New local segmentation model forimages with intensity inhomogeneity

Qiang ZhengEn-Qing Dong

New local segmentation model for images with intensityinhomogeneity

Qiang ZhengEn-Qing Dong,Shandong University at WeihaiSchool of Mechanical, Electrical & Information

EngineeringWeihai 264209, ChinaE-mail: enqdong@sdu.edu.cn

Abstract. A new local segmentation model based on binary level set forimages with intensity inhomogeneity is proposed. The local segmentationproperty is guaranteed with the gradient function of the binary level setfunction (LSF), and the curve evolution precision can be accurate toone-pixel width. Due to utilizing more statistical information which containslocal interior, local exterior, and global interior information, the new seg-mentation energy is more adaptable for local segmentation in the case ofintensity inhomogeneity. Since the contour is initialized as a small shapeinside the desired object and inflated afterwards in local segmentation, amorphological closing operation is used to regularize the binary LSF,which not only can promote curve inflating, but also can maintain the bin-ary property of the LSF. Experiments on medical images show that theproposed model is more effective and robust than the Chan-Vese (CV),local binary fitting (LBF), localizing region-based active contours (LR-AC), and selective binary and Gaussian filtering regularized level setmethod (SB-GFRLS) models in local segmentation for images with inten-sity inhomogeneity. © 2012 Society of Photo-Optical Instrumentation Engineers (SPIE).[DOI: 10.1117/1.OE.51.3.037006]

Subject terms: image segmentation; intensity inhomogeneity; level set.

Paper 110929 received Aug. 3, 2011; revised manuscript received Nov. 8, 2011;accepted for publication Jan. 17, 2012; published online Apr. 4, 2012.

1 IntroductionIn recent years, the level set method proposed by Osherand Sethian1 has been regions of interest. Existing levelset methods can be categorized into two classes: edge-based models2–5 and region-based models.6–14

Edge-based models use image gradients to identify objectboundaries, but have been proven to be sensitive to imagenoise and dependent on initial curve location. Region-basedmodels use global statistical information to attract the contourtoward the object boundaries. Region-based models havemany advantages over edge based models such as insensitiv-ity to image noise, better segmentation performance forimages with weak or no edges, and robustness against initialcurve location, and have therefore been widely used forimage segmentation. One of the most popular region-basedmodels is the Chan-Vese (CV) model,8 which has successfullybeen applied to binary phase segmentation.

Most existing region-based models have an underlyingnotion of homogeneity. In order to tackle intensity inhomo-geneity, piecewise smooth (PS)9 and local binary fitting(LBF) models are proposed. In these models, LSF is drivenby local information. Local information can also be added tosegmentation energy by estimating the brightness of thebackground.12 Moreover, variational frameworks for jointsegmentation and bias field estimation13,14 are also proposedto deal with intensity inhomogeneity.

However, the models in Refs. 6–14 can detect all of thecontours, regardless of where the initial contour is located inthe image. They cannot segment the desired object with asuitable initial contour. An effective approach to realize

local segmentation is to reformulate15–19 region-based seg-mentation energy in a narrow band, but these models eitherhave unstable narrow bands15,16 because of the calculation ofthe signed distance function (SDF) and reinitialization, orcannot segment images with intensity inhomogeneity.17–19

In addition, the thickness of the narrow band is an importantparameter in local segmentation. A large narrow band maylead to unsuccessful local segmentation when images haveadjacent objects. For example, the selective binary and Gaus-sian filtering regularized level set (SB-GFRLS) method19 notonly brings a large curve evolution precision (CEP) whichwill be illustrated in detail in Sec. 3.2, but also restrainscurve inflating to some degree.

In this paper, we propose an effective local segmentationmodel in the case of intensity inhomogeneity, which makesthree main contributions. First, the proposed model has astable narrow band, and the CEP is one pixel-width. Second,we propose a new segmentation energy for intensity inhomo-geneity, which is much more effective than those used in theCV, LBF, localizing region-based active contours (LR-AC),15 and SB-GFRLS models. Third, since the contouris initialized as a small shape inside a target object andinflated afterwards in local segmentation, morphological clos-ing operation is introduced to regularize the LSF, which cannot only promote curve inflating but also maintain the binaryproperty of the LSF, which can guarantee one-pixel-widthCEP in our model.

The rest of the paper is organized as follows. In Sec. 2, wereview some classical models and indicate their limitations.The proposed local segmentation model for images with inten-sity inhomogeneity is presented in Sec. 3. Section 4 containsthe experimental results and some comparisons with otherapproaches, and the paper is summarized in Sec. 5.0091-3286/2012/$25.00 © 2012 SPIE

Optical Engineering 51(3), 037006 (March 2012)

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2 Background

2.1 The CV Model

Chan and Vese8 proposed the CV model which is a sim-plified variant of the Mumford-Shah model.6 LettingIðxÞ∶ Ω → R be a given image and C be a closed curve,the CV model is defined by minimizing the following energyfunctional:

ECV ¼ λ1

ZinsideðCÞ

½IðxÞ − c1�2dx

þ λ2

ZoutsideðCÞ

½IðxÞ − c2�2dx; (1)

with λ1, λ2 ≥ 0. Here, c1 and c2 are two constants usedto approximate the intensities inside and outside thecurve C:

c1ðϕÞ ¼RΩ IðxÞ · HðϕÞdxR

Ω HðϕÞdx ;

c2ðϕÞ ¼RΩ IðxÞ · ½1 − HðϕÞ�dxR

Ω½1 − HðϕÞ�dx :

(2)

The Heaviside function HðϕÞ is defined as

HðϕÞ ¼ 1

2

�1þ 2

πarctan

�ϕ

ε

��: (3)

However, as shown in Ref. 8, the CV model can detect allobjects no matter where the initial contour is located in theimage and has global segmentation properties. Moreover, theCV model cannot segment images with intensity inhomo-geneity.

2.2 The LBF Model

In order to segment images with intensity inhomogeneity, Liet al. the LBF model10,11 by embedding local image informa-tion. LBF is much more efficient than PS when segmentingimages with intensity inhomogeneity. The basic idea is tointroduce a kernel function to the LBF energy functional.Letting IðxÞ∶ Ω → R be a given image and C be a closedcurve, the energy functional can be written as:

ELBF ¼ λ1

ZΩ

ZinsideðCÞ

Kσðx − yÞ½IðyÞ − f 1ðxÞ�2dydx

þ λ2

ZΩ

ZoutsideðCÞ

Kσðx − yÞ½IðyÞ − f 2ðxÞ�2dydx;

(4)

where λ1,λ2 ≥ 0:Kσ is a Gaussian kernel with standard devia-tion σ,

KσðxÞ ¼1

ð2πÞn∕2σn e−jxj2∕2σ2 ; (5)

and f 1 and f 2 are two smooth functions which approximatelocal intensities inside and outside the curve C:

f 1ðxÞ ¼KσðxÞ � ½HðϕÞ · IðxÞ�

KσðxÞ � HðϕÞ ;

f 2ðxÞ ¼KσðxÞ � f½1 − HðϕÞ� · IðxÞg

KσðxÞ � ½1 − HðϕÞ� :

(6)

The Heaviside function HðϕÞ is again defined by Eq. (3).Like the CV model, the LBF model has global segmenta-

tion properties, but the introduction of local informationmakes the LBF model able to segment images with intensityinhomogeneity.

2.3 The LR-AC Model

It has been mentioned above that the effective way to realizelocal segmentation is to reformulate region-based segmenta-tion energy in a narrow band. Lankton et al. proposed15 theLR-AC model, which is a natural framework that allows anyregion-based segmentation energy to be reformulated in anarrow band. Let IðxÞ denote an input image defined onthe domain Ω, and C be a closed curve represented as thezero level set of the SDF, i.e.,C ¼ fxjϕðxÞ ¼ 0g. The narrowband is defined by the Dirac function δðϕÞ:

δðϕðxÞÞ ¼

8>><>>:

1; ϕðxÞ ¼ 0

0; jϕðxÞj > ε12ε

n1þ cos

�πϕðxÞε

�o; otherwise

: (7)

In order to introduce variable y, we define a characteristicfunction in terms of a radius parameter r as:

Bðx; yÞ ¼�1; kx − yk < r0; otherwise

: (8)

Based on the definitions above, the energy functional witha generic force function F is given as:

ELR−AC ¼ λ · lengthðCÞ þZΩx

δ½ϕðxÞ�

×ZΩy

Bðx; yÞ · F½IðyÞ;ϕðyÞ�dydx: (9)

Due to the narrow band controlled by the Dirac functionδ½ϕðxÞ� and the introduction of local information, the LR-ACmodel can realize local segmentation for images with inten-sity inhomogeneity. However, the LSF needs to be re-initialized as an SDF, and the re-initialization will lead toinstability of the narrow band. In addition, re-initializationis a very expensive operation.

2.4 The SB-GFRLS Model

Zhang et al.19 proposed a novel region-based active contourmodel, which has the property of selective local or globalsegmentation, and improves the traditional level set methodby avoiding the computation of SDF and reinitialization. Let-ting IðxÞ∶ Ω → R be a given image, the SB-GFRLS methodcan be summarized as follows:

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(1) Initialize the LSF ϕðxÞ as

ϕðx; t ¼ 0Þ ¼8<:

−ρ; x ∈ Ω0 − ∂Ω0

0; x ∈ ∂Ω0

ρ; x ∈ Ω − Ω0

; (10)

where ρ > 0 is a constant, Ω0 is a subset of the imagedomain Ω, and ∂Ω0 is the boundary of the subset Ω0.

(2) Calculate c1 and c2 respectively by Eq. (2) above andevolve the LSF according to the formulation

∂ϕ∂t

¼ spfðxÞ · αj∇ϕj

¼ IðxÞ − c1þc22

max����IðxÞ − c1þc2

2

���� · αj∇ϕj: (11)

(3) Let ϕðxÞ ¼ 1 if ϕðxÞ > 0; otherwise, ϕðxÞ ¼ −1.(4) Regularize the LSF with a Gaussian filter, i.e.,

ϕðxÞ ¼ ϕðxÞ � Gσ , where σ is the standard deviation.(5) Return to step (2) if the evolution of the LSF has not

converged; otherwise, stop the evolution.

As demonstrated in Ref. 19, the SB-GFRLS model has asimilar drawback to the CV model, the inefficiency intackling intensity inhomogeneity. In addition, though thegradient function j∇ϕj of the binary LSF can obtain athin narrow band, the procedure of regularizing LSF witha Gaussian filter will lead to a thick narrow band afterwards.Therefore, even though step (3) can guarantee the local seg-mentation property, the capability of the SB-GFRLS modelin local segmentation will be restricted by the Gaussian fil-tering procedure, especially for adjacent objects. Moreover,since the contour will be initialized as a small shape inside atarget object in local segmentation, the Gaussian filter willrestrain curve inflating in some degree when the CEPis small.

3 The Proposed Method

3.1 Segmentation Energy Functional with LocalInterior, Local Exterior, and Global InteriorInformation

Intensity inhomogeneity usually appears in real images,especially for medical images. In general, local informationcorresponding to region 1 and region 2 in Fig. 1 can help usto overcome the problem, and one of the most classic modelsis the LBF model proposed by Li al.10,11 However, for someimages, relying only on local information may be not enoughto handle the intensity inhomogeneity problem such as puta-men segmentation from magnetic resonance (MR) brainimages. Thus, to handle the intensity inhomogeneity weneed to utilize more information than just local, and themodel we propose includes more useful information inlocal segmentation, including local interior, local exteriorand global interior statistical information corresponding toregion 2, region 1 and region 3 in Fig. 1, respectively.

Under the same analysis as applied to the LBF model,when the image is intensity inhomogeneous, local versionsin region 1 and region 2 in Fig. 1 can be viewed as intensityhomogeneous, and local versions of means can be used to

roughly approximate the images. However, when the imageis intensity inhomogeneous, the global interior versions ofmeans bounded by curve C in region 3 cannot be viewedas intensity homogeneous, and the global interior versionsof means cannot be used reasonably to approximate theimages because the intensity is inhomogeneous interiorly.It has been pointed out that the intensity inhomogeneity iscaused by a biased field.13,14 If we can eliminate or weakenthe influence of the biased field on intensity inhomogeneity,then the global interior statistical information correspondingto region 3 in Fig. 1 can be used effectively and reasonably tohelp achieve local segmentation for images with intensityinhomogeneity. Based on the above consideration, in orderto utilize this additional information, global interior informa-tion other than local statistics, we use Gaussian filtering toroughly estimate the approximate biased field of the image,and can now subtract the biased field estimate from the ori-ginal image to weaken the influence of the biased field on theintensity inhomogeneity; global interior statistical informationcorresponding to region 3 inside curve C can be utilized effec-tively and reasonably to help realize local segmentation forimages with intensity inhomogeneity.

For medical images, low image contrast often appearsalong with intensity inhomogeneity, and can also sometimesaffects segmentation. Therefore, in order to increase therobustness of the proposed model in handling intensity inho-mogeneity, it is necessary to enhance the contrast of theseimages. Contrast-limited adaptive histogram equalization(CLAHE) is an efficient tool20 for enhancing the contrast.CLAHE operates on local regions in the image, called tiles,rather than the entire image. Compared with the standard his-togram equalization, the adaptive histogram can change thetiles flexibly to satisfy the requirements of enhancing localregions with different sizes, which may be the desired object.

Based on the above consideration, we define our energyfunctional to segment image with intensity inhomogeneityas:

E ¼ μ · lengthðCÞ þ ½α · E1 þ ð1 − αÞ · E2�; (12)

with

E1 ¼ZinsideðCÞ

½IðxÞ − f 1ðxÞ�2dx

þZoutsideðCÞ

½IðxÞ − f 2ðxÞ�2dx; (13)

Fig. 1 Region 1 (local exterior); region 2 (local interior); region 3(global interior).

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E2 ¼ZinsideðCÞ

½IðxÞ − c3ðxÞ�2dx

þZoutsideðCÞ

½IðxÞ − f 4ðxÞ�2dx; (14)

where f 1, f 2, and f 4 are local versions of the means, and c3is a global interior version of the means. The parameter 0 ≤α ≤ 1 is used as a tradeoff between the normalized energyterms E1 and E2.

The energy E1 in Eq. (12) with original image IðxÞ is easyto understand according to the LBF model, which containslocal interior and local exterior energy. The following high-lights the energy E2 and IðxÞ. In Eq. (12), E2 contains globalinterior and local exterior information, and the motivation ofintroducing E2 is to utilize the additional statistical informa-tion to better achieve local segmentation. According to theabove analysis, the image IðxÞ in the energy E2 is defined as:

IðxÞ ¼ adapthisteq½IðxÞ − IðxÞ � GσðxÞ�: (15)

In Eq. (15), the Gaussian filtering can be used to estimatethe approximated biased field of the given image IðxÞ. There-fore, IðxÞ − IðxÞ � Gσ can be regarded as an approximatedintensity homogeneous image in every target object. Basedon the approximation, the global versions of means can beused reasonably. However, the background of the image maystill consist of many regions. Though every region in thebackground may be approximated intensity homogeneous,the entire background, which contains many regions, is stillregarded as intensity inhomogeneous. Therefore, there is alocal exterior energy term in the second energy term E2. More-over, “adapthisteq” is a function for CLAHE implementationin the image processing toolbox of MATLAB, which is usedto handle the low contrast which often appears along withintensity inhomogeneity.

With the level set representation, the energy functional inEq. (12) can be written as

E ¼ μ ·ZΩδðϕÞj∇ϕjdx

þ υ ·

�α ·

�ZΩHðϕÞ½IðxÞ − f 1ðxÞ�2dx

þZΩ½1 − HðϕÞ�½IðxÞ − f 2ðxÞ�2dx

þ ð1 − αÞ ·�Z

ΩHðϕÞ½IðxÞ − c3ðxÞ�2dx

þZΩ½1 − HðϕÞ�½IðxÞ − f 4ðxÞ�2dxÞ

�; (16)

where the Dirac function δðϕÞ is defined as in Eq. (7) and thefollowing Heaviside function HðϕÞ in Eq. (17) is the integra-tion of δðϕÞ.

HðϕðxÞÞ ¼

8>>><>>>:

1; ϕðxÞ > ε0; ϕðxÞ < −ε12

�1þ ϕðxÞ

ε þ 1π sin

hπϕðxÞε

ijϕðxÞj ≤ ε

:

(17)

Minimizing the energy functional in Eq. (16) with respectto the LSF ϕðxÞ, we obtain the gradient descent flow as

∂ϕ∂t

¼ δðϕÞfμ · divð∇ϕ∕j∇ϕjÞ þ υ · ½α · F1 þ ð1 − αÞ · F2�g

¼ δðϕÞ�μ · divð∇ϕ∕j∇ϕjÞ

þ υ · fα · ½ðI − f 2Þ2 − ðI − f 1Þ2�þ ð1 − αÞ · ½ðI − f 4Þ2 − ðI − c3Þ2�g

�: (18)

The approximation of Eq. (18) by forward differencescheme can be simply written as

ϕnþ1 − ϕn

Δt¼ LðϕnÞ; (19)

where LðϕnÞ is the approximation of the right hand side inEq. (18), Δt is the timestep, and Eq. (19) can then be writtenas the following iteration:

ϕnþ1 ¼ ϕn þ Δt · LðϕnÞ: (20)

3.2 Implementation

In this section, in order to improve efficiency, we will onlyimplement Eq. (18) in the narrow band around the zero levelset: first, build a narrow band; second, extract the points inthe narrow band and update the points with the segmentationenergy in Eq. (18); finally, smooth the curve with a suitablecurve smoothing scheme. Therefore, the gradient descentflow in Eq. (18) can be considered from three aspects: narrowband, segmentation energy, and curve smoothing scheme.

3.2.1 Narrow band

For the purpose of analysis, we first give the definitionof CEP:

Definition CEP is the narrow band width that contributes tothe shrink or inflation of the LSF in every iteration, and itshould be pointed out that the smaller the CEP is, themore helpful it is for local segmentation.

For the traditional level set method, the LSF is initializedto be SDF, and the narrow band is controlled by the DiracfunctionδðϕÞ in Eq. (18). However, the LSF then needs to bereinitialized as SDF, and reinitialization will lead to instabil-ity of the narrow band.

To solve these problems, we propose a binary LSF basedmethod. The binary LSF can only take two values 1 and −1:

ϕðxÞ ¼�

1; x ∈ Ω0

−1; x ∈ Ω \ Ω0; (21)

where Ω0 is a subset in the image domain Ω. We define thegradient function j∇ϕj of the binary LSF, which is realizedby central difference. Obviously, the gradient function j∇ϕjequals zero far from the interface of the binary LSF, and onlypoints adjacent to zero level set, i.e., Lout and Lin in Fig. 2(a),have nonzero values. In other words, the gradient functionj∇ϕj builds a two-pixel-wide stable narrow band. The inter-ior Lin with width of one pixel controls the shrink of the LSF,

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and the exterior Lout with width of one pixel controls theinflation of the LSF. That is to say, the CEP is of one-pixel width.

Based on the above consideration, we replace the Diracfunction δðϕÞ in Eq. (18) with the gradient function j∇ϕj ofbinary LSF to build a more effective narrow band. It isimportant to note that Zhao et al. in has replaced21 theDirac function δðϕÞ with the gradient function j∇ϕj of theLSF, but the LSF in that case is an SDF, and the goal isto enlarge the narrow band to achieve global segmentationaccording to the property j∇ϕj ¼ 1 of SDF. However, inour proposed model, the goal is to decrease the narrowband to achieve local segmentation. The methods are funda-mentally different.

3.2.2 Segmentation energy

In Eq. (18), the segmentation energy can be rewritten as:

F ¼ α · F1 þ ð1 − αÞ · F2

¼ α · f½IðxÞ − f 2ðxÞ�2 − ½IðxÞ − f 1ðxÞ�2gþ ð1 − αÞ · f½IðxÞ − f 4ðxÞ�2 − ½IðxÞ − c3ðxÞ�2g: (22)

Assume that the binary LSF has a value of 1 inside thecurve C, and −1 outside. The interior and exterior ofcurve C can then be defined as

HinðxÞ ¼ ðϕðxÞ þ 1Þ∕2 (23)

HoutðxÞ ¼ ð1 − ϕðxÞÞ∕2: (24)

Thus, f 1, f 2, c3, and f 4 can be defined as

f 1ðxÞ ¼RΩy

Bðx; yÞ · HinðyÞ · IðyÞdyRΩy

Bðx; yÞ · HinðyÞdy; (25)

f 2ðxÞ ¼RΩy

Bðx; yÞ · HoutðyÞ · IðyÞdyRΩy

Bðx; yÞ · HoutðyÞdy; (26)

c3ðxÞ ¼RΩy

HinðyÞ · IðyÞdyRΩy

HinðyÞdy; (27)

f 4ðxÞ ¼RΩy

Bðx; yÞ · HoutðyÞ · IðyÞdyRΩy

Bðx; yÞ · HoutðyÞdy; (28)

where Bðx; yÞ is defined by Eq. (8), and IðyÞ is calculatedby Eq. (15).

3.2.3 Curve smoothing scheme

As pointed out in Ref. 19, the regularized termdivð∇ϕ∕j∇ϕjÞ · δðϕÞ in Eq. (18) can be replaced with aGaussian filter based on the property of the SDF and thescale-space theory.22 However, the Gaussian filter regular-ized scheme can destroy the binary property of the LSF.It is worth noting that the binary property of the LSF isan important premise for guaranteeing a stable narrow bandand one-pixel width CEP in the proposed model; the destruc-tion of that property will lead to large CEP. For example, inFig. 2(b), a Gaussian filter sized 3 × 3 will lead to a CEP twopixels wide, and the CEP will increase as the size of theGaussian filtering becomes larger. Thus we must guaranteethe binary property of the LSF in every iteration. Moreover,the Gaussian filter regularized method will prevent the curvefrom inflating in the evolution to some degree when the CEPis small. In Fig. 3(b), though the Gaussian filter pulls thecorners c and d outward, it also pulls the corners a, b, e,f , and g inward simultaneously. If the CEP is small, theGaussian filter may prevent the curve from easily inflatingin the evolution.

In order to solve the problem caused by the Gaussian fil-ter, we propose to replace the Gaussian filter with a morpho-logical closing operation considering the following twopoints. One: the morphological closing operation is a binaryoperation, which can maintain the binary property of LSFeasily. Thus we can guarantee a stable narrow band andone-pixel width CEP. Two: instead of preventing thecurve from inflating, the morphological closing operationwill promote curve inflating even if the CEP is small. InFig. 3(c), the morphological closing operation only has theforce to pull the corners c and d outward, so it will not pre-vent the curve from inflating even if the CEP is small. On thecontrary, the morphological closing operation will facilitatecurve inflation.

Figs. 4(b) and 4(c) consider the example of white mattersegmentation to demonstrate that Gaussian filtering candestroy the binary property of the level set, and leading toa large CEP, which leads to failed local segmentation.

The restraint of the Gaussian filtering and the facilitationof the morphological closing operation to curve inflation inlocal segmentation are shown in Fig. 5.

Fig. 2 (a) j∇ϕj has two-pixel width nonzero narrow band, and theCEP is one-pixel width, (b) j∇ϕj has four-pixel width nonzero narrowband due to 3 × 3 Gaussian filter, and the CEP is two-pixel width.

Fig. 3 (a) Original curve, (b) Gaussian filtering, and (c) morphologicalclosing operation.

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3.2.4 Implementation

Therefore, based on the above consideration, the proceduresof our proposed model are summarized as:

(1) Initialize the curve by binary LSF as in Eq. (21), andthe initial curve should be a small shape inside thetarget object.

(2) Calculate IðxÞ by Eq. (15).(3) Build the narrow band: calculate the gradient function

j∇ϕj of the binary LSF using central difference.(4) Update the points in the narrow band: for every pixel

x with j∇ϕj ≠ 0, calculate f 1, f 2, c3, and f 4 defined inEqs. (25) through (28), respectively, and then evolvethe LSF with the segmentation energy F defined byEq. (22) according to the following formulation:

ϕnþ1 ¼ ϕn þ Δt · υ · F ¼ ϕn þ β · F: (29)

(5) Let ϕðxÞ ¼ 1 if ϕðxÞ ≥ 0; otherwise, ϕðxÞ ¼ −1.(6) Regularize the LSF with a morphological closing

operation.(7) Return to step (3) if the evolution of the LSF has not

converged; otherwise, stop the evolution.

3.2.5 Computational complexity

It should be pointed out that in this section we only comparethe proposed model with two other local segmentation mod-els, the LR-AC model and SB-GFRLS method. The LR-AC

model is an SDF-based model, and costly reinitialization isneeded. Obviously, due to avoiding the reinitialization ofSDF, both the SB-GFRLS method and our proposed modelare less computationally complex than LR-AC model.

In addition, we should compare the proposed modeldirectly with the SB-GFRLS method. If both the F inEq. (29) and the spf ðxÞ in Eq. (11) are not considered,then the only difference between the proposed model andthe SB-GFRLS method is in the curve smoothing scheme(morphological closing operation and Gaussian filtering,respectively). If the sizes of the two curve smoothingschemes are equivalent, then both models have the samecomputational complexity Oðn × NÞ), where n is the sizeof the curve smoothing scheme and N is the image size.However, in the proposed model, local statistical informationfor each of the points in the narrow band must be computedfor each iteration. This increases the complexity of the pro-posed model, and the computation time is beyond that of theSB-GFRLS method, which is a global method. The globalstatistical information should be computed only once for allpoints in each iteration. Therefore, based on this discussion,the proposed model is less efficient than the SB-GFRLSmethod.

Additionally, limited by the CFL (Courant-FriedreicLewy) condition, the CEP of the LR-AC model is much nar-rower than one pixel. Influenced by the Gaussian filtering,the CEP of the SB-GFRLS method is wider than onepixel; for example, if the Gaussian filtering is 5 × 5, theCEP will be three pixels wide. However, the proposedmodel has a stable one-pixel width CEP. Therefore, whensegmenting the same desired object with the same initial con-tour, the proposed model uses fewer iterations than the LR-AC model, but more iterations than the SB-GFRLS method.

Therefore, the proposed model is much more efficientthan the LR-AC model, but less efficient than the SB-GFRLS method. Moreover, when segmenting the desiredobject with the same initial contour, the proposed modeluses fewer iterations than the LR-AC model, but requiresmore iterations than the SB-GFRLS method. In Table 1 cor-responding to Fig. 4, we demonstrate the conclusion aboveby CPU time and number of iterations.

4 Experimental ResultsIn this section, we perform experiments to analyze the per-formance of the proposed method. Our model is implemen-ted in MatlabR2010a using a PC with a 2.99 GHz Intel (R)Core (TM) 2 Duo CPU. Just as a slight clarification, in orderto realize local segmentation, the contour will be initializedas a small square shape inside the target object and inflatedafterwards. In order to make the segmentation results clear,we use a white curve as initial contour and a black curve as

Fig. 4 (a) LR-AC, (b) SB-GFRLS, and (c) the proposed model.

Fig. 5 (a, d) Gaussian filter (w ¼ 5, σ ¼ 0.5); (b, e) Gaussian filter(w ¼ 5, σ ¼ 1); (c, f) Morphological closing operation.

Table 1 CPU times (in seconds) and no. of iterations.

LR-AC Proposed SB-GFRLS

Time (s) 208.7 50.92 6.33

Iterations 2000 200 70

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results for putamen segmentation, and use the black curve asinitial contour and the white curve as results for white mattersegmentation.

4.1 Tuning of the Parameters

In Sec. 3.2.4, the implementation of the proposed model con-tains seven steps. The parameters that need to be tuned in theimplementation are mainly in steps (2), (4) and (6).

4.1.1 Parameters in Step (2)

In step (2), the parameters come from the estimation of IðxÞusing Eq. (15). The estimation of IðxÞ contains two parts,approximate biased field estimation and CLAHE. Theapproximate biased field is estimated by Gaussian filteringwith parameters of size w and standard deviation σ. In theCLAHE implementation, there are also two parameters,“Numtiles”, the number of tiles, and the contrast enhance-ment limit parameter, “ClipLimit”. “Numtiles” specifiesthe number of tiles into which CLAHE divides the image.CLAHE calculates the contrast transform function foreach of these regions individually. “ClipLimit” is a contrastfactor that prevents over saturation of the image specificallyin homogeneous areas.

When tuning the four parameters in local segmentationfor images with intensity inhomogeneity, we should considertwo cases. One: when the desired object is small relative tothe image, the size w of the Gaussian filtering should besmall to estimate the biased field of the desired object,but the number of tiles “Numtiles” should be large to high-light enough desired details. Two: when the desired object is

large relative to the image, the size w of the Gaussian filteringshould be large to estimate the biased field of the desiredobject, but the number of tiles should be small to avoid high-lighting undesired details. In addition, for the entirety ofthese experiments, standard deviation σ should in generalsatisfy σ ¼ w∕3. The contrast enhancement limit parameter“ClipLimit” should be selected from the interval [0, 1]; weset“ClipLimit”¼0.2 in this paper.

For example, in Figs. 6(a) through 6(d) prepared for puta-men segmentation, the image is 256 × 256, and we setw ¼ 15, σ ¼ 5, “Numtiles”¼10 and “CliLimit”¼0.2. Thesegmentation result can be found in Fig. 7(d). In Figs. 6(e)through 6(h) prepared for white matter segmentation, theimage is 238 × 174, and we set w ¼ 45, σ ¼ 15,“Numtiles”¼3 and “CliLimit”¼0.2. The segmentation resultcan be found in Fig. 8(c).

4.1.2 Parameters in Step (4)

In step (4), there are three parameters: the parameter α usedto balance the energies F1 and F2 in the total energy F inEq. (22), the parameter β in Eq. (29) and the neighborhoodradius r in Eq. (8) used to calculate f 1, f 2, c3, and f 4 definedby Eqs. (25) through (28).

In order to tune the parameter α, we normalize theenergies F1 and F2 as F1 ¼ F1∕max½absðF1Þ� and F2 ¼F2∕max½absðF2Þ� in implementation. In general, we set theparameter α to 0.5 in this paper. But when the contrast ofthe image is very low, 0.5 < 1 − α < 1 should be satisfiedto highlight the effect of energy F2.

Because we normalize the energies F1 and F2 in imple-mentation, the parameter β should be large enough to

Fig. 6 (a and e) Original image IðxÞ; (b and f) approximated biased field IðxÞ �GσðxÞ; (c and g) IðxÞ − IðxÞ �GσðxÞ; (d and h) adapthisteqðIðxÞ − IðxÞ �GσðxÞÞ.

Fig. 7 (a) Original image, (b) global, (c) local, and (d) the proposed model.

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guarantee the update of the LSF in Eq. (29). In general, β ≥1 × 105 can satisfy this requirement. In this paper, welet β ¼ 1 × 1015.

The neighborhood radius r in Eq. (8) is used to calculatef 1, f 2, c3, and f 4 defined by Eqs. (25) through (28). It shouldbe determined by the scale of the desired object and theproximity of the surrounding clutter. Small radius is requiredwhen attempting to segment a small object with nearby clut-ter; for example in Fig. 7(d) for putamen segmentation, welet r ¼ 2. In contrast, a large radius should be used whenattempting to segment a large object with less nearby clutter;for example, in Fig. 8(c) for white matter segmentation, welet r ¼ 15.

4.1.3 Parameters in Step (6)

In the morphological closing operation, the radius ρ of thestructuring element is an important parameter. In general,large radius ρ facilitates greater curve inflation and also can-not retain edges convex to the internal. Thus in local segmen-tation, when the desired object is approximated convex, wecan select large radius; for example, in Fig. 7(d) for putamensegmentation fromMR brain images, we let ρ ¼ 5. However,when the desired object is concave, the radiusρ should besmall to retain edges that are convex to the internal; forexample, in Fig. 8(c) for white matter segmentation fromMR brain images, we let ρ ¼ 1.

4.2 Experimental Results

In this section, we report the results of experiments to dem-onstrate the advantages of the proposed model in local seg-mentation for images with intensity inhomogeneity. Weshould point out again that both the CV model and theLBF model mentioned above are global segmentation mod-els, and both the LR-AC model and the SB-GFRLS modelmentioned above are local segmentation models. In Fig. 8,we compare our results with the global segmentation modelsto demonstrate the local segmentation property of our pro-posed model. In Fig. 4, we compare our results with thelocal segmentation models to demonstrate the stability andthe small CEP of the proposed model. In Fig. 7, we compareour results with the global information-based segmentationenergy and the local information-based segmentation energyto demonstrate the model’s segmentation energy’s capabilityof handling intensity inhomogeneity. It is worth noting thatthe global information-based segmentation energy is equiva-lent to those of the CVand SB-GFRLS models, and the localinformation-based segmentation energy is equivalent to thoseof the LBF and LR-AC models. Finally, in Fig. 5, we compareour results with a Gaussian filtering curve smoothing scheme

to demonstrate the effectiveness of the morphological closingoperation for curve inflation.

Based on the discussion in regard to the parametersabove, the parameters in our proposed model for whitematter segmentation from a 238 × 174 MR brain imagein Figs. 8, 4, and 5 are selected as: w ¼ 45, σ ¼ 15,“Numtiles”¼10,“ClipLimit”¼0.2, α ¼ 0.5, β ¼ 1 × 1015,r ¼ 15, and ρ ¼ 1. The parameters in our proposed modelfor putamen segmentation from a 256 × 256 MR brainimage in Figs. 7 and 5 are: w ¼ 15,σ ¼ 5, “Numtiles”¼3,“ClipLimit”¼0.2, α ¼ 0.5, β ¼ 1 × 1015, r ¼ 2, and ρ ¼ 5.

Figure 8 demonstrates the local segmentation property ofour model. Compared with global segmentation models (CVand LBF model), the proposed model can segment thedesired white matter from MR brain images in Fig. 8(c),while the global segmentation model segments all objects.Indeed, in Figs. 8(a) and 8(b), both skull and white matterare segmented by the CV and LBF models.

Figure 4 demonstrates the stability and the small CEP ofour proposed local segmentation model. Figure 4(a) is theresult of the LR-AC model. With the increasing iterations,the error caused by reinitialization of SDF becomes large,and the narrow band becomes instable. Finally, the curve evo-lution is forced to stop for one of the two following reasons.One: no points can be sought in the narrow band; two: only afew points can be sought in the narrow band, but the pointsare smoothed out by curvature term in curve evolution.Figure 4(b) is the result of the SB-GFRLS method. The Gaus-sian filtering regularized scheme in the SB-GFRLS methoddestroys the binary nature of the LSF and creates a largeCEP, which leads to failed local segmentation. Figure 4(b)shows that the SB-GFRLS model cannot segment white mat-ter locally, and the skull is also segmented because of the largeCEP. Figure 4(c) shows that the proposed model can segmentwhite matter well. Comparing Fig. 4(a) with Fig. 4(c), we candemonstrate the narrow band of the proposed model ismuch more stable than that of the LR-AC model. ComparingFig. 4(b) with Fig. 4(c), we can demonstrate that the proposedmodel has smaller CEP than the SB-GFRLS method. The sta-bility and smaller CEP of our model can guarantee successfullocal segmentation.

In addition, in order to verify the conclusion in Sec. 3.2.4about computational complexity, we report the computa-tional time and the number of iterations in Table 1, corre-sponding to Fig. 4 for white matter segmentation. Table 1shows that the computational complexity of the proposedmodel is much lower than that of the LR-AC model, buthigher than that of the SB-GFRLS method.

In order to demonstrate the capability of our model tohandle intensity inhomogeneity and guarantee the fairnessof the comparison, we can replace the proposed segmenta-tion energy F in Eq. (29) with the global information-basedsegmentation energy FGlobal in Eq. (30) as well as localinformation-based segmentation energy FLocal in Eq. (31),respectively in step (4):

FGlobal ¼ ðI − c2Þ2 − ðI − c1Þ2 (30)

FLocal ¼ ðI − f 2Þ2 − ðI − f 1Þ2: (31)

FGlobal is equivalent to that of the CV and SB-GFRLS mod-els, and FLocal is equivalent to that of the LBF and LR-ACmodels; c1, c2, f 1, f 2 can be calculated by Eqs. (2) and (6).

Fig. 8 (a) CV, (b) LBF, and (c) the proposed model.

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Figure 7 considers the example of putamen segmentationto demonstrate our model’s handling of intensity inhomo-geneity. Putamen is a subcortical brain structure in MRbrain images that is obviously intensity inhomogeneous.Figure 7(b) shows that FGlobal cannot account for intensityinhomogeneity. Figure 7(c) shows that though FLocal ismuch better than FGlobal, it is also ineffective for putamensegmentation. Compared with FGlobal and FLocal, the pro-posed segmentation energy F can segment putamen well inFig. 7(d) due to utilizing more helpful information in localsegmentation. Figure 7 demonstrates that the proposedsegmentation energy can more effectively handle intensityinhomogeneity than the global and local information-basedsegmentation energies in the local segmentation methods. Inaddition, Fig. 8 also demonstrates the above conclusion.

In order to demonstrate that a morphological closingoperation can facilitate curve inflation much better thanGaussian filtering when the CEP is small and guaranteethe fairness of the comparison, we can replace step (6) ofthe morphological closing operation in our method withsteps (6-1) and (6-2):

(6-1) Regularize the LSF with a Gaussian filter.(6-2) Let ϕðxÞ ¼ 1 if ϕðxÞ ≥ 0; otherwise, ϕðxÞ ¼ −1.This replacement can guarantee the faithfulness of a

comparison between the two curve smoothing schemesunder the same one-pixel width CEP. Experimental resultsin Figs. 5(a), 5(b), 5(d), and 5(e) demonstrate that Gaussianfiltering can prevent curve inflation when the CEP is small.It should be pointed out that when the standard deviationbecomes larger, the negative effects on curve inflationbecome more serious, but if we decrease the standard devia-tion, the curve-smoothing ability will be weakened. Experi-mental results in Figs. 5(c) and 5(f) demonstrate that themorphological closing operation can facilitate curve inflatingeven if the CEP is small.

4.3 Metrics for the Segmentations

For medical image segmentation, the usual metric for thesegmentations is to compare the segmentation results withthe “gold standard” manually segmented by an expert. Twomethods for comparing segmentation results with “gold stan-dard” were used in Ref. 23:

(1) Coefficient of similarity

ε1 ¼ 1 −jV expert − V algorithmj

Vexpert

(32)

(2) Spatial overlap

ε2 ¼2 � V intersection

V expert þ Valgorithm

. (33)

In Eqs. (32) and (33), V expert (respectively, V algorithm,V intersection) is the true (respectively, computed, intersectionof the true and the computed) area of the desired object;V algorithm and V intersection are the computed area and the inter-section of the computed and true areas, respectively. The twocoefficients will be close to 1 if the segmentation result is cor-rect with respect to the “gold standard”. Cosidering white mat-ter segmentation in Fig. 4 for example, ε1 is 0.5420, 0.8197,

and 0.9958, respectively, and ε2 is 0.6848, 0.8675, and 0.9488respectively for Figs. 4(b)–4(d). From coefficients ε1 and ε2,we can obtain the same conclusion discussed with regard toFig. 7 above.

5 ConclusionsIn this paper, we propose an effective local segmentationmodel for images with intensity inhomogeneity. The pro-posed method has of the ability to perform local segmenta-tion, and the new segmentation energy is more suitable forimages with intensity inhomogeneity because more statisti-cal information is utilized, containing local interior, localexterior, and global interior information. Moreover, a mor-phological closing operation regularized scheme is used topromote the curve evolution, and to maintain the binaryproperty of the LSF. Experiments on synthetic and medicalimages have demonstrated the effectiveness and robustnessof this model over the CV, LBF, LR-AC, and SB-GFRLSmodels. As future work, we would like to extend our methodto segmentation of multiple objects. In addition, we wouldalso like to research applying our method to 3-D segmenta-tion for medical images.

AcknowledgmentsThis work was supported by the Natural Science Foundationof Shandong Province under Grant 2009ZRB01661 andIndependent Innovation Foundation of Shandong University.

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Qiang Zheng received both the BS degree inelectronic information science and technol-ogy and the MS degree in signal and informa-tion processing from Shandong University atWeihai, Shandong, China, in 2008 and 2010,respectively. Currently, he is a PhD candi-date at Shandong University at Weihai,Shandong, China. His current research inter-est is signal processing with applications inMRI image processing.

En-Qing Dong received the BS degree ingeology in 1987 from China University ofMining and Technology, Xuzhou, China,the MS degree in applied geophysics in1993 from Chang an University, Xi’an,China, and the PhD degree in electrical engi-neering in 2002 from Xi’an Jiaotong Univer-sity, China. From July 1993 to September1998, he was an instructor with the Depart-ment of Electrical Engineering, Xi’an Pereo-leum University, China. From July 2002 to

December 2005, he was a full professor with the School of Electricsand Information Engineering, Soochow University, Suzhou, China.From January 2006 to December 2007, he was a visiting professorwith the Broadband Communications Laboratory, Harvard University,Cambridge, USA. Since January 2008, he has been with the Schoolof Mechatronics and Information Engineering, Shandong Universityat Weihai, China, as a full professor. His current research interestsare signal processing with applications in wireless sensor networksand MRI image processing.

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