Research ArticleSegmentation of Brain Tissues from MagneticResonance Images Using Adaptively RegularizedKernel-Based Fuzzy 119862-Means Clustering

Ahmed Elazab123 Changmiao Wang12 Fucang Jia12 Jianhuang Wu12

Guanglin Li12 and Qingmao Hu12

1Shenzhen Institutes of Advanced Technology Chinese Academy of Sciences 1068 Xueyuan Boulevard Shenzhen 518055 China2University of Chinese Academy of Sciences 52 Sanlihe Road Beijing 100864 China3Faculty of Computers and Information Mansoura University Elgomhouria Street Mansoura 35516 Egypt

Correspondence should be addressed to Qingmao Hu qmhusiataccn

Received 29 September 2015 Accepted 23 November 2015

Academic Editor Jesus Pico

Copyright copy 2015 Ahmed Elazab 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

An adaptively regularized kernel-based fuzzy 119862-means clustering framework is proposed for segmentation of brain magneticresonance images The framework can be in the form of three algorithms for the local average grayscale being replaced by thegrayscale of the average filter median filter and devised weighted images respectively The algorithms employ the heterogeneityof grayscales in the neighborhood and exploit this measure for local contextual information and replace the standard Euclideandistance with Gaussian radial basis kernel functions The main advantages are adaptiveness to local context enhanced robustnessto preserve image details independence of clustering parameters and decreased computational costs The algorithms have beenvalidated against both synthetic and clinical magnetic resonance images with different types and levels of noises and comparedwith 6 recent soft clustering algorithms Experimental results show that the proposed algorithms are superior in preserving imagedetails and segmentation accuracy while maintaining a low computational complexity

1 Introduction

Image segmentation is to partition an image into meaningfulnonoverlapping regions with similar features Segmentationof brain magnetic resonance (MR) images is necessary todifferentiate white matter (WM) gray matter (GM) andcerebrospinal fluid (CSF) Such segmentation is essential forstudying anatomical structure changes and brain quantifica-tion [1] It is also a prerequisite for tumor growthmodeling astumors diffuse at different rates according to the surroundingtissues [2] Due to potential existence of noise bias field andpartial volume effect segmentation of brain images remainschallenging

Image segmentation techniques can be roughly catego-rized into [3] thresholding region growing clustering edgedetection and model-based methods Clustering is an unsu-pervised learning strategy that groups similar patterns intoclusters and can be hard or soft Soft clustering is preferred as

every pixel can be assigned to all clusters with different mem-bership values [4 5] The most popular soft clustering meth-ods applied toMR images are [4] fuzzy119862-means (FCM) clus-tering [6 7] mixture modeling and hybrid methods of both

Although the FCM algorithm comes with good accuracyin the absence of noise it is sensitive to noise and otherimaging artifacts Therefore enhancements have been triedto improve its performance by including local spatial andgrayscale information [8ndash14] which will be briefly elaboratedin Section 2

A mixture model composed of a finite number of Gaus-sians has been employed for brain MR image segmentationThe main strategy to incorporate the local information intothe mixture model is to use hiddenMarkov random fields formore accurate segmentation [15] Nikou et al [16] proposeda hierarchical and spatially constrained mixture model thattakes into account spatial information by imposing distinctsmoothness priors on the probabilities of each cluster and

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2015 Article ID 485495 12 pageshttpdxdoiorg1011552015485495

2 Computational and Mathematical Methods in Medicine

pixel neighborhoods In [17] a nonparametric Bayesianmodel for tissue classification of brain MR images known asDirichlet process mixture model was explored Nguyen andWu [18] introduced a way to incorporate spatial informationbetween neighboring pixels into the Gaussian mixture model(GMM)

To be more robust to noise and attain fast convergenceFCM and GMM can be combined Chatzis and Varvarigou[19] embedded the hidden Markov random field model intothe FCM objective function to explore the spatial informa-tion Chatzis [20] introduced a methodology for trainingfinite mixture models under fuzzy clustering principle witha dissimilarity function to incorporate the explicit informa-tion into the fuzzy clustering procedure Recently Ji et al[21] employed robust spatially constrained FCM (RSCFCM)algorithm for brain MR image segmentation by introducinga factor for the spatial direction based on the posteriorprobabilities and prior probabilities

In [22] Li et al presented an algorithm for braintissue classification and bias estimation using a coherentlocal intensity clustering Later they explored multiplicativeintrinsic component optimization (MICO) [23] to improvethe robustness and accuracy of tissue segmentation in thepresence of high level bias field

Generally the current brain MR image segmentationalgorithms suffer from one or more of the following short-comings lack of robustness to outliers [8 9 13] highcomputational cost [8 13 14 16 21] prior adjusting of crucialormany parameters [8ndash11 21] limited segmentation accuracyin the presence of high level noise [8 11 19 22 23] and loss ofsuch image details like CSF [9 13 14 21] In this paper a newsoft clustering framework is to be explored for better handlingof the aforementioned segmentation problems

The rest of this paper is organized as follows Relatedworkof FCM algorithm is presented in Section 2 The proposedframework is then elaborated in Section 3 Experiments onsynthetic and clinical MR images are presented in Section 4Sections 5 and 6 are devoted to discussion and conclusionrespectively

2 Related Work

TheFCMalgorithm in its original form assigns amembershipvalue to each pixel for all clusters in the image spaceFor an image 119868 with set of grayscales 119909

119894at pixel 119894 (119894 =

1 2 119873) 119883 = 1199091 1199092 119909

119873 sub 119877

119896 in 119896-dimensionalspace and cluster centers V = V

1 V2 V

119888 with 119888 being a

positive integer (2 lt 119888 ≪ 119873) there is a membership value 119906119894119895

for each pixel 119894 in the 119895th cluster (119895 = 1 2 119888)The objectivefunction of the FCM algorithm is [7]

119869FCM =

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895

10038171003817100381710038171003817119909119894minus V119895

10038171003817100381710038171003817

2

(1)

where 119898 is a weighting exponent to the degree of fuzzinessthat is 119898 gt 1 and 119909

119894minus V1198952 is the grayscale Euclidean

distance between pixel 119894 and center V119895 The membership 119906

119894119895

should be constrained to the following

forall119894 isin [1119873] 119895 isin [1 119888]

119888

sum

119895=1

119906119894119895= 1 119906

119894119895isin [0 1] 0 le

119873

sum

119894=1

119906119894119895le 119873

(2)

Themembership function and cluster centers are updatediteratively in an alternating process known as alternateoptimization The membership function and cluster centersare

119906119894119895=

1

sum119888

119896=1(10038171003817100381710038171003817119909119894minus V119895

10038171003817100381710038171003817

2

1003817100381710038171003817119909119894 minus V

119896

1003817100381710038171003817

2)

1(119898minus1)

V119895=

sum119873

119894=1119906119898

119894119895119909119894

sum119873

119894=1119906119898

119894119895

(3)

As the objective function in (1) does not include anylocal information the original FCM is very sensitive to noiseand the accuracy of clustering in the presence of noise andimage artifacts will decrease To overcome this problemAhmed et al [8] modified the objective function by addinga term for the spatial information of neighboring pixels Thisalgorithm is denoted as FCM S with the following objectivefunction

119869FCM S =

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895

10038171003817100381710038171003817119909119894minus V119895

10038171003817100381710038171003817

2

+120572

119873119877

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(sum

119903isin119873119894

10038171003817100381710038171003817119909119903minus V119895

10038171003817100381710038171003817

2

)

(4)

where 120572 is a parameter to control the spatial information ofneighbors with 0 lt 120572 le 1119873

119894is the set of pixels around pixel

119894 and119873119877is the cardinality of119873

119894

The FCM S algorithm is computationally expensive asthe local neighborhood term has to be calculated in eachiteration step To overcome this drawback Chen and Zhang[10] replaced the term (1119873

119877) sum119903isin119873119894

119909119903minusV1198952 with 119909

119894minusV1198952

where 119909 is the grayscale of a filtered image that could becalculated once in advance and used kernel function toreplace the Euclidean distance The enhancement could bein two forms that is FCM S1 by using the average filterand FCM S2 by adopting the median filter Their objectivefunction is as follows

119869FCM S12 =119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895

10038171003817100381710038171003817119909119894minus V119895

10038171003817100381710038171003817

2

+ 120572

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895

10038171003817100381710038171003817119909119894minus V119895

10038171003817100381710038171003817

2

(5)

Although the accuracy has been improved it is sensi-tive to high level noises and different types of noises Inaddition the parameter 120572 which has a great impact on theperformance is set manually with care and requires priorinformation about noise

Yang and Tsai [12] proposed a Gaussian kernel-basedFCM method with the parameter 120578

119895calculated in every

Computational and Mathematical Methods in Medicine 3

iteration to replace 120572 for every cluster Similar to FCM S1 andFCM S2 thismethodhas two formsGKFCM1 andGKFCM2for average and median filters respectively The parameter 120578

119895

is estimated using kernel functions

120578119895=

min1198951015840=119895(1 minus 119870 (V 119895 V119895))

max119896(1 minus 119870 (V

119896 119909))

(6)

where 119870 is the kernel function The replacement of 120572 with120578119895could yield better results than FCM S1 and FCM S2

However for good estimation of 120578119895 cluster centers should

be well separated which might not be always true hence thealgorithm has to iterate many times to converge Moreoverthe learning scheme requires a large number of patterns andmany cluster centers to find the optimal value for 120578

119895

To tackle the problem of parameter adjustment Krinidisand Chatzis [13] proposed the FLICM algorithm with afuzzy factor that combined both spatial and grayscale infor-mation of the neighboring pixels The fuzzy factor 119866

119894119895=

sum119896isin119873119894 119894 =119896

(1(1 + 119889119894119896))(1 minus 119906

119894119895)119898119909119894minus V1198952 was embedded into

(1) as follows

119869FLICM =

119873

sum

119894=1

119888

sum

119895=1

[119906119898

119894119895

10038171003817100381710038171003817119909119894minus V119895

10038171003817100381710038171003817

2

+ 119866119894119895] (7)

where pixel 119894 is the center of the local window pixel 119895 is inthe neighborhood and 119889

119894119896is the spatial Euclidean distance

between pixels 119894 and 119896Although FLICMalgorithm enhances robustness to noise

and artifacts it is slow since the fuzzy factor (119866119894119895) has to be

calculated in each iteration Moreover 119866119894119895is heavily affected

by spatial Euclidean distance from the central pixel to itsneighboring pixels to lose small image details due to thesmoothing effect

To enhance the FLICM algorithm Gong et al [14]developed KWFLICM algorithm with a trade-off weightedfuzzy factor to control the local neighbor relationship andreplaced the Euclidean distance with kernel function Theweighted fuzzy factor

119894119895of KWFLICM is

119894119895= sum

119896isin119873119894 119894 =119896

120596119894119896(1 minus 119906

119894119895)119898

(1 minus 119870 (119909119894 V119895)) (8)

where 120596119894119896is the trade-off weighted fuzzy factor of pixel 119896 in

the local window around the central pixel 119894 and 1 minus 119870(119909119894 V119895)

is the kernel metric function The trade-off weighted fuzzyfactor combines both the local spatial and grayscale informa-tion [14] Because of the trade-off weighted fuzzy factor itscomputational cost increases substantially In addition thealgorithm is unable to preserve small image details

In addition to the abovementioned shortcomingsSzilagyi [24] pointed out serious theoretical mistakes inFLICM and KWFLICM It was shown that the iterativeoptimization nature of FLICM and KWFLICM did notminimize their objective functions instead they iterateduntil the partition matrices converged Furthermore theirobjective functions intended to employ local contextualinformation but theoretically failed and were not evensuitable for creating a valid partition [24]

To this end a new way to modify the existing FCMclustering is explored with adaptive regularization for con-textual information The proposed framework utilizes a newparameter to control the effect of pixel neighbors based onthe heterogeneity of local grayscale distribution A weightedimage is devised that combines the local contextual infor-mation with respect to the heterogeneity of local grayscaledistribution and the original grayscale that is calculated oncein advance to reduce the computational cost To improvesegmentation accuracy and robustness to outliers a kernelfunction is employed to replace the Euclidean distance met-ric Validation against both synthetic and clinicalMRdata hasbeen carried out to compare the proposed algorithms with6 recent soft clustering algorithms in terms of segmentationaccuracy and computational costs

3 Proposed Algorithms

We introduce a regularizing parameter to enhance segmenta-tion robustness and preserve image details devise a weightedimage and adopt the Gaussian radial basis function (GRBF)for better accuracy

31 The Introduced Regularization Term The parameter 120572

used in [8ndash10] is usually set in advance to control the desirableamount of contextual information Indeed using a fixed 120572 forevery pixel is not appropriate since noise level differs fromonewindow to another In addition setting such parameter needsprior knowledge about noise which is not always availablein reality Hence adaptive calculation of 120572 is necessaryaccording to the pixel being processed

To be adaptive to noise level of the pixel being processedwe first calculate the local variation coefficient (LVC) toestimate the discrepancy of grayscales in the local window tobe normalized with respect to the local average grayscale Inthe presence of noise to have high heterogeneity between thecentral pixel and its neighbors LVC will increase Consider

LVC119894=

sum119896isin119873119894

(119909119896minus 119909119894)2

119873119877lowast (119909119894)2

(9)

where 119909119896is the grayscale of any pixel falling in the local win-

dow119873119894around the pixel 119894119873

119877is the cardinality of119873

119894 and 119909

119894

is its mean grayscale Next LVC119894is applied to an exponential

function to derive the weights within the local window

120577119894= exp( sum

119896isin119873119894 119894 =119896

LVC119896)

120596119894=

120577119894

sum119896isin119873119894

120577119896

(10)

The ultimate weight assigned to every pixel is associatedwith the average grayscale of the local window

120593119894=

2 + 120596119894 119909119894lt 119909119894

2 minus 120596119894 119909119894gt 119909119894

0 119909119894= 119909119894

(11)

4 Computational and Mathematical Methods in Medicine

(a)

30 30 30 30 30 3030 30 30 30 30 3030 30 30 165 30 3030 30 123 30 30 3030 30 30 30 30 3030 30 30 30 30 30

AB

C

(b)

0000 0000 0000 0000 0000 00000000 0000 1989 1957 1992 00000000 1960 1896 2623 1957 00000000 1952 2085 1896 1989 00000000 1978 1952 1960 0000 00000000 0000 0000 0000 0000 0000

(c)

1000 1000 1000 1000 1000 10001000 1000 1000 1000 1000 10001000 1000 1000 0989 1000 10001000 1000 0999 1000 1000 10001000 1000 1000 1000 1000 10001000 1000 1000 1000 1000 1000

(d)

Figure 1 Calculation and effect of the regularization parameter in different cases (a) Noisy image (b) 6times6 subimage red rectangle from (a)with 3 different windows A B and C (c) Weights associated with each pixel using the proposed method (d) Membership values after threeiterations

The parameter 120593119894assigns higher values for those pixels

with high LVC (for pixel 119894 being brighter than the averagegrayscale of its neighbors 120593

119894will be 2+120596

119894 and120596

119894will be large

when the sum of LVC within its neighborhood is large) andlower values otherwise When the local average grayscale isequal to the grayscale of the central pixel 120593

119894will be zero and

the algorithm will behave as the standard FCM algorithmThe value 2 in (11) is set through experiments to balancebetween the convergence rate and the capability to preservedetails The proposed parameter 120593

119894is embedded into (5) to

replace 120572 Figure 1 shows the calculation of 120593119894with different

cases of noiseHere are some remarks on the parameter 120593

119894

The first point to emphasize is that 120593119894is only relevant

to the grayscales within a specified neighborhood whichis very different from FCM S FLICM and KWFLICMwhere the contextual information is expressed respectivelybysum119909119903isin119873119894

119909119903minus V1198952 119866119894119895 and

119894119895 all containing a loop on the

neighborhood and the cluster center V119895 Due to its irrelevance

to clustering parameters 120593119894could be calculated in advance

before the clustering process which can greatly reduce thecomputational cost On the contrary FCM S FLICM andKWFLICMwill need to update the contextual weights at eachiteration which is the main reason why they have highercomputational cost

The second point is that the contextual informationprovided by 120593

119894is based on the heterogeneity of grayscale

distribution within the local neighborhood which is com-pletely different from existing enhanced versions of FCM tobase the contextual information on the difference betweenthe grayscales of neighboring pixels and cluster centers Asa result the proposed 120593

119894tends to yield a homogeneous

clustering according to local grayscale distribution whileexisting enhanced FCM algorithms tend to make the cluster-ing to have more homogeneous labels by incorporating thecontextual information

32 Devising a Weighted Image In addition to making 119909respectively the grayscale of averagemedian filter of theoriginal image 119909 can also be replaced with the grayscale ofthe newly formed weighted image 120585

120585119894=

1

2 +max (120593119894)(119909119894+

1 +max (120593119894)

119873119877minus 1

sum

119903isin119873119894

119909119903) (12)

where 119909119903and 119873

119894are respectively the grayscale and neigh-

borhood of pixel 119894 and 119873119877is the cardinality of 119873

119894 Formula

(12) is inspired by the weighted image in [9] but utilizes 120593119894

explicitly to make the weighted image free from parametersthat are difficult to adjust

33 Measuring Distance Using Kernel Function The Euclid-ean distance metric is generally simple and computationallyinexpensive but it is sensitive to perturbations and outliersRecently with popular usage of support vector machine anew direction appears to use kernel functions The kernelfunctions are able to project the data into higher dimensionalspace where the data could be more easily separated [25]To do this a so-called kernel trick has been adopted thatcan transform linear algorithm to nonlinear one using dotproduct [26] Using the kernel trick the Euclidean distance

Computational and Mathematical Methods in Medicine 5

term 119909119894minus V1198952 can be replaced with 120601(119909

119894) minus 120601(V

119895)2 that is

defined as10038171003817100381710038171003817120601 (119909119894) minus 120601 (V

119895)10038171003817100381710038171003817

2

= 119870 (119909119894 119909119894) + 119870 (V

119895 V119895)

minus 2119870 (119909119894 V119895)

(13)

where119870 is the kernel functionIn this paper we use GRBF kernel [27]

119870(119909119894 V119895) = exp(minus

10038171003817100381710038171003817119909119894minus V119895

10038171003817100381710038171003817

2

21205902) (14)

where 120590 is the kernel widthUsing GRBF the kernel function in (13) will be

10038171003817100381710038171003817120601 (119909119894) minus 120601 (V

119895)10038171003817100381710038171003817

2

= 2 (1 minus 119870 (119909119894 V119895)) (15)

The choice of the kernel width 120590 is still a problem andhas to be selected carefully If it is large the exponential effectwill be almost linear On the contrary if it is small the clusterboundaries will be sensitive to outliers [27] In [10] 120590 was setto a fixed value of 150 while in [12] the authors used samplevariance to estimate 120590 Similar to [14] we calculate 120590 basedon the distance variances among all pixels

120590 = [

[

sum119873

119894=1(119889119894minus 119889)2

119873 minus 1

]

]

12

(16)

where 119889119894= 119909119894minus 119909 is distance from the grayscale of pixel 119894

to the grayscale average of all pixels and 119889 is the average of alldistances 119889

119894

34 The Proposed Framework The proposed adaptively reg-ularized kernel-based FCM framework is denoted as ARK-FCM First we calculate the adaptive regularization param-eter 120593

119894associated with every pixel to control the contextual

information using (11) The objective function is defined as

119869ARKFCM = 2[

[

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+

119873

sum

119894=1

119888

sum

119895=1

120593119894119906119898

119894119895(1 minus 119870 (119909

119894 V119895))]

]

(17)

Under the conditions specified in (2) the minimizationof 119869ARKFCM(119906 V) can be calculated through an alternateoptimization procedure using (derivation is given in theAppendix)

119906119894119895

=

((1 minus 119870 (119909119894 V119895)) + 120593

119894(1 minus 119870 (119909

119894 V119895)))minus1(119898minus1)

sum119888

119896=1(1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))minus1(119898minus1)

(18)

V119895=

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) 119909119894+ 120593119894119870(119909119894 V119895) 119909119894)

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) + 120593119894119870(119909119894 V119895))

(19)

When 119909 is replaced with the grayscale of the averagemedianfilter of the original image the algorithm is denotedas ARKFCM

1ARKFCM

2 When 119909

119894is replaced with the

weighted image 120585119894defined in (12) the algorithm is denoted

as ARKFCM119908 The main steps for the proposed algorithms

are as follows

(1) Initialize threshold 120576 = 0001 119898 = 2 loop counter119905 = 0 V and 119906

(0)(2) Calculate the adaptive regularization parameter 120593

119894

(3) Calculate 119909119894for ARKFCM

1and ARKFCM

2or 120585 for

ARKFCM119908

(4) Calculate cluster centers V(119905)119895

using 119906(119905) as in (19)

(5) Calculate the membership function 119906(119905+1) with (18)

(6) If max 119906(119905+1)

minus 119906(119905) lt 120576 or 119905 gt 100 then stop

otherwise update 119905 = 119905 + 1 and go to step (4)

4 Experiments

In this section we present the experiments on both syn-thetic and clinical MR images For validation the proposedalgorithms (ARKFCM

1 ARKFCM

2 and ARKFCM

119908) are

compared with 6 recent soft clustering algorithms namelyGKFCM1 [12] GKFCM2 [12] FLICM [13] KWFLICM[14] MICO [21] and RSCFCM [23] As we are unable tohave faithful implementation of RSCFCM [23] RSCFCMis compared only against the common data with resultsreported from [23] The algorithms are implemented usingMATLAB software package (a demo version is freely availableonline (httpwwwmathworkscommatlabcentralfileex-change54141-arkfcm-algorithm)) All the experiments areconducted with window size of 3 times 3 pixels maximumnumber of iterations 119905 = 100 and 120576 = 0001 The accuracyof segmentation is measured using the Jaccard Similarity (JS)metric [28] which is defined as the ratio between the intersec-tion and union of segmented volume 119878

1and ground truth

volume 1198782

JS (1198781 1198782) =

10038161003816100381610038161198781 cap 1198782

100381610038161003816100381610038161003816100381610038161198781 cup 119878

2

1003816100381610038161003816

(20)

41 Experiments on Synthetic Brain MR Images The follow-ing experiments are carried out using Simulated Brain Data-base (SBD) [29] which contains a set of realistic MR volumesproduced by anMR imaging simulator with ground truths ofCSF GM and WM available

The first experiment is to segment a T1-weighted axialslice (number 100) with 217 times 181 pixels corrupted with 7noise and 20 grayscale nonuniformity into WM GM andCSF Figure 2 shows the segmentation results while Table 1summarizes the JS and average running times

The second experiment is to segment a T1-weightedsagittal slice (number 100) with 181 times 217 pixels corruptedwith 7 noise and 20 grayscale nonuniformity This imageis chosen to show the capability of preserving details

6 Computational and Mathematical Methods in Medicine

Table 1 JS and running time on the T1-weighted axial slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO RSCFCM ARKFCM1

ARKFCM2

ARKFCM119908

WM 0930 0933 0941 0937 0894 0898 0940 0941 0940GM 0822 0855 0860 0852 0782 0842 0864 0868 0865CSF 0781 0847 0829 0805 0860 0882 0863 0867 0867Average 0844 0879 0876 0865 0845 0874 0889 0892 0891Time (s) 0911 0586 3403 139470 0673 2530 0356 0282 0329

Table 2 JS and running time on the T1-weighted sagittal slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

WM 0773 0775 0771 0765 0672 0785 0788 0786GM 0796 0806 0794 0791 0669 0815 0816 0816CSF 0834 0852 0835 0824 0869 0871 0872 0875Average 0801 0811 0800 0794 0736 0824 0825 0825Time (s) 1377 1797 5392 166717 0751 0348 0329 0322

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

(g) (h) (i) (j) (k)

Figure 2 Segmentation results on a T1-weighted axial slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity (a)Original image (b) Ground truth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h)RSCFCM results (i) ARKFCM

1results (j) ARKFCM

2results (k) ARKFCM

119908results

The segmentation results and JS are presented in Figure 3 andTable 2 respectively

The third experiment is to check the robustness toRician noise that commonly affects MR images [30] Thesegmentation results and the JS with average running timesof a T1-weighted axial slice (number 91) with 217 times 181 pixelscorrupted with 10 Rician noise are given in Figure 4 andTable 3 respectively

42 Experiments on Clinical Brain MR Images with TumorsWe experimented two T1-weighted axial slices (slices num-bers 80 and 86 denoted resp as Brats1 and Brats2) with240 times 240 pixels respectively fromfiles pat266 1 and pat192 1(available fromMICCAI BRATS 2014 challenge httpswwwvirtualskeletonchBRATSStart2014) (Figures 5 and 6)

From black to white are respectively background CSF GMand WM It should be noted that clustering is carried outonly for CSF GM andWM with the pathology region beingconsidered as background

Since images Brats1 and Brats2 come only with groundtruth for the pathology and have no ground truth for normaltissues a quantificationmeasure other than JS should be usedTo this end an entropy-based metric proposed by Zhanget al [31] is adopted This is to maximize the uniformity ofpixels within each segmented region and to minimize theuniformity across the regions First the entropy of everyregion 119895 is calculated

119867(119877119895) = minus sum

119898isin119881119895

119871119895 (119898)

119878119895

log119871119895 (119898)

119878119895

(21)

Computational and Mathematical Methods in Medicine 7

Table 3 JS and running time on the T1-weighted axial slice (number 91) from SBD corrupted with 10 Rician noise

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

WM 0931 0921 0929 0927 0885 0926 0925 0922GM 0804 0827 0831 0821 0777 0838 0840 0834CSF 0826 0872 0861 0846 0889 0887 0892 0887Average 0850 0874 0874 0865 0850 0884 0886 0881Time (s) 1836 1869 2953 124922 0649 0218 0220 0218

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

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

(i) (j) (k)

Figure 3 Segmentation results on a T1-weighted sagittal slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity (a)Original image (b) Ground truth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h)RSCFCM results (i) ARKFCM

1results (j) ARKFCM

2results (k) ARKFCM

119908results

where119881119895is the set of all possible grayscales in region 119895 119871

119895(119898)

is the number of pixels belonging to region 119895 with grayscale119898 and 119878

119895is the area of region 119895 For the grayscale image 119868 the

119864measure is calculated as

119864 =

119888

sum

119895=1

(119878119895

119878119868

)119867(119877119895) + (minus

119888

sum

119895=1

(119878119895

119878119868

) log119878119895

119878119868

) (22)

where the first term represents the expected region entropy ofthe segmentation and the second term is the layout entropyThe measure 119864 yields smaller value for better segmentationand higher otherwise Table 4 summarizes the segmentationaccuracy in terms of 119864 and running times for Brats1 andBrats2 It should be noted that the 119864 metric is calculatedonly for WM GM and CSF regions without considering thebackground

5 Discussion

FCM clustering is a well-known soft clustering methodthat assigns a membership degree for each pixel to everycluster As the segmentation accuracy of FCM algorithmdecays in the presence of noise artifacts and increasednumber of clusters many investigations have been carriedout on using contextual information to enhance the qualityof segmentation But how the contextual information shall beused effectively is still a challenge

We propose an adaptively regularized kernel-based FCMclustering framework with new parameter 120593

119894that adaptively

controls the contextual information according to the hetero-geneity of grayscale distribution within the local neighbor-hood The new parameter is estimated using local variationcoefficient among pixels within a specified neighborhoodA weighted image is devised that combines the original

8 Computational and Mathematical Methods in Medicine

Table 4 Segmentation performance measure 119864 and running time on the Brats1 and Brats2 images

Image Measure GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

Brats1 E 1311 1288 1392 1424 1309 1274 1270 1279Time (s) 2591 1516 9510 635413 1747 1185 1111 1272

Brats2 E 1307 1297 1336 1340 1298 1273 1271 1279Time (s) 1576 1115 5170 400295 1546 1091 0827 0975

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

(f) (g) (h) (i) (j)

Figure 4 Segmentation results on a T1-weighted axial slice (number 91) from SBD with 10 Rician noise (a) Original image (b) Groundtruth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h) ARKFCM

1results (i)

ARKFCM2results (j) ARKFCM

119908results

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

(f) (g) (h) (i)

Figure 5 Segmentation results on the Brats1 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

Computational and Mathematical Methods in Medicine 9

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

(f) (g) (h) (i)

Figure 6 Segmentation results on the Brats2 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

image and the parameter 120593119894to represent the image contextual

information embedded through the weighting procedureFurthermore a GRBF is adopted to replace Euclidean dis-tance for better partitioning and to be less sensitive to outliersThe proposed framework can be in the form of 3 algorithmsARKFCM

1 ARKFCM

2 and ARKFCM

119908for the local average

grayscale to be replacedwith the grayscale of the average filtermedian filter and the devised weighted image respectivelyExperiments have been carefully carried out to show thesuperiority of the proposed algorithms in comparison with6 recent soft clustering algorithms to be discussed further

51 Segmentation Accuracy The differences in JS betweenthe proposed algorithms and the other 6 algorithms areclear for the axial slice with 7 noise and 20 grayscalenonuniformity (Table 1 and Figure 2) and become moredistinctive in preserving details in the sagittal slice corruptedwith the same noise (Table 2 and Figure 3) The average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0889 0892 and 0891 for the axial slice as shown in Figure 2which are better than the other 6 algorithms with range 13ndash48 (Table 1) For the sagittal slice with 7 noise and 20grayscale nonuniformity shown in Figure 3 the average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0824 0825 and 0825 which are better than the otheralgorithms with range 14ndash89 (Table 2) For the axial slicecorrupted with 10 Rician noise shown in Figure 4 theaverage JSs of ARKFCM

1 ARKFCM

2 and ARKFCM

119908are

respectively 0884 0886 and 0881 which are better than theother algorithms with range 120ndash36 (Table 3) The higherJSs may imply that the proposed algorithms achieve a betterbalance in preserving image details in the presence of noiseand grayscale inhomogeneity

For the clinical brain MR images with tumors (Brats1and Brats2) shown in Figures 5 and 6 the proposed algo-rithms achieve smaller 119864 than other algorithms (Table 4)For Brats1Brats2 image ARKFCM

2attains the lowest 119864 of

12701271 which is smaller than the other 5 algorithmswith ranges 0018ndash0154 and 0025ndash0109 respectively VisiblyGKFCM1 (Figures 5(b) and 6(b)) and GKFCM2 (Figures5(c) and 6(c)) come with good results but they are unableto preserve small details such as CSF being wrongly brokenpossibly due to the difficulty in estimating the parameter120578119895 On the other hand FLICM (Figures 5(d) and 6(d)) and

KWFLICM (Figures 5(e) and 6(e)) yield smooth results anddo not preserve many details which affects the segmentationaccuracy of CSF (CSF is small as compared with surroundingGM and WM) due to the effects of 119866

119894119895and

119894119895 respectively

For MICO algorithm (Figures 5(f) and 6(f)) the edges arenot smooth enough hence the CSF is not well preservedFinally results of ARKFCM

1 ARKFCM

2 and ARKFCM

119908

(Figures 5(g) and 6(g) Figures 5(h) and 6(h) and Figures 5(i)and 6(i)) show good balance between smooth borders andpreserving image details due to the introduction of adaptivelocal contextual information measure 120593

119894to replace the fixed

value of 120572 or the oversmoothing factors 119866119894119895or 119894119895

52 Computational Cost In terms of computational cost theobjective function of FCM algorithm in its original form[6] contains only the difference between the grayscale of thecurrent pixel 119894 and the cluster centers V

119895 This is basically

to cluster grayscales as there is no spatial information so ithas the smallest computational cost and can be implementedbased on grayscale histogram to reduce the computationalcost further [9 11] The enhancement of original FCM isto add the local contextual information to make it robust

10 Computational and Mathematical Methods in Medicine

to noise and image artifacts at the expense of increasedcomputational cost [7ndash14]

The GKFCM1 and GKFCM2 algorithms [12] use theparameter 120578

119895to replace 120572 and need an additional loop on

the number of clusters to be calculated at each pixel toupdate the local contextual information So they have highercomputational cost than the original FCM FCM S1 andFCM S2 at each iteration

The FLICM algorithm [13] introduces 119866119894119895which needs

an additional loop on the neighborhood of the currentpixel to calculate the local information in every iterationthus it has a high computational cost As an extension ofFLICM KWFLICM [14] introduces

119894119895that requires two

additional loops on the neighborhood so it has the highestcomputational cost at each iteration

The RSCFCM algorithm [21] uses a spatial fuzzy factorthat is constructed based on the posterior and prior prob-abilities and takes the spatial direction into account Thatincreases the complexity as many parameters have to beoptimized and consequently the computational complexity

MICO algorithm [23] comes with fast calculations dueto the convexity of its energy function particularly in thepresence of less noise but tends to have many iterations in thepresence of high level noise

The proposed algorithms incorporate the local contextualinformation by introducing LVC which is a measure ofgrayscale heterogeneity and has nothing to dowith the clustercenters So it can be calculated once in advance and hencereduce the complexity of the clustering procedure

The eventual computational cost will be the multipli-cation of the computational cost for each iteration andthe number of iterations for convergence The number ofiterations will be dependent on the initialization as well as theobjective function To make fair comparisons initializationsare all set randomly and the average number of iterationsfor convergence is then recorded for 10 converged timesFrom Figure 7 it can be seen that the average iterationtimes will be data dependent with ARKFCM

2 ARKFCM

119908

and ARKFCM1having the minimum number of iterations

followed by FLICM GKFCM2 and GKFCM1 respectivelyThe running times for the tested images agree well with

the above analysis as illustrated in Figure 8 For SBD imagesgiven in Figures 2 3 and 4 the KWFLICM algorithm takesthe longest time (124ndash166 seconds) followed by FLICM(3ndash54 seconds) RSCFCM (about 25 seconds) GKFCM1GKFCM2 (06ndash19 seconds) and the proposed algorithms(022 to 036 seconds) For the clinical MR images in Figures5 and 6 it takes more time (due to more iterations) asthe images are more complicated but the trend remainsthe same with the proposed algorithms having the lowestcomputational cost (Table 4 and Figure 8)

53 Neighborhood Size The local neighborhood window sizeis a crucial factor to determine the smoothness of clusteringand details to be preserved We have experimented withdifferent window sizes and found that a window size of3 times 3 pixels achieves the best balance between segmentationaccuracy and computational cost Increasing window size to5 times 5 pixels has very small impact on the JS but 7 times 7 pixels

0

20

40

60

80

100

120

GKF

CM1

GKF

CM2

FLIC

M

MIC

O

Num

ber o

f ite

ratio

ns

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 7 Average number of iterations of all the algorithms (exceptRSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a) and 6(a)

002040608

112141618

222242628

GKF

CM1

GKF

CM2

MIC

O

Aver

age r

unni

ng ti

me (

s)

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 8 Average running times of all the algorithms (exceptFLICM RSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a)and 6(a)

or more will significantly decrease the accuracy such aslosing image details like CSF Therefore it is recommendedto use a local window of size 3 times 3 pixels for constructing theneighborhood

54 Limitation From the experiments it was found thatthe proposed algorithms could be sensitive to severe noisein small areas between edges that have width 1 or 2 pixels(eg area between ventricles Figures 4(h) 4(i) and 4(j))Potentially this may be solved by embedding edge detectiontechnique into the clustering process which is yet to beexplored

Computational and Mathematical Methods in Medicine 11

6 Conclusion

An adaptively regularized kernel-based FCM framework hasbeen proposed to enhance the original FCM for highersegmentation accuracy at low computational costThe frame-work can be in the form of three algorithms that employ theheterogeneity of grayscales in the neighborhood employedfor local contextual information The main advantages areadaptiveness to local context enhanced robustness and inde-pendence of clustering parameters to decrease computationalcost GRBF kernel has been adopted as a distance metric Wevalidated the proposed algorithms on synthetic and clinicalMR images with tumors The proposed algorithms attain ahigher JS (Tables 1 2 and 3) and lower entropy measure119864 (Table 4) than the 6 recent soft clustering algorithms andcould preserve small image details (Figures 2 3 4 5 and 6) Inaddition the proposed algorithms have a low computationalcost and are to the best of our knowledge the only algorithmsthat are adaptive to local context and do not include clustercenters Therefore they attain a trade-off between highsegmentation accuracy and low computational costThe pro-posed algorithms can be a potential tool for segmenting brainMR images for further processing and other images as well

Appendix

The objective function 119869ARKFCM given in equation (17) withconditions in (2) is a constrained minimization problemwhich can be solved using Lagrange multiplier method Letthe minimization problem of 119869ARKFCM be written as follows

119871ARKFCM (119906119894119895 V119895 120582119894 120593119894)

= 2

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2

119873

sum

119894=1

119888

sum

119895=1

120593119894119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+

119873

sum

119894=1

120582119894(1 minus

119888

sum

119895=1

119906119894119895)

(A1)

By taking the derivative of 119871ARKFCM with respect to 119906119894119895and

setting the result to zero for119898 gt 1 we have

120597119871ARKFCM120597119906119894119895

= 2119898119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2119898120593119894119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895)) minus 120582

119894

= 0

(A2)

By using (A2) we can calculate 119906119894119895as follows

119906119894119895

= [120582119894

2119898 (1 minus 119870 (119909119894 V119895) + 120593119894(1 minus 119870 (119909

119894 V119895)))

]

1(119898minus1)

(A3)

Since sum119888119896=1

119906119894119896= 1 forallV

119896 from (A3) we can get

119888

sum

119896=1

[120582119894

2119898 (1 minus 119870 (119909119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))

]

1(119898minus1)

= 1 (A4)

120582119894

=1

[sum119888

119896=1(2119898 (1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896))))minus1(119898minus1)

]119898minus1

(A5)

By substituting (A5) into (A3) with necessary calculationsthe final 119906

119894119895is

119906119894119895

=

((1 minus 119870 (119909119894 V119895)) + 120593

119894(1 minus 119870 (119909

119894 V119895)))minus1(119898minus1)

sum119888

119896=1(1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))minus1(119898minus1)

(A6)

Similarly by taking the derivative of 119871ARKFCM with respect toV119895and setting the result to zero we have

120597119871ARKFCM120597V119895

=

119873

sum

119894=1

119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895)

+

119873

sum

119894=1

120593119894119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895) = 0

(A7)

After necessary calculations the final V119895is

V119895=

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) 119909119894+ 120593119894119870(119909119894 V119895) 119909119894)

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) + 120593119894119870(119909119894 V119895))

(A8)

Conflict of Interests

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

Acknowledgments

The authors would like to thank Dr Maoguo Gong andDr Chunming Li for providing their source codes forcomparison This work has been supported by the NationalProgram on Key Basic Research Project (nos 2013CB7338002012CB733803) Key Joint Program of National Natural Sci-ence Foundation and Guangdong Province (no U1201257)and Guangdong Innovative Research Team Program (no201001D0104648280)

References

[1] N Sharma and L M Aggarwal ldquoAutomated medical imagesegmentation techniquesrdquo Journal ofMedical Physics vol 35 no1 pp 3ndash14 2010

[2] A Elazab Q Hu F Jia and X Zhang ldquoContent based modifiedreaction-diffusion equation for modeling tumor growth of lowgrade gliomardquo in Proceedings of the 7th Cairo InternationalBiomedical Engineering Conference (CIBEC rsquo14) pp 107ndash110Giza Egypt December 2014

12 Computational and Mathematical Methods in Medicine

[3] D L PhamCXu and J L Prince ldquoCurrentmethods inmedicalimage segmentationrdquoAnnual Review of Biomedical Engineeringvol 2 pp 315ndash337 2000

[4] I Despotovic B Goossens andW Philips ldquoMRI segmentationof the human brain challenges methods and applicationsrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 450341 23 pages 2015

[5] S Ruan C Jaggi J Xue J Fadili and D Bloyet ldquoBraintissue classification of magnetic resonance images using partialvolume modelingrdquo IEEE Transactions on Medical Imaging vol19 no 12 pp 1172ndash1186 2000

[6] J C Dunn ldquoA fuzzy relative of the ISODATA process and itsuse in detecting compact well-separated clustersrdquo Journal ofCybernetics vol 3 no 3 pp 32ndash57 1973

[7] J C Bezdek Pattern Recognition with Fuzzy Objective FunctionAlgorithms Advanced Applications in Pattern RecognitionSpringer Boston Mass USA 1981

[8] M N Ahmed S M Yamany N Mohamed A A Farag andT Moriarty ldquoA modified fuzzy c-means algorithm for bias fieldestimation and segmentation of MRI datardquo IEEE Transactionson Medical Imaging vol 21 no 3 pp 193ndash199 2002

[9] L Szilagyi Z Benyo S M Szilagyi and H S Adam ldquoMRbrain image segmentation using an enhanced fuzzy C-meansalgorithmrdquo in Proceedings of the 25th Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety pp 724ndash726 Cancun Mexico September 2003

[10] S Chen andD Zhang ldquoRobust image segmentation using FCMwith spatial constraints based on new kernel-induced distancemeasurerdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 4 pp 1907ndash1916 2004

[11] W Cai S Chen and D Zhang ldquoFast and robust fuzzy c-meansclustering algorithms incorporating local information for imagesegmentationrdquo Pattern Recognition vol 40 no 3 pp 825ndash8382007

[12] M-S Yang and H-S Tsai ldquoA Gaussian kernel-based fuzzyc-means algorithm with a spatial bias correctionrdquo PatternRecognition Letters vol 29 no 12 pp 1713ndash1725 2008

[13] S Krinidis and V Chatzis ldquoA robust fuzzy local informationC-means clustering algorithmrdquo IEEE Transactions on ImageProcessing vol 19 no 5 pp 1328ndash1337 2010

[14] M Gong Y Liang J Shi W Ma and J Ma ldquoFuzzy C-meansclustering with local information and kernel metric for imagesegmentationrdquo IEEE Transactions on Image Processing vol 22no 2 pp 573ndash584 2013

[15] M A Balafar ldquoGaussian mixture model based segmentationmethods for brain MRI imagesrdquo Artificial Intelligence Reviewvol 41 no 3 pp 429ndash439 2014

[16] C Nikou N P Galatsanos and A C Likas ldquoA class-adaptivespatially variant mixture model for image segmentationrdquo IEEETransactions on Image Processing vol 16 no 4 pp 1121ndash11302007

[17] A R Ferreira da Silva ldquoA Dirichlet process mixture model forbrain MRI tissue classificationrdquoMedical Image Analysis vol 11no 2 pp 169ndash182 2007

[18] T M Nguyen and Q M J Wu ldquoFast and robust spatiallyconstrained gaussian mixture model for image segmentationrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 23 no 4 pp 621ndash635 2013

[19] S P Chatzis and T A Varvarigou ldquoA fuzzy clustering approachtoward hidden Markov random field models for enhancedspatially constrained image segmentationrdquo IEEE Transactionson Fuzzy Systems vol 16 no 5 pp 1351ndash1361 2008

[20] S Chatzis ldquoA method for training finite mixture models undera fuzzy clustering principlerdquo Fuzzy Sets and Systems vol 161 no23 pp 3000ndash3013 2010

[21] Z Ji J Liu G Cao Q Sun and Q Chen ldquoRobust spatiallyconstrained fuzzy c-means algorithm for brain MR imagesegmentationrdquo Pattern Recognition vol 47 no 7 pp 2454ndash2466 2014

[22] C Li C Xu A Anderson and J C Gore ldquoMRI tissueclassification and bias field estimation based on coherent localintensity clustering a unified energyminimization frameworkrdquoin Information Processing in Medical Imaging 21st InternationalConference IPMI 2009Williamsburg VA USA July 5ndash10 2009Proceedings vol 5636 of Lecture Notes in Computer Science pp288ndash299 Springer Berlin Germany 2009

[23] C Li J C Gore and C Davatzikos ldquoMultiplicative intrinsiccomponent optimization (MICO) for MRI bias field estimationand tissue segmentationrdquoMagnetic Resonance Imaging vol 32no 7 pp 913ndash923 2014

[24] L Szilagyi ldquoLessons to learn from a mistaken optimizationrdquoPattern Recognition Letters vol 36 no 1 pp 29ndash35 2014

[25] T Hofmann B Scholkopf and A J Smola ldquoKernel methodsin machine learningrdquoThe Annals of Statistics vol 36 no 3 pp1171ndash1220 2008

[26] K-R Muller S Mika G Ratsch K Tsuda and B ScholkopfldquoAn introduction to kernel-based learning algorithmsrdquo IEEETransactions onNeural Networks vol 12 no 2 pp 181ndash201 2001

[27] C R SouzaKernel Functions forMachine LearningApplications2010 httpcrsouzacom201003kernel-functions-for-machine-learning-applications

[28] U Vovk F Pernus and B Likar ldquoA review of methods for cor-rection of intensity inhomogeneity in MRIrdquo IEEE Transactionson Medical Imaging vol 26 no 3 pp 405ndash421 2007

[29] C A Cocosco V Kollokian R K-S Kwan and A C EvansldquoBrainWeb Online Interface to a 3D MRI Simulated BrainDatabaserdquo 2011 httpbrainwebbicmnimcgillcabrainweb

[30] L He and I R Greenshields ldquoA non-local maximum likelihoodestimation method for Rician noise reduction in MR imagesrdquoIEEE Transactions on Medical Imaging vol 28 no 2 pp 165ndash172 2009

[31] H Zhang J E Fritts and S A Goldman ldquoAn entropy-basedobjective evaluation method for image segmentationrdquo in Pro-ceedings of the Storage and Retrieval Methods and Applicationsfor Multimedia vol 5307 of Proceedings of SPIE pp 38ndash49January 2004

Computational and Mathematical Methods in Medicine 3

iteration to replace 120572 for every cluster Similar to FCM S1 andFCM S2 thismethodhas two formsGKFCM1 andGKFCM2for average and median filters respectively The parameter 120578

119895

is estimated using kernel functions

120578119895=

min1198951015840=119895(1 minus 119870 (V 119895 V119895))

max119896(1 minus 119870 (V

119896 119909))

(6)

where 119870 is the kernel function The replacement of 120572 with120578119895could yield better results than FCM S1 and FCM S2

However for good estimation of 120578119895 cluster centers should

be well separated which might not be always true hence thealgorithm has to iterate many times to converge Moreoverthe learning scheme requires a large number of patterns andmany cluster centers to find the optimal value for 120578

119895

To tackle the problem of parameter adjustment Krinidisand Chatzis [13] proposed the FLICM algorithm with afuzzy factor that combined both spatial and grayscale infor-mation of the neighboring pixels The fuzzy factor 119866

119894119895=

sum119896isin119873119894 119894 =119896

(1(1 + 119889119894119896))(1 minus 119906

119894119895)119898119909119894minus V1198952 was embedded into

(1) as follows

119869FLICM =

119873

sum

119894=1

119888

sum

119895=1

[119906119898

119894119895

10038171003817100381710038171003817119909119894minus V119895

10038171003817100381710038171003817

2

+ 119866119894119895] (7)

where pixel 119894 is the center of the local window pixel 119895 is inthe neighborhood and 119889

119894119896is the spatial Euclidean distance

between pixels 119894 and 119896Although FLICMalgorithm enhances robustness to noise

and artifacts it is slow since the fuzzy factor (119866119894119895) has to be

calculated in each iteration Moreover 119866119894119895is heavily affected

by spatial Euclidean distance from the central pixel to itsneighboring pixels to lose small image details due to thesmoothing effect

To enhance the FLICM algorithm Gong et al [14]developed KWFLICM algorithm with a trade-off weightedfuzzy factor to control the local neighbor relationship andreplaced the Euclidean distance with kernel function Theweighted fuzzy factor

119894119895of KWFLICM is

119894119895= sum

119896isin119873119894 119894 =119896

120596119894119896(1 minus 119906

119894119895)119898

(1 minus 119870 (119909119894 V119895)) (8)

where 120596119894119896is the trade-off weighted fuzzy factor of pixel 119896 in

the local window around the central pixel 119894 and 1 minus 119870(119909119894 V119895)

is the kernel metric function The trade-off weighted fuzzyfactor combines both the local spatial and grayscale informa-tion [14] Because of the trade-off weighted fuzzy factor itscomputational cost increases substantially In addition thealgorithm is unable to preserve small image details

In addition to the abovementioned shortcomingsSzilagyi [24] pointed out serious theoretical mistakes inFLICM and KWFLICM It was shown that the iterativeoptimization nature of FLICM and KWFLICM did notminimize their objective functions instead they iterateduntil the partition matrices converged Furthermore theirobjective functions intended to employ local contextualinformation but theoretically failed and were not evensuitable for creating a valid partition [24]

To this end a new way to modify the existing FCMclustering is explored with adaptive regularization for con-textual information The proposed framework utilizes a newparameter to control the effect of pixel neighbors based onthe heterogeneity of local grayscale distribution A weightedimage is devised that combines the local contextual infor-mation with respect to the heterogeneity of local grayscaledistribution and the original grayscale that is calculated oncein advance to reduce the computational cost To improvesegmentation accuracy and robustness to outliers a kernelfunction is employed to replace the Euclidean distance met-ric Validation against both synthetic and clinicalMRdata hasbeen carried out to compare the proposed algorithms with6 recent soft clustering algorithms in terms of segmentationaccuracy and computational costs

3 Proposed Algorithms

We introduce a regularizing parameter to enhance segmenta-tion robustness and preserve image details devise a weightedimage and adopt the Gaussian radial basis function (GRBF)for better accuracy

31 The Introduced Regularization Term The parameter 120572

used in [8ndash10] is usually set in advance to control the desirableamount of contextual information Indeed using a fixed 120572 forevery pixel is not appropriate since noise level differs fromonewindow to another In addition setting such parameter needsprior knowledge about noise which is not always availablein reality Hence adaptive calculation of 120572 is necessaryaccording to the pixel being processed

To be adaptive to noise level of the pixel being processedwe first calculate the local variation coefficient (LVC) toestimate the discrepancy of grayscales in the local window tobe normalized with respect to the local average grayscale Inthe presence of noise to have high heterogeneity between thecentral pixel and its neighbors LVC will increase Consider

LVC119894=

sum119896isin119873119894

(119909119896minus 119909119894)2

119873119877lowast (119909119894)2

(9)

where 119909119896is the grayscale of any pixel falling in the local win-

dow119873119894around the pixel 119894119873

119877is the cardinality of119873

119894 and 119909

119894

is its mean grayscale Next LVC119894is applied to an exponential

function to derive the weights within the local window

120577119894= exp( sum

119896isin119873119894 119894 =119896

LVC119896)

120596119894=

120577119894

sum119896isin119873119894

120577119896

(10)

The ultimate weight assigned to every pixel is associatedwith the average grayscale of the local window

120593119894=

2 + 120596119894 119909119894lt 119909119894

2 minus 120596119894 119909119894gt 119909119894

0 119909119894= 119909119894

(11)

4 Computational and Mathematical Methods in Medicine

(a)

30 30 30 30 30 3030 30 30 30 30 3030 30 30 165 30 3030 30 123 30 30 3030 30 30 30 30 3030 30 30 30 30 30

AB

C

(b)

0000 0000 0000 0000 0000 00000000 0000 1989 1957 1992 00000000 1960 1896 2623 1957 00000000 1952 2085 1896 1989 00000000 1978 1952 1960 0000 00000000 0000 0000 0000 0000 0000

(c)

1000 1000 1000 1000 1000 10001000 1000 1000 1000 1000 10001000 1000 1000 0989 1000 10001000 1000 0999 1000 1000 10001000 1000 1000 1000 1000 10001000 1000 1000 1000 1000 1000

(d)

Figure 1 Calculation and effect of the regularization parameter in different cases (a) Noisy image (b) 6times6 subimage red rectangle from (a)with 3 different windows A B and C (c) Weights associated with each pixel using the proposed method (d) Membership values after threeiterations

The parameter 120593119894assigns higher values for those pixels

with high LVC (for pixel 119894 being brighter than the averagegrayscale of its neighbors 120593

119894will be 2+120596

119894 and120596

119894will be large

when the sum of LVC within its neighborhood is large) andlower values otherwise When the local average grayscale isequal to the grayscale of the central pixel 120593

119894will be zero and

the algorithm will behave as the standard FCM algorithmThe value 2 in (11) is set through experiments to balancebetween the convergence rate and the capability to preservedetails The proposed parameter 120593

119894is embedded into (5) to

replace 120572 Figure 1 shows the calculation of 120593119894with different

cases of noiseHere are some remarks on the parameter 120593

119894

The first point to emphasize is that 120593119894is only relevant

to the grayscales within a specified neighborhood whichis very different from FCM S FLICM and KWFLICMwhere the contextual information is expressed respectivelybysum119909119903isin119873119894

119909119903minus V1198952 119866119894119895 and

119894119895 all containing a loop on the

neighborhood and the cluster center V119895 Due to its irrelevance

to clustering parameters 120593119894could be calculated in advance

before the clustering process which can greatly reduce thecomputational cost On the contrary FCM S FLICM andKWFLICMwill need to update the contextual weights at eachiteration which is the main reason why they have highercomputational cost

The second point is that the contextual informationprovided by 120593

119894is based on the heterogeneity of grayscale

distribution within the local neighborhood which is com-pletely different from existing enhanced versions of FCM tobase the contextual information on the difference betweenthe grayscales of neighboring pixels and cluster centers Asa result the proposed 120593

119894tends to yield a homogeneous

clustering according to local grayscale distribution whileexisting enhanced FCM algorithms tend to make the cluster-ing to have more homogeneous labels by incorporating thecontextual information

32 Devising a Weighted Image In addition to making 119909respectively the grayscale of averagemedian filter of theoriginal image 119909 can also be replaced with the grayscale ofthe newly formed weighted image 120585

120585119894=

1

2 +max (120593119894)(119909119894+

1 +max (120593119894)

119873119877minus 1

sum

119903isin119873119894

119909119903) (12)

where 119909119903and 119873

119894are respectively the grayscale and neigh-

borhood of pixel 119894 and 119873119877is the cardinality of 119873

119894 Formula

(12) is inspired by the weighted image in [9] but utilizes 120593119894

explicitly to make the weighted image free from parametersthat are difficult to adjust

33 Measuring Distance Using Kernel Function The Euclid-ean distance metric is generally simple and computationallyinexpensive but it is sensitive to perturbations and outliersRecently with popular usage of support vector machine anew direction appears to use kernel functions The kernelfunctions are able to project the data into higher dimensionalspace where the data could be more easily separated [25]To do this a so-called kernel trick has been adopted thatcan transform linear algorithm to nonlinear one using dotproduct [26] Using the kernel trick the Euclidean distance

Computational and Mathematical Methods in Medicine 5

term 119909119894minus V1198952 can be replaced with 120601(119909

119894) minus 120601(V

119895)2 that is

defined as10038171003817100381710038171003817120601 (119909119894) minus 120601 (V

119895)10038171003817100381710038171003817

2

= 119870 (119909119894 119909119894) + 119870 (V

119895 V119895)

minus 2119870 (119909119894 V119895)

(13)

where119870 is the kernel functionIn this paper we use GRBF kernel [27]

119870(119909119894 V119895) = exp(minus

10038171003817100381710038171003817119909119894minus V119895

10038171003817100381710038171003817

2

21205902) (14)

where 120590 is the kernel widthUsing GRBF the kernel function in (13) will be

10038171003817100381710038171003817120601 (119909119894) minus 120601 (V

119895)10038171003817100381710038171003817

2

= 2 (1 minus 119870 (119909119894 V119895)) (15)

The choice of the kernel width 120590 is still a problem andhas to be selected carefully If it is large the exponential effectwill be almost linear On the contrary if it is small the clusterboundaries will be sensitive to outliers [27] In [10] 120590 was setto a fixed value of 150 while in [12] the authors used samplevariance to estimate 120590 Similar to [14] we calculate 120590 basedon the distance variances among all pixels

120590 = [

[

sum119873

119894=1(119889119894minus 119889)2

119873 minus 1

]

]

12

(16)

where 119889119894= 119909119894minus 119909 is distance from the grayscale of pixel 119894

to the grayscale average of all pixels and 119889 is the average of alldistances 119889

119894

34 The Proposed Framework The proposed adaptively reg-ularized kernel-based FCM framework is denoted as ARK-FCM First we calculate the adaptive regularization param-eter 120593

119894associated with every pixel to control the contextual

information using (11) The objective function is defined as

119869ARKFCM = 2[

[

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+

119873

sum

119894=1

119888

sum

119895=1

120593119894119906119898

119894119895(1 minus 119870 (119909

119894 V119895))]

]

(17)

Under the conditions specified in (2) the minimizationof 119869ARKFCM(119906 V) can be calculated through an alternateoptimization procedure using (derivation is given in theAppendix)

119906119894119895

=

((1 minus 119870 (119909119894 V119895)) + 120593

119894(1 minus 119870 (119909

119894 V119895)))minus1(119898minus1)

sum119888

119896=1(1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))minus1(119898minus1)

(18)

V119895=

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) 119909119894+ 120593119894119870(119909119894 V119895) 119909119894)

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) + 120593119894119870(119909119894 V119895))

(19)

When 119909 is replaced with the grayscale of the averagemedianfilter of the original image the algorithm is denotedas ARKFCM

1ARKFCM

2 When 119909

119894is replaced with the

weighted image 120585119894defined in (12) the algorithm is denoted

as ARKFCM119908 The main steps for the proposed algorithms

are as follows

(1) Initialize threshold 120576 = 0001 119898 = 2 loop counter119905 = 0 V and 119906

(0)(2) Calculate the adaptive regularization parameter 120593

119894

(3) Calculate 119909119894for ARKFCM

1and ARKFCM

2or 120585 for

ARKFCM119908

(4) Calculate cluster centers V(119905)119895

using 119906(119905) as in (19)

(5) Calculate the membership function 119906(119905+1) with (18)

(6) If max 119906(119905+1)

minus 119906(119905) lt 120576 or 119905 gt 100 then stop

otherwise update 119905 = 119905 + 1 and go to step (4)

4 Experiments

In this section we present the experiments on both syn-thetic and clinical MR images For validation the proposedalgorithms (ARKFCM

1 ARKFCM

2 and ARKFCM

119908) are

compared with 6 recent soft clustering algorithms namelyGKFCM1 [12] GKFCM2 [12] FLICM [13] KWFLICM[14] MICO [21] and RSCFCM [23] As we are unable tohave faithful implementation of RSCFCM [23] RSCFCMis compared only against the common data with resultsreported from [23] The algorithms are implemented usingMATLAB software package (a demo version is freely availableonline (httpwwwmathworkscommatlabcentralfileex-change54141-arkfcm-algorithm)) All the experiments areconducted with window size of 3 times 3 pixels maximumnumber of iterations 119905 = 100 and 120576 = 0001 The accuracyof segmentation is measured using the Jaccard Similarity (JS)metric [28] which is defined as the ratio between the intersec-tion and union of segmented volume 119878

1and ground truth

volume 1198782

JS (1198781 1198782) =

10038161003816100381610038161198781 cap 1198782

100381610038161003816100381610038161003816100381610038161198781 cup 119878

2

1003816100381610038161003816

(20)

41 Experiments on Synthetic Brain MR Images The follow-ing experiments are carried out using Simulated Brain Data-base (SBD) [29] which contains a set of realistic MR volumesproduced by anMR imaging simulator with ground truths ofCSF GM and WM available

The first experiment is to segment a T1-weighted axialslice (number 100) with 217 times 181 pixels corrupted with 7noise and 20 grayscale nonuniformity into WM GM andCSF Figure 2 shows the segmentation results while Table 1summarizes the JS and average running times

The second experiment is to segment a T1-weightedsagittal slice (number 100) with 181 times 217 pixels corruptedwith 7 noise and 20 grayscale nonuniformity This imageis chosen to show the capability of preserving details

6 Computational and Mathematical Methods in Medicine

Table 1 JS and running time on the T1-weighted axial slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO RSCFCM ARKFCM1

ARKFCM2

ARKFCM119908

WM 0930 0933 0941 0937 0894 0898 0940 0941 0940GM 0822 0855 0860 0852 0782 0842 0864 0868 0865CSF 0781 0847 0829 0805 0860 0882 0863 0867 0867Average 0844 0879 0876 0865 0845 0874 0889 0892 0891Time (s) 0911 0586 3403 139470 0673 2530 0356 0282 0329

Table 2 JS and running time on the T1-weighted sagittal slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

WM 0773 0775 0771 0765 0672 0785 0788 0786GM 0796 0806 0794 0791 0669 0815 0816 0816CSF 0834 0852 0835 0824 0869 0871 0872 0875Average 0801 0811 0800 0794 0736 0824 0825 0825Time (s) 1377 1797 5392 166717 0751 0348 0329 0322

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

(g) (h) (i) (j) (k)

Figure 2 Segmentation results on a T1-weighted axial slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity (a)Original image (b) Ground truth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h)RSCFCM results (i) ARKFCM

1results (j) ARKFCM

2results (k) ARKFCM

119908results

The segmentation results and JS are presented in Figure 3 andTable 2 respectively

The third experiment is to check the robustness toRician noise that commonly affects MR images [30] Thesegmentation results and the JS with average running timesof a T1-weighted axial slice (number 91) with 217 times 181 pixelscorrupted with 10 Rician noise are given in Figure 4 andTable 3 respectively

42 Experiments on Clinical Brain MR Images with TumorsWe experimented two T1-weighted axial slices (slices num-bers 80 and 86 denoted resp as Brats1 and Brats2) with240 times 240 pixels respectively fromfiles pat266 1 and pat192 1(available fromMICCAI BRATS 2014 challenge httpswwwvirtualskeletonchBRATSStart2014) (Figures 5 and 6)

From black to white are respectively background CSF GMand WM It should be noted that clustering is carried outonly for CSF GM andWM with the pathology region beingconsidered as background

Since images Brats1 and Brats2 come only with groundtruth for the pathology and have no ground truth for normaltissues a quantificationmeasure other than JS should be usedTo this end an entropy-based metric proposed by Zhanget al [31] is adopted This is to maximize the uniformity ofpixels within each segmented region and to minimize theuniformity across the regions First the entropy of everyregion 119895 is calculated

119867(119877119895) = minus sum

119898isin119881119895

119871119895 (119898)

119878119895

log119871119895 (119898)

119878119895

(21)

Computational and Mathematical Methods in Medicine 7

Table 3 JS and running time on the T1-weighted axial slice (number 91) from SBD corrupted with 10 Rician noise

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

WM 0931 0921 0929 0927 0885 0926 0925 0922GM 0804 0827 0831 0821 0777 0838 0840 0834CSF 0826 0872 0861 0846 0889 0887 0892 0887Average 0850 0874 0874 0865 0850 0884 0886 0881Time (s) 1836 1869 2953 124922 0649 0218 0220 0218

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

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

(i) (j) (k)

Figure 3 Segmentation results on a T1-weighted sagittal slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity (a)Original image (b) Ground truth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h)RSCFCM results (i) ARKFCM

1results (j) ARKFCM

2results (k) ARKFCM

119908results

where119881119895is the set of all possible grayscales in region 119895 119871

119895(119898)

is the number of pixels belonging to region 119895 with grayscale119898 and 119878

119895is the area of region 119895 For the grayscale image 119868 the

119864measure is calculated as

119864 =

119888

sum

119895=1

(119878119895

119878119868

)119867(119877119895) + (minus

119888

sum

119895=1

(119878119895

119878119868

) log119878119895

119878119868

) (22)

where the first term represents the expected region entropy ofthe segmentation and the second term is the layout entropyThe measure 119864 yields smaller value for better segmentationand higher otherwise Table 4 summarizes the segmentationaccuracy in terms of 119864 and running times for Brats1 andBrats2 It should be noted that the 119864 metric is calculatedonly for WM GM and CSF regions without considering thebackground

5 Discussion

FCM clustering is a well-known soft clustering methodthat assigns a membership degree for each pixel to everycluster As the segmentation accuracy of FCM algorithmdecays in the presence of noise artifacts and increasednumber of clusters many investigations have been carriedout on using contextual information to enhance the qualityof segmentation But how the contextual information shall beused effectively is still a challenge

We propose an adaptively regularized kernel-based FCMclustering framework with new parameter 120593

119894that adaptively

controls the contextual information according to the hetero-geneity of grayscale distribution within the local neighbor-hood The new parameter is estimated using local variationcoefficient among pixels within a specified neighborhoodA weighted image is devised that combines the original

8 Computational and Mathematical Methods in Medicine

Table 4 Segmentation performance measure 119864 and running time on the Brats1 and Brats2 images

Image Measure GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

Brats1 E 1311 1288 1392 1424 1309 1274 1270 1279Time (s) 2591 1516 9510 635413 1747 1185 1111 1272

Brats2 E 1307 1297 1336 1340 1298 1273 1271 1279Time (s) 1576 1115 5170 400295 1546 1091 0827 0975

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

(f) (g) (h) (i) (j)

Figure 4 Segmentation results on a T1-weighted axial slice (number 91) from SBD with 10 Rician noise (a) Original image (b) Groundtruth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h) ARKFCM

1results (i)

ARKFCM2results (j) ARKFCM

119908results

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

(f) (g) (h) (i)

Figure 5 Segmentation results on the Brats1 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

Computational and Mathematical Methods in Medicine 9

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

(f) (g) (h) (i)

Figure 6 Segmentation results on the Brats2 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

image and the parameter 120593119894to represent the image contextual

information embedded through the weighting procedureFurthermore a GRBF is adopted to replace Euclidean dis-tance for better partitioning and to be less sensitive to outliersThe proposed framework can be in the form of 3 algorithmsARKFCM

1 ARKFCM

2 and ARKFCM

119908for the local average

grayscale to be replacedwith the grayscale of the average filtermedian filter and the devised weighted image respectivelyExperiments have been carefully carried out to show thesuperiority of the proposed algorithms in comparison with6 recent soft clustering algorithms to be discussed further

51 Segmentation Accuracy The differences in JS betweenthe proposed algorithms and the other 6 algorithms areclear for the axial slice with 7 noise and 20 grayscalenonuniformity (Table 1 and Figure 2) and become moredistinctive in preserving details in the sagittal slice corruptedwith the same noise (Table 2 and Figure 3) The average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0889 0892 and 0891 for the axial slice as shown in Figure 2which are better than the other 6 algorithms with range 13ndash48 (Table 1) For the sagittal slice with 7 noise and 20grayscale nonuniformity shown in Figure 3 the average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0824 0825 and 0825 which are better than the otheralgorithms with range 14ndash89 (Table 2) For the axial slicecorrupted with 10 Rician noise shown in Figure 4 theaverage JSs of ARKFCM

1 ARKFCM

2 and ARKFCM

119908are

respectively 0884 0886 and 0881 which are better than theother algorithms with range 120ndash36 (Table 3) The higherJSs may imply that the proposed algorithms achieve a betterbalance in preserving image details in the presence of noiseand grayscale inhomogeneity

For the clinical brain MR images with tumors (Brats1and Brats2) shown in Figures 5 and 6 the proposed algo-rithms achieve smaller 119864 than other algorithms (Table 4)For Brats1Brats2 image ARKFCM

2attains the lowest 119864 of

12701271 which is smaller than the other 5 algorithmswith ranges 0018ndash0154 and 0025ndash0109 respectively VisiblyGKFCM1 (Figures 5(b) and 6(b)) and GKFCM2 (Figures5(c) and 6(c)) come with good results but they are unableto preserve small details such as CSF being wrongly brokenpossibly due to the difficulty in estimating the parameter120578119895 On the other hand FLICM (Figures 5(d) and 6(d)) and

KWFLICM (Figures 5(e) and 6(e)) yield smooth results anddo not preserve many details which affects the segmentationaccuracy of CSF (CSF is small as compared with surroundingGM and WM) due to the effects of 119866

119894119895and

119894119895 respectively

For MICO algorithm (Figures 5(f) and 6(f)) the edges arenot smooth enough hence the CSF is not well preservedFinally results of ARKFCM

1 ARKFCM

2 and ARKFCM

119908

(Figures 5(g) and 6(g) Figures 5(h) and 6(h) and Figures 5(i)and 6(i)) show good balance between smooth borders andpreserving image details due to the introduction of adaptivelocal contextual information measure 120593

119894to replace the fixed

value of 120572 or the oversmoothing factors 119866119894119895or 119894119895

52 Computational Cost In terms of computational cost theobjective function of FCM algorithm in its original form[6] contains only the difference between the grayscale of thecurrent pixel 119894 and the cluster centers V

119895 This is basically

to cluster grayscales as there is no spatial information so ithas the smallest computational cost and can be implementedbased on grayscale histogram to reduce the computationalcost further [9 11] The enhancement of original FCM isto add the local contextual information to make it robust

10 Computational and Mathematical Methods in Medicine

to noise and image artifacts at the expense of increasedcomputational cost [7ndash14]

The GKFCM1 and GKFCM2 algorithms [12] use theparameter 120578

119895to replace 120572 and need an additional loop on

the number of clusters to be calculated at each pixel toupdate the local contextual information So they have highercomputational cost than the original FCM FCM S1 andFCM S2 at each iteration

The FLICM algorithm [13] introduces 119866119894119895which needs

an additional loop on the neighborhood of the currentpixel to calculate the local information in every iterationthus it has a high computational cost As an extension ofFLICM KWFLICM [14] introduces

119894119895that requires two

additional loops on the neighborhood so it has the highestcomputational cost at each iteration

The RSCFCM algorithm [21] uses a spatial fuzzy factorthat is constructed based on the posterior and prior prob-abilities and takes the spatial direction into account Thatincreases the complexity as many parameters have to beoptimized and consequently the computational complexity

MICO algorithm [23] comes with fast calculations dueto the convexity of its energy function particularly in thepresence of less noise but tends to have many iterations in thepresence of high level noise

The proposed algorithms incorporate the local contextualinformation by introducing LVC which is a measure ofgrayscale heterogeneity and has nothing to dowith the clustercenters So it can be calculated once in advance and hencereduce the complexity of the clustering procedure

The eventual computational cost will be the multipli-cation of the computational cost for each iteration andthe number of iterations for convergence The number ofiterations will be dependent on the initialization as well as theobjective function To make fair comparisons initializationsare all set randomly and the average number of iterationsfor convergence is then recorded for 10 converged timesFrom Figure 7 it can be seen that the average iterationtimes will be data dependent with ARKFCM

2 ARKFCM

119908

and ARKFCM1having the minimum number of iterations

followed by FLICM GKFCM2 and GKFCM1 respectivelyThe running times for the tested images agree well with

the above analysis as illustrated in Figure 8 For SBD imagesgiven in Figures 2 3 and 4 the KWFLICM algorithm takesthe longest time (124ndash166 seconds) followed by FLICM(3ndash54 seconds) RSCFCM (about 25 seconds) GKFCM1GKFCM2 (06ndash19 seconds) and the proposed algorithms(022 to 036 seconds) For the clinical MR images in Figures5 and 6 it takes more time (due to more iterations) asthe images are more complicated but the trend remainsthe same with the proposed algorithms having the lowestcomputational cost (Table 4 and Figure 8)

53 Neighborhood Size The local neighborhood window sizeis a crucial factor to determine the smoothness of clusteringand details to be preserved We have experimented withdifferent window sizes and found that a window size of3 times 3 pixels achieves the best balance between segmentationaccuracy and computational cost Increasing window size to5 times 5 pixels has very small impact on the JS but 7 times 7 pixels

0

20

40

60

80

100

120

GKF

CM1

GKF

CM2

FLIC

M

MIC

O

Num

ber o

f ite

ratio

ns

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 7 Average number of iterations of all the algorithms (exceptRSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a) and 6(a)

002040608

112141618

222242628

GKF

CM1

GKF

CM2

MIC

O

Aver

age r

unni

ng ti

me (

s)

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 8 Average running times of all the algorithms (exceptFLICM RSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a)and 6(a)

or more will significantly decrease the accuracy such aslosing image details like CSF Therefore it is recommendedto use a local window of size 3 times 3 pixels for constructing theneighborhood

54 Limitation From the experiments it was found thatthe proposed algorithms could be sensitive to severe noisein small areas between edges that have width 1 or 2 pixels(eg area between ventricles Figures 4(h) 4(i) and 4(j))Potentially this may be solved by embedding edge detectiontechnique into the clustering process which is yet to beexplored

Computational and Mathematical Methods in Medicine 11

6 Conclusion

An adaptively regularized kernel-based FCM framework hasbeen proposed to enhance the original FCM for highersegmentation accuracy at low computational costThe frame-work can be in the form of three algorithms that employ theheterogeneity of grayscales in the neighborhood employedfor local contextual information The main advantages areadaptiveness to local context enhanced robustness and inde-pendence of clustering parameters to decrease computationalcost GRBF kernel has been adopted as a distance metric Wevalidated the proposed algorithms on synthetic and clinicalMR images with tumors The proposed algorithms attain ahigher JS (Tables 1 2 and 3) and lower entropy measure119864 (Table 4) than the 6 recent soft clustering algorithms andcould preserve small image details (Figures 2 3 4 5 and 6) Inaddition the proposed algorithms have a low computationalcost and are to the best of our knowledge the only algorithmsthat are adaptive to local context and do not include clustercenters Therefore they attain a trade-off between highsegmentation accuracy and low computational costThe pro-posed algorithms can be a potential tool for segmenting brainMR images for further processing and other images as well

Appendix

The objective function 119869ARKFCM given in equation (17) withconditions in (2) is a constrained minimization problemwhich can be solved using Lagrange multiplier method Letthe minimization problem of 119869ARKFCM be written as follows

119871ARKFCM (119906119894119895 V119895 120582119894 120593119894)

= 2

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2

119873

sum

119894=1

119888

sum

119895=1

120593119894119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+

119873

sum

119894=1

120582119894(1 minus

119888

sum

119895=1

119906119894119895)

(A1)

By taking the derivative of 119871ARKFCM with respect to 119906119894119895and

setting the result to zero for119898 gt 1 we have

120597119871ARKFCM120597119906119894119895

= 2119898119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2119898120593119894119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895)) minus 120582

119894

= 0

(A2)

By using (A2) we can calculate 119906119894119895as follows

119906119894119895

= [120582119894

2119898 (1 minus 119870 (119909119894 V119895) + 120593119894(1 minus 119870 (119909

119894 V119895)))

]

1(119898minus1)

(A3)

Since sum119888119896=1

119906119894119896= 1 forallV

119896 from (A3) we can get

119888

sum

119896=1

[120582119894

2119898 (1 minus 119870 (119909119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))

]

1(119898minus1)

= 1 (A4)

120582119894

=1

[sum119888

119896=1(2119898 (1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896))))minus1(119898minus1)

]119898minus1

(A5)

By substituting (A5) into (A3) with necessary calculationsthe final 119906

119894119895is

119906119894119895

=

((1 minus 119870 (119909119894 V119895)) + 120593

119894(1 minus 119870 (119909

119894 V119895)))minus1(119898minus1)

sum119888

119896=1(1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))minus1(119898minus1)

(A6)

Similarly by taking the derivative of 119871ARKFCM with respect toV119895and setting the result to zero we have

120597119871ARKFCM120597V119895

=

119873

sum

119894=1

119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895)

+

119873

sum

119894=1

120593119894119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895) = 0

(A7)

After necessary calculations the final V119895is

V119895=

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) 119909119894+ 120593119894119870(119909119894 V119895) 119909119894)

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) + 120593119894119870(119909119894 V119895))

(A8)

Conflict of Interests

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

Acknowledgments

The authors would like to thank Dr Maoguo Gong andDr Chunming Li for providing their source codes forcomparison This work has been supported by the NationalProgram on Key Basic Research Project (nos 2013CB7338002012CB733803) Key Joint Program of National Natural Sci-ence Foundation and Guangdong Province (no U1201257)and Guangdong Innovative Research Team Program (no201001D0104648280)

References

[1] N Sharma and L M Aggarwal ldquoAutomated medical imagesegmentation techniquesrdquo Journal ofMedical Physics vol 35 no1 pp 3ndash14 2010

[2] A Elazab Q Hu F Jia and X Zhang ldquoContent based modifiedreaction-diffusion equation for modeling tumor growth of lowgrade gliomardquo in Proceedings of the 7th Cairo InternationalBiomedical Engineering Conference (CIBEC rsquo14) pp 107ndash110Giza Egypt December 2014

12 Computational and Mathematical Methods in Medicine

[3] D L PhamCXu and J L Prince ldquoCurrentmethods inmedicalimage segmentationrdquoAnnual Review of Biomedical Engineeringvol 2 pp 315ndash337 2000

[4] I Despotovic B Goossens andW Philips ldquoMRI segmentationof the human brain challenges methods and applicationsrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 450341 23 pages 2015

[5] S Ruan C Jaggi J Xue J Fadili and D Bloyet ldquoBraintissue classification of magnetic resonance images using partialvolume modelingrdquo IEEE Transactions on Medical Imaging vol19 no 12 pp 1172ndash1186 2000

[6] J C Dunn ldquoA fuzzy relative of the ISODATA process and itsuse in detecting compact well-separated clustersrdquo Journal ofCybernetics vol 3 no 3 pp 32ndash57 1973

[7] J C Bezdek Pattern Recognition with Fuzzy Objective FunctionAlgorithms Advanced Applications in Pattern RecognitionSpringer Boston Mass USA 1981

[8] M N Ahmed S M Yamany N Mohamed A A Farag andT Moriarty ldquoA modified fuzzy c-means algorithm for bias fieldestimation and segmentation of MRI datardquo IEEE Transactionson Medical Imaging vol 21 no 3 pp 193ndash199 2002

[9] L Szilagyi Z Benyo S M Szilagyi and H S Adam ldquoMRbrain image segmentation using an enhanced fuzzy C-meansalgorithmrdquo in Proceedings of the 25th Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety pp 724ndash726 Cancun Mexico September 2003

[10] S Chen andD Zhang ldquoRobust image segmentation using FCMwith spatial constraints based on new kernel-induced distancemeasurerdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 4 pp 1907ndash1916 2004

[11] W Cai S Chen and D Zhang ldquoFast and robust fuzzy c-meansclustering algorithms incorporating local information for imagesegmentationrdquo Pattern Recognition vol 40 no 3 pp 825ndash8382007

[12] M-S Yang and H-S Tsai ldquoA Gaussian kernel-based fuzzyc-means algorithm with a spatial bias correctionrdquo PatternRecognition Letters vol 29 no 12 pp 1713ndash1725 2008

[13] S Krinidis and V Chatzis ldquoA robust fuzzy local informationC-means clustering algorithmrdquo IEEE Transactions on ImageProcessing vol 19 no 5 pp 1328ndash1337 2010

[14] M Gong Y Liang J Shi W Ma and J Ma ldquoFuzzy C-meansclustering with local information and kernel metric for imagesegmentationrdquo IEEE Transactions on Image Processing vol 22no 2 pp 573ndash584 2013

[15] M A Balafar ldquoGaussian mixture model based segmentationmethods for brain MRI imagesrdquo Artificial Intelligence Reviewvol 41 no 3 pp 429ndash439 2014

[16] C Nikou N P Galatsanos and A C Likas ldquoA class-adaptivespatially variant mixture model for image segmentationrdquo IEEETransactions on Image Processing vol 16 no 4 pp 1121ndash11302007

[17] A R Ferreira da Silva ldquoA Dirichlet process mixture model forbrain MRI tissue classificationrdquoMedical Image Analysis vol 11no 2 pp 169ndash182 2007

[18] T M Nguyen and Q M J Wu ldquoFast and robust spatiallyconstrained gaussian mixture model for image segmentationrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 23 no 4 pp 621ndash635 2013

[19] S P Chatzis and T A Varvarigou ldquoA fuzzy clustering approachtoward hidden Markov random field models for enhancedspatially constrained image segmentationrdquo IEEE Transactionson Fuzzy Systems vol 16 no 5 pp 1351ndash1361 2008

[20] S Chatzis ldquoA method for training finite mixture models undera fuzzy clustering principlerdquo Fuzzy Sets and Systems vol 161 no23 pp 3000ndash3013 2010

[21] Z Ji J Liu G Cao Q Sun and Q Chen ldquoRobust spatiallyconstrained fuzzy c-means algorithm for brain MR imagesegmentationrdquo Pattern Recognition vol 47 no 7 pp 2454ndash2466 2014

[22] C Li C Xu A Anderson and J C Gore ldquoMRI tissueclassification and bias field estimation based on coherent localintensity clustering a unified energyminimization frameworkrdquoin Information Processing in Medical Imaging 21st InternationalConference IPMI 2009Williamsburg VA USA July 5ndash10 2009Proceedings vol 5636 of Lecture Notes in Computer Science pp288ndash299 Springer Berlin Germany 2009

[23] C Li J C Gore and C Davatzikos ldquoMultiplicative intrinsiccomponent optimization (MICO) for MRI bias field estimationand tissue segmentationrdquoMagnetic Resonance Imaging vol 32no 7 pp 913ndash923 2014

[24] L Szilagyi ldquoLessons to learn from a mistaken optimizationrdquoPattern Recognition Letters vol 36 no 1 pp 29ndash35 2014

[25] T Hofmann B Scholkopf and A J Smola ldquoKernel methodsin machine learningrdquoThe Annals of Statistics vol 36 no 3 pp1171ndash1220 2008

[26] K-R Muller S Mika G Ratsch K Tsuda and B ScholkopfldquoAn introduction to kernel-based learning algorithmsrdquo IEEETransactions onNeural Networks vol 12 no 2 pp 181ndash201 2001

[27] C R SouzaKernel Functions forMachine LearningApplications2010 httpcrsouzacom201003kernel-functions-for-machine-learning-applications

[28] U Vovk F Pernus and B Likar ldquoA review of methods for cor-rection of intensity inhomogeneity in MRIrdquo IEEE Transactionson Medical Imaging vol 26 no 3 pp 405ndash421 2007

[29] C A Cocosco V Kollokian R K-S Kwan and A C EvansldquoBrainWeb Online Interface to a 3D MRI Simulated BrainDatabaserdquo 2011 httpbrainwebbicmnimcgillcabrainweb

[30] L He and I R Greenshields ldquoA non-local maximum likelihoodestimation method for Rician noise reduction in MR imagesrdquoIEEE Transactions on Medical Imaging vol 28 no 2 pp 165ndash172 2009

[31] H Zhang J E Fritts and S A Goldman ldquoAn entropy-basedobjective evaluation method for image segmentationrdquo in Pro-ceedings of the Storage and Retrieval Methods and Applicationsfor Multimedia vol 5307 of Proceedings of SPIE pp 38ndash49January 2004

4 Computational and Mathematical Methods in Medicine

(a)

30 30 30 30 30 3030 30 30 30 30 3030 30 30 165 30 3030 30 123 30 30 3030 30 30 30 30 3030 30 30 30 30 30

AB

C

(b)

0000 0000 0000 0000 0000 00000000 0000 1989 1957 1992 00000000 1960 1896 2623 1957 00000000 1952 2085 1896 1989 00000000 1978 1952 1960 0000 00000000 0000 0000 0000 0000 0000

(c)

1000 1000 1000 1000 1000 10001000 1000 1000 1000 1000 10001000 1000 1000 0989 1000 10001000 1000 0999 1000 1000 10001000 1000 1000 1000 1000 10001000 1000 1000 1000 1000 1000

(d)

Figure 1 Calculation and effect of the regularization parameter in different cases (a) Noisy image (b) 6times6 subimage red rectangle from (a)with 3 different windows A B and C (c) Weights associated with each pixel using the proposed method (d) Membership values after threeiterations

The parameter 120593119894assigns higher values for those pixels

with high LVC (for pixel 119894 being brighter than the averagegrayscale of its neighbors 120593

119894will be 2+120596

119894 and120596

119894will be large

when the sum of LVC within its neighborhood is large) andlower values otherwise When the local average grayscale isequal to the grayscale of the central pixel 120593

119894will be zero and

the algorithm will behave as the standard FCM algorithmThe value 2 in (11) is set through experiments to balancebetween the convergence rate and the capability to preservedetails The proposed parameter 120593

119894is embedded into (5) to

replace 120572 Figure 1 shows the calculation of 120593119894with different

cases of noiseHere are some remarks on the parameter 120593

119894

The first point to emphasize is that 120593119894is only relevant

to the grayscales within a specified neighborhood whichis very different from FCM S FLICM and KWFLICMwhere the contextual information is expressed respectivelybysum119909119903isin119873119894

119909119903minus V1198952 119866119894119895 and

119894119895 all containing a loop on the

neighborhood and the cluster center V119895 Due to its irrelevance

to clustering parameters 120593119894could be calculated in advance

before the clustering process which can greatly reduce thecomputational cost On the contrary FCM S FLICM andKWFLICMwill need to update the contextual weights at eachiteration which is the main reason why they have highercomputational cost

The second point is that the contextual informationprovided by 120593

119894is based on the heterogeneity of grayscale

distribution within the local neighborhood which is com-pletely different from existing enhanced versions of FCM tobase the contextual information on the difference betweenthe grayscales of neighboring pixels and cluster centers Asa result the proposed 120593

119894tends to yield a homogeneous

clustering according to local grayscale distribution whileexisting enhanced FCM algorithms tend to make the cluster-ing to have more homogeneous labels by incorporating thecontextual information

32 Devising a Weighted Image In addition to making 119909respectively the grayscale of averagemedian filter of theoriginal image 119909 can also be replaced with the grayscale ofthe newly formed weighted image 120585

120585119894=

1

2 +max (120593119894)(119909119894+

1 +max (120593119894)

119873119877minus 1

sum

119903isin119873119894

119909119903) (12)

where 119909119903and 119873

119894are respectively the grayscale and neigh-

borhood of pixel 119894 and 119873119877is the cardinality of 119873

119894 Formula

(12) is inspired by the weighted image in [9] but utilizes 120593119894

explicitly to make the weighted image free from parametersthat are difficult to adjust

33 Measuring Distance Using Kernel Function The Euclid-ean distance metric is generally simple and computationallyinexpensive but it is sensitive to perturbations and outliersRecently with popular usage of support vector machine anew direction appears to use kernel functions The kernelfunctions are able to project the data into higher dimensionalspace where the data could be more easily separated [25]To do this a so-called kernel trick has been adopted thatcan transform linear algorithm to nonlinear one using dotproduct [26] Using the kernel trick the Euclidean distance

Computational and Mathematical Methods in Medicine 5

term 119909119894minus V1198952 can be replaced with 120601(119909

119894) minus 120601(V

119895)2 that is

defined as10038171003817100381710038171003817120601 (119909119894) minus 120601 (V

119895)10038171003817100381710038171003817

2

= 119870 (119909119894 119909119894) + 119870 (V

119895 V119895)

minus 2119870 (119909119894 V119895)

(13)

where119870 is the kernel functionIn this paper we use GRBF kernel [27]

119870(119909119894 V119895) = exp(minus

10038171003817100381710038171003817119909119894minus V119895

10038171003817100381710038171003817

2

21205902) (14)

where 120590 is the kernel widthUsing GRBF the kernel function in (13) will be

10038171003817100381710038171003817120601 (119909119894) minus 120601 (V

119895)10038171003817100381710038171003817

2

= 2 (1 minus 119870 (119909119894 V119895)) (15)

The choice of the kernel width 120590 is still a problem andhas to be selected carefully If it is large the exponential effectwill be almost linear On the contrary if it is small the clusterboundaries will be sensitive to outliers [27] In [10] 120590 was setto a fixed value of 150 while in [12] the authors used samplevariance to estimate 120590 Similar to [14] we calculate 120590 basedon the distance variances among all pixels

120590 = [

[

sum119873

119894=1(119889119894minus 119889)2

119873 minus 1

]

]

12

(16)

where 119889119894= 119909119894minus 119909 is distance from the grayscale of pixel 119894

to the grayscale average of all pixels and 119889 is the average of alldistances 119889

119894

34 The Proposed Framework The proposed adaptively reg-ularized kernel-based FCM framework is denoted as ARK-FCM First we calculate the adaptive regularization param-eter 120593

119894associated with every pixel to control the contextual

information using (11) The objective function is defined as

119869ARKFCM = 2[

[

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+

119873

sum

119894=1

119888

sum

119895=1

120593119894119906119898

119894119895(1 minus 119870 (119909

119894 V119895))]

]

(17)

Under the conditions specified in (2) the minimizationof 119869ARKFCM(119906 V) can be calculated through an alternateoptimization procedure using (derivation is given in theAppendix)

119906119894119895

=

((1 minus 119870 (119909119894 V119895)) + 120593

119894(1 minus 119870 (119909

119894 V119895)))minus1(119898minus1)

sum119888

119896=1(1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))minus1(119898minus1)

(18)

V119895=

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) 119909119894+ 120593119894119870(119909119894 V119895) 119909119894)

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) + 120593119894119870(119909119894 V119895))

(19)

When 119909 is replaced with the grayscale of the averagemedianfilter of the original image the algorithm is denotedas ARKFCM

1ARKFCM

2 When 119909

119894is replaced with the

weighted image 120585119894defined in (12) the algorithm is denoted

as ARKFCM119908 The main steps for the proposed algorithms

are as follows

(1) Initialize threshold 120576 = 0001 119898 = 2 loop counter119905 = 0 V and 119906

(0)(2) Calculate the adaptive regularization parameter 120593

119894

(3) Calculate 119909119894for ARKFCM

1and ARKFCM

2or 120585 for

ARKFCM119908

(4) Calculate cluster centers V(119905)119895

using 119906(119905) as in (19)

(5) Calculate the membership function 119906(119905+1) with (18)

(6) If max 119906(119905+1)

minus 119906(119905) lt 120576 or 119905 gt 100 then stop

otherwise update 119905 = 119905 + 1 and go to step (4)

4 Experiments

In this section we present the experiments on both syn-thetic and clinical MR images For validation the proposedalgorithms (ARKFCM

1 ARKFCM

2 and ARKFCM

119908) are

compared with 6 recent soft clustering algorithms namelyGKFCM1 [12] GKFCM2 [12] FLICM [13] KWFLICM[14] MICO [21] and RSCFCM [23] As we are unable tohave faithful implementation of RSCFCM [23] RSCFCMis compared only against the common data with resultsreported from [23] The algorithms are implemented usingMATLAB software package (a demo version is freely availableonline (httpwwwmathworkscommatlabcentralfileex-change54141-arkfcm-algorithm)) All the experiments areconducted with window size of 3 times 3 pixels maximumnumber of iterations 119905 = 100 and 120576 = 0001 The accuracyof segmentation is measured using the Jaccard Similarity (JS)metric [28] which is defined as the ratio between the intersec-tion and union of segmented volume 119878

1and ground truth

volume 1198782

JS (1198781 1198782) =

10038161003816100381610038161198781 cap 1198782

100381610038161003816100381610038161003816100381610038161198781 cup 119878

2

1003816100381610038161003816

(20)

41 Experiments on Synthetic Brain MR Images The follow-ing experiments are carried out using Simulated Brain Data-base (SBD) [29] which contains a set of realistic MR volumesproduced by anMR imaging simulator with ground truths ofCSF GM and WM available

The first experiment is to segment a T1-weighted axialslice (number 100) with 217 times 181 pixels corrupted with 7noise and 20 grayscale nonuniformity into WM GM andCSF Figure 2 shows the segmentation results while Table 1summarizes the JS and average running times

The second experiment is to segment a T1-weightedsagittal slice (number 100) with 181 times 217 pixels corruptedwith 7 noise and 20 grayscale nonuniformity This imageis chosen to show the capability of preserving details

6 Computational and Mathematical Methods in Medicine

Table 1 JS and running time on the T1-weighted axial slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO RSCFCM ARKFCM1

ARKFCM2

ARKFCM119908

WM 0930 0933 0941 0937 0894 0898 0940 0941 0940GM 0822 0855 0860 0852 0782 0842 0864 0868 0865CSF 0781 0847 0829 0805 0860 0882 0863 0867 0867Average 0844 0879 0876 0865 0845 0874 0889 0892 0891Time (s) 0911 0586 3403 139470 0673 2530 0356 0282 0329

Table 2 JS and running time on the T1-weighted sagittal slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

WM 0773 0775 0771 0765 0672 0785 0788 0786GM 0796 0806 0794 0791 0669 0815 0816 0816CSF 0834 0852 0835 0824 0869 0871 0872 0875Average 0801 0811 0800 0794 0736 0824 0825 0825Time (s) 1377 1797 5392 166717 0751 0348 0329 0322

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

(g) (h) (i) (j) (k)

Figure 2 Segmentation results on a T1-weighted axial slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity (a)Original image (b) Ground truth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h)RSCFCM results (i) ARKFCM

1results (j) ARKFCM

2results (k) ARKFCM

119908results

The segmentation results and JS are presented in Figure 3 andTable 2 respectively

The third experiment is to check the robustness toRician noise that commonly affects MR images [30] Thesegmentation results and the JS with average running timesof a T1-weighted axial slice (number 91) with 217 times 181 pixelscorrupted with 10 Rician noise are given in Figure 4 andTable 3 respectively

42 Experiments on Clinical Brain MR Images with TumorsWe experimented two T1-weighted axial slices (slices num-bers 80 and 86 denoted resp as Brats1 and Brats2) with240 times 240 pixels respectively fromfiles pat266 1 and pat192 1(available fromMICCAI BRATS 2014 challenge httpswwwvirtualskeletonchBRATSStart2014) (Figures 5 and 6)

From black to white are respectively background CSF GMand WM It should be noted that clustering is carried outonly for CSF GM andWM with the pathology region beingconsidered as background

Since images Brats1 and Brats2 come only with groundtruth for the pathology and have no ground truth for normaltissues a quantificationmeasure other than JS should be usedTo this end an entropy-based metric proposed by Zhanget al [31] is adopted This is to maximize the uniformity ofpixels within each segmented region and to minimize theuniformity across the regions First the entropy of everyregion 119895 is calculated

119867(119877119895) = minus sum

119898isin119881119895

119871119895 (119898)

119878119895

log119871119895 (119898)

119878119895

(21)

Computational and Mathematical Methods in Medicine 7

Table 3 JS and running time on the T1-weighted axial slice (number 91) from SBD corrupted with 10 Rician noise

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

WM 0931 0921 0929 0927 0885 0926 0925 0922GM 0804 0827 0831 0821 0777 0838 0840 0834CSF 0826 0872 0861 0846 0889 0887 0892 0887Average 0850 0874 0874 0865 0850 0884 0886 0881Time (s) 1836 1869 2953 124922 0649 0218 0220 0218

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

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

(i) (j) (k)

Figure 3 Segmentation results on a T1-weighted sagittal slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity (a)Original image (b) Ground truth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h)RSCFCM results (i) ARKFCM

1results (j) ARKFCM

2results (k) ARKFCM

119908results

where119881119895is the set of all possible grayscales in region 119895 119871

119895(119898)

is the number of pixels belonging to region 119895 with grayscale119898 and 119878

119895is the area of region 119895 For the grayscale image 119868 the

119864measure is calculated as

119864 =

119888

sum

119895=1

(119878119895

119878119868

)119867(119877119895) + (minus

119888

sum

119895=1

(119878119895

119878119868

) log119878119895

119878119868

) (22)

where the first term represents the expected region entropy ofthe segmentation and the second term is the layout entropyThe measure 119864 yields smaller value for better segmentationand higher otherwise Table 4 summarizes the segmentationaccuracy in terms of 119864 and running times for Brats1 andBrats2 It should be noted that the 119864 metric is calculatedonly for WM GM and CSF regions without considering thebackground

5 Discussion

FCM clustering is a well-known soft clustering methodthat assigns a membership degree for each pixel to everycluster As the segmentation accuracy of FCM algorithmdecays in the presence of noise artifacts and increasednumber of clusters many investigations have been carriedout on using contextual information to enhance the qualityof segmentation But how the contextual information shall beused effectively is still a challenge

We propose an adaptively regularized kernel-based FCMclustering framework with new parameter 120593

119894that adaptively

controls the contextual information according to the hetero-geneity of grayscale distribution within the local neighbor-hood The new parameter is estimated using local variationcoefficient among pixels within a specified neighborhoodA weighted image is devised that combines the original

8 Computational and Mathematical Methods in Medicine

Table 4 Segmentation performance measure 119864 and running time on the Brats1 and Brats2 images

Image Measure GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

Brats1 E 1311 1288 1392 1424 1309 1274 1270 1279Time (s) 2591 1516 9510 635413 1747 1185 1111 1272

Brats2 E 1307 1297 1336 1340 1298 1273 1271 1279Time (s) 1576 1115 5170 400295 1546 1091 0827 0975

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

(f) (g) (h) (i) (j)

Figure 4 Segmentation results on a T1-weighted axial slice (number 91) from SBD with 10 Rician noise (a) Original image (b) Groundtruth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h) ARKFCM

1results (i)

ARKFCM2results (j) ARKFCM

119908results

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

(f) (g) (h) (i)

Figure 5 Segmentation results on the Brats1 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

Computational and Mathematical Methods in Medicine 9

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

(f) (g) (h) (i)

Figure 6 Segmentation results on the Brats2 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

image and the parameter 120593119894to represent the image contextual

information embedded through the weighting procedureFurthermore a GRBF is adopted to replace Euclidean dis-tance for better partitioning and to be less sensitive to outliersThe proposed framework can be in the form of 3 algorithmsARKFCM

1 ARKFCM

2 and ARKFCM

119908for the local average

grayscale to be replacedwith the grayscale of the average filtermedian filter and the devised weighted image respectivelyExperiments have been carefully carried out to show thesuperiority of the proposed algorithms in comparison with6 recent soft clustering algorithms to be discussed further

51 Segmentation Accuracy The differences in JS betweenthe proposed algorithms and the other 6 algorithms areclear for the axial slice with 7 noise and 20 grayscalenonuniformity (Table 1 and Figure 2) and become moredistinctive in preserving details in the sagittal slice corruptedwith the same noise (Table 2 and Figure 3) The average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0889 0892 and 0891 for the axial slice as shown in Figure 2which are better than the other 6 algorithms with range 13ndash48 (Table 1) For the sagittal slice with 7 noise and 20grayscale nonuniformity shown in Figure 3 the average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0824 0825 and 0825 which are better than the otheralgorithms with range 14ndash89 (Table 2) For the axial slicecorrupted with 10 Rician noise shown in Figure 4 theaverage JSs of ARKFCM

1 ARKFCM

2 and ARKFCM

119908are

respectively 0884 0886 and 0881 which are better than theother algorithms with range 120ndash36 (Table 3) The higherJSs may imply that the proposed algorithms achieve a betterbalance in preserving image details in the presence of noiseand grayscale inhomogeneity

For the clinical brain MR images with tumors (Brats1and Brats2) shown in Figures 5 and 6 the proposed algo-rithms achieve smaller 119864 than other algorithms (Table 4)For Brats1Brats2 image ARKFCM

2attains the lowest 119864 of

12701271 which is smaller than the other 5 algorithmswith ranges 0018ndash0154 and 0025ndash0109 respectively VisiblyGKFCM1 (Figures 5(b) and 6(b)) and GKFCM2 (Figures5(c) and 6(c)) come with good results but they are unableto preserve small details such as CSF being wrongly brokenpossibly due to the difficulty in estimating the parameter120578119895 On the other hand FLICM (Figures 5(d) and 6(d)) and

KWFLICM (Figures 5(e) and 6(e)) yield smooth results anddo not preserve many details which affects the segmentationaccuracy of CSF (CSF is small as compared with surroundingGM and WM) due to the effects of 119866

119894119895and

119894119895 respectively

For MICO algorithm (Figures 5(f) and 6(f)) the edges arenot smooth enough hence the CSF is not well preservedFinally results of ARKFCM

1 ARKFCM

2 and ARKFCM

119908

(Figures 5(g) and 6(g) Figures 5(h) and 6(h) and Figures 5(i)and 6(i)) show good balance between smooth borders andpreserving image details due to the introduction of adaptivelocal contextual information measure 120593

119894to replace the fixed

value of 120572 or the oversmoothing factors 119866119894119895or 119894119895

52 Computational Cost In terms of computational cost theobjective function of FCM algorithm in its original form[6] contains only the difference between the grayscale of thecurrent pixel 119894 and the cluster centers V

119895 This is basically

to cluster grayscales as there is no spatial information so ithas the smallest computational cost and can be implementedbased on grayscale histogram to reduce the computationalcost further [9 11] The enhancement of original FCM isto add the local contextual information to make it robust

10 Computational and Mathematical Methods in Medicine

to noise and image artifacts at the expense of increasedcomputational cost [7ndash14]

The GKFCM1 and GKFCM2 algorithms [12] use theparameter 120578

119895to replace 120572 and need an additional loop on

the number of clusters to be calculated at each pixel toupdate the local contextual information So they have highercomputational cost than the original FCM FCM S1 andFCM S2 at each iteration

The FLICM algorithm [13] introduces 119866119894119895which needs

an additional loop on the neighborhood of the currentpixel to calculate the local information in every iterationthus it has a high computational cost As an extension ofFLICM KWFLICM [14] introduces

119894119895that requires two

additional loops on the neighborhood so it has the highestcomputational cost at each iteration

The RSCFCM algorithm [21] uses a spatial fuzzy factorthat is constructed based on the posterior and prior prob-abilities and takes the spatial direction into account Thatincreases the complexity as many parameters have to beoptimized and consequently the computational complexity

MICO algorithm [23] comes with fast calculations dueto the convexity of its energy function particularly in thepresence of less noise but tends to have many iterations in thepresence of high level noise

The proposed algorithms incorporate the local contextualinformation by introducing LVC which is a measure ofgrayscale heterogeneity and has nothing to dowith the clustercenters So it can be calculated once in advance and hencereduce the complexity of the clustering procedure

The eventual computational cost will be the multipli-cation of the computational cost for each iteration andthe number of iterations for convergence The number ofiterations will be dependent on the initialization as well as theobjective function To make fair comparisons initializationsare all set randomly and the average number of iterationsfor convergence is then recorded for 10 converged timesFrom Figure 7 it can be seen that the average iterationtimes will be data dependent with ARKFCM

2 ARKFCM

119908

and ARKFCM1having the minimum number of iterations

followed by FLICM GKFCM2 and GKFCM1 respectivelyThe running times for the tested images agree well with

the above analysis as illustrated in Figure 8 For SBD imagesgiven in Figures 2 3 and 4 the KWFLICM algorithm takesthe longest time (124ndash166 seconds) followed by FLICM(3ndash54 seconds) RSCFCM (about 25 seconds) GKFCM1GKFCM2 (06ndash19 seconds) and the proposed algorithms(022 to 036 seconds) For the clinical MR images in Figures5 and 6 it takes more time (due to more iterations) asthe images are more complicated but the trend remainsthe same with the proposed algorithms having the lowestcomputational cost (Table 4 and Figure 8)

53 Neighborhood Size The local neighborhood window sizeis a crucial factor to determine the smoothness of clusteringand details to be preserved We have experimented withdifferent window sizes and found that a window size of3 times 3 pixels achieves the best balance between segmentationaccuracy and computational cost Increasing window size to5 times 5 pixels has very small impact on the JS but 7 times 7 pixels

0

20

40

60

80

100

120

GKF

CM1

GKF

CM2

FLIC

M

MIC

O

Num

ber o

f ite

ratio

ns

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 7 Average number of iterations of all the algorithms (exceptRSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a) and 6(a)

002040608

112141618

222242628

GKF

CM1

GKF

CM2

MIC

O

Aver

age r

unni

ng ti

me (

s)

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 8 Average running times of all the algorithms (exceptFLICM RSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a)and 6(a)

or more will significantly decrease the accuracy such aslosing image details like CSF Therefore it is recommendedto use a local window of size 3 times 3 pixels for constructing theneighborhood

54 Limitation From the experiments it was found thatthe proposed algorithms could be sensitive to severe noisein small areas between edges that have width 1 or 2 pixels(eg area between ventricles Figures 4(h) 4(i) and 4(j))Potentially this may be solved by embedding edge detectiontechnique into the clustering process which is yet to beexplored

Computational and Mathematical Methods in Medicine 11

6 Conclusion

An adaptively regularized kernel-based FCM framework hasbeen proposed to enhance the original FCM for highersegmentation accuracy at low computational costThe frame-work can be in the form of three algorithms that employ theheterogeneity of grayscales in the neighborhood employedfor local contextual information The main advantages areadaptiveness to local context enhanced robustness and inde-pendence of clustering parameters to decrease computationalcost GRBF kernel has been adopted as a distance metric Wevalidated the proposed algorithms on synthetic and clinicalMR images with tumors The proposed algorithms attain ahigher JS (Tables 1 2 and 3) and lower entropy measure119864 (Table 4) than the 6 recent soft clustering algorithms andcould preserve small image details (Figures 2 3 4 5 and 6) Inaddition the proposed algorithms have a low computationalcost and are to the best of our knowledge the only algorithmsthat are adaptive to local context and do not include clustercenters Therefore they attain a trade-off between highsegmentation accuracy and low computational costThe pro-posed algorithms can be a potential tool for segmenting brainMR images for further processing and other images as well

Appendix

The objective function 119869ARKFCM given in equation (17) withconditions in (2) is a constrained minimization problemwhich can be solved using Lagrange multiplier method Letthe minimization problem of 119869ARKFCM be written as follows

119871ARKFCM (119906119894119895 V119895 120582119894 120593119894)

= 2

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2

119873

sum

119894=1

119888

sum

119895=1

120593119894119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+

119873

sum

119894=1

120582119894(1 minus

119888

sum

119895=1

119906119894119895)

(A1)

By taking the derivative of 119871ARKFCM with respect to 119906119894119895and

setting the result to zero for119898 gt 1 we have

120597119871ARKFCM120597119906119894119895

= 2119898119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2119898120593119894119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895)) minus 120582

119894

= 0

(A2)

By using (A2) we can calculate 119906119894119895as follows

119906119894119895

= [120582119894

2119898 (1 minus 119870 (119909119894 V119895) + 120593119894(1 minus 119870 (119909

119894 V119895)))

]

1(119898minus1)

(A3)

Since sum119888119896=1

119906119894119896= 1 forallV

119896 from (A3) we can get

119888

sum

119896=1

[120582119894

2119898 (1 minus 119870 (119909119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))

]

1(119898minus1)

= 1 (A4)

120582119894

=1

[sum119888

119896=1(2119898 (1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896))))minus1(119898minus1)

]119898minus1

(A5)

By substituting (A5) into (A3) with necessary calculationsthe final 119906

119894119895is

119906119894119895

=

((1 minus 119870 (119909119894 V119895)) + 120593

119894(1 minus 119870 (119909

119894 V119895)))minus1(119898minus1)

sum119888

119896=1(1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))minus1(119898minus1)

(A6)

Similarly by taking the derivative of 119871ARKFCM with respect toV119895and setting the result to zero we have

120597119871ARKFCM120597V119895

=

119873

sum

119894=1

119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895)

+

119873

sum

119894=1

120593119894119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895) = 0

(A7)

After necessary calculations the final V119895is

V119895=

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) 119909119894+ 120593119894119870(119909119894 V119895) 119909119894)

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) + 120593119894119870(119909119894 V119895))

(A8)

Conflict of Interests

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

Acknowledgments

The authors would like to thank Dr Maoguo Gong andDr Chunming Li for providing their source codes forcomparison This work has been supported by the NationalProgram on Key Basic Research Project (nos 2013CB7338002012CB733803) Key Joint Program of National Natural Sci-ence Foundation and Guangdong Province (no U1201257)and Guangdong Innovative Research Team Program (no201001D0104648280)

References

[1] N Sharma and L M Aggarwal ldquoAutomated medical imagesegmentation techniquesrdquo Journal ofMedical Physics vol 35 no1 pp 3ndash14 2010

[2] A Elazab Q Hu F Jia and X Zhang ldquoContent based modifiedreaction-diffusion equation for modeling tumor growth of lowgrade gliomardquo in Proceedings of the 7th Cairo InternationalBiomedical Engineering Conference (CIBEC rsquo14) pp 107ndash110Giza Egypt December 2014

12 Computational and Mathematical Methods in Medicine

[3] D L PhamCXu and J L Prince ldquoCurrentmethods inmedicalimage segmentationrdquoAnnual Review of Biomedical Engineeringvol 2 pp 315ndash337 2000

[4] I Despotovic B Goossens andW Philips ldquoMRI segmentationof the human brain challenges methods and applicationsrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 450341 23 pages 2015

[5] S Ruan C Jaggi J Xue J Fadili and D Bloyet ldquoBraintissue classification of magnetic resonance images using partialvolume modelingrdquo IEEE Transactions on Medical Imaging vol19 no 12 pp 1172ndash1186 2000

[6] J C Dunn ldquoA fuzzy relative of the ISODATA process and itsuse in detecting compact well-separated clustersrdquo Journal ofCybernetics vol 3 no 3 pp 32ndash57 1973

[7] J C Bezdek Pattern Recognition with Fuzzy Objective FunctionAlgorithms Advanced Applications in Pattern RecognitionSpringer Boston Mass USA 1981

[8] M N Ahmed S M Yamany N Mohamed A A Farag andT Moriarty ldquoA modified fuzzy c-means algorithm for bias fieldestimation and segmentation of MRI datardquo IEEE Transactionson Medical Imaging vol 21 no 3 pp 193ndash199 2002

[9] L Szilagyi Z Benyo S M Szilagyi and H S Adam ldquoMRbrain image segmentation using an enhanced fuzzy C-meansalgorithmrdquo in Proceedings of the 25th Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety pp 724ndash726 Cancun Mexico September 2003

[10] S Chen andD Zhang ldquoRobust image segmentation using FCMwith spatial constraints based on new kernel-induced distancemeasurerdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 4 pp 1907ndash1916 2004

[11] W Cai S Chen and D Zhang ldquoFast and robust fuzzy c-meansclustering algorithms incorporating local information for imagesegmentationrdquo Pattern Recognition vol 40 no 3 pp 825ndash8382007

[12] M-S Yang and H-S Tsai ldquoA Gaussian kernel-based fuzzyc-means algorithm with a spatial bias correctionrdquo PatternRecognition Letters vol 29 no 12 pp 1713ndash1725 2008

[13] S Krinidis and V Chatzis ldquoA robust fuzzy local informationC-means clustering algorithmrdquo IEEE Transactions on ImageProcessing vol 19 no 5 pp 1328ndash1337 2010

[14] M Gong Y Liang J Shi W Ma and J Ma ldquoFuzzy C-meansclustering with local information and kernel metric for imagesegmentationrdquo IEEE Transactions on Image Processing vol 22no 2 pp 573ndash584 2013

[15] M A Balafar ldquoGaussian mixture model based segmentationmethods for brain MRI imagesrdquo Artificial Intelligence Reviewvol 41 no 3 pp 429ndash439 2014

[16] C Nikou N P Galatsanos and A C Likas ldquoA class-adaptivespatially variant mixture model for image segmentationrdquo IEEETransactions on Image Processing vol 16 no 4 pp 1121ndash11302007

[17] A R Ferreira da Silva ldquoA Dirichlet process mixture model forbrain MRI tissue classificationrdquoMedical Image Analysis vol 11no 2 pp 169ndash182 2007

[18] T M Nguyen and Q M J Wu ldquoFast and robust spatiallyconstrained gaussian mixture model for image segmentationrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 23 no 4 pp 621ndash635 2013

[19] S P Chatzis and T A Varvarigou ldquoA fuzzy clustering approachtoward hidden Markov random field models for enhancedspatially constrained image segmentationrdquo IEEE Transactionson Fuzzy Systems vol 16 no 5 pp 1351ndash1361 2008

[20] S Chatzis ldquoA method for training finite mixture models undera fuzzy clustering principlerdquo Fuzzy Sets and Systems vol 161 no23 pp 3000ndash3013 2010

[21] Z Ji J Liu G Cao Q Sun and Q Chen ldquoRobust spatiallyconstrained fuzzy c-means algorithm for brain MR imagesegmentationrdquo Pattern Recognition vol 47 no 7 pp 2454ndash2466 2014

[22] C Li C Xu A Anderson and J C Gore ldquoMRI tissueclassification and bias field estimation based on coherent localintensity clustering a unified energyminimization frameworkrdquoin Information Processing in Medical Imaging 21st InternationalConference IPMI 2009Williamsburg VA USA July 5ndash10 2009Proceedings vol 5636 of Lecture Notes in Computer Science pp288ndash299 Springer Berlin Germany 2009

[23] C Li J C Gore and C Davatzikos ldquoMultiplicative intrinsiccomponent optimization (MICO) for MRI bias field estimationand tissue segmentationrdquoMagnetic Resonance Imaging vol 32no 7 pp 913ndash923 2014

[24] L Szilagyi ldquoLessons to learn from a mistaken optimizationrdquoPattern Recognition Letters vol 36 no 1 pp 29ndash35 2014

[25] T Hofmann B Scholkopf and A J Smola ldquoKernel methodsin machine learningrdquoThe Annals of Statistics vol 36 no 3 pp1171ndash1220 2008

[26] K-R Muller S Mika G Ratsch K Tsuda and B ScholkopfldquoAn introduction to kernel-based learning algorithmsrdquo IEEETransactions onNeural Networks vol 12 no 2 pp 181ndash201 2001

[27] C R SouzaKernel Functions forMachine LearningApplications2010 httpcrsouzacom201003kernel-functions-for-machine-learning-applications

[28] U Vovk F Pernus and B Likar ldquoA review of methods for cor-rection of intensity inhomogeneity in MRIrdquo IEEE Transactionson Medical Imaging vol 26 no 3 pp 405ndash421 2007

[29] C A Cocosco V Kollokian R K-S Kwan and A C EvansldquoBrainWeb Online Interface to a 3D MRI Simulated BrainDatabaserdquo 2011 httpbrainwebbicmnimcgillcabrainweb

[30] L He and I R Greenshields ldquoA non-local maximum likelihoodestimation method for Rician noise reduction in MR imagesrdquoIEEE Transactions on Medical Imaging vol 28 no 2 pp 165ndash172 2009

[31] H Zhang J E Fritts and S A Goldman ldquoAn entropy-basedobjective evaluation method for image segmentationrdquo in Pro-ceedings of the Storage and Retrieval Methods and Applicationsfor Multimedia vol 5307 of Proceedings of SPIE pp 38ndash49January 2004

Computational and Mathematical Methods in Medicine 5

term 119909119894minus V1198952 can be replaced with 120601(119909

119894) minus 120601(V

119895)2 that is

defined as10038171003817100381710038171003817120601 (119909119894) minus 120601 (V

119895)10038171003817100381710038171003817

2

= 119870 (119909119894 119909119894) + 119870 (V

119895 V119895)

minus 2119870 (119909119894 V119895)

(13)

where119870 is the kernel functionIn this paper we use GRBF kernel [27]

119870(119909119894 V119895) = exp(minus

10038171003817100381710038171003817119909119894minus V119895

10038171003817100381710038171003817

2

21205902) (14)

where 120590 is the kernel widthUsing GRBF the kernel function in (13) will be

10038171003817100381710038171003817120601 (119909119894) minus 120601 (V

119895)10038171003817100381710038171003817

2

= 2 (1 minus 119870 (119909119894 V119895)) (15)

The choice of the kernel width 120590 is still a problem andhas to be selected carefully If it is large the exponential effectwill be almost linear On the contrary if it is small the clusterboundaries will be sensitive to outliers [27] In [10] 120590 was setto a fixed value of 150 while in [12] the authors used samplevariance to estimate 120590 Similar to [14] we calculate 120590 basedon the distance variances among all pixels

120590 = [

[

sum119873

119894=1(119889119894minus 119889)2

119873 minus 1

]

]

12

(16)

where 119889119894= 119909119894minus 119909 is distance from the grayscale of pixel 119894

to the grayscale average of all pixels and 119889 is the average of alldistances 119889

119894

34 The Proposed Framework The proposed adaptively reg-ularized kernel-based FCM framework is denoted as ARK-FCM First we calculate the adaptive regularization param-eter 120593

119894associated with every pixel to control the contextual

information using (11) The objective function is defined as

119869ARKFCM = 2[

[

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+

119873

sum

119894=1

119888

sum

119895=1

120593119894119906119898

119894119895(1 minus 119870 (119909

119894 V119895))]

]

(17)

Under the conditions specified in (2) the minimizationof 119869ARKFCM(119906 V) can be calculated through an alternateoptimization procedure using (derivation is given in theAppendix)

119906119894119895

=

((1 minus 119870 (119909119894 V119895)) + 120593

119894(1 minus 119870 (119909

119894 V119895)))minus1(119898minus1)

sum119888

119896=1(1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))minus1(119898minus1)

(18)

V119895=

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) 119909119894+ 120593119894119870(119909119894 V119895) 119909119894)

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) + 120593119894119870(119909119894 V119895))

(19)

When 119909 is replaced with the grayscale of the averagemedianfilter of the original image the algorithm is denotedas ARKFCM

1ARKFCM

2 When 119909

119894is replaced with the

weighted image 120585119894defined in (12) the algorithm is denoted

as ARKFCM119908 The main steps for the proposed algorithms

are as follows

(1) Initialize threshold 120576 = 0001 119898 = 2 loop counter119905 = 0 V and 119906

(0)(2) Calculate the adaptive regularization parameter 120593

119894

(3) Calculate 119909119894for ARKFCM

1and ARKFCM

2or 120585 for

ARKFCM119908

(4) Calculate cluster centers V(119905)119895

using 119906(119905) as in (19)

(5) Calculate the membership function 119906(119905+1) with (18)

(6) If max 119906(119905+1)

minus 119906(119905) lt 120576 or 119905 gt 100 then stop

otherwise update 119905 = 119905 + 1 and go to step (4)

4 Experiments

In this section we present the experiments on both syn-thetic and clinical MR images For validation the proposedalgorithms (ARKFCM

1 ARKFCM

2 and ARKFCM

119908) are

compared with 6 recent soft clustering algorithms namelyGKFCM1 [12] GKFCM2 [12] FLICM [13] KWFLICM[14] MICO [21] and RSCFCM [23] As we are unable tohave faithful implementation of RSCFCM [23] RSCFCMis compared only against the common data with resultsreported from [23] The algorithms are implemented usingMATLAB software package (a demo version is freely availableonline (httpwwwmathworkscommatlabcentralfileex-change54141-arkfcm-algorithm)) All the experiments areconducted with window size of 3 times 3 pixels maximumnumber of iterations 119905 = 100 and 120576 = 0001 The accuracyof segmentation is measured using the Jaccard Similarity (JS)metric [28] which is defined as the ratio between the intersec-tion and union of segmented volume 119878

1and ground truth

volume 1198782

JS (1198781 1198782) =

10038161003816100381610038161198781 cap 1198782

100381610038161003816100381610038161003816100381610038161198781 cup 119878

2

1003816100381610038161003816

(20)

41 Experiments on Synthetic Brain MR Images The follow-ing experiments are carried out using Simulated Brain Data-base (SBD) [29] which contains a set of realistic MR volumesproduced by anMR imaging simulator with ground truths ofCSF GM and WM available

The first experiment is to segment a T1-weighted axialslice (number 100) with 217 times 181 pixels corrupted with 7noise and 20 grayscale nonuniformity into WM GM andCSF Figure 2 shows the segmentation results while Table 1summarizes the JS and average running times

The second experiment is to segment a T1-weightedsagittal slice (number 100) with 181 times 217 pixels corruptedwith 7 noise and 20 grayscale nonuniformity This imageis chosen to show the capability of preserving details

6 Computational and Mathematical Methods in Medicine

Table 1 JS and running time on the T1-weighted axial slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO RSCFCM ARKFCM1

ARKFCM2

ARKFCM119908

WM 0930 0933 0941 0937 0894 0898 0940 0941 0940GM 0822 0855 0860 0852 0782 0842 0864 0868 0865CSF 0781 0847 0829 0805 0860 0882 0863 0867 0867Average 0844 0879 0876 0865 0845 0874 0889 0892 0891Time (s) 0911 0586 3403 139470 0673 2530 0356 0282 0329

Table 2 JS and running time on the T1-weighted sagittal slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

WM 0773 0775 0771 0765 0672 0785 0788 0786GM 0796 0806 0794 0791 0669 0815 0816 0816CSF 0834 0852 0835 0824 0869 0871 0872 0875Average 0801 0811 0800 0794 0736 0824 0825 0825Time (s) 1377 1797 5392 166717 0751 0348 0329 0322

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

(g) (h) (i) (j) (k)

Figure 2 Segmentation results on a T1-weighted axial slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity (a)Original image (b) Ground truth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h)RSCFCM results (i) ARKFCM

1results (j) ARKFCM

2results (k) ARKFCM

119908results

The segmentation results and JS are presented in Figure 3 andTable 2 respectively

The third experiment is to check the robustness toRician noise that commonly affects MR images [30] Thesegmentation results and the JS with average running timesof a T1-weighted axial slice (number 91) with 217 times 181 pixelscorrupted with 10 Rician noise are given in Figure 4 andTable 3 respectively

42 Experiments on Clinical Brain MR Images with TumorsWe experimented two T1-weighted axial slices (slices num-bers 80 and 86 denoted resp as Brats1 and Brats2) with240 times 240 pixels respectively fromfiles pat266 1 and pat192 1(available fromMICCAI BRATS 2014 challenge httpswwwvirtualskeletonchBRATSStart2014) (Figures 5 and 6)

From black to white are respectively background CSF GMand WM It should be noted that clustering is carried outonly for CSF GM andWM with the pathology region beingconsidered as background

Since images Brats1 and Brats2 come only with groundtruth for the pathology and have no ground truth for normaltissues a quantificationmeasure other than JS should be usedTo this end an entropy-based metric proposed by Zhanget al [31] is adopted This is to maximize the uniformity ofpixels within each segmented region and to minimize theuniformity across the regions First the entropy of everyregion 119895 is calculated

119867(119877119895) = minus sum

119898isin119881119895

119871119895 (119898)

119878119895

log119871119895 (119898)

119878119895

(21)

Computational and Mathematical Methods in Medicine 7

Table 3 JS and running time on the T1-weighted axial slice (number 91) from SBD corrupted with 10 Rician noise

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

WM 0931 0921 0929 0927 0885 0926 0925 0922GM 0804 0827 0831 0821 0777 0838 0840 0834CSF 0826 0872 0861 0846 0889 0887 0892 0887Average 0850 0874 0874 0865 0850 0884 0886 0881Time (s) 1836 1869 2953 124922 0649 0218 0220 0218

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

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

(i) (j) (k)

Figure 3 Segmentation results on a T1-weighted sagittal slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity (a)Original image (b) Ground truth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h)RSCFCM results (i) ARKFCM

1results (j) ARKFCM

2results (k) ARKFCM

119908results

where119881119895is the set of all possible grayscales in region 119895 119871

119895(119898)

is the number of pixels belonging to region 119895 with grayscale119898 and 119878

119895is the area of region 119895 For the grayscale image 119868 the

119864measure is calculated as

119864 =

119888

sum

119895=1

(119878119895

119878119868

)119867(119877119895) + (minus

119888

sum

119895=1

(119878119895

119878119868

) log119878119895

119878119868

) (22)

where the first term represents the expected region entropy ofthe segmentation and the second term is the layout entropyThe measure 119864 yields smaller value for better segmentationand higher otherwise Table 4 summarizes the segmentationaccuracy in terms of 119864 and running times for Brats1 andBrats2 It should be noted that the 119864 metric is calculatedonly for WM GM and CSF regions without considering thebackground

5 Discussion

FCM clustering is a well-known soft clustering methodthat assigns a membership degree for each pixel to everycluster As the segmentation accuracy of FCM algorithmdecays in the presence of noise artifacts and increasednumber of clusters many investigations have been carriedout on using contextual information to enhance the qualityof segmentation But how the contextual information shall beused effectively is still a challenge

We propose an adaptively regularized kernel-based FCMclustering framework with new parameter 120593

119894that adaptively

controls the contextual information according to the hetero-geneity of grayscale distribution within the local neighbor-hood The new parameter is estimated using local variationcoefficient among pixels within a specified neighborhoodA weighted image is devised that combines the original

8 Computational and Mathematical Methods in Medicine

Table 4 Segmentation performance measure 119864 and running time on the Brats1 and Brats2 images

Image Measure GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

Brats1 E 1311 1288 1392 1424 1309 1274 1270 1279Time (s) 2591 1516 9510 635413 1747 1185 1111 1272

Brats2 E 1307 1297 1336 1340 1298 1273 1271 1279Time (s) 1576 1115 5170 400295 1546 1091 0827 0975

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

(f) (g) (h) (i) (j)

Figure 4 Segmentation results on a T1-weighted axial slice (number 91) from SBD with 10 Rician noise (a) Original image (b) Groundtruth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h) ARKFCM

1results (i)

ARKFCM2results (j) ARKFCM

119908results

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

(f) (g) (h) (i)

Figure 5 Segmentation results on the Brats1 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

Computational and Mathematical Methods in Medicine 9

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

(f) (g) (h) (i)

Figure 6 Segmentation results on the Brats2 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

image and the parameter 120593119894to represent the image contextual

information embedded through the weighting procedureFurthermore a GRBF is adopted to replace Euclidean dis-tance for better partitioning and to be less sensitive to outliersThe proposed framework can be in the form of 3 algorithmsARKFCM

1 ARKFCM

2 and ARKFCM

119908for the local average

grayscale to be replacedwith the grayscale of the average filtermedian filter and the devised weighted image respectivelyExperiments have been carefully carried out to show thesuperiority of the proposed algorithms in comparison with6 recent soft clustering algorithms to be discussed further

51 Segmentation Accuracy The differences in JS betweenthe proposed algorithms and the other 6 algorithms areclear for the axial slice with 7 noise and 20 grayscalenonuniformity (Table 1 and Figure 2) and become moredistinctive in preserving details in the sagittal slice corruptedwith the same noise (Table 2 and Figure 3) The average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0889 0892 and 0891 for the axial slice as shown in Figure 2which are better than the other 6 algorithms with range 13ndash48 (Table 1) For the sagittal slice with 7 noise and 20grayscale nonuniformity shown in Figure 3 the average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0824 0825 and 0825 which are better than the otheralgorithms with range 14ndash89 (Table 2) For the axial slicecorrupted with 10 Rician noise shown in Figure 4 theaverage JSs of ARKFCM

1 ARKFCM

2 and ARKFCM

119908are

respectively 0884 0886 and 0881 which are better than theother algorithms with range 120ndash36 (Table 3) The higherJSs may imply that the proposed algorithms achieve a betterbalance in preserving image details in the presence of noiseand grayscale inhomogeneity

For the clinical brain MR images with tumors (Brats1and Brats2) shown in Figures 5 and 6 the proposed algo-rithms achieve smaller 119864 than other algorithms (Table 4)For Brats1Brats2 image ARKFCM

2attains the lowest 119864 of

12701271 which is smaller than the other 5 algorithmswith ranges 0018ndash0154 and 0025ndash0109 respectively VisiblyGKFCM1 (Figures 5(b) and 6(b)) and GKFCM2 (Figures5(c) and 6(c)) come with good results but they are unableto preserve small details such as CSF being wrongly brokenpossibly due to the difficulty in estimating the parameter120578119895 On the other hand FLICM (Figures 5(d) and 6(d)) and

KWFLICM (Figures 5(e) and 6(e)) yield smooth results anddo not preserve many details which affects the segmentationaccuracy of CSF (CSF is small as compared with surroundingGM and WM) due to the effects of 119866

119894119895and

119894119895 respectively

For MICO algorithm (Figures 5(f) and 6(f)) the edges arenot smooth enough hence the CSF is not well preservedFinally results of ARKFCM

1 ARKFCM

2 and ARKFCM

119908

(Figures 5(g) and 6(g) Figures 5(h) and 6(h) and Figures 5(i)and 6(i)) show good balance between smooth borders andpreserving image details due to the introduction of adaptivelocal contextual information measure 120593

119894to replace the fixed

value of 120572 or the oversmoothing factors 119866119894119895or 119894119895

52 Computational Cost In terms of computational cost theobjective function of FCM algorithm in its original form[6] contains only the difference between the grayscale of thecurrent pixel 119894 and the cluster centers V

119895 This is basically

to cluster grayscales as there is no spatial information so ithas the smallest computational cost and can be implementedbased on grayscale histogram to reduce the computationalcost further [9 11] The enhancement of original FCM isto add the local contextual information to make it robust

10 Computational and Mathematical Methods in Medicine

to noise and image artifacts at the expense of increasedcomputational cost [7ndash14]

The GKFCM1 and GKFCM2 algorithms [12] use theparameter 120578

119895to replace 120572 and need an additional loop on

the number of clusters to be calculated at each pixel toupdate the local contextual information So they have highercomputational cost than the original FCM FCM S1 andFCM S2 at each iteration

The FLICM algorithm [13] introduces 119866119894119895which needs

an additional loop on the neighborhood of the currentpixel to calculate the local information in every iterationthus it has a high computational cost As an extension ofFLICM KWFLICM [14] introduces

119894119895that requires two

additional loops on the neighborhood so it has the highestcomputational cost at each iteration

The RSCFCM algorithm [21] uses a spatial fuzzy factorthat is constructed based on the posterior and prior prob-abilities and takes the spatial direction into account Thatincreases the complexity as many parameters have to beoptimized and consequently the computational complexity

MICO algorithm [23] comes with fast calculations dueto the convexity of its energy function particularly in thepresence of less noise but tends to have many iterations in thepresence of high level noise

The proposed algorithms incorporate the local contextualinformation by introducing LVC which is a measure ofgrayscale heterogeneity and has nothing to dowith the clustercenters So it can be calculated once in advance and hencereduce the complexity of the clustering procedure

The eventual computational cost will be the multipli-cation of the computational cost for each iteration andthe number of iterations for convergence The number ofiterations will be dependent on the initialization as well as theobjective function To make fair comparisons initializationsare all set randomly and the average number of iterationsfor convergence is then recorded for 10 converged timesFrom Figure 7 it can be seen that the average iterationtimes will be data dependent with ARKFCM

2 ARKFCM

119908

and ARKFCM1having the minimum number of iterations

followed by FLICM GKFCM2 and GKFCM1 respectivelyThe running times for the tested images agree well with

the above analysis as illustrated in Figure 8 For SBD imagesgiven in Figures 2 3 and 4 the KWFLICM algorithm takesthe longest time (124ndash166 seconds) followed by FLICM(3ndash54 seconds) RSCFCM (about 25 seconds) GKFCM1GKFCM2 (06ndash19 seconds) and the proposed algorithms(022 to 036 seconds) For the clinical MR images in Figures5 and 6 it takes more time (due to more iterations) asthe images are more complicated but the trend remainsthe same with the proposed algorithms having the lowestcomputational cost (Table 4 and Figure 8)

53 Neighborhood Size The local neighborhood window sizeis a crucial factor to determine the smoothness of clusteringand details to be preserved We have experimented withdifferent window sizes and found that a window size of3 times 3 pixels achieves the best balance between segmentationaccuracy and computational cost Increasing window size to5 times 5 pixels has very small impact on the JS but 7 times 7 pixels

0

20

40

60

80

100

120

GKF

CM1

GKF

CM2

FLIC

M

MIC

O

Num

ber o

f ite

ratio

ns

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 7 Average number of iterations of all the algorithms (exceptRSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a) and 6(a)

002040608

112141618

222242628

GKF

CM1

GKF

CM2

MIC

O

Aver

age r

unni

ng ti

me (

s)

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 8 Average running times of all the algorithms (exceptFLICM RSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a)and 6(a)

or more will significantly decrease the accuracy such aslosing image details like CSF Therefore it is recommendedto use a local window of size 3 times 3 pixels for constructing theneighborhood

54 Limitation From the experiments it was found thatthe proposed algorithms could be sensitive to severe noisein small areas between edges that have width 1 or 2 pixels(eg area between ventricles Figures 4(h) 4(i) and 4(j))Potentially this may be solved by embedding edge detectiontechnique into the clustering process which is yet to beexplored

Computational and Mathematical Methods in Medicine 11

6 Conclusion

An adaptively regularized kernel-based FCM framework hasbeen proposed to enhance the original FCM for highersegmentation accuracy at low computational costThe frame-work can be in the form of three algorithms that employ theheterogeneity of grayscales in the neighborhood employedfor local contextual information The main advantages areadaptiveness to local context enhanced robustness and inde-pendence of clustering parameters to decrease computationalcost GRBF kernel has been adopted as a distance metric Wevalidated the proposed algorithms on synthetic and clinicalMR images with tumors The proposed algorithms attain ahigher JS (Tables 1 2 and 3) and lower entropy measure119864 (Table 4) than the 6 recent soft clustering algorithms andcould preserve small image details (Figures 2 3 4 5 and 6) Inaddition the proposed algorithms have a low computationalcost and are to the best of our knowledge the only algorithmsthat are adaptive to local context and do not include clustercenters Therefore they attain a trade-off between highsegmentation accuracy and low computational costThe pro-posed algorithms can be a potential tool for segmenting brainMR images for further processing and other images as well

Appendix

The objective function 119869ARKFCM given in equation (17) withconditions in (2) is a constrained minimization problemwhich can be solved using Lagrange multiplier method Letthe minimization problem of 119869ARKFCM be written as follows

119871ARKFCM (119906119894119895 V119895 120582119894 120593119894)

= 2

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2

119873

sum

119894=1

119888

sum

119895=1

120593119894119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+

119873

sum

119894=1

120582119894(1 minus

119888

sum

119895=1

119906119894119895)

(A1)

By taking the derivative of 119871ARKFCM with respect to 119906119894119895and

setting the result to zero for119898 gt 1 we have

120597119871ARKFCM120597119906119894119895

= 2119898119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2119898120593119894119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895)) minus 120582

119894

= 0

(A2)

By using (A2) we can calculate 119906119894119895as follows

119906119894119895

= [120582119894

2119898 (1 minus 119870 (119909119894 V119895) + 120593119894(1 minus 119870 (119909

119894 V119895)))

]

1(119898minus1)

(A3)

Since sum119888119896=1

119906119894119896= 1 forallV

119896 from (A3) we can get

119888

sum

119896=1

[120582119894

2119898 (1 minus 119870 (119909119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))

]

1(119898minus1)

= 1 (A4)

120582119894

=1

[sum119888

119896=1(2119898 (1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896))))minus1(119898minus1)

]119898minus1

(A5)

By substituting (A5) into (A3) with necessary calculationsthe final 119906

119894119895is

119906119894119895

=

((1 minus 119870 (119909119894 V119895)) + 120593

119894(1 minus 119870 (119909

119894 V119895)))minus1(119898minus1)

sum119888

119896=1(1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))minus1(119898minus1)

(A6)

Similarly by taking the derivative of 119871ARKFCM with respect toV119895and setting the result to zero we have

120597119871ARKFCM120597V119895

=

119873

sum

119894=1

119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895)

+

119873

sum

119894=1

120593119894119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895) = 0

(A7)

After necessary calculations the final V119895is

V119895=

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) 119909119894+ 120593119894119870(119909119894 V119895) 119909119894)

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) + 120593119894119870(119909119894 V119895))

(A8)

Conflict of Interests

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

Acknowledgments

The authors would like to thank Dr Maoguo Gong andDr Chunming Li for providing their source codes forcomparison This work has been supported by the NationalProgram on Key Basic Research Project (nos 2013CB7338002012CB733803) Key Joint Program of National Natural Sci-ence Foundation and Guangdong Province (no U1201257)and Guangdong Innovative Research Team Program (no201001D0104648280)

References

[1] N Sharma and L M Aggarwal ldquoAutomated medical imagesegmentation techniquesrdquo Journal ofMedical Physics vol 35 no1 pp 3ndash14 2010

[2] A Elazab Q Hu F Jia and X Zhang ldquoContent based modifiedreaction-diffusion equation for modeling tumor growth of lowgrade gliomardquo in Proceedings of the 7th Cairo InternationalBiomedical Engineering Conference (CIBEC rsquo14) pp 107ndash110Giza Egypt December 2014

12 Computational and Mathematical Methods in Medicine

[3] D L PhamCXu and J L Prince ldquoCurrentmethods inmedicalimage segmentationrdquoAnnual Review of Biomedical Engineeringvol 2 pp 315ndash337 2000

[4] I Despotovic B Goossens andW Philips ldquoMRI segmentationof the human brain challenges methods and applicationsrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 450341 23 pages 2015

[5] S Ruan C Jaggi J Xue J Fadili and D Bloyet ldquoBraintissue classification of magnetic resonance images using partialvolume modelingrdquo IEEE Transactions on Medical Imaging vol19 no 12 pp 1172ndash1186 2000

[6] J C Dunn ldquoA fuzzy relative of the ISODATA process and itsuse in detecting compact well-separated clustersrdquo Journal ofCybernetics vol 3 no 3 pp 32ndash57 1973

[7] J C Bezdek Pattern Recognition with Fuzzy Objective FunctionAlgorithms Advanced Applications in Pattern RecognitionSpringer Boston Mass USA 1981

[8] M N Ahmed S M Yamany N Mohamed A A Farag andT Moriarty ldquoA modified fuzzy c-means algorithm for bias fieldestimation and segmentation of MRI datardquo IEEE Transactionson Medical Imaging vol 21 no 3 pp 193ndash199 2002

[9] L Szilagyi Z Benyo S M Szilagyi and H S Adam ldquoMRbrain image segmentation using an enhanced fuzzy C-meansalgorithmrdquo in Proceedings of the 25th Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety pp 724ndash726 Cancun Mexico September 2003

[10] S Chen andD Zhang ldquoRobust image segmentation using FCMwith spatial constraints based on new kernel-induced distancemeasurerdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 4 pp 1907ndash1916 2004

[11] W Cai S Chen and D Zhang ldquoFast and robust fuzzy c-meansclustering algorithms incorporating local information for imagesegmentationrdquo Pattern Recognition vol 40 no 3 pp 825ndash8382007

[12] M-S Yang and H-S Tsai ldquoA Gaussian kernel-based fuzzyc-means algorithm with a spatial bias correctionrdquo PatternRecognition Letters vol 29 no 12 pp 1713ndash1725 2008

[13] S Krinidis and V Chatzis ldquoA robust fuzzy local informationC-means clustering algorithmrdquo IEEE Transactions on ImageProcessing vol 19 no 5 pp 1328ndash1337 2010

[14] M Gong Y Liang J Shi W Ma and J Ma ldquoFuzzy C-meansclustering with local information and kernel metric for imagesegmentationrdquo IEEE Transactions on Image Processing vol 22no 2 pp 573ndash584 2013

[15] M A Balafar ldquoGaussian mixture model based segmentationmethods for brain MRI imagesrdquo Artificial Intelligence Reviewvol 41 no 3 pp 429ndash439 2014

[16] C Nikou N P Galatsanos and A C Likas ldquoA class-adaptivespatially variant mixture model for image segmentationrdquo IEEETransactions on Image Processing vol 16 no 4 pp 1121ndash11302007

[17] A R Ferreira da Silva ldquoA Dirichlet process mixture model forbrain MRI tissue classificationrdquoMedical Image Analysis vol 11no 2 pp 169ndash182 2007

[18] T M Nguyen and Q M J Wu ldquoFast and robust spatiallyconstrained gaussian mixture model for image segmentationrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 23 no 4 pp 621ndash635 2013

[19] S P Chatzis and T A Varvarigou ldquoA fuzzy clustering approachtoward hidden Markov random field models for enhancedspatially constrained image segmentationrdquo IEEE Transactionson Fuzzy Systems vol 16 no 5 pp 1351ndash1361 2008

[20] S Chatzis ldquoA method for training finite mixture models undera fuzzy clustering principlerdquo Fuzzy Sets and Systems vol 161 no23 pp 3000ndash3013 2010

[21] Z Ji J Liu G Cao Q Sun and Q Chen ldquoRobust spatiallyconstrained fuzzy c-means algorithm for brain MR imagesegmentationrdquo Pattern Recognition vol 47 no 7 pp 2454ndash2466 2014

[22] C Li C Xu A Anderson and J C Gore ldquoMRI tissueclassification and bias field estimation based on coherent localintensity clustering a unified energyminimization frameworkrdquoin Information Processing in Medical Imaging 21st InternationalConference IPMI 2009Williamsburg VA USA July 5ndash10 2009Proceedings vol 5636 of Lecture Notes in Computer Science pp288ndash299 Springer Berlin Germany 2009

[23] C Li J C Gore and C Davatzikos ldquoMultiplicative intrinsiccomponent optimization (MICO) for MRI bias field estimationand tissue segmentationrdquoMagnetic Resonance Imaging vol 32no 7 pp 913ndash923 2014

[24] L Szilagyi ldquoLessons to learn from a mistaken optimizationrdquoPattern Recognition Letters vol 36 no 1 pp 29ndash35 2014

[25] T Hofmann B Scholkopf and A J Smola ldquoKernel methodsin machine learningrdquoThe Annals of Statistics vol 36 no 3 pp1171ndash1220 2008

[26] K-R Muller S Mika G Ratsch K Tsuda and B ScholkopfldquoAn introduction to kernel-based learning algorithmsrdquo IEEETransactions onNeural Networks vol 12 no 2 pp 181ndash201 2001

[27] C R SouzaKernel Functions forMachine LearningApplications2010 httpcrsouzacom201003kernel-functions-for-machine-learning-applications

[28] U Vovk F Pernus and B Likar ldquoA review of methods for cor-rection of intensity inhomogeneity in MRIrdquo IEEE Transactionson Medical Imaging vol 26 no 3 pp 405ndash421 2007

[29] C A Cocosco V Kollokian R K-S Kwan and A C EvansldquoBrainWeb Online Interface to a 3D MRI Simulated BrainDatabaserdquo 2011 httpbrainwebbicmnimcgillcabrainweb

[30] L He and I R Greenshields ldquoA non-local maximum likelihoodestimation method for Rician noise reduction in MR imagesrdquoIEEE Transactions on Medical Imaging vol 28 no 2 pp 165ndash172 2009

[31] H Zhang J E Fritts and S A Goldman ldquoAn entropy-basedobjective evaluation method for image segmentationrdquo in Pro-ceedings of the Storage and Retrieval Methods and Applicationsfor Multimedia vol 5307 of Proceedings of SPIE pp 38ndash49January 2004

6 Computational and Mathematical Methods in Medicine

Table 1 JS and running time on the T1-weighted axial slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO RSCFCM ARKFCM1

ARKFCM2

ARKFCM119908

WM 0930 0933 0941 0937 0894 0898 0940 0941 0940GM 0822 0855 0860 0852 0782 0842 0864 0868 0865CSF 0781 0847 0829 0805 0860 0882 0863 0867 0867Average 0844 0879 0876 0865 0845 0874 0889 0892 0891Time (s) 0911 0586 3403 139470 0673 2530 0356 0282 0329

Table 2 JS and running time on the T1-weighted sagittal slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

WM 0773 0775 0771 0765 0672 0785 0788 0786GM 0796 0806 0794 0791 0669 0815 0816 0816CSF 0834 0852 0835 0824 0869 0871 0872 0875Average 0801 0811 0800 0794 0736 0824 0825 0825Time (s) 1377 1797 5392 166717 0751 0348 0329 0322

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

(g) (h) (i) (j) (k)

Figure 2 Segmentation results on a T1-weighted axial slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity (a)Original image (b) Ground truth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h)RSCFCM results (i) ARKFCM

1results (j) ARKFCM

2results (k) ARKFCM

119908results

The segmentation results and JS are presented in Figure 3 andTable 2 respectively

The third experiment is to check the robustness toRician noise that commonly affects MR images [30] Thesegmentation results and the JS with average running timesof a T1-weighted axial slice (number 91) with 217 times 181 pixelscorrupted with 10 Rician noise are given in Figure 4 andTable 3 respectively

42 Experiments on Clinical Brain MR Images with TumorsWe experimented two T1-weighted axial slices (slices num-bers 80 and 86 denoted resp as Brats1 and Brats2) with240 times 240 pixels respectively fromfiles pat266 1 and pat192 1(available fromMICCAI BRATS 2014 challenge httpswwwvirtualskeletonchBRATSStart2014) (Figures 5 and 6)

From black to white are respectively background CSF GMand WM It should be noted that clustering is carried outonly for CSF GM andWM with the pathology region beingconsidered as background

Since images Brats1 and Brats2 come only with groundtruth for the pathology and have no ground truth for normaltissues a quantificationmeasure other than JS should be usedTo this end an entropy-based metric proposed by Zhanget al [31] is adopted This is to maximize the uniformity ofpixels within each segmented region and to minimize theuniformity across the regions First the entropy of everyregion 119895 is calculated

119867(119877119895) = minus sum

119898isin119881119895

119871119895 (119898)

119878119895

log119871119895 (119898)

119878119895

(21)

Computational and Mathematical Methods in Medicine 7

Table 3 JS and running time on the T1-weighted axial slice (number 91) from SBD corrupted with 10 Rician noise

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

WM 0931 0921 0929 0927 0885 0926 0925 0922GM 0804 0827 0831 0821 0777 0838 0840 0834CSF 0826 0872 0861 0846 0889 0887 0892 0887Average 0850 0874 0874 0865 0850 0884 0886 0881Time (s) 1836 1869 2953 124922 0649 0218 0220 0218

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

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

(i) (j) (k)

Figure 3 Segmentation results on a T1-weighted sagittal slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity (a)Original image (b) Ground truth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h)RSCFCM results (i) ARKFCM

1results (j) ARKFCM

2results (k) ARKFCM

119908results

where119881119895is the set of all possible grayscales in region 119895 119871

119895(119898)

is the number of pixels belonging to region 119895 with grayscale119898 and 119878

119895is the area of region 119895 For the grayscale image 119868 the

119864measure is calculated as

119864 =

119888

sum

119895=1

(119878119895

119878119868

)119867(119877119895) + (minus

119888

sum

119895=1

(119878119895

119878119868

) log119878119895

119878119868

) (22)

where the first term represents the expected region entropy ofthe segmentation and the second term is the layout entropyThe measure 119864 yields smaller value for better segmentationand higher otherwise Table 4 summarizes the segmentationaccuracy in terms of 119864 and running times for Brats1 andBrats2 It should be noted that the 119864 metric is calculatedonly for WM GM and CSF regions without considering thebackground

5 Discussion

FCM clustering is a well-known soft clustering methodthat assigns a membership degree for each pixel to everycluster As the segmentation accuracy of FCM algorithmdecays in the presence of noise artifacts and increasednumber of clusters many investigations have been carriedout on using contextual information to enhance the qualityof segmentation But how the contextual information shall beused effectively is still a challenge

We propose an adaptively regularized kernel-based FCMclustering framework with new parameter 120593

119894that adaptively

controls the contextual information according to the hetero-geneity of grayscale distribution within the local neighbor-hood The new parameter is estimated using local variationcoefficient among pixels within a specified neighborhoodA weighted image is devised that combines the original

8 Computational and Mathematical Methods in Medicine

Table 4 Segmentation performance measure 119864 and running time on the Brats1 and Brats2 images

Image Measure GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

Brats1 E 1311 1288 1392 1424 1309 1274 1270 1279Time (s) 2591 1516 9510 635413 1747 1185 1111 1272

Brats2 E 1307 1297 1336 1340 1298 1273 1271 1279Time (s) 1576 1115 5170 400295 1546 1091 0827 0975

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

(f) (g) (h) (i) (j)

Figure 4 Segmentation results on a T1-weighted axial slice (number 91) from SBD with 10 Rician noise (a) Original image (b) Groundtruth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h) ARKFCM

1results (i)

ARKFCM2results (j) ARKFCM

119908results

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

(f) (g) (h) (i)

Figure 5 Segmentation results on the Brats1 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

Computational and Mathematical Methods in Medicine 9

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

(f) (g) (h) (i)

Figure 6 Segmentation results on the Brats2 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

image and the parameter 120593119894to represent the image contextual

information embedded through the weighting procedureFurthermore a GRBF is adopted to replace Euclidean dis-tance for better partitioning and to be less sensitive to outliersThe proposed framework can be in the form of 3 algorithmsARKFCM

1 ARKFCM

2 and ARKFCM

119908for the local average

grayscale to be replacedwith the grayscale of the average filtermedian filter and the devised weighted image respectivelyExperiments have been carefully carried out to show thesuperiority of the proposed algorithms in comparison with6 recent soft clustering algorithms to be discussed further

51 Segmentation Accuracy The differences in JS betweenthe proposed algorithms and the other 6 algorithms areclear for the axial slice with 7 noise and 20 grayscalenonuniformity (Table 1 and Figure 2) and become moredistinctive in preserving details in the sagittal slice corruptedwith the same noise (Table 2 and Figure 3) The average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0889 0892 and 0891 for the axial slice as shown in Figure 2which are better than the other 6 algorithms with range 13ndash48 (Table 1) For the sagittal slice with 7 noise and 20grayscale nonuniformity shown in Figure 3 the average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0824 0825 and 0825 which are better than the otheralgorithms with range 14ndash89 (Table 2) For the axial slicecorrupted with 10 Rician noise shown in Figure 4 theaverage JSs of ARKFCM

1 ARKFCM

2 and ARKFCM

119908are

respectively 0884 0886 and 0881 which are better than theother algorithms with range 120ndash36 (Table 3) The higherJSs may imply that the proposed algorithms achieve a betterbalance in preserving image details in the presence of noiseand grayscale inhomogeneity

For the clinical brain MR images with tumors (Brats1and Brats2) shown in Figures 5 and 6 the proposed algo-rithms achieve smaller 119864 than other algorithms (Table 4)For Brats1Brats2 image ARKFCM

2attains the lowest 119864 of

12701271 which is smaller than the other 5 algorithmswith ranges 0018ndash0154 and 0025ndash0109 respectively VisiblyGKFCM1 (Figures 5(b) and 6(b)) and GKFCM2 (Figures5(c) and 6(c)) come with good results but they are unableto preserve small details such as CSF being wrongly brokenpossibly due to the difficulty in estimating the parameter120578119895 On the other hand FLICM (Figures 5(d) and 6(d)) and

KWFLICM (Figures 5(e) and 6(e)) yield smooth results anddo not preserve many details which affects the segmentationaccuracy of CSF (CSF is small as compared with surroundingGM and WM) due to the effects of 119866

119894119895and

119894119895 respectively

For MICO algorithm (Figures 5(f) and 6(f)) the edges arenot smooth enough hence the CSF is not well preservedFinally results of ARKFCM

1 ARKFCM

2 and ARKFCM

119908

(Figures 5(g) and 6(g) Figures 5(h) and 6(h) and Figures 5(i)and 6(i)) show good balance between smooth borders andpreserving image details due to the introduction of adaptivelocal contextual information measure 120593

119894to replace the fixed

value of 120572 or the oversmoothing factors 119866119894119895or 119894119895

52 Computational Cost In terms of computational cost theobjective function of FCM algorithm in its original form[6] contains only the difference between the grayscale of thecurrent pixel 119894 and the cluster centers V

119895 This is basically

to cluster grayscales as there is no spatial information so ithas the smallest computational cost and can be implementedbased on grayscale histogram to reduce the computationalcost further [9 11] The enhancement of original FCM isto add the local contextual information to make it robust

10 Computational and Mathematical Methods in Medicine

to noise and image artifacts at the expense of increasedcomputational cost [7ndash14]

The GKFCM1 and GKFCM2 algorithms [12] use theparameter 120578

119895to replace 120572 and need an additional loop on

the number of clusters to be calculated at each pixel toupdate the local contextual information So they have highercomputational cost than the original FCM FCM S1 andFCM S2 at each iteration

The FLICM algorithm [13] introduces 119866119894119895which needs

an additional loop on the neighborhood of the currentpixel to calculate the local information in every iterationthus it has a high computational cost As an extension ofFLICM KWFLICM [14] introduces

119894119895that requires two

additional loops on the neighborhood so it has the highestcomputational cost at each iteration

The RSCFCM algorithm [21] uses a spatial fuzzy factorthat is constructed based on the posterior and prior prob-abilities and takes the spatial direction into account Thatincreases the complexity as many parameters have to beoptimized and consequently the computational complexity

MICO algorithm [23] comes with fast calculations dueto the convexity of its energy function particularly in thepresence of less noise but tends to have many iterations in thepresence of high level noise

The proposed algorithms incorporate the local contextualinformation by introducing LVC which is a measure ofgrayscale heterogeneity and has nothing to dowith the clustercenters So it can be calculated once in advance and hencereduce the complexity of the clustering procedure

The eventual computational cost will be the multipli-cation of the computational cost for each iteration andthe number of iterations for convergence The number ofiterations will be dependent on the initialization as well as theobjective function To make fair comparisons initializationsare all set randomly and the average number of iterationsfor convergence is then recorded for 10 converged timesFrom Figure 7 it can be seen that the average iterationtimes will be data dependent with ARKFCM

2 ARKFCM

119908

and ARKFCM1having the minimum number of iterations

followed by FLICM GKFCM2 and GKFCM1 respectivelyThe running times for the tested images agree well with

the above analysis as illustrated in Figure 8 For SBD imagesgiven in Figures 2 3 and 4 the KWFLICM algorithm takesthe longest time (124ndash166 seconds) followed by FLICM(3ndash54 seconds) RSCFCM (about 25 seconds) GKFCM1GKFCM2 (06ndash19 seconds) and the proposed algorithms(022 to 036 seconds) For the clinical MR images in Figures5 and 6 it takes more time (due to more iterations) asthe images are more complicated but the trend remainsthe same with the proposed algorithms having the lowestcomputational cost (Table 4 and Figure 8)

53 Neighborhood Size The local neighborhood window sizeis a crucial factor to determine the smoothness of clusteringand details to be preserved We have experimented withdifferent window sizes and found that a window size of3 times 3 pixels achieves the best balance between segmentationaccuracy and computational cost Increasing window size to5 times 5 pixels has very small impact on the JS but 7 times 7 pixels

0

20

40

60

80

100

120

GKF

CM1

GKF

CM2

FLIC

M

MIC

O

Num

ber o

f ite

ratio

ns

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 7 Average number of iterations of all the algorithms (exceptRSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a) and 6(a)

002040608

112141618

222242628

GKF

CM1

GKF

CM2

MIC

O

Aver

age r

unni

ng ti

me (

s)

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 8 Average running times of all the algorithms (exceptFLICM RSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a)and 6(a)

or more will significantly decrease the accuracy such aslosing image details like CSF Therefore it is recommendedto use a local window of size 3 times 3 pixels for constructing theneighborhood

54 Limitation From the experiments it was found thatthe proposed algorithms could be sensitive to severe noisein small areas between edges that have width 1 or 2 pixels(eg area between ventricles Figures 4(h) 4(i) and 4(j))Potentially this may be solved by embedding edge detectiontechnique into the clustering process which is yet to beexplored

Computational and Mathematical Methods in Medicine 11

6 Conclusion

An adaptively regularized kernel-based FCM framework hasbeen proposed to enhance the original FCM for highersegmentation accuracy at low computational costThe frame-work can be in the form of three algorithms that employ theheterogeneity of grayscales in the neighborhood employedfor local contextual information The main advantages areadaptiveness to local context enhanced robustness and inde-pendence of clustering parameters to decrease computationalcost GRBF kernel has been adopted as a distance metric Wevalidated the proposed algorithms on synthetic and clinicalMR images with tumors The proposed algorithms attain ahigher JS (Tables 1 2 and 3) and lower entropy measure119864 (Table 4) than the 6 recent soft clustering algorithms andcould preserve small image details (Figures 2 3 4 5 and 6) Inaddition the proposed algorithms have a low computationalcost and are to the best of our knowledge the only algorithmsthat are adaptive to local context and do not include clustercenters Therefore they attain a trade-off between highsegmentation accuracy and low computational costThe pro-posed algorithms can be a potential tool for segmenting brainMR images for further processing and other images as well

Appendix

The objective function 119869ARKFCM given in equation (17) withconditions in (2) is a constrained minimization problemwhich can be solved using Lagrange multiplier method Letthe minimization problem of 119869ARKFCM be written as follows

119871ARKFCM (119906119894119895 V119895 120582119894 120593119894)

= 2

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2

119873

sum

119894=1

119888

sum

119895=1

120593119894119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+

119873

sum

119894=1

120582119894(1 minus

119888

sum

119895=1

119906119894119895)

(A1)

By taking the derivative of 119871ARKFCM with respect to 119906119894119895and

setting the result to zero for119898 gt 1 we have

120597119871ARKFCM120597119906119894119895

= 2119898119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2119898120593119894119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895)) minus 120582

119894

= 0

(A2)

By using (A2) we can calculate 119906119894119895as follows

119906119894119895

= [120582119894

2119898 (1 minus 119870 (119909119894 V119895) + 120593119894(1 minus 119870 (119909

119894 V119895)))

]

1(119898minus1)

(A3)

Since sum119888119896=1

119906119894119896= 1 forallV

119896 from (A3) we can get

119888

sum

119896=1

[120582119894

2119898 (1 minus 119870 (119909119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))

]

1(119898minus1)

= 1 (A4)

120582119894

=1

[sum119888

119896=1(2119898 (1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896))))minus1(119898minus1)

]119898minus1

(A5)

By substituting (A5) into (A3) with necessary calculationsthe final 119906

119894119895is

119906119894119895

=

((1 minus 119870 (119909119894 V119895)) + 120593

119894(1 minus 119870 (119909

119894 V119895)))minus1(119898minus1)

sum119888

119896=1(1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))minus1(119898minus1)

(A6)

Similarly by taking the derivative of 119871ARKFCM with respect toV119895and setting the result to zero we have

120597119871ARKFCM120597V119895

=

119873

sum

119894=1

119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895)

+

119873

sum

119894=1

120593119894119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895) = 0

(A7)

After necessary calculations the final V119895is

V119895=

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) 119909119894+ 120593119894119870(119909119894 V119895) 119909119894)

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) + 120593119894119870(119909119894 V119895))

(A8)

Conflict of Interests

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

Acknowledgments

The authors would like to thank Dr Maoguo Gong andDr Chunming Li for providing their source codes forcomparison This work has been supported by the NationalProgram on Key Basic Research Project (nos 2013CB7338002012CB733803) Key Joint Program of National Natural Sci-ence Foundation and Guangdong Province (no U1201257)and Guangdong Innovative Research Team Program (no201001D0104648280)

References

[1] N Sharma and L M Aggarwal ldquoAutomated medical imagesegmentation techniquesrdquo Journal ofMedical Physics vol 35 no1 pp 3ndash14 2010

[2] A Elazab Q Hu F Jia and X Zhang ldquoContent based modifiedreaction-diffusion equation for modeling tumor growth of lowgrade gliomardquo in Proceedings of the 7th Cairo InternationalBiomedical Engineering Conference (CIBEC rsquo14) pp 107ndash110Giza Egypt December 2014

12 Computational and Mathematical Methods in Medicine

[3] D L PhamCXu and J L Prince ldquoCurrentmethods inmedicalimage segmentationrdquoAnnual Review of Biomedical Engineeringvol 2 pp 315ndash337 2000

[4] I Despotovic B Goossens andW Philips ldquoMRI segmentationof the human brain challenges methods and applicationsrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 450341 23 pages 2015

[5] S Ruan C Jaggi J Xue J Fadili and D Bloyet ldquoBraintissue classification of magnetic resonance images using partialvolume modelingrdquo IEEE Transactions on Medical Imaging vol19 no 12 pp 1172ndash1186 2000

[6] J C Dunn ldquoA fuzzy relative of the ISODATA process and itsuse in detecting compact well-separated clustersrdquo Journal ofCybernetics vol 3 no 3 pp 32ndash57 1973

[7] J C Bezdek Pattern Recognition with Fuzzy Objective FunctionAlgorithms Advanced Applications in Pattern RecognitionSpringer Boston Mass USA 1981

[8] M N Ahmed S M Yamany N Mohamed A A Farag andT Moriarty ldquoA modified fuzzy c-means algorithm for bias fieldestimation and segmentation of MRI datardquo IEEE Transactionson Medical Imaging vol 21 no 3 pp 193ndash199 2002

[9] L Szilagyi Z Benyo S M Szilagyi and H S Adam ldquoMRbrain image segmentation using an enhanced fuzzy C-meansalgorithmrdquo in Proceedings of the 25th Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety pp 724ndash726 Cancun Mexico September 2003

[10] S Chen andD Zhang ldquoRobust image segmentation using FCMwith spatial constraints based on new kernel-induced distancemeasurerdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 4 pp 1907ndash1916 2004

[11] W Cai S Chen and D Zhang ldquoFast and robust fuzzy c-meansclustering algorithms incorporating local information for imagesegmentationrdquo Pattern Recognition vol 40 no 3 pp 825ndash8382007

[12] M-S Yang and H-S Tsai ldquoA Gaussian kernel-based fuzzyc-means algorithm with a spatial bias correctionrdquo PatternRecognition Letters vol 29 no 12 pp 1713ndash1725 2008

[13] S Krinidis and V Chatzis ldquoA robust fuzzy local informationC-means clustering algorithmrdquo IEEE Transactions on ImageProcessing vol 19 no 5 pp 1328ndash1337 2010

[14] M Gong Y Liang J Shi W Ma and J Ma ldquoFuzzy C-meansclustering with local information and kernel metric for imagesegmentationrdquo IEEE Transactions on Image Processing vol 22no 2 pp 573ndash584 2013

[15] M A Balafar ldquoGaussian mixture model based segmentationmethods for brain MRI imagesrdquo Artificial Intelligence Reviewvol 41 no 3 pp 429ndash439 2014

[16] C Nikou N P Galatsanos and A C Likas ldquoA class-adaptivespatially variant mixture model for image segmentationrdquo IEEETransactions on Image Processing vol 16 no 4 pp 1121ndash11302007

[17] A R Ferreira da Silva ldquoA Dirichlet process mixture model forbrain MRI tissue classificationrdquoMedical Image Analysis vol 11no 2 pp 169ndash182 2007

[18] T M Nguyen and Q M J Wu ldquoFast and robust spatiallyconstrained gaussian mixture model for image segmentationrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 23 no 4 pp 621ndash635 2013

[19] S P Chatzis and T A Varvarigou ldquoA fuzzy clustering approachtoward hidden Markov random field models for enhancedspatially constrained image segmentationrdquo IEEE Transactionson Fuzzy Systems vol 16 no 5 pp 1351ndash1361 2008

[20] S Chatzis ldquoA method for training finite mixture models undera fuzzy clustering principlerdquo Fuzzy Sets and Systems vol 161 no23 pp 3000ndash3013 2010

[21] Z Ji J Liu G Cao Q Sun and Q Chen ldquoRobust spatiallyconstrained fuzzy c-means algorithm for brain MR imagesegmentationrdquo Pattern Recognition vol 47 no 7 pp 2454ndash2466 2014

[22] C Li C Xu A Anderson and J C Gore ldquoMRI tissueclassification and bias field estimation based on coherent localintensity clustering a unified energyminimization frameworkrdquoin Information Processing in Medical Imaging 21st InternationalConference IPMI 2009Williamsburg VA USA July 5ndash10 2009Proceedings vol 5636 of Lecture Notes in Computer Science pp288ndash299 Springer Berlin Germany 2009

[23] C Li J C Gore and C Davatzikos ldquoMultiplicative intrinsiccomponent optimization (MICO) for MRI bias field estimationand tissue segmentationrdquoMagnetic Resonance Imaging vol 32no 7 pp 913ndash923 2014

[24] L Szilagyi ldquoLessons to learn from a mistaken optimizationrdquoPattern Recognition Letters vol 36 no 1 pp 29ndash35 2014

[25] T Hofmann B Scholkopf and A J Smola ldquoKernel methodsin machine learningrdquoThe Annals of Statistics vol 36 no 3 pp1171ndash1220 2008

[26] K-R Muller S Mika G Ratsch K Tsuda and B ScholkopfldquoAn introduction to kernel-based learning algorithmsrdquo IEEETransactions onNeural Networks vol 12 no 2 pp 181ndash201 2001

[27] C R SouzaKernel Functions forMachine LearningApplications2010 httpcrsouzacom201003kernel-functions-for-machine-learning-applications

[28] U Vovk F Pernus and B Likar ldquoA review of methods for cor-rection of intensity inhomogeneity in MRIrdquo IEEE Transactionson Medical Imaging vol 26 no 3 pp 405ndash421 2007

[29] C A Cocosco V Kollokian R K-S Kwan and A C EvansldquoBrainWeb Online Interface to a 3D MRI Simulated BrainDatabaserdquo 2011 httpbrainwebbicmnimcgillcabrainweb

[30] L He and I R Greenshields ldquoA non-local maximum likelihoodestimation method for Rician noise reduction in MR imagesrdquoIEEE Transactions on Medical Imaging vol 28 no 2 pp 165ndash172 2009

[31] H Zhang J E Fritts and S A Goldman ldquoAn entropy-basedobjective evaluation method for image segmentationrdquo in Pro-ceedings of the Storage and Retrieval Methods and Applicationsfor Multimedia vol 5307 of Proceedings of SPIE pp 38ndash49January 2004

Computational and Mathematical Methods in Medicine 7

Table 3 JS and running time on the T1-weighted axial slice (number 91) from SBD corrupted with 10 Rician noise

Algorithm GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

WM 0931 0921 0929 0927 0885 0926 0925 0922GM 0804 0827 0831 0821 0777 0838 0840 0834CSF 0826 0872 0861 0846 0889 0887 0892 0887Average 0850 0874 0874 0865 0850 0884 0886 0881Time (s) 1836 1869 2953 124922 0649 0218 0220 0218

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

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

(i) (j) (k)

Figure 3 Segmentation results on a T1-weighted sagittal slice (number 100) from SBD with 7 noise and 20 grayscale nonuniformity (a)Original image (b) Ground truth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h)RSCFCM results (i) ARKFCM

1results (j) ARKFCM

2results (k) ARKFCM

119908results

where119881119895is the set of all possible grayscales in region 119895 119871

119895(119898)

is the number of pixels belonging to region 119895 with grayscale119898 and 119878

119895is the area of region 119895 For the grayscale image 119868 the

119864measure is calculated as

119864 =

119888

sum

119895=1

(119878119895

119878119868

)119867(119877119895) + (minus

119888

sum

119895=1

(119878119895

119878119868

) log119878119895

119878119868

) (22)

where the first term represents the expected region entropy ofthe segmentation and the second term is the layout entropyThe measure 119864 yields smaller value for better segmentationand higher otherwise Table 4 summarizes the segmentationaccuracy in terms of 119864 and running times for Brats1 andBrats2 It should be noted that the 119864 metric is calculatedonly for WM GM and CSF regions without considering thebackground

5 Discussion

FCM clustering is a well-known soft clustering methodthat assigns a membership degree for each pixel to everycluster As the segmentation accuracy of FCM algorithmdecays in the presence of noise artifacts and increasednumber of clusters many investigations have been carriedout on using contextual information to enhance the qualityof segmentation But how the contextual information shall beused effectively is still a challenge

We propose an adaptively regularized kernel-based FCMclustering framework with new parameter 120593

119894that adaptively

controls the contextual information according to the hetero-geneity of grayscale distribution within the local neighbor-hood The new parameter is estimated using local variationcoefficient among pixels within a specified neighborhoodA weighted image is devised that combines the original

8 Computational and Mathematical Methods in Medicine

Table 4 Segmentation performance measure 119864 and running time on the Brats1 and Brats2 images

Image Measure GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

Brats1 E 1311 1288 1392 1424 1309 1274 1270 1279Time (s) 2591 1516 9510 635413 1747 1185 1111 1272

Brats2 E 1307 1297 1336 1340 1298 1273 1271 1279Time (s) 1576 1115 5170 400295 1546 1091 0827 0975

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

(f) (g) (h) (i) (j)

Figure 4 Segmentation results on a T1-weighted axial slice (number 91) from SBD with 10 Rician noise (a) Original image (b) Groundtruth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h) ARKFCM

1results (i)

ARKFCM2results (j) ARKFCM

119908results

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

(f) (g) (h) (i)

Figure 5 Segmentation results on the Brats1 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

Computational and Mathematical Methods in Medicine 9

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

(f) (g) (h) (i)

Figure 6 Segmentation results on the Brats2 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

image and the parameter 120593119894to represent the image contextual

information embedded through the weighting procedureFurthermore a GRBF is adopted to replace Euclidean dis-tance for better partitioning and to be less sensitive to outliersThe proposed framework can be in the form of 3 algorithmsARKFCM

1 ARKFCM

2 and ARKFCM

119908for the local average

grayscale to be replacedwith the grayscale of the average filtermedian filter and the devised weighted image respectivelyExperiments have been carefully carried out to show thesuperiority of the proposed algorithms in comparison with6 recent soft clustering algorithms to be discussed further

51 Segmentation Accuracy The differences in JS betweenthe proposed algorithms and the other 6 algorithms areclear for the axial slice with 7 noise and 20 grayscalenonuniformity (Table 1 and Figure 2) and become moredistinctive in preserving details in the sagittal slice corruptedwith the same noise (Table 2 and Figure 3) The average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0889 0892 and 0891 for the axial slice as shown in Figure 2which are better than the other 6 algorithms with range 13ndash48 (Table 1) For the sagittal slice with 7 noise and 20grayscale nonuniformity shown in Figure 3 the average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0824 0825 and 0825 which are better than the otheralgorithms with range 14ndash89 (Table 2) For the axial slicecorrupted with 10 Rician noise shown in Figure 4 theaverage JSs of ARKFCM

1 ARKFCM

2 and ARKFCM

119908are

respectively 0884 0886 and 0881 which are better than theother algorithms with range 120ndash36 (Table 3) The higherJSs may imply that the proposed algorithms achieve a betterbalance in preserving image details in the presence of noiseand grayscale inhomogeneity

For the clinical brain MR images with tumors (Brats1and Brats2) shown in Figures 5 and 6 the proposed algo-rithms achieve smaller 119864 than other algorithms (Table 4)For Brats1Brats2 image ARKFCM

2attains the lowest 119864 of

12701271 which is smaller than the other 5 algorithmswith ranges 0018ndash0154 and 0025ndash0109 respectively VisiblyGKFCM1 (Figures 5(b) and 6(b)) and GKFCM2 (Figures5(c) and 6(c)) come with good results but they are unableto preserve small details such as CSF being wrongly brokenpossibly due to the difficulty in estimating the parameter120578119895 On the other hand FLICM (Figures 5(d) and 6(d)) and

KWFLICM (Figures 5(e) and 6(e)) yield smooth results anddo not preserve many details which affects the segmentationaccuracy of CSF (CSF is small as compared with surroundingGM and WM) due to the effects of 119866

119894119895and

119894119895 respectively

For MICO algorithm (Figures 5(f) and 6(f)) the edges arenot smooth enough hence the CSF is not well preservedFinally results of ARKFCM

1 ARKFCM

2 and ARKFCM

119908

(Figures 5(g) and 6(g) Figures 5(h) and 6(h) and Figures 5(i)and 6(i)) show good balance between smooth borders andpreserving image details due to the introduction of adaptivelocal contextual information measure 120593

119894to replace the fixed

value of 120572 or the oversmoothing factors 119866119894119895or 119894119895

52 Computational Cost In terms of computational cost theobjective function of FCM algorithm in its original form[6] contains only the difference between the grayscale of thecurrent pixel 119894 and the cluster centers V

119895 This is basically

to cluster grayscales as there is no spatial information so ithas the smallest computational cost and can be implementedbased on grayscale histogram to reduce the computationalcost further [9 11] The enhancement of original FCM isto add the local contextual information to make it robust

10 Computational and Mathematical Methods in Medicine

to noise and image artifacts at the expense of increasedcomputational cost [7ndash14]

The GKFCM1 and GKFCM2 algorithms [12] use theparameter 120578

119895to replace 120572 and need an additional loop on

the number of clusters to be calculated at each pixel toupdate the local contextual information So they have highercomputational cost than the original FCM FCM S1 andFCM S2 at each iteration

The FLICM algorithm [13] introduces 119866119894119895which needs

an additional loop on the neighborhood of the currentpixel to calculate the local information in every iterationthus it has a high computational cost As an extension ofFLICM KWFLICM [14] introduces

119894119895that requires two

additional loops on the neighborhood so it has the highestcomputational cost at each iteration

The RSCFCM algorithm [21] uses a spatial fuzzy factorthat is constructed based on the posterior and prior prob-abilities and takes the spatial direction into account Thatincreases the complexity as many parameters have to beoptimized and consequently the computational complexity

MICO algorithm [23] comes with fast calculations dueto the convexity of its energy function particularly in thepresence of less noise but tends to have many iterations in thepresence of high level noise

The proposed algorithms incorporate the local contextualinformation by introducing LVC which is a measure ofgrayscale heterogeneity and has nothing to dowith the clustercenters So it can be calculated once in advance and hencereduce the complexity of the clustering procedure

The eventual computational cost will be the multipli-cation of the computational cost for each iteration andthe number of iterations for convergence The number ofiterations will be dependent on the initialization as well as theobjective function To make fair comparisons initializationsare all set randomly and the average number of iterationsfor convergence is then recorded for 10 converged timesFrom Figure 7 it can be seen that the average iterationtimes will be data dependent with ARKFCM

2 ARKFCM

119908

and ARKFCM1having the minimum number of iterations

followed by FLICM GKFCM2 and GKFCM1 respectivelyThe running times for the tested images agree well with

the above analysis as illustrated in Figure 8 For SBD imagesgiven in Figures 2 3 and 4 the KWFLICM algorithm takesthe longest time (124ndash166 seconds) followed by FLICM(3ndash54 seconds) RSCFCM (about 25 seconds) GKFCM1GKFCM2 (06ndash19 seconds) and the proposed algorithms(022 to 036 seconds) For the clinical MR images in Figures5 and 6 it takes more time (due to more iterations) asthe images are more complicated but the trend remainsthe same with the proposed algorithms having the lowestcomputational cost (Table 4 and Figure 8)

53 Neighborhood Size The local neighborhood window sizeis a crucial factor to determine the smoothness of clusteringand details to be preserved We have experimented withdifferent window sizes and found that a window size of3 times 3 pixels achieves the best balance between segmentationaccuracy and computational cost Increasing window size to5 times 5 pixels has very small impact on the JS but 7 times 7 pixels

0

20

40

60

80

100

120

GKF

CM1

GKF

CM2

FLIC

M

MIC

O

Num

ber o

f ite

ratio

ns

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 7 Average number of iterations of all the algorithms (exceptRSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a) and 6(a)

002040608

112141618

222242628

GKF

CM1

GKF

CM2

MIC

O

Aver

age r

unni

ng ti

me (

s)

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 8 Average running times of all the algorithms (exceptFLICM RSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a)and 6(a)

or more will significantly decrease the accuracy such aslosing image details like CSF Therefore it is recommendedto use a local window of size 3 times 3 pixels for constructing theneighborhood

54 Limitation From the experiments it was found thatthe proposed algorithms could be sensitive to severe noisein small areas between edges that have width 1 or 2 pixels(eg area between ventricles Figures 4(h) 4(i) and 4(j))Potentially this may be solved by embedding edge detectiontechnique into the clustering process which is yet to beexplored

Computational and Mathematical Methods in Medicine 11

6 Conclusion

An adaptively regularized kernel-based FCM framework hasbeen proposed to enhance the original FCM for highersegmentation accuracy at low computational costThe frame-work can be in the form of three algorithms that employ theheterogeneity of grayscales in the neighborhood employedfor local contextual information The main advantages areadaptiveness to local context enhanced robustness and inde-pendence of clustering parameters to decrease computationalcost GRBF kernel has been adopted as a distance metric Wevalidated the proposed algorithms on synthetic and clinicalMR images with tumors The proposed algorithms attain ahigher JS (Tables 1 2 and 3) and lower entropy measure119864 (Table 4) than the 6 recent soft clustering algorithms andcould preserve small image details (Figures 2 3 4 5 and 6) Inaddition the proposed algorithms have a low computationalcost and are to the best of our knowledge the only algorithmsthat are adaptive to local context and do not include clustercenters Therefore they attain a trade-off between highsegmentation accuracy and low computational costThe pro-posed algorithms can be a potential tool for segmenting brainMR images for further processing and other images as well

Appendix

The objective function 119869ARKFCM given in equation (17) withconditions in (2) is a constrained minimization problemwhich can be solved using Lagrange multiplier method Letthe minimization problem of 119869ARKFCM be written as follows

119871ARKFCM (119906119894119895 V119895 120582119894 120593119894)

= 2

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2

119873

sum

119894=1

119888

sum

119895=1

120593119894119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+

119873

sum

119894=1

120582119894(1 minus

119888

sum

119895=1

119906119894119895)

(A1)

By taking the derivative of 119871ARKFCM with respect to 119906119894119895and

setting the result to zero for119898 gt 1 we have

120597119871ARKFCM120597119906119894119895

= 2119898119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2119898120593119894119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895)) minus 120582

119894

= 0

(A2)

By using (A2) we can calculate 119906119894119895as follows

119906119894119895

= [120582119894

2119898 (1 minus 119870 (119909119894 V119895) + 120593119894(1 minus 119870 (119909

119894 V119895)))

]

1(119898minus1)

(A3)

Since sum119888119896=1

119906119894119896= 1 forallV

119896 from (A3) we can get

119888

sum

119896=1

[120582119894

2119898 (1 minus 119870 (119909119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))

]

1(119898minus1)

= 1 (A4)

120582119894

=1

[sum119888

119896=1(2119898 (1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896))))minus1(119898minus1)

]119898minus1

(A5)

By substituting (A5) into (A3) with necessary calculationsthe final 119906

119894119895is

119906119894119895

=

((1 minus 119870 (119909119894 V119895)) + 120593

119894(1 minus 119870 (119909

119894 V119895)))minus1(119898minus1)

sum119888

119896=1(1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))minus1(119898minus1)

(A6)

Similarly by taking the derivative of 119871ARKFCM with respect toV119895and setting the result to zero we have

120597119871ARKFCM120597V119895

=

119873

sum

119894=1

119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895)

+

119873

sum

119894=1

120593119894119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895) = 0

(A7)

After necessary calculations the final V119895is

V119895=

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) 119909119894+ 120593119894119870(119909119894 V119895) 119909119894)

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) + 120593119894119870(119909119894 V119895))

(A8)

Conflict of Interests

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

Acknowledgments

The authors would like to thank Dr Maoguo Gong andDr Chunming Li for providing their source codes forcomparison This work has been supported by the NationalProgram on Key Basic Research Project (nos 2013CB7338002012CB733803) Key Joint Program of National Natural Sci-ence Foundation and Guangdong Province (no U1201257)and Guangdong Innovative Research Team Program (no201001D0104648280)

References

[1] N Sharma and L M Aggarwal ldquoAutomated medical imagesegmentation techniquesrdquo Journal ofMedical Physics vol 35 no1 pp 3ndash14 2010

[2] A Elazab Q Hu F Jia and X Zhang ldquoContent based modifiedreaction-diffusion equation for modeling tumor growth of lowgrade gliomardquo in Proceedings of the 7th Cairo InternationalBiomedical Engineering Conference (CIBEC rsquo14) pp 107ndash110Giza Egypt December 2014

12 Computational and Mathematical Methods in Medicine

[3] D L PhamCXu and J L Prince ldquoCurrentmethods inmedicalimage segmentationrdquoAnnual Review of Biomedical Engineeringvol 2 pp 315ndash337 2000

[4] I Despotovic B Goossens andW Philips ldquoMRI segmentationof the human brain challenges methods and applicationsrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 450341 23 pages 2015

[5] S Ruan C Jaggi J Xue J Fadili and D Bloyet ldquoBraintissue classification of magnetic resonance images using partialvolume modelingrdquo IEEE Transactions on Medical Imaging vol19 no 12 pp 1172ndash1186 2000

[6] J C Dunn ldquoA fuzzy relative of the ISODATA process and itsuse in detecting compact well-separated clustersrdquo Journal ofCybernetics vol 3 no 3 pp 32ndash57 1973

[7] J C Bezdek Pattern Recognition with Fuzzy Objective FunctionAlgorithms Advanced Applications in Pattern RecognitionSpringer Boston Mass USA 1981

[8] M N Ahmed S M Yamany N Mohamed A A Farag andT Moriarty ldquoA modified fuzzy c-means algorithm for bias fieldestimation and segmentation of MRI datardquo IEEE Transactionson Medical Imaging vol 21 no 3 pp 193ndash199 2002

[9] L Szilagyi Z Benyo S M Szilagyi and H S Adam ldquoMRbrain image segmentation using an enhanced fuzzy C-meansalgorithmrdquo in Proceedings of the 25th Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety pp 724ndash726 Cancun Mexico September 2003

[10] S Chen andD Zhang ldquoRobust image segmentation using FCMwith spatial constraints based on new kernel-induced distancemeasurerdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 4 pp 1907ndash1916 2004

[11] W Cai S Chen and D Zhang ldquoFast and robust fuzzy c-meansclustering algorithms incorporating local information for imagesegmentationrdquo Pattern Recognition vol 40 no 3 pp 825ndash8382007

[12] M-S Yang and H-S Tsai ldquoA Gaussian kernel-based fuzzyc-means algorithm with a spatial bias correctionrdquo PatternRecognition Letters vol 29 no 12 pp 1713ndash1725 2008

[13] S Krinidis and V Chatzis ldquoA robust fuzzy local informationC-means clustering algorithmrdquo IEEE Transactions on ImageProcessing vol 19 no 5 pp 1328ndash1337 2010

[14] M Gong Y Liang J Shi W Ma and J Ma ldquoFuzzy C-meansclustering with local information and kernel metric for imagesegmentationrdquo IEEE Transactions on Image Processing vol 22no 2 pp 573ndash584 2013

[15] M A Balafar ldquoGaussian mixture model based segmentationmethods for brain MRI imagesrdquo Artificial Intelligence Reviewvol 41 no 3 pp 429ndash439 2014

[16] C Nikou N P Galatsanos and A C Likas ldquoA class-adaptivespatially variant mixture model for image segmentationrdquo IEEETransactions on Image Processing vol 16 no 4 pp 1121ndash11302007

[17] A R Ferreira da Silva ldquoA Dirichlet process mixture model forbrain MRI tissue classificationrdquoMedical Image Analysis vol 11no 2 pp 169ndash182 2007

[18] T M Nguyen and Q M J Wu ldquoFast and robust spatiallyconstrained gaussian mixture model for image segmentationrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 23 no 4 pp 621ndash635 2013

[19] S P Chatzis and T A Varvarigou ldquoA fuzzy clustering approachtoward hidden Markov random field models for enhancedspatially constrained image segmentationrdquo IEEE Transactionson Fuzzy Systems vol 16 no 5 pp 1351ndash1361 2008

[20] S Chatzis ldquoA method for training finite mixture models undera fuzzy clustering principlerdquo Fuzzy Sets and Systems vol 161 no23 pp 3000ndash3013 2010

[21] Z Ji J Liu G Cao Q Sun and Q Chen ldquoRobust spatiallyconstrained fuzzy c-means algorithm for brain MR imagesegmentationrdquo Pattern Recognition vol 47 no 7 pp 2454ndash2466 2014

[22] C Li C Xu A Anderson and J C Gore ldquoMRI tissueclassification and bias field estimation based on coherent localintensity clustering a unified energyminimization frameworkrdquoin Information Processing in Medical Imaging 21st InternationalConference IPMI 2009Williamsburg VA USA July 5ndash10 2009Proceedings vol 5636 of Lecture Notes in Computer Science pp288ndash299 Springer Berlin Germany 2009

[23] C Li J C Gore and C Davatzikos ldquoMultiplicative intrinsiccomponent optimization (MICO) for MRI bias field estimationand tissue segmentationrdquoMagnetic Resonance Imaging vol 32no 7 pp 913ndash923 2014

[24] L Szilagyi ldquoLessons to learn from a mistaken optimizationrdquoPattern Recognition Letters vol 36 no 1 pp 29ndash35 2014

[25] T Hofmann B Scholkopf and A J Smola ldquoKernel methodsin machine learningrdquoThe Annals of Statistics vol 36 no 3 pp1171ndash1220 2008

[26] K-R Muller S Mika G Ratsch K Tsuda and B ScholkopfldquoAn introduction to kernel-based learning algorithmsrdquo IEEETransactions onNeural Networks vol 12 no 2 pp 181ndash201 2001

[27] C R SouzaKernel Functions forMachine LearningApplications2010 httpcrsouzacom201003kernel-functions-for-machine-learning-applications

[28] U Vovk F Pernus and B Likar ldquoA review of methods for cor-rection of intensity inhomogeneity in MRIrdquo IEEE Transactionson Medical Imaging vol 26 no 3 pp 405ndash421 2007

[29] C A Cocosco V Kollokian R K-S Kwan and A C EvansldquoBrainWeb Online Interface to a 3D MRI Simulated BrainDatabaserdquo 2011 httpbrainwebbicmnimcgillcabrainweb

[30] L He and I R Greenshields ldquoA non-local maximum likelihoodestimation method for Rician noise reduction in MR imagesrdquoIEEE Transactions on Medical Imaging vol 28 no 2 pp 165ndash172 2009

[31] H Zhang J E Fritts and S A Goldman ldquoAn entropy-basedobjective evaluation method for image segmentationrdquo in Pro-ceedings of the Storage and Retrieval Methods and Applicationsfor Multimedia vol 5307 of Proceedings of SPIE pp 38ndash49January 2004

8 Computational and Mathematical Methods in Medicine

Table 4 Segmentation performance measure 119864 and running time on the Brats1 and Brats2 images

Image Measure GKFCM1 GKFCM2 FLICM KWFLICM MICO ARKFCM1

ARKFCM2

ARKFCM119908

Brats1 E 1311 1288 1392 1424 1309 1274 1270 1279Time (s) 2591 1516 9510 635413 1747 1185 1111 1272

Brats2 E 1307 1297 1336 1340 1298 1273 1271 1279Time (s) 1576 1115 5170 400295 1546 1091 0827 0975

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

(f) (g) (h) (i) (j)

Figure 4 Segmentation results on a T1-weighted axial slice (number 91) from SBD with 10 Rician noise (a) Original image (b) Groundtruth (c) GKFCM1 results (d) GKFCM2 results (e) FLICM results (f) KWFLICM results (g) MICO results (h) ARKFCM

1results (i)

ARKFCM2results (j) ARKFCM

119908results

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

(f) (g) (h) (i)

Figure 5 Segmentation results on the Brats1 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

Computational and Mathematical Methods in Medicine 9

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

(f) (g) (h) (i)

Figure 6 Segmentation results on the Brats2 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

image and the parameter 120593119894to represent the image contextual

information embedded through the weighting procedureFurthermore a GRBF is adopted to replace Euclidean dis-tance for better partitioning and to be less sensitive to outliersThe proposed framework can be in the form of 3 algorithmsARKFCM

1 ARKFCM

2 and ARKFCM

119908for the local average

grayscale to be replacedwith the grayscale of the average filtermedian filter and the devised weighted image respectivelyExperiments have been carefully carried out to show thesuperiority of the proposed algorithms in comparison with6 recent soft clustering algorithms to be discussed further

51 Segmentation Accuracy The differences in JS betweenthe proposed algorithms and the other 6 algorithms areclear for the axial slice with 7 noise and 20 grayscalenonuniformity (Table 1 and Figure 2) and become moredistinctive in preserving details in the sagittal slice corruptedwith the same noise (Table 2 and Figure 3) The average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0889 0892 and 0891 for the axial slice as shown in Figure 2which are better than the other 6 algorithms with range 13ndash48 (Table 1) For the sagittal slice with 7 noise and 20grayscale nonuniformity shown in Figure 3 the average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0824 0825 and 0825 which are better than the otheralgorithms with range 14ndash89 (Table 2) For the axial slicecorrupted with 10 Rician noise shown in Figure 4 theaverage JSs of ARKFCM

1 ARKFCM

2 and ARKFCM

119908are

respectively 0884 0886 and 0881 which are better than theother algorithms with range 120ndash36 (Table 3) The higherJSs may imply that the proposed algorithms achieve a betterbalance in preserving image details in the presence of noiseand grayscale inhomogeneity

For the clinical brain MR images with tumors (Brats1and Brats2) shown in Figures 5 and 6 the proposed algo-rithms achieve smaller 119864 than other algorithms (Table 4)For Brats1Brats2 image ARKFCM

2attains the lowest 119864 of

12701271 which is smaller than the other 5 algorithmswith ranges 0018ndash0154 and 0025ndash0109 respectively VisiblyGKFCM1 (Figures 5(b) and 6(b)) and GKFCM2 (Figures5(c) and 6(c)) come with good results but they are unableto preserve small details such as CSF being wrongly brokenpossibly due to the difficulty in estimating the parameter120578119895 On the other hand FLICM (Figures 5(d) and 6(d)) and

KWFLICM (Figures 5(e) and 6(e)) yield smooth results anddo not preserve many details which affects the segmentationaccuracy of CSF (CSF is small as compared with surroundingGM and WM) due to the effects of 119866

119894119895and

119894119895 respectively

For MICO algorithm (Figures 5(f) and 6(f)) the edges arenot smooth enough hence the CSF is not well preservedFinally results of ARKFCM

1 ARKFCM

2 and ARKFCM

119908

(Figures 5(g) and 6(g) Figures 5(h) and 6(h) and Figures 5(i)and 6(i)) show good balance between smooth borders andpreserving image details due to the introduction of adaptivelocal contextual information measure 120593

119894to replace the fixed

value of 120572 or the oversmoothing factors 119866119894119895or 119894119895

52 Computational Cost In terms of computational cost theobjective function of FCM algorithm in its original form[6] contains only the difference between the grayscale of thecurrent pixel 119894 and the cluster centers V

119895 This is basically

to cluster grayscales as there is no spatial information so ithas the smallest computational cost and can be implementedbased on grayscale histogram to reduce the computationalcost further [9 11] The enhancement of original FCM isto add the local contextual information to make it robust

10 Computational and Mathematical Methods in Medicine

to noise and image artifacts at the expense of increasedcomputational cost [7ndash14]

The GKFCM1 and GKFCM2 algorithms [12] use theparameter 120578

119895to replace 120572 and need an additional loop on

the number of clusters to be calculated at each pixel toupdate the local contextual information So they have highercomputational cost than the original FCM FCM S1 andFCM S2 at each iteration

The FLICM algorithm [13] introduces 119866119894119895which needs

an additional loop on the neighborhood of the currentpixel to calculate the local information in every iterationthus it has a high computational cost As an extension ofFLICM KWFLICM [14] introduces

119894119895that requires two

additional loops on the neighborhood so it has the highestcomputational cost at each iteration

The RSCFCM algorithm [21] uses a spatial fuzzy factorthat is constructed based on the posterior and prior prob-abilities and takes the spatial direction into account Thatincreases the complexity as many parameters have to beoptimized and consequently the computational complexity

MICO algorithm [23] comes with fast calculations dueto the convexity of its energy function particularly in thepresence of less noise but tends to have many iterations in thepresence of high level noise

The proposed algorithms incorporate the local contextualinformation by introducing LVC which is a measure ofgrayscale heterogeneity and has nothing to dowith the clustercenters So it can be calculated once in advance and hencereduce the complexity of the clustering procedure

The eventual computational cost will be the multipli-cation of the computational cost for each iteration andthe number of iterations for convergence The number ofiterations will be dependent on the initialization as well as theobjective function To make fair comparisons initializationsare all set randomly and the average number of iterationsfor convergence is then recorded for 10 converged timesFrom Figure 7 it can be seen that the average iterationtimes will be data dependent with ARKFCM

2 ARKFCM

119908

and ARKFCM1having the minimum number of iterations

followed by FLICM GKFCM2 and GKFCM1 respectivelyThe running times for the tested images agree well with

the above analysis as illustrated in Figure 8 For SBD imagesgiven in Figures 2 3 and 4 the KWFLICM algorithm takesthe longest time (124ndash166 seconds) followed by FLICM(3ndash54 seconds) RSCFCM (about 25 seconds) GKFCM1GKFCM2 (06ndash19 seconds) and the proposed algorithms(022 to 036 seconds) For the clinical MR images in Figures5 and 6 it takes more time (due to more iterations) asthe images are more complicated but the trend remainsthe same with the proposed algorithms having the lowestcomputational cost (Table 4 and Figure 8)

53 Neighborhood Size The local neighborhood window sizeis a crucial factor to determine the smoothness of clusteringand details to be preserved We have experimented withdifferent window sizes and found that a window size of3 times 3 pixels achieves the best balance between segmentationaccuracy and computational cost Increasing window size to5 times 5 pixels has very small impact on the JS but 7 times 7 pixels

0

20

40

60

80

100

120

GKF

CM1

GKF

CM2

FLIC

M

MIC

O

Num

ber o

f ite

ratio

ns

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 7 Average number of iterations of all the algorithms (exceptRSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a) and 6(a)

002040608

112141618

222242628

GKF

CM1

GKF

CM2

MIC

O

Aver

age r

unni

ng ti

me (

s)

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 8 Average running times of all the algorithms (exceptFLICM RSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a)and 6(a)

or more will significantly decrease the accuracy such aslosing image details like CSF Therefore it is recommendedto use a local window of size 3 times 3 pixels for constructing theneighborhood

54 Limitation From the experiments it was found thatthe proposed algorithms could be sensitive to severe noisein small areas between edges that have width 1 or 2 pixels(eg area between ventricles Figures 4(h) 4(i) and 4(j))Potentially this may be solved by embedding edge detectiontechnique into the clustering process which is yet to beexplored

Computational and Mathematical Methods in Medicine 11

6 Conclusion

An adaptively regularized kernel-based FCM framework hasbeen proposed to enhance the original FCM for highersegmentation accuracy at low computational costThe frame-work can be in the form of three algorithms that employ theheterogeneity of grayscales in the neighborhood employedfor local contextual information The main advantages areadaptiveness to local context enhanced robustness and inde-pendence of clustering parameters to decrease computationalcost GRBF kernel has been adopted as a distance metric Wevalidated the proposed algorithms on synthetic and clinicalMR images with tumors The proposed algorithms attain ahigher JS (Tables 1 2 and 3) and lower entropy measure119864 (Table 4) than the 6 recent soft clustering algorithms andcould preserve small image details (Figures 2 3 4 5 and 6) Inaddition the proposed algorithms have a low computationalcost and are to the best of our knowledge the only algorithmsthat are adaptive to local context and do not include clustercenters Therefore they attain a trade-off between highsegmentation accuracy and low computational costThe pro-posed algorithms can be a potential tool for segmenting brainMR images for further processing and other images as well

Appendix

The objective function 119869ARKFCM given in equation (17) withconditions in (2) is a constrained minimization problemwhich can be solved using Lagrange multiplier method Letthe minimization problem of 119869ARKFCM be written as follows

119871ARKFCM (119906119894119895 V119895 120582119894 120593119894)

= 2

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2

119873

sum

119894=1

119888

sum

119895=1

120593119894119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+

119873

sum

119894=1

120582119894(1 minus

119888

sum

119895=1

119906119894119895)

(A1)

By taking the derivative of 119871ARKFCM with respect to 119906119894119895and

setting the result to zero for119898 gt 1 we have

120597119871ARKFCM120597119906119894119895

= 2119898119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2119898120593119894119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895)) minus 120582

119894

= 0

(A2)

By using (A2) we can calculate 119906119894119895as follows

119906119894119895

= [120582119894

2119898 (1 minus 119870 (119909119894 V119895) + 120593119894(1 minus 119870 (119909

119894 V119895)))

]

1(119898minus1)

(A3)

Since sum119888119896=1

119906119894119896= 1 forallV

119896 from (A3) we can get

119888

sum

119896=1

[120582119894

2119898 (1 minus 119870 (119909119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))

]

1(119898minus1)

= 1 (A4)

120582119894

=1

[sum119888

119896=1(2119898 (1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896))))minus1(119898minus1)

]119898minus1

(A5)

By substituting (A5) into (A3) with necessary calculationsthe final 119906

119894119895is

119906119894119895

=

((1 minus 119870 (119909119894 V119895)) + 120593

119894(1 minus 119870 (119909

119894 V119895)))minus1(119898minus1)

sum119888

119896=1(1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))minus1(119898minus1)

(A6)

Similarly by taking the derivative of 119871ARKFCM with respect toV119895and setting the result to zero we have

120597119871ARKFCM120597V119895

=

119873

sum

119894=1

119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895)

+

119873

sum

119894=1

120593119894119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895) = 0

(A7)

After necessary calculations the final V119895is

V119895=

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) 119909119894+ 120593119894119870(119909119894 V119895) 119909119894)

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) + 120593119894119870(119909119894 V119895))

(A8)

Conflict of Interests

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

Acknowledgments

The authors would like to thank Dr Maoguo Gong andDr Chunming Li for providing their source codes forcomparison This work has been supported by the NationalProgram on Key Basic Research Project (nos 2013CB7338002012CB733803) Key Joint Program of National Natural Sci-ence Foundation and Guangdong Province (no U1201257)and Guangdong Innovative Research Team Program (no201001D0104648280)

References

[1] N Sharma and L M Aggarwal ldquoAutomated medical imagesegmentation techniquesrdquo Journal ofMedical Physics vol 35 no1 pp 3ndash14 2010

[2] A Elazab Q Hu F Jia and X Zhang ldquoContent based modifiedreaction-diffusion equation for modeling tumor growth of lowgrade gliomardquo in Proceedings of the 7th Cairo InternationalBiomedical Engineering Conference (CIBEC rsquo14) pp 107ndash110Giza Egypt December 2014

12 Computational and Mathematical Methods in Medicine

[3] D L PhamCXu and J L Prince ldquoCurrentmethods inmedicalimage segmentationrdquoAnnual Review of Biomedical Engineeringvol 2 pp 315ndash337 2000

[4] I Despotovic B Goossens andW Philips ldquoMRI segmentationof the human brain challenges methods and applicationsrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 450341 23 pages 2015

[5] S Ruan C Jaggi J Xue J Fadili and D Bloyet ldquoBraintissue classification of magnetic resonance images using partialvolume modelingrdquo IEEE Transactions on Medical Imaging vol19 no 12 pp 1172ndash1186 2000

[6] J C Dunn ldquoA fuzzy relative of the ISODATA process and itsuse in detecting compact well-separated clustersrdquo Journal ofCybernetics vol 3 no 3 pp 32ndash57 1973

[7] J C Bezdek Pattern Recognition with Fuzzy Objective FunctionAlgorithms Advanced Applications in Pattern RecognitionSpringer Boston Mass USA 1981

[8] M N Ahmed S M Yamany N Mohamed A A Farag andT Moriarty ldquoA modified fuzzy c-means algorithm for bias fieldestimation and segmentation of MRI datardquo IEEE Transactionson Medical Imaging vol 21 no 3 pp 193ndash199 2002

[9] L Szilagyi Z Benyo S M Szilagyi and H S Adam ldquoMRbrain image segmentation using an enhanced fuzzy C-meansalgorithmrdquo in Proceedings of the 25th Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety pp 724ndash726 Cancun Mexico September 2003

[10] S Chen andD Zhang ldquoRobust image segmentation using FCMwith spatial constraints based on new kernel-induced distancemeasurerdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 4 pp 1907ndash1916 2004

[11] W Cai S Chen and D Zhang ldquoFast and robust fuzzy c-meansclustering algorithms incorporating local information for imagesegmentationrdquo Pattern Recognition vol 40 no 3 pp 825ndash8382007

[12] M-S Yang and H-S Tsai ldquoA Gaussian kernel-based fuzzyc-means algorithm with a spatial bias correctionrdquo PatternRecognition Letters vol 29 no 12 pp 1713ndash1725 2008

[13] S Krinidis and V Chatzis ldquoA robust fuzzy local informationC-means clustering algorithmrdquo IEEE Transactions on ImageProcessing vol 19 no 5 pp 1328ndash1337 2010

[14] M Gong Y Liang J Shi W Ma and J Ma ldquoFuzzy C-meansclustering with local information and kernel metric for imagesegmentationrdquo IEEE Transactions on Image Processing vol 22no 2 pp 573ndash584 2013

[15] M A Balafar ldquoGaussian mixture model based segmentationmethods for brain MRI imagesrdquo Artificial Intelligence Reviewvol 41 no 3 pp 429ndash439 2014

[16] C Nikou N P Galatsanos and A C Likas ldquoA class-adaptivespatially variant mixture model for image segmentationrdquo IEEETransactions on Image Processing vol 16 no 4 pp 1121ndash11302007

[17] A R Ferreira da Silva ldquoA Dirichlet process mixture model forbrain MRI tissue classificationrdquoMedical Image Analysis vol 11no 2 pp 169ndash182 2007

[18] T M Nguyen and Q M J Wu ldquoFast and robust spatiallyconstrained gaussian mixture model for image segmentationrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 23 no 4 pp 621ndash635 2013

[19] S P Chatzis and T A Varvarigou ldquoA fuzzy clustering approachtoward hidden Markov random field models for enhancedspatially constrained image segmentationrdquo IEEE Transactionson Fuzzy Systems vol 16 no 5 pp 1351ndash1361 2008

[20] S Chatzis ldquoA method for training finite mixture models undera fuzzy clustering principlerdquo Fuzzy Sets and Systems vol 161 no23 pp 3000ndash3013 2010

[21] Z Ji J Liu G Cao Q Sun and Q Chen ldquoRobust spatiallyconstrained fuzzy c-means algorithm for brain MR imagesegmentationrdquo Pattern Recognition vol 47 no 7 pp 2454ndash2466 2014

[22] C Li C Xu A Anderson and J C Gore ldquoMRI tissueclassification and bias field estimation based on coherent localintensity clustering a unified energyminimization frameworkrdquoin Information Processing in Medical Imaging 21st InternationalConference IPMI 2009Williamsburg VA USA July 5ndash10 2009Proceedings vol 5636 of Lecture Notes in Computer Science pp288ndash299 Springer Berlin Germany 2009

[23] C Li J C Gore and C Davatzikos ldquoMultiplicative intrinsiccomponent optimization (MICO) for MRI bias field estimationand tissue segmentationrdquoMagnetic Resonance Imaging vol 32no 7 pp 913ndash923 2014

[24] L Szilagyi ldquoLessons to learn from a mistaken optimizationrdquoPattern Recognition Letters vol 36 no 1 pp 29ndash35 2014

[25] T Hofmann B Scholkopf and A J Smola ldquoKernel methodsin machine learningrdquoThe Annals of Statistics vol 36 no 3 pp1171ndash1220 2008

[26] K-R Muller S Mika G Ratsch K Tsuda and B ScholkopfldquoAn introduction to kernel-based learning algorithmsrdquo IEEETransactions onNeural Networks vol 12 no 2 pp 181ndash201 2001

[27] C R SouzaKernel Functions forMachine LearningApplications2010 httpcrsouzacom201003kernel-functions-for-machine-learning-applications

[28] U Vovk F Pernus and B Likar ldquoA review of methods for cor-rection of intensity inhomogeneity in MRIrdquo IEEE Transactionson Medical Imaging vol 26 no 3 pp 405ndash421 2007

[29] C A Cocosco V Kollokian R K-S Kwan and A C EvansldquoBrainWeb Online Interface to a 3D MRI Simulated BrainDatabaserdquo 2011 httpbrainwebbicmnimcgillcabrainweb

[30] L He and I R Greenshields ldquoA non-local maximum likelihoodestimation method for Rician noise reduction in MR imagesrdquoIEEE Transactions on Medical Imaging vol 28 no 2 pp 165ndash172 2009

[31] H Zhang J E Fritts and S A Goldman ldquoAn entropy-basedobjective evaluation method for image segmentationrdquo in Pro-ceedings of the Storage and Retrieval Methods and Applicationsfor Multimedia vol 5307 of Proceedings of SPIE pp 38ndash49January 2004

Computational and Mathematical Methods in Medicine 9

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

(f) (g) (h) (i)

Figure 6 Segmentation results on the Brats2 image (a) Original image (b) GKFCM1 results (c) GKFCM2 results (d) FLICM results (e)KWFLICM results (f) MICO results (g) ARKFCM

1results (h) ARKFCM

2results (i) ARKFCM

119908results

image and the parameter 120593119894to represent the image contextual

information embedded through the weighting procedureFurthermore a GRBF is adopted to replace Euclidean dis-tance for better partitioning and to be less sensitive to outliersThe proposed framework can be in the form of 3 algorithmsARKFCM

1 ARKFCM

2 and ARKFCM

119908for the local average

grayscale to be replacedwith the grayscale of the average filtermedian filter and the devised weighted image respectivelyExperiments have been carefully carried out to show thesuperiority of the proposed algorithms in comparison with6 recent soft clustering algorithms to be discussed further

51 Segmentation Accuracy The differences in JS betweenthe proposed algorithms and the other 6 algorithms areclear for the axial slice with 7 noise and 20 grayscalenonuniformity (Table 1 and Figure 2) and become moredistinctive in preserving details in the sagittal slice corruptedwith the same noise (Table 2 and Figure 3) The average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0889 0892 and 0891 for the axial slice as shown in Figure 2which are better than the other 6 algorithms with range 13ndash48 (Table 1) For the sagittal slice with 7 noise and 20grayscale nonuniformity shown in Figure 3 the average JSsof ARKFCM

1 ARKFCM

2 and ARKFCM

119908are respectively

0824 0825 and 0825 which are better than the otheralgorithms with range 14ndash89 (Table 2) For the axial slicecorrupted with 10 Rician noise shown in Figure 4 theaverage JSs of ARKFCM

1 ARKFCM

2 and ARKFCM

119908are

respectively 0884 0886 and 0881 which are better than theother algorithms with range 120ndash36 (Table 3) The higherJSs may imply that the proposed algorithms achieve a betterbalance in preserving image details in the presence of noiseand grayscale inhomogeneity

For the clinical brain MR images with tumors (Brats1and Brats2) shown in Figures 5 and 6 the proposed algo-rithms achieve smaller 119864 than other algorithms (Table 4)For Brats1Brats2 image ARKFCM

2attains the lowest 119864 of

12701271 which is smaller than the other 5 algorithmswith ranges 0018ndash0154 and 0025ndash0109 respectively VisiblyGKFCM1 (Figures 5(b) and 6(b)) and GKFCM2 (Figures5(c) and 6(c)) come with good results but they are unableto preserve small details such as CSF being wrongly brokenpossibly due to the difficulty in estimating the parameter120578119895 On the other hand FLICM (Figures 5(d) and 6(d)) and

KWFLICM (Figures 5(e) and 6(e)) yield smooth results anddo not preserve many details which affects the segmentationaccuracy of CSF (CSF is small as compared with surroundingGM and WM) due to the effects of 119866

119894119895and

119894119895 respectively

For MICO algorithm (Figures 5(f) and 6(f)) the edges arenot smooth enough hence the CSF is not well preservedFinally results of ARKFCM

1 ARKFCM

2 and ARKFCM

119908

(Figures 5(g) and 6(g) Figures 5(h) and 6(h) and Figures 5(i)and 6(i)) show good balance between smooth borders andpreserving image details due to the introduction of adaptivelocal contextual information measure 120593

119894to replace the fixed

value of 120572 or the oversmoothing factors 119866119894119895or 119894119895

52 Computational Cost In terms of computational cost theobjective function of FCM algorithm in its original form[6] contains only the difference between the grayscale of thecurrent pixel 119894 and the cluster centers V

119895 This is basically

to cluster grayscales as there is no spatial information so ithas the smallest computational cost and can be implementedbased on grayscale histogram to reduce the computationalcost further [9 11] The enhancement of original FCM isto add the local contextual information to make it robust

10 Computational and Mathematical Methods in Medicine

to noise and image artifacts at the expense of increasedcomputational cost [7ndash14]

The GKFCM1 and GKFCM2 algorithms [12] use theparameter 120578

119895to replace 120572 and need an additional loop on

the number of clusters to be calculated at each pixel toupdate the local contextual information So they have highercomputational cost than the original FCM FCM S1 andFCM S2 at each iteration

The FLICM algorithm [13] introduces 119866119894119895which needs

an additional loop on the neighborhood of the currentpixel to calculate the local information in every iterationthus it has a high computational cost As an extension ofFLICM KWFLICM [14] introduces

119894119895that requires two

additional loops on the neighborhood so it has the highestcomputational cost at each iteration

The RSCFCM algorithm [21] uses a spatial fuzzy factorthat is constructed based on the posterior and prior prob-abilities and takes the spatial direction into account Thatincreases the complexity as many parameters have to beoptimized and consequently the computational complexity

MICO algorithm [23] comes with fast calculations dueto the convexity of its energy function particularly in thepresence of less noise but tends to have many iterations in thepresence of high level noise

The proposed algorithms incorporate the local contextualinformation by introducing LVC which is a measure ofgrayscale heterogeneity and has nothing to dowith the clustercenters So it can be calculated once in advance and hencereduce the complexity of the clustering procedure

The eventual computational cost will be the multipli-cation of the computational cost for each iteration andthe number of iterations for convergence The number ofiterations will be dependent on the initialization as well as theobjective function To make fair comparisons initializationsare all set randomly and the average number of iterationsfor convergence is then recorded for 10 converged timesFrom Figure 7 it can be seen that the average iterationtimes will be data dependent with ARKFCM

2 ARKFCM

119908

and ARKFCM1having the minimum number of iterations

followed by FLICM GKFCM2 and GKFCM1 respectivelyThe running times for the tested images agree well with

the above analysis as illustrated in Figure 8 For SBD imagesgiven in Figures 2 3 and 4 the KWFLICM algorithm takesthe longest time (124ndash166 seconds) followed by FLICM(3ndash54 seconds) RSCFCM (about 25 seconds) GKFCM1GKFCM2 (06ndash19 seconds) and the proposed algorithms(022 to 036 seconds) For the clinical MR images in Figures5 and 6 it takes more time (due to more iterations) asthe images are more complicated but the trend remainsthe same with the proposed algorithms having the lowestcomputational cost (Table 4 and Figure 8)

53 Neighborhood Size The local neighborhood window sizeis a crucial factor to determine the smoothness of clusteringand details to be preserved We have experimented withdifferent window sizes and found that a window size of3 times 3 pixels achieves the best balance between segmentationaccuracy and computational cost Increasing window size to5 times 5 pixels has very small impact on the JS but 7 times 7 pixels

0

20

40

60

80

100

120

GKF

CM1

GKF

CM2

FLIC

M

MIC

O

Num

ber o

f ite

ratio

ns

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 7 Average number of iterations of all the algorithms (exceptRSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a) and 6(a)

002040608

112141618

222242628

GKF

CM1

GKF

CM2

MIC

O

Aver

age r

unni

ng ti

me (

s)

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 8 Average running times of all the algorithms (exceptFLICM RSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a)and 6(a)

or more will significantly decrease the accuracy such aslosing image details like CSF Therefore it is recommendedto use a local window of size 3 times 3 pixels for constructing theneighborhood

54 Limitation From the experiments it was found thatthe proposed algorithms could be sensitive to severe noisein small areas between edges that have width 1 or 2 pixels(eg area between ventricles Figures 4(h) 4(i) and 4(j))Potentially this may be solved by embedding edge detectiontechnique into the clustering process which is yet to beexplored

Computational and Mathematical Methods in Medicine 11

6 Conclusion

An adaptively regularized kernel-based FCM framework hasbeen proposed to enhance the original FCM for highersegmentation accuracy at low computational costThe frame-work can be in the form of three algorithms that employ theheterogeneity of grayscales in the neighborhood employedfor local contextual information The main advantages areadaptiveness to local context enhanced robustness and inde-pendence of clustering parameters to decrease computationalcost GRBF kernel has been adopted as a distance metric Wevalidated the proposed algorithms on synthetic and clinicalMR images with tumors The proposed algorithms attain ahigher JS (Tables 1 2 and 3) and lower entropy measure119864 (Table 4) than the 6 recent soft clustering algorithms andcould preserve small image details (Figures 2 3 4 5 and 6) Inaddition the proposed algorithms have a low computationalcost and are to the best of our knowledge the only algorithmsthat are adaptive to local context and do not include clustercenters Therefore they attain a trade-off between highsegmentation accuracy and low computational costThe pro-posed algorithms can be a potential tool for segmenting brainMR images for further processing and other images as well

Appendix

The objective function 119869ARKFCM given in equation (17) withconditions in (2) is a constrained minimization problemwhich can be solved using Lagrange multiplier method Letthe minimization problem of 119869ARKFCM be written as follows

119871ARKFCM (119906119894119895 V119895 120582119894 120593119894)

= 2

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2

119873

sum

119894=1

119888

sum

119895=1

120593119894119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+

119873

sum

119894=1

120582119894(1 minus

119888

sum

119895=1

119906119894119895)

(A1)

By taking the derivative of 119871ARKFCM with respect to 119906119894119895and

setting the result to zero for119898 gt 1 we have

120597119871ARKFCM120597119906119894119895

= 2119898119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2119898120593119894119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895)) minus 120582

119894

= 0

(A2)

By using (A2) we can calculate 119906119894119895as follows

119906119894119895

= [120582119894

2119898 (1 minus 119870 (119909119894 V119895) + 120593119894(1 minus 119870 (119909

119894 V119895)))

]

1(119898minus1)

(A3)

Since sum119888119896=1

119906119894119896= 1 forallV

119896 from (A3) we can get

119888

sum

119896=1

[120582119894

2119898 (1 minus 119870 (119909119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))

]

1(119898minus1)

= 1 (A4)

120582119894

=1

[sum119888

119896=1(2119898 (1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896))))minus1(119898minus1)

]119898minus1

(A5)

By substituting (A5) into (A3) with necessary calculationsthe final 119906

119894119895is

119906119894119895

=

((1 minus 119870 (119909119894 V119895)) + 120593

119894(1 minus 119870 (119909

119894 V119895)))minus1(119898minus1)

sum119888

119896=1(1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))minus1(119898minus1)

(A6)

Similarly by taking the derivative of 119871ARKFCM with respect toV119895and setting the result to zero we have

120597119871ARKFCM120597V119895

=

119873

sum

119894=1

119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895)

+

119873

sum

119894=1

120593119894119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895) = 0

(A7)

After necessary calculations the final V119895is

V119895=

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) 119909119894+ 120593119894119870(119909119894 V119895) 119909119894)

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) + 120593119894119870(119909119894 V119895))

(A8)

Conflict of Interests

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

Acknowledgments

The authors would like to thank Dr Maoguo Gong andDr Chunming Li for providing their source codes forcomparison This work has been supported by the NationalProgram on Key Basic Research Project (nos 2013CB7338002012CB733803) Key Joint Program of National Natural Sci-ence Foundation and Guangdong Province (no U1201257)and Guangdong Innovative Research Team Program (no201001D0104648280)

References

[1] N Sharma and L M Aggarwal ldquoAutomated medical imagesegmentation techniquesrdquo Journal ofMedical Physics vol 35 no1 pp 3ndash14 2010

[2] A Elazab Q Hu F Jia and X Zhang ldquoContent based modifiedreaction-diffusion equation for modeling tumor growth of lowgrade gliomardquo in Proceedings of the 7th Cairo InternationalBiomedical Engineering Conference (CIBEC rsquo14) pp 107ndash110Giza Egypt December 2014

12 Computational and Mathematical Methods in Medicine

[3] D L PhamCXu and J L Prince ldquoCurrentmethods inmedicalimage segmentationrdquoAnnual Review of Biomedical Engineeringvol 2 pp 315ndash337 2000

[4] I Despotovic B Goossens andW Philips ldquoMRI segmentationof the human brain challenges methods and applicationsrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 450341 23 pages 2015

[5] S Ruan C Jaggi J Xue J Fadili and D Bloyet ldquoBraintissue classification of magnetic resonance images using partialvolume modelingrdquo IEEE Transactions on Medical Imaging vol19 no 12 pp 1172ndash1186 2000

[6] J C Dunn ldquoA fuzzy relative of the ISODATA process and itsuse in detecting compact well-separated clustersrdquo Journal ofCybernetics vol 3 no 3 pp 32ndash57 1973

[7] J C Bezdek Pattern Recognition with Fuzzy Objective FunctionAlgorithms Advanced Applications in Pattern RecognitionSpringer Boston Mass USA 1981

[8] M N Ahmed S M Yamany N Mohamed A A Farag andT Moriarty ldquoA modified fuzzy c-means algorithm for bias fieldestimation and segmentation of MRI datardquo IEEE Transactionson Medical Imaging vol 21 no 3 pp 193ndash199 2002

[9] L Szilagyi Z Benyo S M Szilagyi and H S Adam ldquoMRbrain image segmentation using an enhanced fuzzy C-meansalgorithmrdquo in Proceedings of the 25th Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety pp 724ndash726 Cancun Mexico September 2003

[10] S Chen andD Zhang ldquoRobust image segmentation using FCMwith spatial constraints based on new kernel-induced distancemeasurerdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 4 pp 1907ndash1916 2004

[11] W Cai S Chen and D Zhang ldquoFast and robust fuzzy c-meansclustering algorithms incorporating local information for imagesegmentationrdquo Pattern Recognition vol 40 no 3 pp 825ndash8382007

[12] M-S Yang and H-S Tsai ldquoA Gaussian kernel-based fuzzyc-means algorithm with a spatial bias correctionrdquo PatternRecognition Letters vol 29 no 12 pp 1713ndash1725 2008

[13] S Krinidis and V Chatzis ldquoA robust fuzzy local informationC-means clustering algorithmrdquo IEEE Transactions on ImageProcessing vol 19 no 5 pp 1328ndash1337 2010

[14] M Gong Y Liang J Shi W Ma and J Ma ldquoFuzzy C-meansclustering with local information and kernel metric for imagesegmentationrdquo IEEE Transactions on Image Processing vol 22no 2 pp 573ndash584 2013

[15] M A Balafar ldquoGaussian mixture model based segmentationmethods for brain MRI imagesrdquo Artificial Intelligence Reviewvol 41 no 3 pp 429ndash439 2014

[16] C Nikou N P Galatsanos and A C Likas ldquoA class-adaptivespatially variant mixture model for image segmentationrdquo IEEETransactions on Image Processing vol 16 no 4 pp 1121ndash11302007

[17] A R Ferreira da Silva ldquoA Dirichlet process mixture model forbrain MRI tissue classificationrdquoMedical Image Analysis vol 11no 2 pp 169ndash182 2007

[18] T M Nguyen and Q M J Wu ldquoFast and robust spatiallyconstrained gaussian mixture model for image segmentationrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 23 no 4 pp 621ndash635 2013

[19] S P Chatzis and T A Varvarigou ldquoA fuzzy clustering approachtoward hidden Markov random field models for enhancedspatially constrained image segmentationrdquo IEEE Transactionson Fuzzy Systems vol 16 no 5 pp 1351ndash1361 2008

[20] S Chatzis ldquoA method for training finite mixture models undera fuzzy clustering principlerdquo Fuzzy Sets and Systems vol 161 no23 pp 3000ndash3013 2010

[21] Z Ji J Liu G Cao Q Sun and Q Chen ldquoRobust spatiallyconstrained fuzzy c-means algorithm for brain MR imagesegmentationrdquo Pattern Recognition vol 47 no 7 pp 2454ndash2466 2014

[22] C Li C Xu A Anderson and J C Gore ldquoMRI tissueclassification and bias field estimation based on coherent localintensity clustering a unified energyminimization frameworkrdquoin Information Processing in Medical Imaging 21st InternationalConference IPMI 2009Williamsburg VA USA July 5ndash10 2009Proceedings vol 5636 of Lecture Notes in Computer Science pp288ndash299 Springer Berlin Germany 2009

[23] C Li J C Gore and C Davatzikos ldquoMultiplicative intrinsiccomponent optimization (MICO) for MRI bias field estimationand tissue segmentationrdquoMagnetic Resonance Imaging vol 32no 7 pp 913ndash923 2014

[24] L Szilagyi ldquoLessons to learn from a mistaken optimizationrdquoPattern Recognition Letters vol 36 no 1 pp 29ndash35 2014

[25] T Hofmann B Scholkopf and A J Smola ldquoKernel methodsin machine learningrdquoThe Annals of Statistics vol 36 no 3 pp1171ndash1220 2008

[26] K-R Muller S Mika G Ratsch K Tsuda and B ScholkopfldquoAn introduction to kernel-based learning algorithmsrdquo IEEETransactions onNeural Networks vol 12 no 2 pp 181ndash201 2001

[27] C R SouzaKernel Functions forMachine LearningApplications2010 httpcrsouzacom201003kernel-functions-for-machine-learning-applications

[28] U Vovk F Pernus and B Likar ldquoA review of methods for cor-rection of intensity inhomogeneity in MRIrdquo IEEE Transactionson Medical Imaging vol 26 no 3 pp 405ndash421 2007

[29] C A Cocosco V Kollokian R K-S Kwan and A C EvansldquoBrainWeb Online Interface to a 3D MRI Simulated BrainDatabaserdquo 2011 httpbrainwebbicmnimcgillcabrainweb

[30] L He and I R Greenshields ldquoA non-local maximum likelihoodestimation method for Rician noise reduction in MR imagesrdquoIEEE Transactions on Medical Imaging vol 28 no 2 pp 165ndash172 2009

[31] H Zhang J E Fritts and S A Goldman ldquoAn entropy-basedobjective evaluation method for image segmentationrdquo in Pro-ceedings of the Storage and Retrieval Methods and Applicationsfor Multimedia vol 5307 of Proceedings of SPIE pp 38ndash49January 2004

10 Computational and Mathematical Methods in Medicine

to noise and image artifacts at the expense of increasedcomputational cost [7ndash14]

The GKFCM1 and GKFCM2 algorithms [12] use theparameter 120578

119895to replace 120572 and need an additional loop on

the number of clusters to be calculated at each pixel toupdate the local contextual information So they have highercomputational cost than the original FCM FCM S1 andFCM S2 at each iteration

The FLICM algorithm [13] introduces 119866119894119895which needs

an additional loop on the neighborhood of the currentpixel to calculate the local information in every iterationthus it has a high computational cost As an extension ofFLICM KWFLICM [14] introduces

119894119895that requires two

additional loops on the neighborhood so it has the highestcomputational cost at each iteration

The RSCFCM algorithm [21] uses a spatial fuzzy factorthat is constructed based on the posterior and prior prob-abilities and takes the spatial direction into account Thatincreases the complexity as many parameters have to beoptimized and consequently the computational complexity

MICO algorithm [23] comes with fast calculations dueto the convexity of its energy function particularly in thepresence of less noise but tends to have many iterations in thepresence of high level noise

The proposed algorithms incorporate the local contextualinformation by introducing LVC which is a measure ofgrayscale heterogeneity and has nothing to dowith the clustercenters So it can be calculated once in advance and hencereduce the complexity of the clustering procedure

The eventual computational cost will be the multipli-cation of the computational cost for each iteration andthe number of iterations for convergence The number ofiterations will be dependent on the initialization as well as theobjective function To make fair comparisons initializationsare all set randomly and the average number of iterationsfor convergence is then recorded for 10 converged timesFrom Figure 7 it can be seen that the average iterationtimes will be data dependent with ARKFCM

2 ARKFCM

119908

and ARKFCM1having the minimum number of iterations

followed by FLICM GKFCM2 and GKFCM1 respectivelyThe running times for the tested images agree well with

the above analysis as illustrated in Figure 8 For SBD imagesgiven in Figures 2 3 and 4 the KWFLICM algorithm takesthe longest time (124ndash166 seconds) followed by FLICM(3ndash54 seconds) RSCFCM (about 25 seconds) GKFCM1GKFCM2 (06ndash19 seconds) and the proposed algorithms(022 to 036 seconds) For the clinical MR images in Figures5 and 6 it takes more time (due to more iterations) asthe images are more complicated but the trend remainsthe same with the proposed algorithms having the lowestcomputational cost (Table 4 and Figure 8)

53 Neighborhood Size The local neighborhood window sizeis a crucial factor to determine the smoothness of clusteringand details to be preserved We have experimented withdifferent window sizes and found that a window size of3 times 3 pixels achieves the best balance between segmentationaccuracy and computational cost Increasing window size to5 times 5 pixels has very small impact on the JS but 7 times 7 pixels

0

20

40

60

80

100

120

GKF

CM1

GKF

CM2

FLIC

M

MIC

O

Num

ber o

f ite

ratio

ns

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 7 Average number of iterations of all the algorithms (exceptRSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a) and 6(a)

002040608

112141618

222242628

GKF

CM1

GKF

CM2

MIC

O

Aver

age r

unni

ng ti

me (

s)

Segmentation algorithms

Figure 6Figure 5

Figure 4Figure 3Figure 2

ARK

FCM

w

ARK

FCM

2

ARK

FCM

1

Figure 8 Average running times of all the algorithms (exceptFLICM RSCFCM and KWFLICM) on Figures 2(a) 3(a) 4(a) 5(a)and 6(a)

or more will significantly decrease the accuracy such aslosing image details like CSF Therefore it is recommendedto use a local window of size 3 times 3 pixels for constructing theneighborhood

54 Limitation From the experiments it was found thatthe proposed algorithms could be sensitive to severe noisein small areas between edges that have width 1 or 2 pixels(eg area between ventricles Figures 4(h) 4(i) and 4(j))Potentially this may be solved by embedding edge detectiontechnique into the clustering process which is yet to beexplored

Computational and Mathematical Methods in Medicine 11

6 Conclusion

An adaptively regularized kernel-based FCM framework hasbeen proposed to enhance the original FCM for highersegmentation accuracy at low computational costThe frame-work can be in the form of three algorithms that employ theheterogeneity of grayscales in the neighborhood employedfor local contextual information The main advantages areadaptiveness to local context enhanced robustness and inde-pendence of clustering parameters to decrease computationalcost GRBF kernel has been adopted as a distance metric Wevalidated the proposed algorithms on synthetic and clinicalMR images with tumors The proposed algorithms attain ahigher JS (Tables 1 2 and 3) and lower entropy measure119864 (Table 4) than the 6 recent soft clustering algorithms andcould preserve small image details (Figures 2 3 4 5 and 6) Inaddition the proposed algorithms have a low computationalcost and are to the best of our knowledge the only algorithmsthat are adaptive to local context and do not include clustercenters Therefore they attain a trade-off between highsegmentation accuracy and low computational costThe pro-posed algorithms can be a potential tool for segmenting brainMR images for further processing and other images as well

Appendix

The objective function 119869ARKFCM given in equation (17) withconditions in (2) is a constrained minimization problemwhich can be solved using Lagrange multiplier method Letthe minimization problem of 119869ARKFCM be written as follows

119871ARKFCM (119906119894119895 V119895 120582119894 120593119894)

= 2

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2

119873

sum

119894=1

119888

sum

119895=1

120593119894119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+

119873

sum

119894=1

120582119894(1 minus

119888

sum

119895=1

119906119894119895)

(A1)

By taking the derivative of 119871ARKFCM with respect to 119906119894119895and

setting the result to zero for119898 gt 1 we have

120597119871ARKFCM120597119906119894119895

= 2119898119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2119898120593119894119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895)) minus 120582

119894

= 0

(A2)

By using (A2) we can calculate 119906119894119895as follows

119906119894119895

= [120582119894

2119898 (1 minus 119870 (119909119894 V119895) + 120593119894(1 minus 119870 (119909

119894 V119895)))

]

1(119898minus1)

(A3)

Since sum119888119896=1

119906119894119896= 1 forallV

119896 from (A3) we can get

119888

sum

119896=1

[120582119894

2119898 (1 minus 119870 (119909119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))

]

1(119898minus1)

= 1 (A4)

120582119894

=1

[sum119888

119896=1(2119898 (1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896))))minus1(119898minus1)

]119898minus1

(A5)

By substituting (A5) into (A3) with necessary calculationsthe final 119906

119894119895is

119906119894119895

=

((1 minus 119870 (119909119894 V119895)) + 120593

119894(1 minus 119870 (119909

119894 V119895)))minus1(119898minus1)

sum119888

119896=1(1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))minus1(119898minus1)

(A6)

Similarly by taking the derivative of 119871ARKFCM with respect toV119895and setting the result to zero we have

120597119871ARKFCM120597V119895

=

119873

sum

119894=1

119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895)

+

119873

sum

119894=1

120593119894119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895) = 0

(A7)

After necessary calculations the final V119895is

V119895=

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) 119909119894+ 120593119894119870(119909119894 V119895) 119909119894)

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) + 120593119894119870(119909119894 V119895))

(A8)

Conflict of Interests

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

Acknowledgments

The authors would like to thank Dr Maoguo Gong andDr Chunming Li for providing their source codes forcomparison This work has been supported by the NationalProgram on Key Basic Research Project (nos 2013CB7338002012CB733803) Key Joint Program of National Natural Sci-ence Foundation and Guangdong Province (no U1201257)and Guangdong Innovative Research Team Program (no201001D0104648280)

References

[1] N Sharma and L M Aggarwal ldquoAutomated medical imagesegmentation techniquesrdquo Journal ofMedical Physics vol 35 no1 pp 3ndash14 2010

[2] A Elazab Q Hu F Jia and X Zhang ldquoContent based modifiedreaction-diffusion equation for modeling tumor growth of lowgrade gliomardquo in Proceedings of the 7th Cairo InternationalBiomedical Engineering Conference (CIBEC rsquo14) pp 107ndash110Giza Egypt December 2014

12 Computational and Mathematical Methods in Medicine

[3] D L PhamCXu and J L Prince ldquoCurrentmethods inmedicalimage segmentationrdquoAnnual Review of Biomedical Engineeringvol 2 pp 315ndash337 2000

[4] I Despotovic B Goossens andW Philips ldquoMRI segmentationof the human brain challenges methods and applicationsrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 450341 23 pages 2015

[5] S Ruan C Jaggi J Xue J Fadili and D Bloyet ldquoBraintissue classification of magnetic resonance images using partialvolume modelingrdquo IEEE Transactions on Medical Imaging vol19 no 12 pp 1172ndash1186 2000

[6] J C Dunn ldquoA fuzzy relative of the ISODATA process and itsuse in detecting compact well-separated clustersrdquo Journal ofCybernetics vol 3 no 3 pp 32ndash57 1973

[7] J C Bezdek Pattern Recognition with Fuzzy Objective FunctionAlgorithms Advanced Applications in Pattern RecognitionSpringer Boston Mass USA 1981

[8] M N Ahmed S M Yamany N Mohamed A A Farag andT Moriarty ldquoA modified fuzzy c-means algorithm for bias fieldestimation and segmentation of MRI datardquo IEEE Transactionson Medical Imaging vol 21 no 3 pp 193ndash199 2002

[9] L Szilagyi Z Benyo S M Szilagyi and H S Adam ldquoMRbrain image segmentation using an enhanced fuzzy C-meansalgorithmrdquo in Proceedings of the 25th Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety pp 724ndash726 Cancun Mexico September 2003

[10] S Chen andD Zhang ldquoRobust image segmentation using FCMwith spatial constraints based on new kernel-induced distancemeasurerdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 4 pp 1907ndash1916 2004

[11] W Cai S Chen and D Zhang ldquoFast and robust fuzzy c-meansclustering algorithms incorporating local information for imagesegmentationrdquo Pattern Recognition vol 40 no 3 pp 825ndash8382007

[12] M-S Yang and H-S Tsai ldquoA Gaussian kernel-based fuzzyc-means algorithm with a spatial bias correctionrdquo PatternRecognition Letters vol 29 no 12 pp 1713ndash1725 2008

[13] S Krinidis and V Chatzis ldquoA robust fuzzy local informationC-means clustering algorithmrdquo IEEE Transactions on ImageProcessing vol 19 no 5 pp 1328ndash1337 2010

[14] M Gong Y Liang J Shi W Ma and J Ma ldquoFuzzy C-meansclustering with local information and kernel metric for imagesegmentationrdquo IEEE Transactions on Image Processing vol 22no 2 pp 573ndash584 2013

[15] M A Balafar ldquoGaussian mixture model based segmentationmethods for brain MRI imagesrdquo Artificial Intelligence Reviewvol 41 no 3 pp 429ndash439 2014

[16] C Nikou N P Galatsanos and A C Likas ldquoA class-adaptivespatially variant mixture model for image segmentationrdquo IEEETransactions on Image Processing vol 16 no 4 pp 1121ndash11302007

[17] A R Ferreira da Silva ldquoA Dirichlet process mixture model forbrain MRI tissue classificationrdquoMedical Image Analysis vol 11no 2 pp 169ndash182 2007

[18] T M Nguyen and Q M J Wu ldquoFast and robust spatiallyconstrained gaussian mixture model for image segmentationrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 23 no 4 pp 621ndash635 2013

[19] S P Chatzis and T A Varvarigou ldquoA fuzzy clustering approachtoward hidden Markov random field models for enhancedspatially constrained image segmentationrdquo IEEE Transactionson Fuzzy Systems vol 16 no 5 pp 1351ndash1361 2008

[20] S Chatzis ldquoA method for training finite mixture models undera fuzzy clustering principlerdquo Fuzzy Sets and Systems vol 161 no23 pp 3000ndash3013 2010

[21] Z Ji J Liu G Cao Q Sun and Q Chen ldquoRobust spatiallyconstrained fuzzy c-means algorithm for brain MR imagesegmentationrdquo Pattern Recognition vol 47 no 7 pp 2454ndash2466 2014

[22] C Li C Xu A Anderson and J C Gore ldquoMRI tissueclassification and bias field estimation based on coherent localintensity clustering a unified energyminimization frameworkrdquoin Information Processing in Medical Imaging 21st InternationalConference IPMI 2009Williamsburg VA USA July 5ndash10 2009Proceedings vol 5636 of Lecture Notes in Computer Science pp288ndash299 Springer Berlin Germany 2009

[23] C Li J C Gore and C Davatzikos ldquoMultiplicative intrinsiccomponent optimization (MICO) for MRI bias field estimationand tissue segmentationrdquoMagnetic Resonance Imaging vol 32no 7 pp 913ndash923 2014

[24] L Szilagyi ldquoLessons to learn from a mistaken optimizationrdquoPattern Recognition Letters vol 36 no 1 pp 29ndash35 2014

[25] T Hofmann B Scholkopf and A J Smola ldquoKernel methodsin machine learningrdquoThe Annals of Statistics vol 36 no 3 pp1171ndash1220 2008

[26] K-R Muller S Mika G Ratsch K Tsuda and B ScholkopfldquoAn introduction to kernel-based learning algorithmsrdquo IEEETransactions onNeural Networks vol 12 no 2 pp 181ndash201 2001

[27] C R SouzaKernel Functions forMachine LearningApplications2010 httpcrsouzacom201003kernel-functions-for-machine-learning-applications

[28] U Vovk F Pernus and B Likar ldquoA review of methods for cor-rection of intensity inhomogeneity in MRIrdquo IEEE Transactionson Medical Imaging vol 26 no 3 pp 405ndash421 2007

[29] C A Cocosco V Kollokian R K-S Kwan and A C EvansldquoBrainWeb Online Interface to a 3D MRI Simulated BrainDatabaserdquo 2011 httpbrainwebbicmnimcgillcabrainweb

[30] L He and I R Greenshields ldquoA non-local maximum likelihoodestimation method for Rician noise reduction in MR imagesrdquoIEEE Transactions on Medical Imaging vol 28 no 2 pp 165ndash172 2009

[31] H Zhang J E Fritts and S A Goldman ldquoAn entropy-basedobjective evaluation method for image segmentationrdquo in Pro-ceedings of the Storage and Retrieval Methods and Applicationsfor Multimedia vol 5307 of Proceedings of SPIE pp 38ndash49January 2004

Computational and Mathematical Methods in Medicine 11

6 Conclusion

An adaptively regularized kernel-based FCM framework hasbeen proposed to enhance the original FCM for highersegmentation accuracy at low computational costThe frame-work can be in the form of three algorithms that employ theheterogeneity of grayscales in the neighborhood employedfor local contextual information The main advantages areadaptiveness to local context enhanced robustness and inde-pendence of clustering parameters to decrease computationalcost GRBF kernel has been adopted as a distance metric Wevalidated the proposed algorithms on synthetic and clinicalMR images with tumors The proposed algorithms attain ahigher JS (Tables 1 2 and 3) and lower entropy measure119864 (Table 4) than the 6 recent soft clustering algorithms andcould preserve small image details (Figures 2 3 4 5 and 6) Inaddition the proposed algorithms have a low computationalcost and are to the best of our knowledge the only algorithmsthat are adaptive to local context and do not include clustercenters Therefore they attain a trade-off between highsegmentation accuracy and low computational costThe pro-posed algorithms can be a potential tool for segmenting brainMR images for further processing and other images as well

Appendix

The objective function 119869ARKFCM given in equation (17) withconditions in (2) is a constrained minimization problemwhich can be solved using Lagrange multiplier method Letthe minimization problem of 119869ARKFCM be written as follows

119871ARKFCM (119906119894119895 V119895 120582119894 120593119894)

= 2

119873

sum

119894=1

119888

sum

119895=1

119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2

119873

sum

119894=1

119888

sum

119895=1

120593119894119906119898

119894119895(1 minus 119870 (119909

119894 V119895))

+

119873

sum

119894=1

120582119894(1 minus

119888

sum

119895=1

119906119894119895)

(A1)

By taking the derivative of 119871ARKFCM with respect to 119906119894119895and

setting the result to zero for119898 gt 1 we have

120597119871ARKFCM120597119906119894119895

= 2119898119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895))

+ 2119898120593119894119906119898minus1

119894119895(1 minus 119870 (119909

119894 V119895)) minus 120582

119894

= 0

(A2)

By using (A2) we can calculate 119906119894119895as follows

119906119894119895

= [120582119894

2119898 (1 minus 119870 (119909119894 V119895) + 120593119894(1 minus 119870 (119909

119894 V119895)))

]

1(119898minus1)

(A3)

Since sum119888119896=1

119906119894119896= 1 forallV

119896 from (A3) we can get

119888

sum

119896=1

[120582119894

2119898 (1 minus 119870 (119909119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))

]

1(119898minus1)

= 1 (A4)

120582119894

=1

[sum119888

119896=1(2119898 (1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896))))minus1(119898minus1)

]119898minus1

(A5)

By substituting (A5) into (A3) with necessary calculationsthe final 119906

119894119895is

119906119894119895

=

((1 minus 119870 (119909119894 V119895)) + 120593

119894(1 minus 119870 (119909

119894 V119895)))minus1(119898minus1)

sum119888

119896=1(1 minus 119870 (119909

119894 V119896) + 120593119894(1 minus 119870 (119909

119894 V119896)))minus1(119898minus1)

(A6)

Similarly by taking the derivative of 119871ARKFCM with respect toV119895and setting the result to zero we have

120597119871ARKFCM120597V119895

=

119873

sum

119894=1

119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895)

+

119873

sum

119894=1

120593119894119906119898

119894119895119870(119909119894 V119895) (119909119894minus V119895) = 0

(A7)

After necessary calculations the final V119895is

V119895=

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) 119909119894+ 120593119894119870(119909119894 V119895) 119909119894)

sum119873

119894=1119906119898

119894119895(119870 (119909

119894 V119895) + 120593119894119870(119909119894 V119895))

(A8)

Conflict of Interests

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

Acknowledgments

The authors would like to thank Dr Maoguo Gong andDr Chunming Li for providing their source codes forcomparison This work has been supported by the NationalProgram on Key Basic Research Project (nos 2013CB7338002012CB733803) Key Joint Program of National Natural Sci-ence Foundation and Guangdong Province (no U1201257)and Guangdong Innovative Research Team Program (no201001D0104648280)

References

[1] N Sharma and L M Aggarwal ldquoAutomated medical imagesegmentation techniquesrdquo Journal ofMedical Physics vol 35 no1 pp 3ndash14 2010

[2] A Elazab Q Hu F Jia and X Zhang ldquoContent based modifiedreaction-diffusion equation for modeling tumor growth of lowgrade gliomardquo in Proceedings of the 7th Cairo InternationalBiomedical Engineering Conference (CIBEC rsquo14) pp 107ndash110Giza Egypt December 2014

12 Computational and Mathematical Methods in Medicine

[3] D L PhamCXu and J L Prince ldquoCurrentmethods inmedicalimage segmentationrdquoAnnual Review of Biomedical Engineeringvol 2 pp 315ndash337 2000

[4] I Despotovic B Goossens andW Philips ldquoMRI segmentationof the human brain challenges methods and applicationsrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 450341 23 pages 2015

[5] S Ruan C Jaggi J Xue J Fadili and D Bloyet ldquoBraintissue classification of magnetic resonance images using partialvolume modelingrdquo IEEE Transactions on Medical Imaging vol19 no 12 pp 1172ndash1186 2000

[6] J C Dunn ldquoA fuzzy relative of the ISODATA process and itsuse in detecting compact well-separated clustersrdquo Journal ofCybernetics vol 3 no 3 pp 32ndash57 1973

[7] J C Bezdek Pattern Recognition with Fuzzy Objective FunctionAlgorithms Advanced Applications in Pattern RecognitionSpringer Boston Mass USA 1981

[8] M N Ahmed S M Yamany N Mohamed A A Farag andT Moriarty ldquoA modified fuzzy c-means algorithm for bias fieldestimation and segmentation of MRI datardquo IEEE Transactionson Medical Imaging vol 21 no 3 pp 193ndash199 2002

[9] L Szilagyi Z Benyo S M Szilagyi and H S Adam ldquoMRbrain image segmentation using an enhanced fuzzy C-meansalgorithmrdquo in Proceedings of the 25th Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety pp 724ndash726 Cancun Mexico September 2003

[10] S Chen andD Zhang ldquoRobust image segmentation using FCMwith spatial constraints based on new kernel-induced distancemeasurerdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 4 pp 1907ndash1916 2004

[11] W Cai S Chen and D Zhang ldquoFast and robust fuzzy c-meansclustering algorithms incorporating local information for imagesegmentationrdquo Pattern Recognition vol 40 no 3 pp 825ndash8382007

[12] M-S Yang and H-S Tsai ldquoA Gaussian kernel-based fuzzyc-means algorithm with a spatial bias correctionrdquo PatternRecognition Letters vol 29 no 12 pp 1713ndash1725 2008

[13] S Krinidis and V Chatzis ldquoA robust fuzzy local informationC-means clustering algorithmrdquo IEEE Transactions on ImageProcessing vol 19 no 5 pp 1328ndash1337 2010

[14] M Gong Y Liang J Shi W Ma and J Ma ldquoFuzzy C-meansclustering with local information and kernel metric for imagesegmentationrdquo IEEE Transactions on Image Processing vol 22no 2 pp 573ndash584 2013

[15] M A Balafar ldquoGaussian mixture model based segmentationmethods for brain MRI imagesrdquo Artificial Intelligence Reviewvol 41 no 3 pp 429ndash439 2014

[16] C Nikou N P Galatsanos and A C Likas ldquoA class-adaptivespatially variant mixture model for image segmentationrdquo IEEETransactions on Image Processing vol 16 no 4 pp 1121ndash11302007

[17] A R Ferreira da Silva ldquoA Dirichlet process mixture model forbrain MRI tissue classificationrdquoMedical Image Analysis vol 11no 2 pp 169ndash182 2007

[18] T M Nguyen and Q M J Wu ldquoFast and robust spatiallyconstrained gaussian mixture model for image segmentationrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 23 no 4 pp 621ndash635 2013

[19] S P Chatzis and T A Varvarigou ldquoA fuzzy clustering approachtoward hidden Markov random field models for enhancedspatially constrained image segmentationrdquo IEEE Transactionson Fuzzy Systems vol 16 no 5 pp 1351ndash1361 2008

[20] S Chatzis ldquoA method for training finite mixture models undera fuzzy clustering principlerdquo Fuzzy Sets and Systems vol 161 no23 pp 3000ndash3013 2010

[21] Z Ji J Liu G Cao Q Sun and Q Chen ldquoRobust spatiallyconstrained fuzzy c-means algorithm for brain MR imagesegmentationrdquo Pattern Recognition vol 47 no 7 pp 2454ndash2466 2014

[22] C Li C Xu A Anderson and J C Gore ldquoMRI tissueclassification and bias field estimation based on coherent localintensity clustering a unified energyminimization frameworkrdquoin Information Processing in Medical Imaging 21st InternationalConference IPMI 2009Williamsburg VA USA July 5ndash10 2009Proceedings vol 5636 of Lecture Notes in Computer Science pp288ndash299 Springer Berlin Germany 2009

[23] C Li J C Gore and C Davatzikos ldquoMultiplicative intrinsiccomponent optimization (MICO) for MRI bias field estimationand tissue segmentationrdquoMagnetic Resonance Imaging vol 32no 7 pp 913ndash923 2014

[24] L Szilagyi ldquoLessons to learn from a mistaken optimizationrdquoPattern Recognition Letters vol 36 no 1 pp 29ndash35 2014

[25] T Hofmann B Scholkopf and A J Smola ldquoKernel methodsin machine learningrdquoThe Annals of Statistics vol 36 no 3 pp1171ndash1220 2008

[26] K-R Muller S Mika G Ratsch K Tsuda and B ScholkopfldquoAn introduction to kernel-based learning algorithmsrdquo IEEETransactions onNeural Networks vol 12 no 2 pp 181ndash201 2001

[27] C R SouzaKernel Functions forMachine LearningApplications2010 httpcrsouzacom201003kernel-functions-for-machine-learning-applications

[28] U Vovk F Pernus and B Likar ldquoA review of methods for cor-rection of intensity inhomogeneity in MRIrdquo IEEE Transactionson Medical Imaging vol 26 no 3 pp 405ndash421 2007

[29] C A Cocosco V Kollokian R K-S Kwan and A C EvansldquoBrainWeb Online Interface to a 3D MRI Simulated BrainDatabaserdquo 2011 httpbrainwebbicmnimcgillcabrainweb

[30] L He and I R Greenshields ldquoA non-local maximum likelihoodestimation method for Rician noise reduction in MR imagesrdquoIEEE Transactions on Medical Imaging vol 28 no 2 pp 165ndash172 2009

[31] H Zhang J E Fritts and S A Goldman ldquoAn entropy-basedobjective evaluation method for image segmentationrdquo in Pro-ceedings of the Storage and Retrieval Methods and Applicationsfor Multimedia vol 5307 of Proceedings of SPIE pp 38ndash49January 2004

12 Computational and Mathematical Methods in Medicine

[3] D L PhamCXu and J L Prince ldquoCurrentmethods inmedicalimage segmentationrdquoAnnual Review of Biomedical Engineeringvol 2 pp 315ndash337 2000

[4] I Despotovic B Goossens andW Philips ldquoMRI segmentationof the human brain challenges methods and applicationsrdquoComputational and Mathematical Methods in Medicine vol2015 Article ID 450341 23 pages 2015

[5] S Ruan C Jaggi J Xue J Fadili and D Bloyet ldquoBraintissue classification of magnetic resonance images using partialvolume modelingrdquo IEEE Transactions on Medical Imaging vol19 no 12 pp 1172ndash1186 2000

[6] J C Dunn ldquoA fuzzy relative of the ISODATA process and itsuse in detecting compact well-separated clustersrdquo Journal ofCybernetics vol 3 no 3 pp 32ndash57 1973

[7] J C Bezdek Pattern Recognition with Fuzzy Objective FunctionAlgorithms Advanced Applications in Pattern RecognitionSpringer Boston Mass USA 1981

[8] M N Ahmed S M Yamany N Mohamed A A Farag andT Moriarty ldquoA modified fuzzy c-means algorithm for bias fieldestimation and segmentation of MRI datardquo IEEE Transactionson Medical Imaging vol 21 no 3 pp 193ndash199 2002

[9] L Szilagyi Z Benyo S M Szilagyi and H S Adam ldquoMRbrain image segmentation using an enhanced fuzzy C-meansalgorithmrdquo in Proceedings of the 25th Annual InternationalConference of the IEEE Engineering in Medicine and BiologySociety pp 724ndash726 Cancun Mexico September 2003

[10] S Chen andD Zhang ldquoRobust image segmentation using FCMwith spatial constraints based on new kernel-induced distancemeasurerdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 34 no 4 pp 1907ndash1916 2004

[11] W Cai S Chen and D Zhang ldquoFast and robust fuzzy c-meansclustering algorithms incorporating local information for imagesegmentationrdquo Pattern Recognition vol 40 no 3 pp 825ndash8382007

[12] M-S Yang and H-S Tsai ldquoA Gaussian kernel-based fuzzyc-means algorithm with a spatial bias correctionrdquo PatternRecognition Letters vol 29 no 12 pp 1713ndash1725 2008

[13] S Krinidis and V Chatzis ldquoA robust fuzzy local informationC-means clustering algorithmrdquo IEEE Transactions on ImageProcessing vol 19 no 5 pp 1328ndash1337 2010

[14] M Gong Y Liang J Shi W Ma and J Ma ldquoFuzzy C-meansclustering with local information and kernel metric for imagesegmentationrdquo IEEE Transactions on Image Processing vol 22no 2 pp 573ndash584 2013

[15] M A Balafar ldquoGaussian mixture model based segmentationmethods for brain MRI imagesrdquo Artificial Intelligence Reviewvol 41 no 3 pp 429ndash439 2014

[16] C Nikou N P Galatsanos and A C Likas ldquoA class-adaptivespatially variant mixture model for image segmentationrdquo IEEETransactions on Image Processing vol 16 no 4 pp 1121ndash11302007

[17] A R Ferreira da Silva ldquoA Dirichlet process mixture model forbrain MRI tissue classificationrdquoMedical Image Analysis vol 11no 2 pp 169ndash182 2007

[18] T M Nguyen and Q M J Wu ldquoFast and robust spatiallyconstrained gaussian mixture model for image segmentationrdquoIEEE Transactions on Circuits and Systems for Video Technologyvol 23 no 4 pp 621ndash635 2013

[19] S P Chatzis and T A Varvarigou ldquoA fuzzy clustering approachtoward hidden Markov random field models for enhancedspatially constrained image segmentationrdquo IEEE Transactionson Fuzzy Systems vol 16 no 5 pp 1351ndash1361 2008

[20] S Chatzis ldquoA method for training finite mixture models undera fuzzy clustering principlerdquo Fuzzy Sets and Systems vol 161 no23 pp 3000ndash3013 2010

[21] Z Ji J Liu G Cao Q Sun and Q Chen ldquoRobust spatiallyconstrained fuzzy c-means algorithm for brain MR imagesegmentationrdquo Pattern Recognition vol 47 no 7 pp 2454ndash2466 2014

[22] C Li C Xu A Anderson and J C Gore ldquoMRI tissueclassification and bias field estimation based on coherent localintensity clustering a unified energyminimization frameworkrdquoin Information Processing in Medical Imaging 21st InternationalConference IPMI 2009Williamsburg VA USA July 5ndash10 2009Proceedings vol 5636 of Lecture Notes in Computer Science pp288ndash299 Springer Berlin Germany 2009

[23] C Li J C Gore and C Davatzikos ldquoMultiplicative intrinsiccomponent optimization (MICO) for MRI bias field estimationand tissue segmentationrdquoMagnetic Resonance Imaging vol 32no 7 pp 913ndash923 2014

[24] L Szilagyi ldquoLessons to learn from a mistaken optimizationrdquoPattern Recognition Letters vol 36 no 1 pp 29ndash35 2014

[25] T Hofmann B Scholkopf and A J Smola ldquoKernel methodsin machine learningrdquoThe Annals of Statistics vol 36 no 3 pp1171ndash1220 2008

[26] K-R Muller S Mika G Ratsch K Tsuda and B ScholkopfldquoAn introduction to kernel-based learning algorithmsrdquo IEEETransactions onNeural Networks vol 12 no 2 pp 181ndash201 2001

[27] C R SouzaKernel Functions forMachine LearningApplications2010 httpcrsouzacom201003kernel-functions-for-machine-learning-applications

[28] U Vovk F Pernus and B Likar ldquoA review of methods for cor-rection of intensity inhomogeneity in MRIrdquo IEEE Transactionson Medical Imaging vol 26 no 3 pp 405ndash421 2007

[29] C A Cocosco V Kollokian R K-S Kwan and A C EvansldquoBrainWeb Online Interface to a 3D MRI Simulated BrainDatabaserdquo 2011 httpbrainwebbicmnimcgillcabrainweb

[30] L He and I R Greenshields ldquoA non-local maximum likelihoodestimation method for Rician noise reduction in MR imagesrdquoIEEE Transactions on Medical Imaging vol 28 no 2 pp 165ndash172 2009

[31] H Zhang J E Fritts and S A Goldman ldquoAn entropy-basedobjective evaluation method for image segmentationrdquo in Pro-ceedings of the Storage and Retrieval Methods and Applicationsfor Multimedia vol 5307 of Proceedings of SPIE pp 38ndash49January 2004