research article a fast region-based segmentation model ...[0,1] is a parameter evolution curve, and...

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2013 Article ID 501628 7 pageshttpdxdoiorg1011552013501628

Research ArticleA Fast Region-Based Segmentation Model withGaussian Kernel of Fractional Order

Bo Chen Qing-Hua Zou Wen-Sheng Chen and Yan Li

College of Mathematics and Computational Science Shenzhen University Shenzhen 518060 China

Correspondence should be addressed to Wen-Sheng Chen chenwsszueducn

Received 11 August 2013 Revised 3 September 2013 Accepted 3 September 2013

Academic Editor Ming Li

Copyright copy 2013 Bo Chen et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

By summarizing some classical active contourmodels from the view of level set representation a simple energy function expressionwith the Gaussian kernel of fractional order is proposed and then a novel region-based geometric active contour model isestablished In this proposed model the energy function with value of [minus1 1] is built the local mean and global mean of theinside and outside of the evolution curve are employed and the segmentation results are obtained by controlling the expansionand contraction of the evolution curve The model is simple and easy to implement it can also protect weak edges because ofconsidering more statistical information Experimental results on synthetic and natural images show that the proposed model ismuch more effective in dealing with the images with weak or blurred edges and it takes less time

1 Introduction

Image segmentation is a basic and important topic in thefields of image processing Accurate image segmentation canprovide more important information for the follow-up appli-cation such as machine vision and motion tracking How-ever segmental results are always affected by low contrast andthe problems of intensity inhomogeneity The main idea ofimage segmentation is to extract the concerned regions andtheir contours from the whole image There have been thou-sands of image segmentation algorithms proposed in recentdecades Some researchers put forward the edge detectionbased on the gradient derivatives or Canny edge detectionand so on Edge detection is good for simple image but notsuitable for the clutter target boundary extraction The mainreasons are as follows Firstly edge extracted for compleximage is often not corresponding to the target boundary Sec-ondly the extracted edge is discontinuous but the goal oftenneeds closed boundary to separate the object from the wholeimage In addition edge detection is dependent on the localinformation near pixel it has advantages sometimes but inmany cases overall appearance of the target is the key so theconcepts of the image segmentation and edge detection arenot one and the same

Regional growth is a simple technique to provide seg-mental region the algorithm begins with some seed pointsand found pixels near the seed which has similar image char-acteristics such as gray scale and color characteristics Thisalgorithm has been applied to Mumford-Shah function [1]Another region-based method is active contour (AC) model[2] Active contour model is 2D or 3D surface contourdescription which involves the contour evolution under anappropriate energy in order to get a satisfactory segmentationresult such as the target boundary with the closed contourOver the past decade researchers have proposedmany differ-ent active contour models which are mainly divided into twocategories namely parametric active contour models andgeometric active contour models In parametric active con-tour models the parameter equation of the curve is 119862(119901) =

[119909(119901) 119910(119901)] in which 0 le 119901 le 1 The parametric active con-tour model essentially depends on the energy function ratherthan the geometric figures of the contour Therefore thismodel cannot handle topology changes when it detectsmulti-ple targets but geometric active contour model can deal withtopological changes because it uses the structure of levelset in which the curve 119862 is zero level set function 120601(119909 119905)

119877

119899

times [0infin) rarr 119877 119899 = 2 3 for example 119862 = 119909 isin 119877

119899

120601(119909 119905) = 0 The first type of geometric active contour model

2 Advances in Mathematical Physics

is introduced by Caselles et al [3] its main idea is to use cur-vature and normal direction forcing curve movement so thatit stops on the edge with the edge function 119892(119909) = 119892(|nabla119868|

2

)where nabla119868 means the gradient of the given image 119868 whichhas a property that it equals zero on the border and othersequal one For example 119892(119909) = 119890

minus(11205902

119890)

|nabla119866

120590lowast 119868(119909)|

2 where120590

119890is a scale factor 119866

120590= (1

radic

2120587120590)119890

minus((119909minus120583)21205902) is a Gaussian

kernel in which 120590 denotes the standard deviation of thegiven image and 120583 denotes the expectation of the givenimage Another type of geometric active contourmodel is theGeodesic active contour model [4] which can search for theminimum length of the edge weights under the energy func-tionThismodel is similar to the former geometricmodel butthere is a big difference a vector filed term is employed inGeodesic active contour model to stop the motion curveon the weak edges Paragios et al put forward the famousgradient vector flow (GVF) instead of nabla119892 to increase therange of results called GVF geometric AC [5] Chan andVese[6] proposed a new model CV AC and Li et al [7] proposeda new model (LBF) which uses energy function to overcomethe problemof nonhomogeneity LBFmodel can dealwith theimage of different gray levels by adding the kernel functionand it can employ local gray level information effectively

Many structures of different level set evolution modelshave been summarized before The level set evolution of theabove energy functions [8] can be represented as

120597120601

120597119905

= 120572

119896119870

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

+ 119881

119873

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

+

119873 sdot nabla120601

(1)

where 119870 is the Euclidean curvature and 120572

119896 119881119873 and

119873 arethree parameters which decide the speed and direction ofthe evolution The term based on curvature vector is usedto smooth the curve The normal direction is used to controlshrinkage and expansion of the curve and force the curve tomove along the direction vector Details are shown in Table 1where 120578 120583 120582

1 and 120582

2are constant [ V] is the GVF and 119896(119909)

is a function based on the normal curvature and GVF [9] Atthe same time 119868(119909) in Table 1 is original gray level image 119862inand 119862out are average gray values of 119868(119909) inside and outside ofcurve and 119890

1and 119890

2are the weighted average gray values of

119868(119909) inside and outside of curve in the Gaussian windowThe rest of the paper is organized as follows in the

next two sections we will review classical existing geometricmodels Chan-Vese model and LBF model The new modelis introduced in Section 4 Some experimental results areshown in Section 5 We conclude the paper in Section 6

2 Chan-Vese Model

In Chan-Vese (CV) model we considered the simplest typeof segmentation which divided the image into the target andthe background and the distributions of the gray values oftarget and background are approximately constant values CVmodel is based on the evolution of the level set and can dealwith curve topology changes better for the curve which is

Table 1 The analysis table of energy function model

Model 120572

119896 119881

119873

119873

Geometric AC [3] 119892(119909)119903119892(119909)

0

Geodesic AC [4] 119892(119909)

119903119892(119909)

nabla119892(119909)

GVF Geo AC [5] 119892(119909)

119903119896(119909)119892(119909)

119892(119909) (1 minus |119896 (119909)| []])

CV AC [6] 120578

120583 + (119868 minus 119888in)2

minus(119868 minus 119888out)2

0

LBF AC [7] 120578

120582

1119890

1minus 120582

2119890

2

0

expressed by the level set function The energy function ofCV model is

119864 (119862 119888

1 119888

2) = 120583

0Length (119862)

+ 120582

1int

inside(119862)

1003816

1003816

1003816

1003816

119906

0(119909 119910) minus 119888

1

1003816

1003816

1003816

1003816

2

119889119909 119889119910

+ 120582

2int

outside(119862)

1003816

1003816

1003816

1003816

119906

0(119909 119910) minus 119888

2

1003816

1003816

1003816

1003816

2

119889119909 119889119910

(2)

where 1199060is the given image 119862 is the evolution curve 119862(119904)

[0 1] rarr 119877 is a parameter evolution curve and 120583

0is the

weight coefficient In the energy function the first item is thelength of the curve evolution and it can regularize the curveThe last two items are global binary fitting items The basicidea of themodel is tominimize the fitting item119865

1+119865

2 where

119865

1= int

inside(119862)

1003816

1003816

1003816

1003816

119906

0minus 119888

1

1003816

1003816

1003816

1003816

2

119889119909 119889119910

119865

2= int

outside(119862)

1003816

1003816

1003816

1003816

119906

0minus 119888

2

1003816

1003816

1003816

1003816

2

119889119909 119889119910

(3)

The level set function 120601 is defined as

120601 (119909 119910) gt 0 (119909 119910) isin in (119862)

120601 (119909 119910) lt 0 (119909 119910) isin out (119862)

120601 (119909 119910) = 0 (119909 119910) isin (119862)

(4)

The following are Dirac function and Heaviside functionrespectively

120575 (120601) =

119889119867 (120601)

119889120601

119867 (120601) =

1 120601 ge 0

0 120601 lt 0

(5)

Because function119867(120601) cannot directly take the derivativeof 120601 we can replace119867 with119867

120576(120601) in the CV model where

119867

120576(120601) =

1

2

(1 +

2

120587

arctan(120601

120576

)) (6)

Advances in Mathematical Physics 3

The level set function of the CV model is

119864 (120601 119888

1 119888

2) = 120583int

Ω

120575

120576(120601)

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

119889119909 119889119910

+ 120582

1int

Ω

1003816

1003816

1003816

1003816

119906

0(119909 119910) minus 119888

1

1003816

1003816

1003816

1003816

2

119867

120576(120601) 119889119909 119889119910

+ 120582

2int

Ω

1003816

1003816

1003816

1003816

119906

0(119909 119910) minus 119888

2

1003816

1003816

1003816

1003816

2

(1 minus 119867

120576(120601)) 119889119909 119889119910

(7)

where

120575

120576(120601) = 119867

1015840

120576(120601) =

1

120587

120576

120576

2+ 120587

2 (8)

From the definition of the level set function 120601 expressions of119888

1 1198882are respectively as follows

119888

1=

int

Ω

119906

0(119909 119910)119867 (120601 (119909 119910)) 119889119909 119889119910

int

Ω

119867(120601 (119909 119910)) 119889119909 119889119910

119888

2=

int

Ω

119906

0(119909 119910) (1 minus 119867 (120601 (119909 119910))) 119889119909 119889119910

int

Ω

(1 minus 119867 (120601 (119909 119910))) 119889119909 119889119910

(9)

According to Euler-Lagrange equation the level set expres-sion of CV model is obtained as120597120601

120597119905

= 120575 (120601) [120583 div(nabla120601

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

) minus 120582

1(119906

0minus 119888

1)

2

+ 120582

2(119906

0minus 119888

2)

2

]

(10)

3 LBF Model

LBFmodel defines a local binary fitting energy item which isactually a kernel function the model is as follows

119864 (119862 119891

1 119891

2)

= 120582

1int

Ω

int

inside(119862)119870

120590(119909 minus 119910)

1003816

1003816

1003816

1003816

119868 (119910) minus 119891

1(119909)

1003816

1003816

1003816

1003816

2

119889119910119889119909

+ 120582

2int

Ω

int

outside(119862)119870

120590(119909 minus 119910)

1003816

1003816

1003816

1003816

119868 (119910) minus 119891

2(119909)

1003816

1003816

1003816

1003816

2

119889119910119889119909

(11)

where 119868 Ω rarr 119877 is the original image (119909 119910) isin Ω 119870120590=

(12120587120590

2

)119890

minus((119909minus120583119909)2+(119910minus120583

119910)21205902) and is a Gaussian kernel func-

tion 120583119909and 120583119910are its expectancies and 120590 is its standard devi-

ation1198911and1198912are the image fitting function of the local gray

level inside and outside of the contourThe variational level set function of (11) which is got by

Euler-Lagrange equation is as follows120597120601

120597119905

= minus120575 (120601) (120582

1119890

1minus 120582

2119890

2)

(12)

By introducing a sign distance constraint and lengthconstraint item the level set evolution equation is

120597120601

120597119905

= 120583(Δ120601 minus div(nabla120601

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

)) + V120575 (120601) div(nabla120601

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

)

minus 120575 (120601) (120582

1119890

1minus 120582

2119890

2)

(13)

where functions 1198901 1198902and 119891

1 1198912are as follows respectively

119890

1(119909) = int

Ω

119870

120590(119910 minus 119909)

1003816

1003816

1003816

1003816

119868 (119909) minus 119891

1(119910)

1003816

1003816

1003816

1003816

2

119889119910

119890

2(119909) = int

Ω

119870

120590(119910 minus 119909)

1003816

1003816

1003816

1003816

119868 (119909) minus 119891

2(119910)

1003816

1003816

1003816

1003816

2

119889119910

(14)

119891

1=

119870

120590lowast [119867 (120601) 119868 (119909)]

119870

120590lowast 119867 (120601)

119891

2=

119870

120590lowast [(1 minus 119867 (120601)) 119868 (119909)]

119870

120590lowast ((1 minus 119867 (120601)))

(15)

Equation (15) shows that 1198911and 1198912are the weighted aver-

age gray values with the Gaussian window inside and outsideof contour Obviously they share the local characteristics sothat the segmentation of original image by LBFmodel ismoreaccurate

4 Proposed Region-Based Model withGaussian Kernel of Fractional Order

In order to get better image segmentation results effectivelyand construct a fast region-based segmentation model weshould keep the energy functional as simple as possible andenergy informationmust be used effectively Based on the lawof some classical energy function expressions of active con-tour models summarized in Table 1 we know that the term119881

119873in energy function is very important and many models

made a breakthrough on it At the same time the item

119873 ofthe models is often set as

0 and item 120572

119896is only ordinary

parameters Therefore only keeping item 119881

119873in the new

model will simplify the expression of energy functionIn order to avoid jumping internally level set function

initialized by symbolic distance function (SDF) in the tradi-tional level set method but it usually needs to be reinitializedThis will lead to the fact that it is hard to decide when toreinitialize and how to reinitialize as it is hard to find bound-ary when the zero level set is away from the inner regionSo reinitialization is a very complex operation problem Tosolve this problem we propose a new level set method At thesame time fractional systems [10 11] gain increasing attentionin applied sciences and functions of fractional order aremore flexible so the new method uses a Gaussian filter withfractional order to regularize binary level set functionThe traditional level set method uses curvature itemdiv(nabla120601|nabla120601|)|nabla120601| to regularize the level set function and let-ting |nabla120601| = 1 [12] it can replace the regular items with Lapla-cian Based on scale space theory in [13] a function with theLaplacian evolution is equivalent to using a Gaussian filterThen we filter the initial conditions of the level set functionwith Gaussian kernel filters of the level set function and 120590

controls the regular strength similar to the item 120578 in Table 1With Gaussian kernel function the item div(nabla120601|nabla120601|)|nabla120601|(similar to the item 120572

119896in Table 1) can be removed so the key

of whole model is the choice of item 119881

119873

From the view of level set function we need to find afunction that can adjust the pressure inside and outside of theinterest areas It drives the curve to contract when the curve


(a) (b) (c)

(d) (e) (f)

Figure 1 The segmentation results of the aircraft composite image (a) and (d) results using CV model (b) and (e) results using LBF modeland (c) and (f) results using the new model

is outside the target and expands when the curve is within thetarget Based on SPF function with value of [minus1 1] defined in[14] we can construct a function as follow

119891 (119868 (119909)) =

(119888

1+ 119888

2+ 119891

1+ 119891

2) 4 minus 119868 (119909)

max (100381610038161003816

1003816

(119888

1+ 119888

2+ 119891

1+ 119891

2) 4 minus 119868 (119909)

1003816

1003816

1003816

1003816

)

119909 isin Ω

(16)

where

Min (119868 (119909)) le 119888

1 119888

2 119891

1 119891

2le Max (119868 (119909))

Min (119868 (119909)) le(119888

1+ 119888

2+ 119891

1+ 119891

2)

4

le Max (119868 (119909)) (17)

119891

1=

119870

120590lowast [119867 (120601) 119868 (119909)]

119870

120590lowast 119867 (120601)

119891

2=

119870

120590lowast [(1 minus 119867 (120601)) 119868 (119909)]

119870

120590lowast (1 minus 119867 (120601))

(18)

119870

120590(120572) =

1

2120587120590

2119890

minus(119909minus1205831205721205902)

(19)

where 119870120590(120572) is the Gaussian kernel of fractional order 120572 It

takes the ordinary Gaussian kernel as its special case when120572 = 2Thus it ismore flexible than the ordinary oneWemustemphasize that the parameter 120572may be different from fractalparameters [15 16] In this paper we call120572 likely informal thefractional order of the Gaussian kernel expressed by (19) Inthe following experiments we try different values of the frac-tal order 120572 in the evolution level set functionThe value of thefunction is between minus1 and 1 It drives curve to contract exter-nally the target and expands when the curve is within the tar-get According to the summary of the classic model of general

expression (1) we only keep item 119881

119873 so we obtain the corre-

sponding variational level set formulation as follows

120597120601

120597119905

= 119891 (119868 (119909))

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

119909 isin Ω

(20)

By adding a parameter the final level set equation of the newmodel is

120597120601

120597119905

= 120578119891 (119868 (119909))

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

119909 isin Ω

(21)

The main algorithm of the new model is as followsStep 1 initialize the level set function 120601Step 2 compute themean and the weighted average values of

inside and outside of the curve 1198881 1198882 1198911 and 119891

2

Step 3 compute the evolution of level set function by (21)Step 4 use Gaussian filter to regularize level set function 120601 =

120601 lowast 119866

120590

Step 5 repeat Steps 2 to 4 until convergence

5 Experimental Results

In this section we will show some experimental results of theproposed model on synthetic image and nature image theresults also will be compared with those got by the conven-tional CV model and LBF model Our algorithm is imple-mented in Windows 7 Operating System i3 Dual Core CPU213GHz and 2GB RAM The initial value and parameterssuch as time step take different values in the specific exper-iments in this paper

Figures 1(d)ndash1(f) got with 2000 1500 40 iterations sepa-rately and consume time 119905 = 2176 s 2326 s 138 s in turnWecan see that Chan-Vese model and LBF model cannot get


(a) (b) (c)

(d) (e) (f)

Figure 2The segmentation results of the natural image (a) and (d) results using CVmodel (b) and (e) results using LBFmodel and (c) and(f) results using the new model

(a) (b) (c)

(d) (e) (f)


satisfactory result While here the proposed model gets satis-factory result meanwhile the proposedmodel costs less time

Figure 2 shows the experimental results of natural figurewith objects having inhomogeneous background CVmodelLBF model and the proposed model share the same envi-ronment of the initial value We can find that the gray levelof the natural star figure is extremely inhomogeneous easily

The first line shows the segmentation result of CVmodelThesecond line shows the segmentation result of LBFmodelThethird line shows the segmentation result of new model Thesegmentation images reveal that the proposed model gets themost ideal segmentation result

Figure 3 shows the experimental results of natural figurewith CV model LBF model and the proposed model gray


(a) (b) (c)

(d) (e) (f)

Figure 4 The segmentation results of the blood vessels image (a) and (d) results using CV model (b) and (e) results using LBF model and(c) and (f) results using the new model

(a) (b) (c)

(d) (e) (f)

Figure 5The segmentation results of the CT image (a) and (d) results using CVmodel (b) and (e) results using LBF model and (c) and (f)results using the new model

level of the image is extremely inhomogeneous The threemodels share the same initial value Figures 3(a) and 3(d)show the segmentation result of CV model Figures 3(b) and3(e) show the segmentation result of LBF model Figures 3(c)and 3(f) show the segmentation result of the new model Thesegmentation results show that the proposed model gets thesatisfactory complete result while the CV model and LBFmodel have some redundant and inaccurate segmentation

In Figure 4 the experiment shows the segmentationresults of a gray matter blood vessels image The first columnshows the segmentation results of Chan-Vese model Thesecond column shows the results of LBFmodelThe third col-umn shows the result of proposed model Figures 4(d)ndash4(f)got with 1000 5000 40 iterations separately and consumetime 119905 = 792 s 1742 s61 s in turnBecause themean informa-tion of Chan-Vese model is very sensitive to inhomogeneity


image it fails to extract the accurate contour The LBF modelis better than Chan-Vese model but it gets many itera-tions and it is very sensitive to initial value As shown in thelow left of the segmental contour the proposed model got abetter segmentation result

In Figure 5 experimental results show the segmentationresults of CT bone images The initialization is a single circleThe first column shows the segmentation result of Chan-Vese model The second column shows the result of the LBFmodel The third row shows the result of proposed modelFigures 5(d)ndash5(f) are the corresponding contours to Fig-ures 5(a)ndash5(c) Figures 5(d)ndash5(f) got with 50 1000 80iterations separately and consume time 119905 = 40 s 346 s 125 sin turn As seen from Figure 5 the newmethod can get morecomplete contour than Chan-Vese model and the LBF modeland take less time

6 Conclusion

Inspired by the idea of some classical energy function expres-sions of active contourmodel from the view of level set repre-sentation a novel fast region-based segmentationmodel withGaussian kernel of fractional order is proposedThemodel issimple and easy to be implementated and it can protect weakedges because of considering more statistical informationThe experimental results on synthetic images and naturalimages show that the proposed model is superior to thetraditional methods The new model is much more effectivein dealing with the images with weak or blurred edges and ittakes less time

Acknowledgments

This paper is partially supported by NSFC (61272252)and Science and Technology Planning Project ofShenzhen City (JC201105130461A JCYJ20120613102415154ZYC201105130115A and JCYJ20130326111024546) Theauthors also would like to thank the Key Laboratory of Medi-cal Image Processing in Southern Medical University forproviding original medical images for experiment

References

[1] DMumford and J Shah ldquoOptimal approximations by piecewisesmooth functions and associated variational problemsrdquo Com-munications on Pure and AppliedMathematics vol 42 no 5 pp577ndash685 1989

[2] MKass AWitkin andD Terzopoulos ldquoSnakes active contourmodelsrdquo International Journal of Computer Vision vol 1 no 4pp 321ndash331 1988

[3] V Caselles F Catte T Coll and F Dibos ldquoA geometric modelfor active contours in image processingrdquo Numerische Mathe-matik vol 66 no 1 pp 1ndash31 1993

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

[5] N Paragios O Mellina-Gottardo and V Ramesh ldquoGradientvector flow fast geometric active contoursrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 26 no 3 pp402ndash407 2004

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

[7] C Li C-Y Kao J C Gore and Z Ding ldquoImplicit active con-tours driven by local binary fitting energyrdquo in Proceedings of theIEEE Computer Society Conference on Computer Vision and Pat-tern Recognition (CVPR rsquo07) Minneapolis Minn USA June2007

[8] P T H Truc T-S Kim S Lee and Y-K Lee ldquoHomogeneity-and density distance-driven active contours for medical imagesegmentationrdquoComputers in Biology andMedicine vol 41 no 5pp 292ndash301 2011

[9] A Vasilevskiy and K Siddiqi ldquoFlux maximizing geometricflowsrdquo IEEETransactions on PatternAnalysis andMachine Intel-ligence vol 24 no 12 pp 1565ndash1578 2002

[10] M Li ldquoApproximating ideal filters by systems of fractionalorderrdquo Computational and Mathematical Methods in Medicinevol 2012 Article ID 365054 6 pages 2012

[11] M Li S C Lim and S Chen ldquoExact solution of impulseresponse to a class of fractional oscillators and its stabilityrdquoMathematical Problems in Engineering vol 2011 Article ID657839 9 pages 2011

[12] S Osher and R Fedkiw Level SetMethods andDynamic ImplicitSurfaces vol 153 of Applied Mathematical Sciences SpringerNew York NY USA 2003

[13] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990

[14] K Z Zhang L ZhangHH Song andWG Zhou ldquoActive con-tours with selective local or global segmentation a new formu-lation and level set methedrdquo Image and Vision Computing vol28 no 4 pp 668ndash676 2010

[15] M Li and W Zhao ldquoQuantitatively investigating locally weakstationarity of modified multifractional Gaussian noiserdquo Phys-ica A vol 391 no 24 pp 6268ndash6278 2012

[16] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 23 pages 2012

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


is introduced by Caselles et al [3] its main idea is to use cur-vature and normal direction forcing curve movement so thatit stops on the edge with the edge function 119892(119909) = 119892(|nabla119868|

2

)where nabla119868 means the gradient of the given image 119868 whichhas a property that it equals zero on the border and othersequal one For example 119892(119909) = 119890

minus(11205902

119890)

|nabla119866

120590lowast 119868(119909)|

2 where120590

119890is a scale factor 119866

120590= (1

radic

2120587120590)119890

minus((119909minus120583)21205902) is a Gaussian

kernel in which 120590 denotes the standard deviation of thegiven image and 120583 denotes the expectation of the givenimage Another type of geometric active contourmodel is theGeodesic active contour model [4] which can search for theminimum length of the edge weights under the energy func-tionThismodel is similar to the former geometricmodel butthere is a big difference a vector filed term is employed inGeodesic active contour model to stop the motion curveon the weak edges Paragios et al put forward the famousgradient vector flow (GVF) instead of nabla119892 to increase therange of results called GVF geometric AC [5] Chan andVese[6] proposed a new model CV AC and Li et al [7] proposeda new model (LBF) which uses energy function to overcomethe problemof nonhomogeneity LBFmodel can dealwith theimage of different gray levels by adding the kernel functionand it can employ local gray level information effectively

Many structures of different level set evolution modelshave been summarized before The level set evolution of theabove energy functions [8] can be represented as

120597120601

120597119905

= 120572

119896119870

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

+ 119881

119873

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

+

119873 sdot nabla120601

(1)

where 119870 is the Euclidean curvature and 120572

119896 119881119873 and

119873 arethree parameters which decide the speed and direction ofthe evolution The term based on curvature vector is usedto smooth the curve The normal direction is used to controlshrinkage and expansion of the curve and force the curve tomove along the direction vector Details are shown in Table 1where 120578 120583 120582

1 and 120582

2are constant [ V] is the GVF and 119896(119909)

is a function based on the normal curvature and GVF [9] Atthe same time 119868(119909) in Table 1 is original gray level image 119862inand 119862out are average gray values of 119868(119909) inside and outside ofcurve and 119890

1and 119890

2are the weighted average gray values of

119868(119909) inside and outside of curve in the Gaussian windowThe rest of the paper is organized as follows in the

next two sections we will review classical existing geometricmodels Chan-Vese model and LBF model The new modelis introduced in Section 4 Some experimental results areshown in Section 5 We conclude the paper in Section 6

2 Chan-Vese Model

In Chan-Vese (CV) model we considered the simplest typeof segmentation which divided the image into the target andthe background and the distributions of the gray values oftarget and background are approximately constant values CVmodel is based on the evolution of the level set and can dealwith curve topology changes better for the curve which is

Table 1 The analysis table of energy function model

Model 120572

119896 119881

119873

119873

Geometric AC [3] 119892(119909)119903119892(119909)

0

Geodesic AC [4] 119892(119909)

119903119892(119909)

nabla119892(119909)

GVF Geo AC [5] 119892(119909)

119903119896(119909)119892(119909)

119892(119909) (1 minus |119896 (119909)| []])

CV AC [6] 120578

120583 + (119868 minus 119888in)2

minus(119868 minus 119888out)2

0

LBF AC [7] 120578

120582

1119890

1minus 120582

2119890

2

0

expressed by the level set function The energy function ofCV model is

119864 (119862 119888

1 119888

2) = 120583

0Length (119862)

+ 120582

1int

inside(119862)

1003816

1003816

1003816

1003816

119906

0(119909 119910) minus 119888

1

1003816

1003816

1003816

1003816

2

119889119909 119889119910

+ 120582

2int

outside(119862)

1003816

1003816

1003816

1003816

119906

0(119909 119910) minus 119888

2

1003816

1003816

1003816

1003816

2

119889119909 119889119910

(2)

where 1199060is the given image 119862 is the evolution curve 119862(119904)

[0 1] rarr 119877 is a parameter evolution curve and 120583

0is the

weight coefficient In the energy function the first item is thelength of the curve evolution and it can regularize the curveThe last two items are global binary fitting items The basicidea of themodel is tominimize the fitting item119865

1+119865

2 where

119865

1= int

inside(119862)

1003816

1003816

1003816

1003816

119906

0minus 119888

1

1003816

1003816

1003816

1003816

2

119889119909 119889119910

119865

2= int

outside(119862)

1003816

1003816

1003816

1003816

119906

0minus 119888

2

1003816

1003816

1003816

1003816

2

119889119909 119889119910

(3)

The level set function 120601 is defined as

120601 (119909 119910) gt 0 (119909 119910) isin in (119862)

120601 (119909 119910) lt 0 (119909 119910) isin out (119862)

120601 (119909 119910) = 0 (119909 119910) isin (119862)

(4)

The following are Dirac function and Heaviside functionrespectively

120575 (120601) =

119889119867 (120601)

119889120601

119867 (120601) =

1 120601 ge 0

0 120601 lt 0

(5)

Because function119867(120601) cannot directly take the derivativeof 120601 we can replace119867 with119867

120576(120601) in the CV model where

119867

120576(120601) =

1

2

(1 +

2

120587

arctan(120601

120576

)) (6)



119864 (120601 119888

1 119888

2) = 120583int

Ω

120575

120576(120601)

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

119889119909 119889119910

+ 120582

1int

Ω

1003816

1003816

1003816

1003816

119906

0(119909 119910) minus 119888

1

1003816

1003816

1003816

1003816

2

119867

120576(120601) 119889119909 119889119910

+ 120582

2int

Ω

1003816

1003816

1003816

1003816

119906

0(119909 119910) minus 119888

2

1003816

1003816

1003816

1003816

2

(1 minus 119867

120576(120601)) 119889119909 119889119910

(7)

where

120575

120576(120601) = 119867

1015840

120576(120601) =

1

120587

120576

120576

2+ 120587

2 (8)



119888

1=

int

Ω

119906

0(119909 119910)119867 (120601 (119909 119910)) 119889119909 119889119910

int

Ω

119867(120601 (119909 119910)) 119889119909 119889119910

119888

2=

int

Ω

119906

0(119909 119910) (1 minus 119867 (120601 (119909 119910))) 119889119909 119889119910

int

Ω

(1 minus 119867 (120601 (119909 119910))) 119889119909 119889119910

(9)


120597119905

= 120575 (120601) [120583 div(nabla120601

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

) minus 120582

1(119906

0minus 119888

1)

2

+ 120582

2(119906

0minus 119888

2)

2

]

(10)

3 LBF Model


119864 (119862 119891

1 119891

2)

= 120582

1int

Ω

int

inside(119862)119870

120590(119909 minus 119910)

1003816

1003816

1003816

1003816

119868 (119910) minus 119891

1(119909)

1003816

1003816

1003816

1003816

2

119889119910119889119909

+ 120582

2int

Ω

int

outside(119862)119870

120590(119909 minus 119910)

1003816

1003816

1003816

1003816

119868 (119910) minus 119891

2(119909)

1003816

1003816

1003816

1003816

2

119889119910119889119909

(11)


(12120587120590

2

)119890







120597119905

= minus120575 (120601) (120582

1119890

1minus 120582

2119890

2)

(12)


120597120601

120597119905

= 120583(Δ120601 minus div(nabla120601

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

)) + V120575 (120601) div(nabla120601

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

)

minus 120575 (120601) (120582

1119890

1minus 120582

2119890

2)

(13)



119890

1(119909) = int

Ω

119870

120590(119910 minus 119909)

1003816

1003816

1003816

1003816

119868 (119909) minus 119891

1(119910)

1003816

1003816

1003816

1003816

2

119889119910

119890

2(119909) = int

Ω

119870

120590(119910 minus 119909)

1003816

1003816

1003816

1003816

119868 (119909) minus 119891

2(119910)

1003816

1003816

1003816

1003816

2

119889119910

(14)

119891

1=

119870

120590lowast [119867 (120601) 119868 (119909)]

119870

120590lowast 119867 (120601)

119891

2=

119870

120590lowast [(1 minus 119867 (120601)) 119868 (119909)]

119870

120590lowast ((1 minus 119867 (120601)))

(15)








0 and item 120572



119873in the new






119873



(a) (b) (c)

(d) (e) (f)



119891 (119868 (119909)) =

(119888

1+ 119888

2+ 119891

1+ 119891

2) 4 minus 119868 (119909)

max (100381610038161003816

1003816

(119888

1+ 119888

2+ 119891

1+ 119891

2) 4 minus 119868 (119909)

1003816

1003816

1003816

1003816

)

119909 isin Ω

(16)

where

Min (119868 (119909)) le 119888

1 119888

2 119891

1 119891

2le Max (119868 (119909))

Min (119868 (119909)) le(119888

1+ 119888

2+ 119891

1+ 119891

2)

4

le Max (119868 (119909)) (17)

119891

1=

119870

120590lowast [119867 (120601) 119868 (119909)]

119870

120590lowast 119867 (120601)

119891

2=

119870

120590lowast [(1 minus 119867 (120601)) 119868 (119909)]

119870

120590lowast (1 minus 119867 (120601))

(18)

119870

120590(120572) =

1

2120587120590

2119890

minus(119909minus1205831205721205902)

(19)






120597120601

120597119905

= 119891 (119868 (119909))

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

119909 isin Ω

(20)


120597120601

120597119905

= 120578119891 (119868 (119909))

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

119909 isin Ω

(21)



2


120601 lowast 119866

120590






(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)







(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)







6 Conclusion


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119864 (120601 119888

1 119888

2) = 120583int

Ω

120575

120576(120601)

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

119889119909 119889119910

+ 120582

1int

Ω

1003816

1003816

1003816

1003816

119906

0(119909 119910) minus 119888

1

1003816

1003816

1003816

1003816

2

119867

120576(120601) 119889119909 119889119910

+ 120582

2int

Ω

1003816

1003816

1003816

1003816

119906

0(119909 119910) minus 119888

2

1003816

1003816

1003816

1003816

2

(1 minus 119867

120576(120601)) 119889119909 119889119910

(7)

where

120575

120576(120601) = 119867

1015840

120576(120601) =

1

120587

120576

120576

2+ 120587

2 (8)



119888

1=

int

Ω

119906

0(119909 119910)119867 (120601 (119909 119910)) 119889119909 119889119910

int

Ω

119867(120601 (119909 119910)) 119889119909 119889119910

119888

2=

int

Ω

119906

0(119909 119910) (1 minus 119867 (120601 (119909 119910))) 119889119909 119889119910

int

Ω

(1 minus 119867 (120601 (119909 119910))) 119889119909 119889119910

(9)


120597119905

= 120575 (120601) [120583 div(nabla120601

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

) minus 120582

1(119906

0minus 119888

1)

2

+ 120582

2(119906

0minus 119888

2)

2

]

(10)

3 LBF Model


119864 (119862 119891

1 119891

2)

= 120582

1int

Ω

int

inside(119862)119870

120590(119909 minus 119910)

1003816

1003816

1003816

1003816

119868 (119910) minus 119891

1(119909)

1003816

1003816

1003816

1003816

2

119889119910119889119909

+ 120582

2int

Ω

int

outside(119862)119870

120590(119909 minus 119910)

1003816

1003816

1003816

1003816

119868 (119910) minus 119891

2(119909)

1003816

1003816

1003816

1003816

2

119889119910119889119909

(11)


(12120587120590

2

)119890







120597119905

= minus120575 (120601) (120582

1119890

1minus 120582

2119890

2)

(12)


120597120601

120597119905

= 120583(Δ120601 minus div(nabla120601

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

)) + V120575 (120601) div(nabla120601

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

)

minus 120575 (120601) (120582

1119890

1minus 120582

2119890

2)

(13)



119890

1(119909) = int

Ω

119870

120590(119910 minus 119909)

1003816

1003816

1003816

1003816

119868 (119909) minus 119891

1(119910)

1003816

1003816

1003816

1003816

2

119889119910

119890

2(119909) = int

Ω

119870

120590(119910 minus 119909)

1003816

1003816

1003816

1003816

119868 (119909) minus 119891

2(119910)

1003816

1003816

1003816

1003816

2

119889119910

(14)

119891

1=

119870

120590lowast [119867 (120601) 119868 (119909)]

119870

120590lowast 119867 (120601)

119891

2=

119870

120590lowast [(1 minus 119867 (120601)) 119868 (119909)]

119870

120590lowast ((1 minus 119867 (120601)))

(15)








0 and item 120572



119873in the new






119873



(a) (b) (c)

(d) (e) (f)



119891 (119868 (119909)) =

(119888

1+ 119888

2+ 119891

1+ 119891

2) 4 minus 119868 (119909)

max (100381610038161003816

1003816

(119888

1+ 119888

2+ 119891

1+ 119891

2) 4 minus 119868 (119909)

1003816

1003816

1003816

1003816

)

119909 isin Ω

(16)

where

Min (119868 (119909)) le 119888

1 119888

2 119891

1 119891

2le Max (119868 (119909))

Min (119868 (119909)) le(119888

1+ 119888

2+ 119891

1+ 119891

2)

4

le Max (119868 (119909)) (17)

119891

1=

119870

120590lowast [119867 (120601) 119868 (119909)]

119870

120590lowast 119867 (120601)

119891

2=

119870

120590lowast [(1 minus 119867 (120601)) 119868 (119909)]

119870

120590lowast (1 minus 119867 (120601))

(18)

119870

120590(120572) =

1

2120587120590

2119890

minus(119909minus1205831205721205902)

(19)






120597120601

120597119905

= 119891 (119868 (119909))

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

119909 isin Ω

(20)


120597120601

120597119905

= 120578119891 (119868 (119909))

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

119909 isin Ω

(21)



2


120601 lowast 119866

120590






(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)







(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)







6 Conclusion


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)

(d) (e) (f)



119891 (119868 (119909)) =

(119888

1+ 119888

2+ 119891

1+ 119891

2) 4 minus 119868 (119909)

max (100381610038161003816

1003816

(119888

1+ 119888

2+ 119891

1+ 119891

2) 4 minus 119868 (119909)

1003816

1003816

1003816

1003816

)

119909 isin Ω

(16)

where

Min (119868 (119909)) le 119888

1 119888

2 119891

1 119891

2le Max (119868 (119909))

Min (119868 (119909)) le(119888

1+ 119888

2+ 119891

1+ 119891

2)

4

le Max (119868 (119909)) (17)

119891

1=

119870

120590lowast [119867 (120601) 119868 (119909)]

119870

120590lowast 119867 (120601)

119891

2=

119870

120590lowast [(1 minus 119867 (120601)) 119868 (119909)]

119870

120590lowast (1 minus 119867 (120601))

(18)

119870

120590(120572) =

1

2120587120590

2119890

minus(119909minus1205831205721205902)

(19)






120597120601

120597119905

= 119891 (119868 (119909))

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

119909 isin Ω

(20)


120597120601

120597119905

= 120578119891 (119868 (119909))

1003816

1003816

1003816

1003816

nabla120601

1003816

1003816

1003816

1003816

119909 isin Ω

(21)



2


120601 lowast 119866

120590






(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)







(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)







6 Conclusion


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)







(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)







6 Conclusion


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)

(d) (e) (f)


(a) (b) (c)

(d) (e) (f)







6 Conclusion


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












6 Conclusion


Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a fast region-based segmentation model ...[0,1] is a parameter evolution curve, and...

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