (b) Result by LIF method (c) Result by the IGAC model (d) Result by fuzzy based approach

the smoothness of the zero level curve and it can be chosenfrom 1 to 30 The coefficient 120574 of the weighted area termshould be a positive value so that the contours can shrinkfaster Also 120572 = 15Δ119905 120573 = 6 and 120574 = 3 All parameters aredetermined by experiments

In experiment 1 both the object and background arehomogenous We can see that the LIF method does not workwell as shown in Figure 2(b) the IGAC model can performrelativelywell withminor errors as shown in Figure 2(c) how-ever the proposed method can detect the boundaries evenbetter as shown in Figure 2(d)

In experiment 2 a more complex image with inhomoge-neous intensities is tested The background is homogeneousand the object is inhomogeneous with stepwise gray valuesExperiment 2 demonstrates that the proposed method per-forms better than IGAC model on inhomogeneous imagesThe LIF method completely failed and cannot converge asshown in Figure 3(b) In Figure 3(c) four regions of the objectare wrongly segmented This is due to the fact that IGACmodel tends to drive the zero level curve towards the bound-aries corresponding to the gradients and to stop evolvingwiththe strongest boundary response However in many cases

the real boundary may not have the strongest response andIGAC model cannot have sufficient global knowledge tocapture the real boundary

In experiment 3 the proposed approach LIFmethod andIGAC method are applied to a real image from AmsterdamLibrary ofObject Images (ALOI) [28]The result of LIF is alsovery poor as shown in Figure 4(b) After applying the IGACmethod the ill-defined border of the box is not connectedwell due to the leakages occurring in the weak edges Theresult of the proposed approach is shown in Figure 4(d)where the border is well connected and correctly detected asshown in Figure 4(d)

We have also tested many images with low contrast andnonuniform illuminations selected from Amsterdam Libraryof Object Images (ALOI)We can observe from Figures 5 and6 that the proposed method produces good results and theshapes and edges of the objects can be extracted much betterThe IGAC method tends to converge to the interior of theobjects and obtains wrong boundariesThe leakages occurredin the week edges The LIF method performs the poorestamong these methods as shown in corresponding Figures5(b) and 6(b)


(a) (b)

(c) (d)

Figure 9 Results of BUS image 1 (a) Original image (b) Result by LIF method (c) Result by the IGAC model (d) Result by fuzzy basedapproach

In experiment 4 methods are applied to the real imagesfrom other resources LIF method generates too many seg-ments as shown in Figures 7(b) and 8(b) More backgroundregions are wrongly covered when using the IGAC methodas shown in Figure 7(c) The proposed method can capturethe complex boundaries more accurately and achieve betterperformance than both the IGAC and LIF methods

In experiment 5 we use real breast ultrasound (BUS)images [29] to evaluate IGAC LIF and the proposed meth-ods The images are very noisy with low contrast and inho-mogeneous Due to high level of inherent speckle noise LIFproduces oversegments as shown in Figures 9(b) and 10(b) InFigure 9(c) IGAC converges to a false boundary and becauseof that the image is noisy and has blurry boundary of thetumor In Figure 10(c) although the tumor boundary is quiteclear IGAC still achieves wrong segmentation due to leakageThe proposedmethod can obtain accurate results as shown inFigures 9(d) and 10(d)

For evaluating segmentation results three area errormet-rics were used the true positive (TP) ratio the false positive(FP) ratio and the similarity (SI) [30 31] They are popularlyused for evaluating the performance of segmentation Let119860

120572

be the object region selected by the algorithm and let 119860119898be

the corresponding real object region the three error metricsare

TP =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

FP =1003816100381610038161003816119860119898 cup 119860119886 minus 119860119898

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

SI =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898 cup 119860119886

1003816100381610038161003816

(13)

The object regions obtained by the algorithms 119860120572s are

compared with manual delineations 119860119898s which are consid-

ered as the grand truthsWhen the TP ratio is higher itmeansthat more real object region 119860

119898is covered by 119860

120572 and when

the FP ratio is lower it means that less background region iscovered by 119860

120572 Meanwhile the higher SI ratio implies that

119860120572is more similar to 119860

119898 that is the overall performance

is better Since LIF oversegments all the images and cannotfind the major regions in the background and objects thefollowing discussion will not utilize the results of LIF Theperformances of the IGAC model and proposed method arelisted in Table 1

The TP ratios of the proposed method are much higherthan that of the IGAC model (especially in the secondand third rows in Table 1) and they indicate that the realobject regions in all images were segmented by the proposedmethodmore accurately Because of low contrast of the edgesthere are many local minima and the IGAC model mayconverge to some local minima and its TP ratios could beextremely low (Table 1) and the FP ratios of the IGACmodelare much higher than that of the proposed method It means


Table 1 Performance of IGAC method and fuzzy based approach

TP () FP () SI ()

Experiment 1 Leaf IGAC method 9922 185 9897The proposed method 9985 084 9976

Experiment 2 Rabbit IGAC method 2493 163 2401The proposed method 9979 120 9927

Experiment 3

Box IGAC method 2432 174 2415The proposed method 9883 022 9853

Diabolo IGAC method 6837 005 6835The proposed method 9968 004 9966

Coffee can IGAC method 7219 003 7217The proposed method 9991 002 9990

Experiment 4Swan IGAC method 9473 1424 8351

The proposed method 9726 353 9474

Boat IGAC method 9103 544 8601The proposed method 9675 342 9411

Experiment 5BUS image 1 IGAC method 100 1753 8785


BUS image 2 IGAC method 7521 033 7475The proposed method 9896 048 9803

(a) (b)

(c) (d)



that many background regions are included in the objectregions generated by the IGAC model In addition theunsuitable regions cannot be cut off easily and the resultsdirectly influence the subsequent analysis The proposedmethod can handle the blurry and weak boundaries well andthe segmentation results are more accurate and reliable Inthe last row of Table 1 the FP ratio of the proposed methodis a little higher than that of the IGAC model This is dueto the weak edges and blurry boundaries and the evolvingfunction of IGACmethodwill tend to converge to the interiorof the object therefore even if it has lower FP ratio it achievesseverally wrong segmentation Nevertheless the proposedmethod has much higher SI ratios than those of the IGACmodel that demonstrate that the overall performance of theproposed method is much better

4 Conclusions

In this paper we have developed a novel level set active con-tour method based on fuzzy logic and variation theory Theproposed approach is more efficient than the level set meth-ods in performing image segmentation due to its capability inhandling fuzziness and uncertainty Three popular area errormetrics are used for evaluating segmentation performanceThe proposed method and other popular methods (IGACmodel and LIF method) are applied to the same images forcomparison The experimental results demonstrate that theproposedmethod ismore accurate and robust evenwithweakboundaries noise and inhomogeneous intensities This isbecause the proposed approach takes the advantages of bothlevel set theory and fuzzy logic It may find wide applicationsin the related areas

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work is supported in part by National Natural ScienceFoundation of China and the Civil Aviation Administrationof China (Grant no U1433103)

References

[1] YWu andCHe ldquoA convex variational level setmodel for imagesegmentationrdquo Signal Processing vol 106 pp 123ndash133 2015

[2] D Lui C Scharfenberger K Fergani A Wong and D AClausi ldquoEnhanced decoupled active contour using structuraland textural variation energy functionalsrdquo IEEETransactions onImage Processing vol 23 no 2 pp 855ndash869 2014

[3] X Gao B Wang D Tao and X Li ldquoA relay level set method forautomatic image segmentationrdquo IEEE Transactions on SystemsMan and Cybernetics Part B Cybernetics vol 41 no 2 pp 518ndash525 2011

[4] 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 Vision

and Pattern Recognition (CVPR rsquo05) pp 430ndash436 San DiegoCalif USA June 2005

[5] K Zhang H Song and L Zhang ldquoActive contours driven bylocal image fitting energyrdquo Pattern Recognition vol 43 no 4pp 1199ndash1206 2010

[6] J Lie M Lysaker and X-C Tai ldquoA binary level set modeland some applications to Mumford-Shah image segmentationrdquoIEEE Transactions on Image Processing vol 15 no 5 pp 1171ndash1181 2006

[7] Z Lu G Carneiro and A P Bradley ldquoAn improved jointoptimization of multiple level set functions for the segmenta-tion of overlapping cervical cellsrdquo IEEE Transactions on ImageProcessing vol 24 no 4 pp 1261ndash1272 2015

[8] L Wang C Li Q Sun D Xia and C-Y Kao ldquoActive contoursdriven by local and global intensity fitting energy with applica-tion to brain MR image segmentationrdquo Computerized MedicalImaging and Graphics vol 33 no 7 pp 520ndash531 2009

[9] ADubrovina-Karni G Rosman andRKimmel ldquoMulti-regionactive contours with a single level set functionrdquo IEEE Transac-tions on Pattern Analysis and Machine Intelligence vol 37 no 8pp 1585ndash1601 2015

[10] R Ronfard ldquoRegion-based strategies for active contour mod-elsrdquo International Journal of Computer Vision vol 13 no 2 pp229ndash251 1994

[11] C Samson L Blanc-Feraud G Aubert and J Zerubia ldquoAvariationalmodel for image classification and restorationrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol22 no 5 pp 460ndash472 2000

[12] C Li C Xu C Gui and M D Fox ldquoDistance regularized levelset evolution and its application to image segmentationrdquo IEEETransactions on Image Processing vol 19 no 12 pp 3243ndash32542010

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

[14] A Vasilevskiy and K Siddiqi ldquoFlux maximizing geometricflowsrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 24 no 12 pp 1565ndash1578 2002

[15] M Rousson and N Paragios ldquoShape priors for level set rep-resentationsrdquo in Proceedings of the 7th European Conference onComputer Vision (ECCV rsquo02) pp 416ndash418 IEEE CopenhagenDenmark 2002

[16] T Chan andWZhu ldquoLevel set based shape prior segmentationrdquoin Proceedings of the IEEE Computer Society Conference onComputer Vision and Pattern Recognition (CVPR rsquo05) pp 1164ndash1170 San Diego Calif USA June 2005

[17] K-F Yang C-Y Li and Y-J Li ldquoMultifeature-based surroundinhibition improves contour detection in natural imagesrdquo IEEETransactions on Image Processing vol 23 no 12 pp 5020ndash50322014

[18] C Li R Huang Z Ding J Gatenby D N Metaxas and JC Gore ldquoA level set method for image segmentation in thepresence of intensity inhomogeneities with application toMRIrdquoIEEE Transactions on Image Processing vol 20 no 7 pp 2007ndash2016 2011

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

[20] A Tsai A Yezzi Jr and A S Willsky ldquoCurve evolution imple-mentation of the Mumford-Shah functional for image seg-mentation denoising interpolation and magnificationrdquo IEEE


Transactions on Image Processing vol 10 no 8 pp 1169ndash11862001

[21] L A Vese and T F Chan ldquoA multiphase level set frameworkfor image segmentation using the Mumford and Shah modelrdquoInternational Journal of Computer Vision vol 50 no 3 pp 271ndash293 2002

[22] B Liu H D Cheng J Huang J Tian X Tang and J LiuldquoProbability density difference-based active contour for ultra-sound image segmentationrdquo Pattern Recognition vol 43 no 6pp 2028ndash2042 2010

[23] Y Zhang H D Cheng J Tian J Huang and X Tang ldquoFrac-tional subpixel diffusion and fuzzy logic approach for ultra-sound speckle reductionrdquo Pattern Recognition vol 43 no 8 pp2962ndash2970 2010

[24] H D Cheng and J-R Chen ldquoAutomatically determine themembership function based on the maximum entropy princi-plerdquo Information Sciences vol 96 no 3-4 pp 163ndash182 1997

[25] Q D Katatbeh J Martınez-Aroza J F Gomez-Lopera andD Blanco-Navarro ldquoAn optimal segmentation method usingjensenndashshannon divergence via a multi-size sliding windowtechniquerdquo Entropy vol 17 no 12 pp 7996ndash8006 2015

[26] F Y Nie ldquoTsallis cross-entropy based framework for image seg-mentation with histogram thresholdingrdquo Journal of ElectronicImaging vol 24 no 1 Article ID 013002 2015

[27] O A Kittaneh M A Khan M Akbar and H A BayoudldquoAverage entropy a new uncertainty measure with applicationto image segmentationrdquoTheAmerican Statistician vol 70 no 1pp 18ndash24 2016

[28] J-M Geusebroek G J Burghouts and A W M SmeuldersldquoTheAmsterdam library of object imagesrdquo International Journalof Computer Vision vol 61 no 1 pp 103ndash112 2005

[29] M Xian Y Zhang and H D Cheng ldquoFully automatic segmen-tation of breast ultrasound images based on breast characteris-tics in space and frequency domainsrdquo Pattern Recognition vol48 no 2 pp 485ndash497 2015

[30] J Shan H D Cheng and Y Wang ldquoCompletely automatedsegmentation approach for breast ultrasound images usingmultiple-domain featuresrdquoUltrasound inMedicine and Biologyvol 38 no 2 pp 262ndash275 2012

[31] H Shao Y ZhangM Xian andHD Cheng ldquoA saliencymodelfor automated tumor detection in breast ultrasound imagesrdquoin Proceedings of the IEEE International Conference on ImageProcessing pp 1424ndash1428 Quebec City Canada September2015

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


shapes and defines a probability distribution over the vari-ances of a set of training shapes Obviously such methodcannot work well if the prior shapes of the objects areunknown

Edge-based approaches do not need to assume the homo-geneity of image intensities hence they can even be appliedto images with inhomogeneous intensities [12] and has beenapplied to many segmentation tasks An edge representationis used to find the boundary curve with strong edge response[13]However these approaches suffer from serious boundaryleakage problems especially when the objects have inhomo-geneous intensities and they cannot converge on the realboundaries of the objects in many cases either [22] This willbe discussed in detail in the Experimental Results as well

Multifeature-based approaches use the surround inhi-bition weights of individual features including orientationluminance and luminance contrast Features are combinedaccording to a scale-guided strategy and the combinedweights are then used to modulate the final surround inhi-bition of the objects [17]

The blurry and weak edges the noise and the inhomoge-neous intensities can cause uncertainty and fuzziness whichwill result in poor outcomes of segmentation In fact imageshave uncertainty and fuzziness due to the following (1)Whenperforming 3D to 2D projection some information was lost(2) Some definitions of images such as edges and contrast areuncertain and fuzzy For instance an edge is defined as thereis intensity difference between a pixel and its neighbor pixelshowever how large the difference should be is not preciselyspecified and defined and eventually it is task-dependentTherefore we should use fuzzy logic to handle the uncertaintyand fuzziness of the images [23 24]

Many methods try to detect contour by calculating highgradients of color or gray levels As a result they are verysensitive to noise and textures Entropies can be used as ameasure of dissimilarity or inverse cohesion between two (ormore) probability distributions [25] For example in [26]a thresholding scheme is proposed to minimize the Tsalliscross-entropy between the original image and the thresh-olded image Then the contours of the objects are obtainedAverage Entropy (AE) is defined as a new informationmeasurement of regions in image [27]

In this paper a novel fuzzy level set active contour modelis proposed It will combine the advantages of fuzzy logicand level set theory and can generate much better resultsin segmentation In experiments a series of images areemployed for evaluating the proposed method and compar-ing with existing segmentation algorithmsThe experimentalresults demonstrate that the proposed approach can segmentboth synthetic and real images satisfactorily Furthermore itmakes the evolving function converge to the real boundarieseven with low contrast inhomogeneous intensities andblurry edges

The rest of the paper is organized as follows In Section 2the proposed fuzzy level set approach for segmentation isdescribed A variety of images have been tested and validatedthe proposed approach (however due to page limit only a fewof them are shown here) and the performance is evaluated

in Section 3 Finally the conclusions are summarized inSection 4

2 Proposed Fuzzy Level Set Approach

The proposed fuzzy edge-based level set approach consists ofthe following major components fuzzification fuzzy energyfunction and evolution equation

21 Image Fuzzification Assume that the size of image 119868 is119872times119873 and 119868(119909 119910) is the gray level of the pixel at coordinates(119909 119910) In order to apply fuzzy logic to deal with the fuzzinessand uncertainty of the image a suitablemembership functionis necessaryThemost commonly usedmembership functionis the standard 119878-function [24]

120583 (119868 (119909 119910)) = 119878 (119868 (119909 119910) 119886 119887 119888)

=

0 119868 (119909 119910) le 119886

(119868 (119909 119910) minus 119886)2

(119887 minus 119886) (119888 minus 119886)119886 le 119868 (119909 119910) le 119887

1 minus(119868 (119909 119910) minus 119888)

2

(119888 minus 119887) (119888 minus 119886)119887 le 119868 (119909 119910) le 119888

1 119868 (119909 119910) ge 119888

(1)

The value of 120583(119868(119909 119910)) represents the membership of119868(119909 119910) which is simplified as 120583(119909 119910) and the fuzzified imageis denoted by 120583 Parameters 119886 119887 and 119888 determine the shapeof the 119878-function Based on information theory the maximumentropy corresponds to the maximum information We use themaximum fuzzy entropy principle to determine parameters119886 119887 and 119888 We will find the combination of the parameterscorresponding to the maximum entropy

119867(119868) =1

119872119873

119872minus1

sum

119909=0

119873minus1

sum

119910=0

119878119899(120583 (119909 119910)) (2)

where 119867(119868) is the entropy of the image and 119878119899(sdot) is the

Shannon function

119878119899(120583 (119909 119910)) = minus120583 (119909 119910) log

2120583 (119909 119910)

minus (1 minus 120583 (119909 119910)) log2(1 minus 120583 (119909 119910))

(3)

There aremanyways to find themaximumvalue of (2) such assimulated annealing neural networks and genetic algorithmIn this paper we use simulated annealing algorithm [24]to find the optimum values of (119886opt 119887opt 119888opt) and to avoidsticking at local minimum

119867max (119868 119886opt 119887opt 119888opt)

= max 119867 (119868 119886 119887 119888) | 119868min le 119886 lt 119887 lt 119888 le 119868max (4)

where 119868min and 119868max are the minimum and maximum inten-sity values of the image respectively

Using above fuzzification process we can have the max-imum information when transforming image from space


Begin




120583

Get 120601n+1120583

from 120601n

120583by (12)

Yesn lt nmax

No

End


(a) (b)

(c) (d)



(a) (b)

(c) (d)









119864 (120601120583) = 120572119875 (120601

120583) + 120573119871

119892(120601120583) + 120574119860

119892(120601120583) (5)





119875 (120601120583) = intΩ

1

2(10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816minus 1)2

119889119909 119889119910 (6)




120583is defined as

119871119892(120601120583) = intΩ

119892120575 (120601120583)10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816119889119909 119889119910

119860119892(120601120583) = intΩ

119892119867(minus120601120583) 119889119909 119889119910

(7)



119892(120601120583)


119892 =1

1 +10038161003816100381610038161003816nabla119866120590lowast 120601120583

10038161003816100381610038161003816

2 (8)




(a) (b)

(c) (d)


(a) (b)

(c) (d)



(a) (b)

(c) (d)



120597120601120583

120597119905= 120572[Δ120601

120583minus div(

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)]

+ 120573120575 (120601120583) div(119892 sdot

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

+ 120574119892120575 (120601120583)

120601120583(119909 119910 0) = 120601

0

120583(119909 119910) in Ω

120575 (120601120583)

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

sdot120597120601120583

120597= 0 on 120597Ω

(9)



1206010

120583(119909119910)=

minus1198890 (119909 119910) isin Ω

0minus 120597Ω0

0 (119909 119910) isin 120597Ω0

1198890 (119909 119910) isin Ω minus Ω

0

(10)




120575120576(119911) =

0 |119911| gt 120576

1

2120576[1 + cos(120587119911

120576)] |119911| le 120576

(11)



120601119899+1

120583(119894119895)minus 120601119899

120583(119894119895)

Δ119905= 120572 (119875

119894119895minus 119870119894119895) + 120573120575

120576(120601120583) sdot 119876119894119895+ 120574119892

sdot 120575120576(120601120583)

119875119894119895= Δ120601120583= 1198630119909119909

119894119895120601120583+ 1198630119910119910

119894119895120601120583

119870119894119895= div(

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

= 1198630119909

119894119895

1198630119909

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2


(a) (b)

(c) (d)


+ 1198630119910

119894119895

1198630119910

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

119876119894119895= div(119892 sdot

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

= 119892 sdot 119870119894119895+

1198630119909

119894119895119892 sdot 1198630119909

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

+

1198630119910

119894119895119892 sdot 1198630119910

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

(12)






(4) Calculate 120601120583

119899+1 from 120601120583

119899 by (12)






(a) (b)

(c) (d)









(a) (b)

(c) (d)





120572



TP =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

FP =1003816100381610038161003816119860119898 cup 119860119886 minus 119860119898

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

SI =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898 cup 119860119886

1003816100381610038161003816

(13)





120572 and when









TP () FP () SI ()



Experiment 3










(a) (b)

(c) (d)




4 Conclusions


Competing Interests


Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Begin




120583

Get 120601n+1120583

from 120601n

120583by (12)

Yesn lt nmax

No

End


(a) (b)

(c) (d)



(a) (b)

(c) (d)









119864 (120601120583) = 120572119875 (120601

120583) + 120573119871

119892(120601120583) + 120574119860

119892(120601120583) (5)





119875 (120601120583) = intΩ

1

2(10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816minus 1)2

119889119909 119889119910 (6)




120583is defined as

119871119892(120601120583) = intΩ

119892120575 (120601120583)10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816119889119909 119889119910

119860119892(120601120583) = intΩ

119892119867(minus120601120583) 119889119909 119889119910

(7)



119892(120601120583)


119892 =1

1 +10038161003816100381610038161003816nabla119866120590lowast 120601120583

10038161003816100381610038161003816

2 (8)




(a) (b)

(c) (d)


(a) (b)

(c) (d)



(a) (b)

(c) (d)



120597120601120583

120597119905= 120572[Δ120601

120583minus div(

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)]

+ 120573120575 (120601120583) div(119892 sdot

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

+ 120574119892120575 (120601120583)

120601120583(119909 119910 0) = 120601

0

120583(119909 119910) in Ω

120575 (120601120583)

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

sdot120597120601120583

120597= 0 on 120597Ω

(9)



1206010

120583(119909119910)=

minus1198890 (119909 119910) isin Ω

0minus 120597Ω0

0 (119909 119910) isin 120597Ω0

1198890 (119909 119910) isin Ω minus Ω

0

(10)




120575120576(119911) =

0 |119911| gt 120576

1

2120576[1 + cos(120587119911

120576)] |119911| le 120576

(11)



120601119899+1

120583(119894119895)minus 120601119899

120583(119894119895)

Δ119905= 120572 (119875

119894119895minus 119870119894119895) + 120573120575

120576(120601120583) sdot 119876119894119895+ 120574119892

sdot 120575120576(120601120583)

119875119894119895= Δ120601120583= 1198630119909119909

119894119895120601120583+ 1198630119910119910

119894119895120601120583

119870119894119895= div(

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

= 1198630119909

119894119895

1198630119909

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2


(a) (b)

(c) (d)


+ 1198630119910

119894119895

1198630119910

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

119876119894119895= div(119892 sdot

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

= 119892 sdot 119870119894119895+

1198630119909

119894119895119892 sdot 1198630119909

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

+

1198630119910

119894119895119892 sdot 1198630119910

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

(12)






(4) Calculate 120601120583

119899+1 from 120601120583

119899 by (12)






(a) (b)

(c) (d)









(a) (b)

(c) (d)





120572



TP =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

FP =1003816100381610038161003816119860119898 cup 119860119886 minus 119860119898

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

SI =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898 cup 119860119886

1003816100381610038161003816

(13)





120572 and when









TP () FP () SI ()



Experiment 3










(a) (b)

(c) (d)




4 Conclusions


Competing Interests


Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)









119864 (120601120583) = 120572119875 (120601

120583) + 120573119871

119892(120601120583) + 120574119860

119892(120601120583) (5)





119875 (120601120583) = intΩ

1

2(10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816minus 1)2

119889119909 119889119910 (6)




120583is defined as

119871119892(120601120583) = intΩ

119892120575 (120601120583)10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816119889119909 119889119910

119860119892(120601120583) = intΩ

119892119867(minus120601120583) 119889119909 119889119910

(7)



119892(120601120583)


119892 =1

1 +10038161003816100381610038161003816nabla119866120590lowast 120601120583

10038161003816100381610038161003816

2 (8)




(a) (b)

(c) (d)


(a) (b)

(c) (d)



(a) (b)

(c) (d)



120597120601120583

120597119905= 120572[Δ120601

120583minus div(

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)]

+ 120573120575 (120601120583) div(119892 sdot

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

+ 120574119892120575 (120601120583)

120601120583(119909 119910 0) = 120601

0

120583(119909 119910) in Ω

120575 (120601120583)

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

sdot120597120601120583

120597= 0 on 120597Ω

(9)



1206010

120583(119909119910)=

minus1198890 (119909 119910) isin Ω

0minus 120597Ω0

0 (119909 119910) isin 120597Ω0

1198890 (119909 119910) isin Ω minus Ω

0

(10)




120575120576(119911) =

0 |119911| gt 120576

1

2120576[1 + cos(120587119911

120576)] |119911| le 120576

(11)



120601119899+1

120583(119894119895)minus 120601119899

120583(119894119895)

Δ119905= 120572 (119875

119894119895minus 119870119894119895) + 120573120575

120576(120601120583) sdot 119876119894119895+ 120574119892

sdot 120575120576(120601120583)

119875119894119895= Δ120601120583= 1198630119909119909

119894119895120601120583+ 1198630119910119910

119894119895120601120583

119870119894119895= div(

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

= 1198630119909

119894119895

1198630119909

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2


(a) (b)

(c) (d)


+ 1198630119910

119894119895

1198630119910

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

119876119894119895= div(119892 sdot

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

= 119892 sdot 119870119894119895+

1198630119909

119894119895119892 sdot 1198630119909

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

+

1198630119910

119894119895119892 sdot 1198630119910

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

(12)






(4) Calculate 120601120583

119899+1 from 120601120583

119899 by (12)






(a) (b)

(c) (d)









(a) (b)

(c) (d)





120572



TP =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

FP =1003816100381610038161003816119860119898 cup 119860119886 minus 119860119898

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

SI =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898 cup 119860119886

1003816100381610038161003816

(13)





120572 and when









TP () FP () SI ()



Experiment 3










(a) (b)

(c) (d)




4 Conclusions


Competing Interests


Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)


(a) (b)

(c) (d)



(a) (b)

(c) (d)



120597120601120583

120597119905= 120572[Δ120601

120583minus div(

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)]

+ 120573120575 (120601120583) div(119892 sdot

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

+ 120574119892120575 (120601120583)

120601120583(119909 119910 0) = 120601

0

120583(119909 119910) in Ω

120575 (120601120583)

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

sdot120597120601120583

120597= 0 on 120597Ω

(9)



1206010

120583(119909119910)=

minus1198890 (119909 119910) isin Ω

0minus 120597Ω0

0 (119909 119910) isin 120597Ω0

1198890 (119909 119910) isin Ω minus Ω

0

(10)




120575120576(119911) =

0 |119911| gt 120576

1

2120576[1 + cos(120587119911

120576)] |119911| le 120576

(11)



120601119899+1

120583(119894119895)minus 120601119899

120583(119894119895)

Δ119905= 120572 (119875

119894119895minus 119870119894119895) + 120573120575

120576(120601120583) sdot 119876119894119895+ 120574119892

sdot 120575120576(120601120583)

119875119894119895= Δ120601120583= 1198630119909119909

119894119895120601120583+ 1198630119910119910

119894119895120601120583

119870119894119895= div(

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

= 1198630119909

119894119895

1198630119909

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2


(a) (b)

(c) (d)


+ 1198630119910

119894119895

1198630119910

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

119876119894119895= div(119892 sdot

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

= 119892 sdot 119870119894119895+

1198630119909

119894119895119892 sdot 1198630119909

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

+

1198630119910

119894119895119892 sdot 1198630119910

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

(12)






(4) Calculate 120601120583

119899+1 from 120601120583

119899 by (12)






(a) (b)

(c) (d)









(a) (b)

(c) (d)





120572



TP =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

FP =1003816100381610038161003816119860119898 cup 119860119886 minus 119860119898

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

SI =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898 cup 119860119886

1003816100381610038161003816

(13)





120572 and when









TP () FP () SI ()



Experiment 3










(a) (b)

(c) (d)




4 Conclusions


Competing Interests


Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)



120597120601120583

120597119905= 120572[Δ120601

120583minus div(

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)]

+ 120573120575 (120601120583) div(119892 sdot

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

+ 120574119892120575 (120601120583)

120601120583(119909 119910 0) = 120601

0

120583(119909 119910) in Ω

120575 (120601120583)

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

sdot120597120601120583

120597= 0 on 120597Ω

(9)



1206010

120583(119909119910)=

minus1198890 (119909 119910) isin Ω

0minus 120597Ω0

0 (119909 119910) isin 120597Ω0

1198890 (119909 119910) isin Ω minus Ω

0

(10)




120575120576(119911) =

0 |119911| gt 120576

1

2120576[1 + cos(120587119911

120576)] |119911| le 120576

(11)



120601119899+1

120583(119894119895)minus 120601119899

120583(119894119895)

Δ119905= 120572 (119875

119894119895minus 119870119894119895) + 120573120575

120576(120601120583) sdot 119876119894119895+ 120574119892

sdot 120575120576(120601120583)

119875119894119895= Δ120601120583= 1198630119909119909

119894119895120601120583+ 1198630119910119910

119894119895120601120583

119870119894119895= div(

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

= 1198630119909

119894119895

1198630119909

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2


(a) (b)

(c) (d)


+ 1198630119910

119894119895

1198630119910

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

119876119894119895= div(119892 sdot

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

= 119892 sdot 119870119894119895+

1198630119909

119894119895119892 sdot 1198630119909

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

+

1198630119910

119894119895119892 sdot 1198630119910

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

(12)






(4) Calculate 120601120583

119899+1 from 120601120583

119899 by (12)






(a) (b)

(c) (d)









(a) (b)

(c) (d)





120572



TP =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

FP =1003816100381610038161003816119860119898 cup 119860119886 minus 119860119898

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

SI =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898 cup 119860119886

1003816100381610038161003816

(13)





120572 and when









TP () FP () SI ()



Experiment 3










(a) (b)

(c) (d)




4 Conclusions


Competing Interests


Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)


+ 1198630119910

119894119895

1198630119910

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

119876119894119895= div(119892 sdot

nabla120601120583

10038161003816100381610038161003816nabla120601120583

10038161003816100381610038161003816

)

= 119892 sdot 119870119894119895+

1198630119909

119894119895119892 sdot 1198630119909

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

+

1198630119910

119894119895119892 sdot 1198630119910

119894119895120601120583

radic(1198630119909

119894119895120601120583)2

+ (1198630119910

119894119895120601120583)2

(12)






(4) Calculate 120601120583

119899+1 from 120601120583

119899 by (12)






(a) (b)

(c) (d)









(a) (b)

(c) (d)





120572



TP =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

FP =1003816100381610038161003816119860119898 cup 119860119886 minus 119860119898

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

SI =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898 cup 119860119886

1003816100381610038161003816

(13)





120572 and when









TP () FP () SI ()



Experiment 3










(a) (b)

(c) (d)




4 Conclusions


Competing Interests


Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)









(a) (b)

(c) (d)





120572



TP =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

FP =1003816100381610038161003816119860119898 cup 119860119886 minus 119860119898

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

SI =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898 cup 119860119886

1003816100381610038161003816

(13)





120572 and when









TP () FP () SI ()



Experiment 3










(a) (b)

(c) (d)




4 Conclusions


Competing Interests


Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)





120572



TP =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

FP =1003816100381610038161003816119860119898 cup 119860119886 minus 119860119898

10038161003816100381610038161003816100381610038161003816119860119898

1003816100381610038161003816

SI =1003816100381610038161003816119860119898 cap 119860119886

10038161003816100381610038161003816100381610038161003816119860119898 cup 119860119886

1003816100381610038161003816

(13)





120572 and when









TP () FP () SI ()



Experiment 3










(a) (b)

(c) (d)




4 Conclusions


Competing Interests


Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











TP () FP () SI ()



Experiment 3










(a) (b)

(c) (d)




4 Conclusions


Competing Interests


Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











4 Conclusions


Competing Interests


Acknowledgments


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a novel fuzzy level set approach for...

Documents