Review Article FCM Clustering Algorithms for Segmentation...

Review ArticleFCM Clustering Algorithms for Segmentation ofBrain MR Images

Yogita K Dubey and Milind M Mushrif

Department of Electronics and Telecommunication Yeshwantrao Chavan College of Engineering WanadongriHingna Road Nagpur Maharashtra 441110 India

Correspondence should be addressed to Yogita K Dubey yogeetakdubeyyahoocoin

Received 8 November 2015 Revised 16 February 2016 Accepted 17 February 2016

Academic Editor RustomM Mamlook

Copyright copy 2016 Y K Dubey and M M Mushrif This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The study of brain disorders requires accurate tissue segmentation of magnetic resonance (MR) brain images which is veryimportant for detecting tumors edema and necrotic tissues Segmentation of brain images especially into three main tissue typesCerebrospinal Fluid (CSF) Gray Matter (GM) and White Matter (WM) has important role in computer aided neurosurgery anddiagnosis Brain images mostly contain noise intensity inhomogeneity and weak boundaries Therefore accurate segmentation ofbrain images is still a challenging area of researchThis paper presents a review of fuzzy 119888-means (FCM) clustering algorithms for thesegmentation of brain MR images The review covers the detailed analysis of FCM based algorithms with intensity inhomogeneitycorrection and noise robustness Different methods for the modification of standard fuzzy objective function with updating ofmembership and cluster centroid are also discussed

1 Introduction

The purpose of image segmentation is to partition imageto different regions based on given criteria for futureprocessing Image segmentation plays an important role inmedical applications such as abnormality detection quanti-tative analysis and postsurgical assessment Due to unknownnoise intensity inhomogeneity and partial volume effecttheir precise segmentation is a difficult task A variety offuzzy techniques [1ndash3] have been reported in the literaturefor image segmentation These methods fail to deal withlocal spatial property of images which leads to strong noisesensibility Brain MR images mainly suffer from intensityinhomogeneity and noise caused due to radio frequency coilused in image acquisition [4 5] Therefore correction ofintensity inhomogeneity as well as removal of noise is alwaysdesirable before segmentation of brain MR images

Segmentation ofMR images is still a challenging problembecause they are affected by multiple factors such as

(1) noise caused in image acquisition(2) poor contrast and intensity inhomogeneity physically

linked to the radio frequency MR signal

(3) partial volume effect being the mixture of severaltissue signals in the same pixel induced by the imageresolution

Few review papers for the segmentation of brain MRimages with intensity inhomogeneity correction and noiserobustness have been published In [6] the performancesof four methods (a phantom correction method an imagesmoothing technique homomorphic filtering and surfacefitting approach) have been investigated In [7] quantita-tive and qualitative evaluation of six retrospective methods(low pass filtering homomorphic filtering Fourier domainnonparametric histogram based and model based methods)have been reported In [8] review of intensity correctionhas been categorized in model based methods such as lowfrequency hyper surface and statistical models Reviews dis-cussed in [9ndash11] aremainly based on intensity inhomogeneitycorrection methods that are categorized as prospective andretrospective methods only

In [9] retrospective methods are divided into two cat-egories such as gray scale based and transformed domainbased Review discussed different surface fitting (polynomialfitting spline fitting) spatial filtering (low pass filtering and

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2016 Article ID 3406406 14 pageshttpdxdoiorg10115520163406406

2 Advances in Fuzzy Systems

homomorphic filtering) and statistical methods (Markovrandom fields (MRF) and FCM) under the category ofgray scale based methods and probability density functionFourier domain and wavelet domain methods under thecategory of transform based methods

Prospective methods like phantom multicoil and specialsequences and retrospective methods such as filtering meth-ods (homomorphic and unsharp masking) surface fitting(intensity and gradient) segmentation based (maximumlikelihood (ML) maximum a posteriori (MAP) FCM andnonparametric methods) and histogram based (high fre-quency maximization information minimization and his-togrammatching) methods are discussed in [10] This reviewalso analysed numerous publications to assess major trendspopularity and applications of intensity inhomogeneity cor-rection methods

Different brain image denoising methods such as aniso-tropic nonlinear diffusion MRF wavelet based analyticalcorrection and nonlocal methods are discussed in [11] Thisreview also discussed prospective methods like phantommulticoil and special sequences and retrospective methodssuch as surface fitting methods segmentation based (MLMAP and FCM) histogram based and filtering methodsMain focus of this paper is as follows

(1) Comprehensive review of FCM clustering based algo-rithms for the segmentation of brain MR imageswith intensity inhomogeneity corrections and noiserobustness is presented

(2) Complete mathematical analysis for the formulationof objective function of FCM clustering based algo-rithms for intensity inhomogeneity correction andnoise robustness is carried out

(3) Algorithms are compared on the basis of updating ofmembership function and cluster centroid

(4) Computational complexity and noise robustness ofthese algorithms are discussed

(5) Quantitative measures used by researcher are alsodiscussed

Rest of the paper is organized as follows Intensity inho-mogeneity in brain MR images and its correction methodsare described in Section 2 Noise in brain MR images anddenoising methods are briefed in Section 3 FCM clusteringalgorithm and its drawback for the segmentation of brainMRimage are briefly explained in Section 4 Detailed analysisof FCM clustering based algorithms for the segmentation ofbrain MR images with intensity inhomogeneity correctionand noise robustness is presented in Sections 5 6 and 7Section 8 describes the different validation methods whichare used for comparison Information about the dataset isgiven in Section 9 followed by the conclusion of the studyin Section 10

2 Intensity Inhomogeneity in Brain MRImages and Its Correction Methods

In brain MR images the observed signal is modelled as aproduct of the true signal and spatial varying factor calledbias field

119884119896 = 119883119896119866119896 1 le 119896 le 119873 (1)

where 119883119896 and 119884119896 are the true and observed intensities atthe 119896th pixel respectively 119866119896 is the bias field and 119873 isthe total number of pixels in a brain MR image Intensityinhomogeneity or bias is the slowly changing and smoothvariation in signal intensity For a bias field of magnitude40 the signal is multiplied by a field with values rangingfrom 08 to 12 (ie with values between 20 below trueintensity and 20 above true intensity) The application oflogarithmic transformation to the intensities allows artifactto be modelled as additive bias field [12] as given below

119910119896 = 119909119896 + 120573119896 1 le 119896 le 119873 (2)

Figure 1 depicts the effect of intensity inhomogeneityon brain MR images Original brain MR image [13] isshown in Figure 1(a) and images altered by 40 80 and100 intensity inhomogeneity respectively are shown inFigures 1(b) 1(c) and 1(d) respectively

Ahmed et al [14] used sinusoidal gain field of higherspatial frequency as bias field In [15ndash17] the bias field isestimated by a linear combination of a set of basis functionsas

119887119896 =

119872

sum

1

120596119896119892119896 (119896) = 120596119879119866 (119870) (3)

where 120596119879 = (1205961 1205962 120596119872)119879 120596119896 isin R 119896 = 1 2 119872 are

the combination coefficients119866(119896) = (1198921(119896) 1198922(119896) 119892119872(119896))119879

are the set of basis functions In [18 19] weighted bias field isused with initialization of weights as 120596119896 isin (0 1) for examplewith 10 data items 1205961 = 0008 increase 0001 1205962 1205963 12059610

21 FCM Clustering Methods with Spatial Constraints TheFCM based methods are modified by incorporating spatialconstraint in the objective function of FCM These methods[12 20] are used for intensity inhomogeneity correction andpartial volume segmentation of brain MR images Ahmedet al [14] proposed bias corrected FCM (BCFCM) bymodifying the objective function of FCM to compensatefor the intensity inhomogeneity But BCFCM is very time-consuming since it computes the neighbourhood term ineach iteration step This drawback is eliminated in spatiallyconstrained kernelized FCM (SKFCM) [21] where differentpenalty terms containing spatial neighbourhood informationin the objective function are used They also replaced thesimilarity measurement in the FCM by a kernel induceddistance

22 Modified FCM Based Methods Improved and enhancedFCM clustering algorithms [22ndash27] have been used to

Advances in Fuzzy Systems 3

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

Figure 1 Intensity inhomogeneity in brain MR images (a) Original brain MR image The images altered by 40 80 and 100 intensityinhomogeneity in (b) (c) and (d) respectively

accelerate the image segmentation process and to correct theintensity inhomogeneity during segmentation Integration offuzzy spatial relations in deformable models is proposed in[28 29] for brain MRI segmentation

23 Other Methods In recent years multilevel methods [3031] multidimensional approach [32] multichannel [33 34]model based approaches [35ndash38] and level set approaches[39ndash42] have been widely used for brain image segmentationand tumor detection

24 Generalized FCM Based Methods Modified fast FCMalgorithms [15 43 44] rough set based FCM clustering algo-rithms [17 45 46] possibilistic FCM clustering algorithm[16] and FCM based algorithms [18 19 47ndash49] are also usedfor brain image segmentation Recently intuitionistic fuzzyset based clustering [50 51] approaches have been used forbrain image segmentation

3 Noise in Brain MR Imagesand Denoising Methods

Image segmentation algorithms are sensitive to noise Thepresence of noise produces undesirable visual quality andlowers the visibility of low contrast objects Effect of noise onbrain MR images is shown in Figure 2 Original brain MRimage [13] is shown in Figure 2(a) and images corruptedby 3 7 and 11 noise are shown in Figures 2(b) 2(c)and 2(d) respectively Thus image denoising is one of theimportant steps as a preprocessing in various applications Itsobjective is to recover the best estimate of the original imagefrom its noisy version The performance of segmentationalgorithms can be improved by image filtering Linear lowpass filters update value of a pixel by average of its neigh-bourhoods These filters reduce noise but produce blurredimages and fail to preserve the edges in the presence oflarge amount of noise Nonlinear filters have better per-formance in edge preserving but degrade fine structuretherefore the resolution of the image is reduced Waveletfiltering can overcome this drawback Wavelet transform at

high frequency gives good spatial resolution while at lowfrequency it gives good frequency resolution Thus differentimage denoisingmethods have advantages and disadvantagesin terms of computation cost quality of denoising andboundary preserving Therefore denoising is still an openissue and needs an improvement

31 Nonlocal Filters Coupe et al [52] proposed optimizedblock-wise nonlocal means denoising filter for 3D magneticresonance images which uses the natural redundancy ofinformation in image to remove the noise Liu et al [53]proposed enhanced nonlocal means denoising filter for 3Dmagnetic resonance imagesThe filter was designed consider-ing characteristics of Rician noise in the MR images Manjonet al [54] proposed adaptive nonlocal means denoisingmethod forMR images with spatially varying noise levels Huet al [55] presented improved DCT-based nonlocal meansfilter for MR image denoising

32 Anisotropic Diffusion Filters The anisotropic diffusionfiltering is widely used for MR image enhancement It is ageneral scale-space approach to edge detection introducedin [56] Fully automatic method for the segmentation ofMR brain images is proposed in [57] The method usedprobabilistic anisotropic diffusion to suppress the influenceof extracerebral tissue in brain MR images and multiscalewatersheds for image segmentation Samsonov and Johnson[58] proposed noise-adaptive nonlinear diffusion filteringof MR images with spatially varying noise levels Theyuse a priori information regarding the image noise levelspatial distribution for the local adjustment of the anisotropicdiffusion filter Krissian and Aja-Fernandez [59] presentednoise-driven anisotropic diffusion filtering to remove Riciannoise from magnetic resonance images This filter relies on arobust estimation of the standard deviation of the noise

33 Wavelet BasedMethods Wavelet basedmethods are pro-posed by Nowak [60] Alexander et al [61] Bao and Zhang[62] and Anand and Sahambi [63] for noise suppressionin magnetic resonance images Xue et al [64] presented


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

Figure 2 Noise in brain MR images (a) Original brain MR image The images corrupted by 3 7 and 11 noise in (b) (c) and (d)respectively

an integrated method of the adaptive enhancement for anunsupervised global-to-local segmentation of brain tissues inthree-dimensional (3D) MR images where wavelet filter isused to denoise the image and segmentation is carried outby combining spatial feature based fuzzy clustering

4 FCM Clustering

FCM is the most effective algorithm for data clustering FCMwas proposed by Dunn [65] and later on it was modifiedby Bezdek [66] The standard FCM objective function forpartitioning the data 119909119896

119873

119896=1into 119888 clusters is given as

119869FCM = (119880119881) =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2 (4)

where 119881 = V119894119888

119894=1are the prototype of cluster and array 119880 =

120583119894119896 represents the partitionmatrix 119888 is the number of clustercentroids119873 is the number of pixels or data points 119909119896 is the119896th pixel and V119894 is the centroid of 119894th cluster 119909119896 minus V119894

2=

119889119894119896 = 119889(119909119896 V119894) is the distance measure between cluster centerV119894 and the pixel 119909119896120583119894119896 is the fuzzymembership of 119896th pixel 119894thcluster This membership value satisfies the conditions 120583119894119896 isin[0 1] 1 le 119894 le 119888 1 le 119896 le 119873 0 lt sum

119873

119896=1120583119894119896 lt 119873 1 le 119894 le 119888

and sum119888119894=1

120583119894119896 = 1 1 le 119896 le 119873Parameter 119901 isin (1infin) is a weighing exponent on each

membership (1 for hard clustering and increasing for fuzzyclustering) It determines the amount of fuzziness of theresulting classification and is usually set as 2 The FCMobjective function is minimized when high membershipvalues are assigned to pixels which are close to the centroid ofits particular class and low membership values are assignedwhen pixels are away from the centroid [67] The partitionmatrix and cluster centroid are updated as

120583119894119896 =

1

sum119888

119895=1(1198892

1198941198961198892

119895119896)

1119901

V119894 =sum119873

119896=1120583119901

119894119896119909119896

sum119873

119896=1120583119901

119894119896

(5)

The drawback of FCM clustering for image segmentationis that its objective function does not take into considerationany spatial dependence among pixels of image but deals withimages the same as separate points Second drawback of FCMclustering method is that the membership function is mostlydecided by 119889(119909119896 V119894) which measures the similarity betweenthe pixel intensity and the cluster center Higher membershipdepends on closer intensity values to the cluster center Hencemembership function is highly sensitive to noise In MRimage with noise and intensity inhomogeneity this results inimproper segmentation

The result of FCM on brain MR images corrupted bynoise and intensity inhomogeneity fromMcGill database [13]is shown in Figure 3 Figure 3(a) shows original image with3 noise and 0 intensity inhomogeneity Region of CSFRegion of GM and Region of WM respectively Figure 3(b)shows original image with 0 noise and 40 intensity inho-mogeneity Region of CSF Region ofGM andRegion ofWMrespectively In case of noisy images tissue class may differand appears as salt and pepper noise For example few GMpixels are shown in the homogeneous region ofWMas shownin Figure 3(a) In case of images corrupted by intensityinhomogeneity significant size of pixels in GM and WMmay be erroneously classified as another class as shown inFigure 3(b) In order to deal with these issues many variantsof FCMare proposed In these algorithms spatial informationis incorporated in objective function of original FCM algo-rithm to improve the performance of image segmentation

In the next sections the comparative analysis of FCMbasedmethods is presentedThesemethods are categorized asmethods dealingwith intensity inhomogeneity onlymethodsdealing with noise only and methods dealing with intensityinhomogeneity and noise

5 FCM Based Methods with IntensityInhomogeneity Correction

This section gives the mathematical analysis of FCM basedmethods for the segmentation of brain MR images withintensity inhomogeneity correction


(a)

(b)

Figure 3 Result of FCM on brain MR images corrupted by noise and intensity variation (a) Original image with 3 noise and 0 intensityinhomogeneity Region of CSF Region of GM and Region of WM respectively (b) Original image with 0 noise and 40 intensityinhomogeneity Region of CSF Region of GM and Region of WM respectively

51 Bias Corrected FCM (BCFCM) Ahmed et al [14] pro-posed a modification in standard FCM objective function todeal with intensity inhomogeneity of brain MR images byintroducing a term that allows the labelling of pixel to beinfluenced by the intensity of its immediate neighbourhoodNeighbourhood acts as a regularizer and biases the solutiontowards piece-wise homogeneous labelling and is useful insegmentingMR scan corrupted by salt and pepper noiseThemodified objective function is given as

119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119910119896 minus 120573119896 minus V119894

1003817100381710038171003817

2

+

120572

119873119877

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896( sum

119910119903isin119873119896

1003817100381710038171003817119910119903 minus 120573119903 minus V119894

1003817100381710038171003817

2)

(6)

where 119910119896 is the observed log-transformed intensities at the119896th pixel and 119873119896 stands for set of neighbours that existsin a window around 119909119896 and is the cardinality of 119873119877 Theeffect of the neighbours term is controlled by parameter 120572The relative importance of the regularizing term is inverselyproportional to signal-to-noise ratio (SNR) of theMRI signalLower SNR would require a higher value of the parameter 120572

The membership function centroid and bias field areupdated as follows

120583lowast

119894119896

=

1

sum119888

119895=1((119863119894119896 + (120572119873119877) 120574119894) (119863119895119896 + (120572119873119877) 120574119895))

1(119901minus1)

Vlowast119894=

sum119873

119896=1120583119901

119894119896((119910119896 minus 120573119896) + (120572119873119877)sum119910

119903isin119873119896

(119910119903 minus 120573119896))

(1 + 120572)sum119873

119896=1120583119901

119894119896

120573lowast

119896= 119910119896 minus

sum119888

119894=1120583119901

119894119896V119894

sum119888

119894=1120583119901

119894119896

(7)

where 119863119894119896 = 119910119896 minus 120573119896 minus V1198942 and 120574119894 = sum

119910119903isin119873119896

119910119903 minus

120573119896 minus V1198942 The BCFCM outperformed the FCM on both

simulated and real MR images In noisy images the BCFCMtechnique produced better results and compensates for noiseby including a regularization term

52 FCM with Spatial Constraints The BCFCM is computa-tionally inefficient due to introduction of spatial constraintsinsufficient robustness to outliers and difficulty in clusteringnon-Euclidean structure data Chen andZhang [21] proposed


a modified objective function which is computationallyefficient as compared to BCFCM and is given as

119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2+ 120572

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1003817100381710038171003817119896 minus V119894

1003817100381710038171003817

2) (8)

where 119896 is the sample mean of neighbouring pixels lyingwithin awindow around pixel119909(119896)which can be computed inadvanceThe idea is that the term (1119873119877) sum119909

119903isin119873119896

119909119903minusV1198942 can

be written as (1119873119877) sum119909119903isin119873119896

119909119903 minus 1198962+ 119896 minus V119894

2 Thus themodified FCM algorithm with spatial constraints is iteratedfor minimizing objective function with the following updateequations for membership function and centroid

120583119894119896 =

(1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2+ 120572

1003817100381710038171003817119896 minus V119894

1003817100381710038171003817

2)

minus1(119901minus1)

sum119888

119895=1(

10038171003817100381710038171003817119909119896 minus V119895

10038171003817100381710038171003817

2

+ 120572

10038171003817100381710038171003817119896 minus V119895

10038171003817100381710038171003817

2

)


V119894 =sum119873

119896=1120583119901

119894119896(119909119896 + 119896)

(1 + 120572)sum119873

119896=1120583119901

119894119896

(9)

Above equations are computationally simpler than thoseobtained from [14] Further to enhance robustness of cluster-ing 119896 can be taken as median of neighbours with specifiedwindow around 119909119896 Chen and Zhang [21] named thesealgorithms with mean and median filtering as FCM S1 andFCM S2 respectively In FCM S1 and FCM S2 the Euclideandistance 119909119896 minus V119894

2 was replaced with Gaussian kernelinduced distance 1 minus119870(119909119896 V119894) = 1 minus exp(minus119909119896 minus V119894

21205902) and

the kernel based objective function was modified as

119869119872 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119909119896 V119894))

+ 120572

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119896 V119894))

(10)

The necessary conditions for minimizing objective functionare given

120583119894119896

=

(1 minus 119870 (119909119896 V119894)) + 120572 (1 minus 119870 (119896 V119894))minus1(119901minus1)

sum119888

119895=1((1 minus 119870 (119909119896 V119895)) + 120572 (1 minus 119870 (119896 V119895)))


V119894 =sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) 119909119896 + 120572119870 (119896 V119894) 119896)

sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) + 120572119870 (119896 V119894))

(11)

These algorithms with mean and median filter with Gaus-sian kernel induced distance are named as KFCM S1 andKFCM S2 This method can also be used to improve theperformance of other FCM-like algorithms based on addingsome type of penalty terms to the original FCM objectivefunction

6 FCM Based Methods with Noise Robustness

This section gives the mathematical analysis of FCM basedmethods for the segmentation of brain MR images withrobustness to noise level in brain MR images

61 Improved FCM Segmentation (IFS) Shen et al [26]proposed the improved fuzzy 119888-means segmentation (IFS)algorithm to overcome noise effects in MR images Insteadof modifying the objective function of traditional FCMalgorithm it improves the similarity measurement of thepixel intensity and the cluster center by considering neigh-bourhood attraction The objective function is expressed by

119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896119889 (119909119896 V119894) (12)

where 119889(119909119896 V119894) is a similarity measurement between the pixelintensity and the cluster centers and is defined as follows

119889 (119909119896 V119894) =1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2(1 minus 120582119867119894119896 minus 120585119865119894119896)

(13)

where 120582 and 120585 adjust the degree of attraction and havea magnitude between 0 and 1 Here 119867119894119896 is called featureattraction and 119865119894119896 is called distance attraction given by

119867119894119896 =

sum119878

119895=1120583119894119895119892119896119895

sum119878

119895=1119892119896119895

119865119894119896 =

sum119878

119895=11205832

1198941198951199022

119896119895

sum119878

119895=11199022

119896119895

(14)

and here 119892119896119895 = 119909119896 minus 119909119895 is the intensity difference betweenstudy 119896th pixel and its neighbouring 119895th pixel and 119902119896119895 is therelation location between 119896th pixel and its neighbouring 119895thpixel 119878 is the number of neighbouring pixels and 120583119894119895 is themembership of neighbouring 119895th pixel to the 119894th cluster Thebias field is estimated by the following equation

120573119896 = 119909119896 minus 119901119896 minus

sum119873

119896=1(119909119896 minus 119901119896)

sum119888

119894=1120583119901

1198941198961198722sum119873

119896=1(sum119888

119894=1120583119901

1198941198961198722)

(15)

where 119901119896 = sum119888

119894=1120583119901

1198941198961198722V119894sum

119888

119894=1120583119901

1198941198961198722 and 119872 = 1 minus 120582119867119894119896 minus

120585119865119894119896 The objective function is minimized with updates formembership value and cluster given as

120583119894119896 =

1

sum119888

119895=1(119889 (119909119896 V119894) 119889 (119909119896 V119895))

2(119901minus1)

V119894 =sum119873

119896=1120583119901

119894119896(119909119896 minus 120573119896)

sum119873

119896=1120583119901

119894119896

(16)

If a pixel has a very similar intensity to one of its neigh-bours the attraction between them is stronger than theattraction between the pixel and another neighbour with


different intensities A spatially closer neighbouring pixelhas a stronger attraction than a neighbour which is spatiallydistant Segmentation using IFCM is decided by the pixelitself and by its neighbouring pixels which improves thesegmentation results

62 Fast Generalized FCM (FGFCM) Algorithm FCM S1and FCM S2 proposed in [21] as two extensions to FCM Shave yielded effective segmentation for images but bothstill lack enough robustness to noise and outliers in absenceof prior knowledge of the noise Parameter 120572 is used toachieve balance between robustness to noise and effectivenessof preserving the details of the image its selection has tobe made by experimentation which is difficult Also thetime of segmenting an image is heavily dependent on theimage size Motivated by individual strengths of FCM S1 andFCM S2 Cai et al [68] proposed a novel fast and robustFCM framework for image segmentation called FGFCMclustering algorithms incorporating local information Theyintroduced a novel factor 119878119896119895 incorporating both the localspatial relationship (called 119878119904 119896119895) and the local gray-levelrelationship (called 119878119892 119896119895) to replace parameter 120572 and makeit play a more important role in clustering Its definition isgiven by

119878119896119895 =

S119904 119896119895 times 119878119892 119896119895 119895 = 119896

0 119895 = 119896

(17)

where the 119896th pixel is the center of the local window (eg3 times 3) and 119895th pixels are the set of the neighbours falling intoa window around the 119894th pixel given by

119878119904 119896119895 = exp(minusmax (1003816100381610038161003816

1003816119901119895 minus 119901119896

10038161003816100381610038161003816

10038161003816100381610038161003816119902119895 minus 119902119896

10038161003816100381610038161003816)

120582119904

) (18)

where (119901119896 119902119896) is a spatial coordinate of the 119896th pixel 120582119904denotes the scale factor of the spread of 119878119904 119896119895 determiningits change characteristic and 119878119904 119896119895 reflects the damping extentof the neighbours with the spatial distances from the centralpixel The local gray-level similarity measure 119878119892 119896119895 is given by

119878119892 119896119895 = exp(minus

10038171003817100381710038171003817119909119896 minus 119909119895

10038171003817100381710038171003817

2

1205821198921205902

119892 119896

) (19)

where 119909119896 is gray value of the central pixel within a specialwindow 119909119895 is gray value of the 119895th pixels in the same windowand 120582119892 denotes the global scale factor of the spread of 119878119892 119896119895The function of the local density surrounding the central pixel120590119892 119896 is given by

120590119892 119896 = sqrt(sum119895isin119873119896

10038171003817100381710038171003817119909119895 minus 119909119896

10038171003817100381710038171003817

2

119873119877

) (20)

The value of parameter 120590119892 119896 reflects gray value homogeneitydegree of the local window The smaller its value is the more

homogeneous the local window is and vice versa 120590119892 119896 canchange automatically with different gray-levels of the pixelsover an image and thus reflects the damping extent ingray values FGFCM clustering algorithm incorporates localspatial and gray-level information into its objective functionwhich is given by

119869 =

119888

sum

119894=1

119902

sum

119896=1

120574119896120583119901

119894119896(120585119896 minus V119894)

2 (21)

where V119894 represents the prototype of the 119894th cluster andrepresents the fuzzy membership of gray value 119896with respectto cluster 119894 120574119896 is the number of the pixels having the grayvalue equal to 119896 where 119896 = 1 2 119902 and sum119902

119896=1120574119896 = 119873 and

119902 denotes the number of the gray-levels of the given imagewhich is generally much smaller than119873 The new generatedimage is computed in terms of

120585119896 =

sum119895isin119873119896

119878119896119895119909119895

sum119895isin119873119896

119878119896119895

(22)

where 119909119895 is gray value of the neighbours of 119909119896 (windowcenter) 119873119896 is set of neighbours falling in the local windowand 119878119896119895 is local similarity measure between the 119896th pixel andthe 119895th pixel 119878119896119895 can be considered as the weight of the 119895thpixel and 120585119896 can be considered as the 119896th pixel of the linearlyweighted summed image The partition matrix and clustercentroid are updated by

120583119894119896 =

(120585119896 minus V119894)minus2(119901minus1)

sum119888

119895=1((120585119896 minus V119895)


)

V119894 =sum119902

119896=1120574119896120583119901

119894119896120585119896

sum119902

119896=1120574119896120583119901

119894119896

(23)

FGFCM introduces a new factor 119878119896119895 as a local (spatial andgray) similarity measure with robustness to noise and detail-preserving for image and removes the empirically adjustedparameter 120572 FGFCM produces fast clustering for givenimage which is attributed to its dependence only on thenumber of the gray-levels 119902 rather than the size 119873 of theimage which reduces its time complexity of clustering

63 A Gaussian Kernel Based FCM (GKFCM) Yang and Tsai[27] modified objective function given by Chen and Zhang[21] to make it independent of parameter 120572 The modifiedobjective function is given by

119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119909119896 V119894))

+

119888

sum

119894=1

119873

sum

119896=1

120578119894120583119901

119894119896(1 minus 119870 (119896 V119894))

(24)

where119870(119909 119910) = exp(minus119909 minus 11991021205902) 1205902 = sum119873

119896=1(119909119896 minus 119909

2120578119894)

with119909 = sum119873

119895=1(119909119896119873) and 120578119894 = min1198941015840 =119894(1minus119870(V

1015840

119894 V119894))max119895(1minus

119870(V119895 119909))


The membership function and cluster prototype areupdated as follows

120583119894119896

=

((1 minus 119870 (119909119896 V119894)) + 120578119894 (1 minus 119870 ( V119894)))minus1(119901minus1)

sum119888

119895=1((1 minus 119870 (119909119896 V119895)) + 120578119894 (1 minus 119870 ( V119895)))


V119894 =sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) 119909119896 + 120578119894119870(119896 V119894) 119896)

sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) + 120578119894119870(119896 V119894))

(25)

GKFCM automatically learn the parameters by theprototype-driven learning schemeThe results of GKFCMaremore robust to noise and outliers than BCFCM KFCM S1and KFCM S2 especially in segmenting MR images

64 AModified FCMMethod Using Multiscale Fuzzy 119888-Means(MsFCM) Wang and Fei [44] proposed multiscale fuzzy 119888-means (MsFCM) algorithm for the classification of brainMRimages which performs classification from the coarsest to thefinest scale that is the original imageThe classification resultat a coarser level 119905 + 1 was used to initialize the classificationat a higher scale level 119905 The final classification is the resultat scale level 0 During the classification processing at level

119905+1 the pixelswith the highestmembership above a thresholdare identified and assigned to the corresponding class Thesepixels are labelled as training data for the next level 119905 Theobjective function of the MsFCM at level 119905 is given by

119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

+

120572

119873119877

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896( sum

119910119903isin119873119896

1003817100381710038171003817119910119903 minus V119894

1003817100381710038171003817

2)

+ 120573

119888

sum

119894=1

119873

sum

119896=1

(120583119894119896 minus 1205831015840

119894119896)

119901 1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

(26)

and here 120572 and 120573 are scaling factors 1205831015840119894119896is the membership

obtained from the classification in the previous scale and isdetermined by

1205831015840

119894119896=

1205831015840

119894119896 if max (120583119905+1

119894119896) gt 119870

0 otherwise(27)

where119870 is the threshold to determine the pixels with a knownclass in the next scale classification and is set as 085 Thepartition matrix and class centers are updated by

120583119894119896 =

1 + 120573sum119888

119895=1((1205831015840

1198941198961198892

119894119896minus 1205831015840

1198941198951198892

119894119896) ((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119894119896)))

sum119888

119895=1(((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)) ((1 + 120573) 119889

2

119895119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)))

V119894 =sum119873

119896=11205832

119894119896(119909119896 + (120572119873119877)sum119909

119903isin119873119896

119909119903) + 120573sum119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)

2

119909119896

(1 + 120572)sum119873

119896=11205832

119894119896+ 120573sum

119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)2

(28)

The multiscale scheme improves the speed of the classifica-tion and robustness The centroids of the initial classes fromthe coarser image improve the convergence of the classifica-tion algorithm A pixel with a high probability of belongingto one class in the coarse image is expected to belong to thesame class in the next fine image Threshold 119870 is used toselect these pixels at the coarse level and thus to constrainthe classification in the next level A smaller threshold meansmore reliable classification in the coarse imageThis thresholddepends on noise levels and preprocessing such as diffusionfiltering A smaller threshold is expected for images withbetter image quality

7 FCM Methods with Noise Robustness andIntensity Inhomogeneity Correction

This section includes the mathematical analysis of FCMbased method for the segmentation of brain MR images withintensity inhomogeneity correction and robustness to noise

71 A Framework with Modified Fast FCM (MFCM) Algo-rithm Ji et al [15] proposed a new automated method todetermine the initial value of centroid and also an adaptivemethod to incorporate the local spatial continuity in thesegmentation of brain MR image The objective function formodified FCM algorithm is given by

119869 =

119888

sum

119894=1

119873

sum

119896=1

[(1 minus 120572) 120583119901

119894119896(119910119896 minus 119887119896 minus V119894)

2

+ 120572 sum

119903isin119873119896

120583119901

119894119896120596119896119903 (119910119903 minus 119887119903 minus V119894)

2] +

119873

sum

119896=1

120572119896

119888

sum

119894=1

120583119894119896 (1

minus 120583119901minus1

119894119896)

(29)

where 120596119896119903 is the weight of pixels in the neighbourhoodcentered at 119896th pixel 120572119896 is the control parameter of the termwhich rewards the crispness membership degrees In thisalgorithm the initialization of centroid was done automat-ically using histon and adaptive method was proposed toincorporate the local spatial continuity to overcome the noise


effectively and prevent the edges from blurringThe intensityinhomogeneity is estimated by linear combination of theset of basis functions and regularization term is added inobjective function to reduce the number of iteration stepsThe partition matrix and cluster centroid and bias field areupdated as

120583119894119896 = (

119888

sum

119895=1

(

(1 minus 120572) 119889119896119894 + 120572sum119903isin119873119896

120596119896119903119889119903119894 minus 120572119896

(1 minus 120572) 119889119896119895 + 120572sum119903isin119873119896

120596119896119903119889119903119895 minus 120572119896

))

minus1

V119894

=

sum119873

119896=1120583119901

119894119896(1 minus 120572) (119910119896 minus 119887119896) + 120572sum119903isin119873

119896

120596119896119903 (119910119903 minus 119887119903)

sum119873

119896=1120583119901

119894119896

119887119896 =

119872

sum

119896=1

120596119896119892119896 (119896) = 120596119879119866 (119896)

(30)

where 119889119896119894 = (119910119896 minus 119887119896 minus V119894)2 120596 = (1205961 1205962 120596119872)

119879 and119866(119896) = (1198921(119896) 1198922(119896) 119892119872(119896))

119879 and here 120596119896 isin R 119896 =

1 2 119872 are the combination coefficients The orthogonalpolynomials are used as the basis functions which satisfy thecondition given as follows

int

Ω

119892119894 (119909) 119892119895 (119909) = 120575119894119895 (31)

where 120575119894119895 = 0 119894 = 119895 120575119894119895 = 1 119894 = 119895 120596 = 119860minus1119861 119860 =

sum119896isinΩ

119866(119896)119866(119896)119879sum119888

119894=1120583119901

119894119896 and 119861 = sum

119896isinΩ119866(119896)sum

119888

119894=1120583119901

119894119896(119910119896 minus

V119894) 119860 is 119872 times 119872 is matrix and 119861 is 119872 times 1 matrix with 119872

being the number of the basis functions MFCM methodovercomes the threemajor artifacts of brain images (intensityinhomogeneity noise and partial volume effect) at the sametime The initial values of the centroids are determinedautomatically Then an adaptive method to incorporatethe local spatial continuity is used to overcome the noiseeffectively and prevent the edge from blurring The intensityinhomogeneity is estimated by a linear combination of a setof basis functions A regularization term is added to reducethe iteration steps and accelerate the algorithm The weightsof the regularization terms are all automatically computed toavoid the manually tuned parameter

72 Modified Robust FCM Algorithm with Weighted Bias Esti-mation (MRFCM-wBE) Ramathilagam et al [19] proposeda modified robust fuzzy 119888-means algorithm with specialweighted bias estimation (MRFCM-wBE) for segmentationof brain MR images To reduce the number of iterationsthe proposed robust algorithm initializes the centroid usingdist-max initialization algorithm before the execution ofalgorithm iterativelyThe objective function ofMRFCM-wBEis given by

119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+ (1 +

119888

sum

119894=1

120583119901

119894119896) (32)

where = (sum119896119910119896)119873 and 120596119896 isin (0 1) is the weight of 119896th

pixel The partition matrix centroid cluster and bias field areupdated as follows

120583lowast

119894119896=

119888

sum

119895=1

((

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+

10038171003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119895

10038171003817100381710038171003817

2

+

)

1(119901minus1)

)

minus1

Vlowast119894=

sum119873

119896=1120583119901

119894119896(119910119896 minus 120596119896120573119896)

sum119873

119896=1120583119901

119894119896

120573lowast

119896=

1

120596119896

(119910119896 minus

sum119888

119894=1120583119901

119894119896V119894

sum119888

119894=1120583119901

119894119896

)

(33)

Experimental results of MRFCM-wBE indicate that the algo-rithm is more robust to the noises and faster thanmany othersegmentation algorithms

73 Efficient Inhomogeneity Compensation Using FCM Clus-teringModels The compensation of intensity inhomogeneityartifacts is a computationally costly problem which demandshighly efficient design and implementation Szilagyi et al[69] demonstrated intensity inhomogeneity compensationon a single-channel intensity image via 119888-means clusteringmodels The operations performed during the iterations ofthe alternating optimization (AO) scheme are separated intoglobally working ones and locally applied ones and theirexecution is optimized according to their necessities Globalcriteria are applied to gray intensities instead of individualpixels which makes a drastic reduction of the computationalload This formulation and improved clustering models arecombined with multistage INU compensation to obtain highaccuracy and efficiency The objective function of efficientFCMmethod is given by

119869FCM 119902119887 =119888

sum

119894=1

sum

119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897(1 minus V119894)

2

(34)

and here ℎ119905119897is the number of pixels for which the compen-

sated intensity in iteration 119905 satisfies 119910119896 minus 119887119896 = 1 Ω(119905) is therange of possible values of 119910119896 minus 119887119896 and sum119897isinΩ(119905) ℎ

(119905)

119897= 119899 And

hence ℎ119905119897with 119897 isin Ω

(119905) represents the intensity histogram ofthe compensated image in iteration 119905 Thus for any compen-sated intensity 119897 isin Ω(119905) and any cluster indexed 119894 = 1 2 119888the partition matrix and centroid cluster are updated asfollows

120583119894119897 =


sum119888

119895=1(1 minus V119895)


V119894 =sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897119897

sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897

(35)

For any 119897 isin Ω(119905) auxiliary variables are defined and organizedin look-up table as 119902119897 = sum

119888

119894=1120583119901

119894119897V119894sum119888

119894=1120583119901

119894119897 For any pixel with

index 119896 = 1 119899 the bias field is estimated as119887119896 = 119910119896 minus 119902119897

119896

(36)

with 119897119896 = 119910119896 minus 119887119905minus1

119896


8 Validation Methods

Quantitative evaluation is essential for objective comparisonof the results of different correction methods Commonlyused metrics for inhomogeneity correction through segmen-tation are as follows

(1) Segmentation accuracy (SA) used in [14] is defined as

SA = (

(Number of Correctly Classified Pixels)(Total Number of Pixels)

)

times 100

(37)

(2) Three evaluation parameters used in [26 44] aredefined as follows

(a) Undersegmentation representing the of neg-ative false segmentation119873119891119901119873119899

(b) Oversegmentation representing the of posi-tive false segmentation119873119891119899119873119901

(c) Incorrect segmentation representing the total of false segmentation (119873119891119899 + 119873119891119901)119873

where119873119891119901 is the number of pixels that do not belongto a cluster and are segmented into the cluster119873119891119899 isthe number of pixels that belong to a cluster and arenot segmented into the cluster 119873119901 is the number ofall pixels that belong to a cluster and 119873119899 is the totalnumber of pixels that do not belong to a cluster

(3) Jaccard similarity and the Dice coefficient estimatethe segmentation of voxels of one segmented tissueThe Jaccard similarity used in [15ndash17] is definedas the ratio between intersection and union of twosets and representing the obtained and gold standardsegmentations respectively

119869 (1198781 1198782) =

1198781 cap 1198782

1198781 cup 1198782

(38)

The authors also used the Dice coefficient which is aspecial case of the index defined as

120581 (1198781 1198782) =

210038161003816100381610038161198781 cap 1198782

1003816100381610038161003816

10038161003816100381610038161198781

1003816100381610038161003816+10038161003816100381610038161198782

1003816100381610038161003816

(39)

In comparison to the Dice coefficient the Jaccardsimilarity is more sensitive when sets are more sim-ilar Jaccard similarity is referred to as segmentationmeasure or comparison score in [21 68]

(4) The silhouette width 119878(119894) of the object 119894 used in [19] isobtained using the equation

119878 (119894) =

119887 (119894) minus 119886 (119894)

max 119887 (119894) 119886 (119894) (40)

In the above equation 119886(119894) is the average distancebetween the 119894th data and all other data in the cluster

119887(119894) is the smallest average distance between the 119894thdata and all other data of other clusters All thesesegmentation based measures provide quantitativeinformation on segmentation accuracy assuming thatgold standard segmentations are available

9 Dataset Used for Validation

An objective validation method may require a ground truthbased on strong prior knowledge about the real structureof the object of interest For datasets acquired in clinicalsituation this ground truth is usually incomplete leading to asubjective quality assessment Consequently other solutionshave been proposed for validation purpose leading to theuse of numerical data either synthetic or simulated or theacquisition of images of physical phantoms with knowncharacteristics

(1) Numerical datasets Numerical datasets are com-monly used for validation because of the ideal priorknowledge they provide They allow both qualita-tive and quantitative evaluation Typical numericaldatasets are either synthetic or simulated

(2) Synthetic datasets These datasets carry no realisticanatomical or physical information They are usuallyused as a first step for objectively evaluating animage processing method with their characteristicsbeing already known Since a synthetic dataset doesnot have realistic behaviour simulated datasets havebecome necessary With the increase in computa-tional power such realistic datasets can be simulatedAn MRI simulator and simulated brain datasets areavailable from the McConnell Brain Imaging Centreat the Montreal Neurological Institute and Hospital[13]

(3) Real datasets The Internet Brain SegmentationRepository (IBSR) [70] is aWorldWideWeb resourceproviding access to magnetic resonance brain imagedata and segmentation results contributed and uti-lized by researchers

A comparison of these algorithmic approaches for brainMR image segmentation with intensity inhomogeneity cor-rection using fuzzy clustering in terms of advantages andlimitations is summarized in Table 1

10 Conclusion

In this paper a comprehensive review of FCM clusteringalgorithms for the segmentation of MR brain images withintensity inhomogeneity corrections and noise robustness ispresented The algorithms are analysed according to variousfeatures like modification of standard fuzzy objective func-tion and updating of fuzzy membership function and clustercenter A number of important issues have been empha-sized like algorithmic parameter selection computationalcomplexity and noise robustness indicating that intensityinhomogeneity correction as well as noise removal is still achallenging task Because of intensity inhomogeneity noise


Table1Com

paris

onof

thea

lgorith

msu

sedforsegmentatio

nforb

rain

MRim

ages

with

intensity

inho

mogeneitycorrectio

nin

term

sofadvantagesa

ndlim

itatio

ns

Algorith

mAd

vantages

Limitatio

nsQuantitativ

eevaluation

Bias

correctedFC

M(BCF

CM)

[14]

BCFC

Mcompensatefor

theg

ray(in

tensity

)levels

inho

mogeneityandallowthelabellin

gof

apixel

tobe

influ

encedby

thelabels

inits

immediate

neighb

ourhoo

d

BCFC

Mcompu

testhe

neighb

ourhoo

dterm

ineach

iteratio

nste

pwhich

isvery

compu

tatio

nally

time-consum

ingandlackse

noug

hrobu

stnessto

noise

andou

tliersa

ndisdifficultto

cluste

rno

n-Eu

clidean

structure

data

Segm

entatio

naccuracy

(SA)

Spatially

constrainedkernelized

FCM

(FCM

S1FCM

S2and

KFCM

S1K

FCM

S2)[21]

Clustersthen

on-Euclid

eanstr

ucturesindata

Robu

stnesso

fthe

originalclu

sterin

galgorithm

sto

noise

andou

tliersw

hileretaining

compu

tatio

nalsim

plicity

KFCM

S1andKF

CMS2

areh

eavilyaffectedby

theirp

aram

eters

Segm

entatio

naccuracy

andsim

ilaritymeasure

Improved

FCM

(IFC

M)[26]

IFCM

perfo

rmsb

etterthanthetraditio

nalF

CMSensitive

toalgorithm

icparameter

values

election

Und

ersegm

entatio

n(U

nS)oversegm

entatio

n(O

vS)incorrectsegmentatio

n(In

S)and

comparis

onscore

Fastgeneralized

FCM

(FGFC

M)

[68]

FGFC

Mlessensthe

disadvantageso

ffuzzy

119888-m

eans

(FCM

)algorith

msw

ithspatial

constraints(FC

MS)

andatthes

ametim

eenhances

thec

luste

ringperfo

rmanceTh

eintro

ductionof

localspatia

linformationto

the

correspo

ndingob

jectivefun

ctions

enhances

their

insensitivenessto

noise

Lack

enou

ghrobu

stnesstono

iseandou

tliers

especiallyin

absenceo

fprio

rkno

wledgeo

fthe

noise

Thetim

eofsegmentin

gan

imageis

depend

ento

ntheimages

izeandhencethe

larger

thes

izeo

fthe

imagethem

orethe

segm

entatio

ntim

e

Segm

entatio

naccuracy

andsim

ilaritymeasure

GaussianKe

rnelized

FCM

(GKF

CM)[27]

Itisag

eneralized

type

ofFC

MB

CFCM

KF

CMS1and

KFCM

S2algorithm

sand

ismore

efficientand

robu

stwith

good

parameter

learning

schemeincomparis

onwith

FCM

andits

varia

nt

Forthe

case

ofGaussianno

iseof

10level

KFCM

Shasb

etterresultthanGKF

CMSegm

entatio

naccuracy

Mod

ified

FCM

classificatio

nmetho

dusingmultiscaleFu

zzy

119888-m

eans

(MsFCM

)[44

]

Them

etho

disrobu

stforn

oise

andlowcontrast

MRim

ages

becauseo

fitsmultiscalediffu

sion

filterin

gscheme

Thethresho

ldvalueo

fparam

eter119896used

inthe

algorithm

may

need

tobe

adjuste

dwhenthis

metho

disappliedto

otherimages

with

adifferent

noise

level

Dicec

oefficientand

overlap

ratio

Mod

ified

fastFC

M(M

FCM)for

brainMRim

ages

segm

entatio

n[15]

Intensity

inho

mogeneitynoiseand

partial

volume(PV

)effectsa

retakeninto

accoun

tfor

images

egmentatio

nAu

tomated

metho

dto

determ

inethe

initialvalues

ofthec

entro

idsa

ndadaptiv

emetho

dto

incorporatethe

localspatia

lcontinuityto

overcomethe

noise

effectiv

elyand

preventthe

edge

from

blurrin

g

Quantitativ

eaccuracydecreasesw

henthen

oise

levelinbrainMRim

ages

increasesLarger

radius

ofneighb

ourhoo

dleadstothelosso

ftextureso

CSFcann

otbe

segm

entedaccurately

Jaccardsim

ilarityandDicec

oefficient

Mod

ified

robu

stFC

Malgorithm

with

weightedbias

estim

ation

(MRF

CM-w

BE)[19]

Initializes

thec

entro

idsu

singdist-

max

initializationalgorithm

toredu

cethen

umbero

fiteratio

ns

Thea

lgorith

mwas

teste

don

T1-T2weighted

images

andsim

ulated

brainim

ages

corrup

tedby

Gaussianno

iseon

lySilhou

ettevalue


and weak boundaries magnetic resonance brain segmenta-tion is still a challenging area of research and there is a needfor future research to improve the accuracy precision andspeed of segmentation methods

Competing Interests

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

References

[1] T L Huntsherger C L Jacobs and R L Cannon ldquoIterativefuzzy image segmentationrdquo Pattern Recognition vol 18 no 2pp 131ndash138 1985

[2] M Trivedi and J C Bezdek ldquoLow level segmentation of aerialimages with fuzzy clusteringrdquo IEEE Transaction on SystemMan Cybernate vol 16 no 4 pp 589ndash598 1986

[3] F Masulli and A Schenone ldquoA fuzzy clustering based seg-mentation system as support to diagnosis in medical imagingrdquoArtificial Intelligence inMedicine vol 16 no 2 pp 129ndash147 1999

[4] A Simmons P S Tofts G J Barker and S R ArridgeldquoSources of intensity nonuniformity in spin echo images at15 TrdquoMagnetic Resonance inMedicine vol 32 no 1 pp 121ndash1281994

[5] J G Sled and G B Pike ldquoUnderstanding intensity non-uni-formity in MRIrdquo in Medical Image Computing and Computer-Assisted InterventationmdashMICCAI rsquo98 First International Con-ference Cambridge MA USA October 11ndash13 1998 Proceedingsvol 1496 of Lecture Notes in Computer Science pp 614ndash622Springer Berlin Germany 1998

[6] R P Velthuizen J J Heine A B Cantor H Lin L M Fletcherand L P Clarke ldquoReview and evaluation ofMRI nonuniformitycorrections for brain tumor response measurementsrdquo MedicalPhysics vol 25 no 9 pp 1655ndash1666 1998

[7] J B Arnold J-S Liow K A Schaper et al ldquoQualitative andquantitative evaluation of six algorithms for correcting intensitynonuniformity effectsrdquo NeuroImage vol 13 no 5 pp 931ndash9432001

[8] Z Hou ldquoA review on MR image intensity inhomogeneitycorrectionrdquo International Journal of Biomedical Imaging vol2006 Article ID 49515 11 pages 2006

[9] B Belaroussi J Milles S Carme Y M Zhu and H Benoit-Cattin ldquoIntensity non-uniformity correction in MRI existingmethods and their validationrdquo Medical Image Analysis vol 10no 2 pp 234ndash246 2006

[10] 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

[11] M A Balafar A R Ramli M I Saripan and S MashohorldquoReview of brain MRI image segmentation methodsrdquo ArtificialIntelligence Review vol 33 no 3 pp 261ndash274 2010

[12] W M Wells III W E L Crimson R Kikinis and F A JoleszldquoAdaptive segmentation of mri datardquo IEEE Transactions onMedical Imaging vol 15 no 4 pp 429ndash442 1996

[13] Brainweb httpwwwbicmnimcgillcabrainweb[14] M N Ahmed S M Yamany N Mohamed A A Farag and

T 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

[15] Z-X Ji Q-S Sun and D-S Xia ldquoA framework with modifiedfast FCM for brain MR images segmentationrdquo Pattern Recogni-tion vol 44 no 5 pp 999ndash1013 2011

[16] Z-X Ji Q-S Sun and D-S Xia ldquoA modified possibilisticfuzzy c-means clustering algorithm for bias field estimationand segmentation of brain MR imagerdquo Computerized MedicalImaging and Graphics vol 35 no 5 pp 383ndash397 2011

[17] Z Ji Q Sun Y Xia Q Chen D Xia and D Feng ldquoGeneralizedrough fuzzy c-means algorithm for brain MR image segmenta-tionrdquo Computer Methods and Programs in Biomedicine vol 108no 2 pp 644ndash655 2012

[18] S R Kannan ldquoA new segmentation system for brainMR imagesbased on fuzzy techniquesrdquoApplied SoftComputing Journal vol8 no 4 pp 1599ndash1606 2008

[19] S Ramathilagam R Pandiyarajan A Sathya R Devi and S RKannan ldquoModified fuzzy c-means algorithm for segmentationof T1ndashT2-weighted brain MRIrdquo Journal of Computational andApplied Mathematics vol 235 no 6 pp 1578ndash1586 2011

[20] D L Pham and J L Prince ldquoAn adaptive fuzzy C-meansalgorithm for image segmentation in the presence of intensityinhomogeneitiesrdquo Pattern Recognition Letters vol 20 no 1 pp57ndash68 1999

[21] 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

[22] Y A Tolias and S M Panas ldquoImage segmentation by afuzzy clustering algorithm using adaptive spatially constrainedmembership functionsrdquo IEEE Transactions on Systems Manand CyberneticsmdashPart A Systems and Humans vol 28 no 3pp 359ndash369 1998

[23] Y A Tolias and S M Panas ldquoOn applying spatial constraints infuzzy image clustering using a fuzzy rule-based systemrdquo IEEESignal Processing Letters vol 5 no 10 pp 245ndash247 1998

[24] L Szilagyi Z Benyo S M Szilagyii 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 vol 1 pp 724ndash726 IEEE Buenos Aires ArgentinaSeptember 2003

[25] J Liu J K Udupa D Odhner D Hackney and G MoonisldquoA system for brain tumor volume estimation via MR imagingand fuzzy connectednessrdquo Computerized Medical Imaging andGraphics vol 29 no 1 pp 21ndash34 2005

[26] S Shen W Sandham M Granat and A Sterr ldquoMRI fuzzysegmentation of brain tissue using neighborhood attractionwith neural-network optimizationrdquo IEEE Transactions on Infor-mation Technology in Biomedicine vol 9 no 3 pp 459ndash4672005

[27] M-S Yang and H-S Tsai ldquoA Gaussian kernel-based fuzzy c-means algorithm with a spatial bias correctionrdquo Pattern Rec-ognition Letters vol 29 no 12 pp 1713ndash1725 2008

[28] M S Atkins and B T Mackiewich ldquoFully automatic segmenta-tion of the brain inMRIrdquo IEEETransactions onMedical Imagingvol 17 no 1 pp 98ndash107 1998

[29] O Colliot O Camara and I Bloch ldquoIntegration of fuzzyspatial relations in deformable modelsmdashapplication to brainMRI segmentationrdquo Pattern Recognition vol 39 no 8 pp 1401ndash1414 2006

[30] A Zavaljevski A P Dhawan M Gaskil W Ball and J D John-son ldquoMulti-level adaptive segmentation ofmulti-parameterMR


brain imagesrdquoComputerizedMedical Imaging andGraphics vol24 no 2 pp 87ndash98 2000

[31] J J Corso E Sharon S Dube S El-Saden U Sinha andA Yuille ldquoEfficient multilevel brain tumor segmentation withintegrated bayesian model classificationrdquo IEEE Transactions onMedical Imaging vol 27 no 5 pp 629ndash640 2008

[32] R Cardenes R de Luis-Garcıa and M Bach-Cuadra ldquoAmultidimensional segmentation evaluation for medical imagedatardquo Computer Methods and Programs in Biomedicine vol 96no 2 pp 108ndash124 2009

[33] A Akselrod-Ballin M Galun J M Gomori et al ldquoAutomaticsegmentation and classification of multiple sclerosis in multi-channel MRIrdquo IEEE Transactions on Biomedical Engineeringvol 56 no 10 pp 2461ndash2469 2009

[34] S Datta and P A Narayana ldquoA comprehensive approach tothe segmentation of multichannel three-dimensional MR brainimages in multiple sclerosisrdquo NeuroImage Clinical vol 2 no 1pp 184ndash196 2013

[35] S Bricq C Collet and J P Armspach ldquoUnifying framework formultimodal brain MRI segmentation based on HiddenMarkovChainsrdquo Medical Image Analysis vol 12 no 6 pp 639ndash6522008

[36] A R Ferreira da Silva ldquoBayesian mixture models of variabledimension for image segmentationrdquo Computer Methods andPrograms in Biomedicine vol 94 no 1 pp 1ndash14 2009

[37] B Scherrer F Forbes C Garbay and M Dojat ldquoDistributedlocal MRFmodels for tissue and structure brain segmentationrdquoIEEE Transactions on Medical Imaging vol 28 no 8 pp 1278ndash1295 2009

[38] J Nie Z Xue T Liu et al ldquoAutomated brain tumor segmen-tation using spatial accuracy-weighted hidden markov randomfieldrdquo Computerized Medical Imaging and Graphics vol 33 no6 pp 431ndash441 2009

[39] AHuang RAbugharbieh andR Tam ldquoAhybrid geometricsta-tistical deformable model for automated 3-D segmentation inbrain MRIrdquo IEEE Transactions on Biomedical Engineering vol56 no 7 pp 1838ndash1848 2009

[40] L Wang Y Chen X Pan X Hong and D Xia ldquoLevel set seg-mentation of brain magnetic resonance images based on localGaussian distribution fitting energyrdquo Journal of NeuroscienceMethods vol 188 no 2 pp 316ndash325 2010

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

[42] K Thapaliya J-Y Pyun C-S Park and G-R Kwon ldquoLevel setmethod with automatic selective local statistics for brain tumorsegmentation in MR imagesrdquo Computerized Medical Imagingand Graphics vol 37 no 7-8 pp 522ndash537 2013

[43] Z M Wang Y C Soh Q Song and K Sim ldquoAdaptive spatialinformation-theoretic clustering for image segmentationrdquo Pat-tern Recognition vol 42 no 9 pp 2029ndash2044 2009

[44] H Wang and B Fei ldquoA modified fuzzy c-means classificationmethod using a multiscale diffusion filtering schemerdquo MedicalImage Analysis vol 13 no 2 pp 193ndash202 2009

[45] P Maji and S K Pal ldquoRough set based generalized fuzzy C-means algorithm and quantitative indicesrdquo IEEE Transactionson Systems Man and Cybernetics Part B Cybernetics vol 37no 6 pp 1529ndash1540 2007

[46] S Mitra W Pedrycz and B Barman ldquoShadowed c-meansintegrating fuzzy and rough clusteringrdquo Pattern Recognitionvol 43 no 4 pp 1282ndash1291 2010

[47] R He S Datta B R Sajja and P A Narayana ldquoGeneralizedfuzzy clustering for segmentation of multi-spectral magneticresonance imagesrdquo Computerized Medical Imaging and Graph-ics vol 32 no 5 pp 353ndash366 2008

[48] P Hore L O Hall D B Goldgof Y Gu A A Maudsley andA Darkazanli ldquoA scalable framework for segmenting magneticresonance imagesrdquo Journal of Signal Processing Systems vol 54no 1ndash3 pp 183ndash203 2009

[49] B Caldairou N Passat P A Habas C Studholme and FRousseau ldquoA non-local fuzzy segmentation method applica-tion to brain MRIrdquo Pattern Recognition vol 44 no 9 pp 1916ndash1927 2011

[50] T Chaira ldquoA novel intuitionistic fuzzy C means clusteringalgorithm and its application to medical imagesrdquo Applied SoftComputing Journal vol 11 no 2 pp 1711ndash1717 2011

[51] Y K Dubey and M M Mushrif ldquoSegmentation of brainMR images using intuitionistic fuzzy clustering algorithmrdquo inProceedings of the 8th Indian Conference on Computer VisionGraphics and Image Processing (ICVGIP rsquo12) ACM MumbaiIndia December 2012

[52] P Coupe P Yger S Prima P Hellier C Kervrann and CBarillot ldquoAn optimized blockwise nonlocal means denoisingfilter for 3-Dmagnetic resonance imagesrdquo IEEE Transactions onMedical Imaging vol 27 no 4 pp 425ndash441 2008

[53] H Liu C Yang N Pan E Song and R Green ldquoDenoising 3DMR images by the enhanced non-local means filter for Riciannoiserdquo Magnetic Resonance Imaging vol 28 no 10 pp 1485ndash1496 2010

[54] J V Manjon P Coupe L Martı-Bonmatı D L Collinsand M Robles ldquoAdaptive non-local means denoising of MRimages with spatially varying noise levelsrdquo Journal of MagneticResonance Imaging vol 31 no 1 pp 192ndash203 2010

[55] J Hu Y Pu X Wu Y Zhang and J Zhou ldquoImprovedDCT-based nonlocal means filter for MR images denoisingrdquoComputational and Mathematical Methods in Medicine vol2012 Article ID 232685 14 pages 2012

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

[57] C Undeman and T Lindeberg ldquoFully automatic segmentationof mri brain images using probabilistic anisotropic diffusionand multi-scale watershedsrdquo in Scale Space Methods in Com-puter Vision vol 2695 of Lecture Notes in Computer Science pp641ndash656 Springer Berlin Germany 2003

[58] A A Samsonov and C R Johnson ldquoNoise-adaptive nonlineardiffusion filtering of MR images with spatially varying noiselevelsrdquoMagnetic Resonance in Medicine vol 52 no 4 pp 798ndash806 2004

[59] K Krissian and S Aja-Fernandez ldquoNoise-driven anisotropicdiffusion filtering of MRIrdquo IEEE Transactions on Image Process-ing vol 18 no 10 pp 2265ndash2274 2009

[60] R D Nowak ldquoWavelet-based rician noise removal formagneticresonance imagingrdquo IEEETransactions on Image Processing vol8 no 10 pp 1408ndash1419 1999

[61] M E Alexander R Baumgartner A R Summers et al ldquoAwavelet-based method for improving signal-to-noise ratio andcontrast in MR imagesrdquo Magnetic Resonance Imaging vol 18no 2 pp 169ndash180 2000


[62] P Bao and L Zhang ldquoNoise reduction for magnetic resonanceimages via adaptive multiscale products thresholdingrdquo IEEETransactions on Medical Imaging vol 22 no 9 pp 1089ndash10992003

[63] C S Anand and J S Sahambi ldquoWavelet domain non-linearfiltering for MRI denoisingrdquo Magnetic Resonance Imaging vol28 no 6 pp 842ndash861 2010

[64] J-H Xue A PizuricaW Philips E Kerre R VanDeWalle andI Lemahieu ldquoAn integrated method of adaptive enhancementfor unsupervised segmentation of MRI brain imagesrdquo PatternRecognition Letters vol 24 no 15 pp 2549ndash2560 2003

[65] 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 1974

[66] J C Bezdek ldquoA convergence theorem for the fuzzy ISODATAclustering algorithmsrdquo IEEE Transaction on Pattern Analysisand Machine Intelligence vol 2 no 1 pp 1ndash8 1980

[67] J C Bezdek R Ehrlich and W Full ldquoFCM the fuzzy c-meansclustering algorithmrdquoComputers and Geosciences vol 10 no 2-3 pp 191ndash203 1984

[68] 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

[69] L Szilagyi S M Szilagyi and B Benyo ldquoEfficient inhomo-geneity compensation using fuzzy c-means clustering modelsrdquoComputer Methods and Programs in Biomedicine vol 108 no 1pp 80ndash89 2012

[70] IBSR httpwwwcmamghharvardeduibsr

Submit your manuscripts athttpwwwhindawicom

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homomorphic filtering) and statistical methods (Markovrandom fields (MRF) and FCM) under the category ofgray scale based methods and probability density functionFourier domain and wavelet domain methods under thecategory of transform based methods

Prospective methods like phantom multicoil and specialsequences and retrospective methods such as filtering meth-ods (homomorphic and unsharp masking) surface fitting(intensity and gradient) segmentation based (maximumlikelihood (ML) maximum a posteriori (MAP) FCM andnonparametric methods) and histogram based (high fre-quency maximization information minimization and his-togrammatching) methods are discussed in [10] This reviewalso analysed numerous publications to assess major trendspopularity and applications of intensity inhomogeneity cor-rection methods

Different brain image denoising methods such as aniso-tropic nonlinear diffusion MRF wavelet based analyticalcorrection and nonlocal methods are discussed in [11] Thisreview also discussed prospective methods like phantommulticoil and special sequences and retrospective methodssuch as surface fitting methods segmentation based (MLMAP and FCM) histogram based and filtering methodsMain focus of this paper is as follows

(1) Comprehensive review of FCM clustering based algo-rithms for the segmentation of brain MR imageswith intensity inhomogeneity corrections and noiserobustness is presented

(2) Complete mathematical analysis for the formulationof objective function of FCM clustering based algo-rithms for intensity inhomogeneity correction andnoise robustness is carried out

(3) Algorithms are compared on the basis of updating ofmembership function and cluster centroid

(4) Computational complexity and noise robustness ofthese algorithms are discussed

(5) Quantitative measures used by researcher are alsodiscussed

Rest of the paper is organized as follows Intensity inho-mogeneity in brain MR images and its correction methodsare described in Section 2 Noise in brain MR images anddenoising methods are briefed in Section 3 FCM clusteringalgorithm and its drawback for the segmentation of brainMRimage are briefly explained in Section 4 Detailed analysisof FCM clustering based algorithms for the segmentation ofbrain MR images with intensity inhomogeneity correctionand noise robustness is presented in Sections 5 6 and 7Section 8 describes the different validation methods whichare used for comparison Information about the dataset isgiven in Section 9 followed by the conclusion of the studyin Section 10

2 Intensity Inhomogeneity in Brain MRImages and Its Correction Methods

In brain MR images the observed signal is modelled as aproduct of the true signal and spatial varying factor calledbias field

119884119896 = 119883119896119866119896 1 le 119896 le 119873 (1)

where 119883119896 and 119884119896 are the true and observed intensities atthe 119896th pixel respectively 119866119896 is the bias field and 119873 isthe total number of pixels in a brain MR image Intensityinhomogeneity or bias is the slowly changing and smoothvariation in signal intensity For a bias field of magnitude40 the signal is multiplied by a field with values rangingfrom 08 to 12 (ie with values between 20 below trueintensity and 20 above true intensity) The application oflogarithmic transformation to the intensities allows artifactto be modelled as additive bias field [12] as given below

119910119896 = 119909119896 + 120573119896 1 le 119896 le 119873 (2)

Figure 1 depicts the effect of intensity inhomogeneityon brain MR images Original brain MR image [13] isshown in Figure 1(a) and images altered by 40 80 and100 intensity inhomogeneity respectively are shown inFigures 1(b) 1(c) and 1(d) respectively

Ahmed et al [14] used sinusoidal gain field of higherspatial frequency as bias field In [15ndash17] the bias field isestimated by a linear combination of a set of basis functionsas

119887119896 =

119872

sum

1

120596119896119892119896 (119896) = 120596119879119866 (119870) (3)

where 120596119879 = (1205961 1205962 120596119872)119879 120596119896 isin R 119896 = 1 2 119872 are

the combination coefficients119866(119896) = (1198921(119896) 1198922(119896) 119892119872(119896))119879

are the set of basis functions In [18 19] weighted bias field isused with initialization of weights as 120596119896 isin (0 1) for examplewith 10 data items 1205961 = 0008 increase 0001 1205962 1205963 12059610

21 FCM Clustering Methods with Spatial Constraints TheFCM based methods are modified by incorporating spatialconstraint in the objective function of FCM These methods[12 20] are used for intensity inhomogeneity correction andpartial volume segmentation of brain MR images Ahmedet al [14] proposed bias corrected FCM (BCFCM) bymodifying the objective function of FCM to compensatefor the intensity inhomogeneity But BCFCM is very time-consuming since it computes the neighbourhood term ineach iteration step This drawback is eliminated in spatiallyconstrained kernelized FCM (SKFCM) [21] where differentpenalty terms containing spatial neighbourhood informationin the objective function are used They also replaced thesimilarity measurement in the FCM by a kernel induceddistance

22 Modified FCM Based Methods Improved and enhancedFCM clustering algorithms [22ndash27] have been used to


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












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



4 FCM Clustering


119873


119869FCM = (119880119881) =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2 (4)

where 119881 = V119894119888



2=


119873

119896=1120583119894119896 lt 119873 1 le 119894 le 119888

and sum119888119894=1



120583119894119896 =

1

sum119888

119895=1(1198892

1198941198961198892

119895119896)

1119901

V119894 =sum119873

119896=1120583119901

119894119896119909119896

sum119873

119896=1120583119901

119894119896

(5)







(a)

(b)



119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119910119896 minus 120573119896 minus V119894

1003817100381710038171003817

2

+

120572

119873119877

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896( sum

119910119903isin119873119896

1003817100381710038171003817119910119903 minus 120573119903 minus V119894

1003817100381710038171003817

2)

(6)



120583lowast

119894119896

=

1

sum119888

119895=1((119863119894119896 + (120572119873119877) 120574119894) (119863119895119896 + (120572119873119877) 120574119895))

1(119901minus1)

Vlowast119894=

sum119873

119896=1120583119901

119894119896((119910119896 minus 120573119896) + (120572119873119877)sum119910

119903isin119873119896

(119910119903 minus 120573119896))

(1 + 120572)sum119873

119896=1120583119901

119894119896

120573lowast

119896= 119910119896 minus

sum119888

119894=1120583119901

119894119896V119894

sum119888

119894=1120583119901

119894119896

(7)


119910119903isin119873119896

119910119903 minus






119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2+ 120572

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1003817100381710038171003817119896 minus V119894

1003817100381710038171003817

2) (8)


119903isin119873119896

119909119903minusV1198942 can


119909119903 minus 1198962+ 119896 minus V119894


120583119894119896 =

(1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2+ 120572

1003817100381710038171003817119896 minus V119894

1003817100381710038171003817

2)


sum119888

119895=1(

10038171003817100381710038171003817119909119896 minus V119895

10038171003817100381710038171003817

2

+ 120572

10038171003817100381710038171003817119896 minus V119895

10038171003817100381710038171003817

2

)


V119894 =sum119873

119896=1120583119901

119894119896(119909119896 + 119896)

(1 + 120572)sum119873

119896=1120583119901

119894119896

(9)



21205902) and


119869119872 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119909119896 V119894))

+ 120572

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119896 V119894))

(10)


120583119894119896

=


sum119888

119895=1((1 minus 119870 (119909119896 V119895)) + 120572 (1 minus 119870 (119896 V119895)))


V119894 =sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) 119909119896 + 120572119870 (119896 V119894) 119896)

sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) + 120572119870 (119896 V119894))

(11)





119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896119889 (119909119896 V119894) (12)


119889 (119909119896 V119894) =1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2(1 minus 120582119867119894119896 minus 120585119865119894119896)

(13)


119867119894119896 =

sum119878

119895=1120583119894119895119892119896119895

sum119878

119895=1119892119896119895

119865119894119896 =

sum119878

119895=11205832

1198941198951199022

119896119895

sum119878

119895=11199022

119896119895

(14)


120573119896 = 119909119896 minus 119901119896 minus

sum119873

119896=1(119909119896 minus 119901119896)

sum119888

119894=1120583119901

1198941198961198722sum119873

119896=1(sum119888

119894=1120583119901

1198941198961198722)

(15)

where 119901119896 = sum119888

119894=1120583119901

1198941198961198722V119894sum

119888

119894=1120583119901

1198941198961198722 and 119872 = 1 minus 120582119867119894119896 minus


120583119894119896 =

1

sum119888

119895=1(119889 (119909119896 V119894) 119889 (119909119896 V119895))

2(119901minus1)

V119894 =sum119873

119896=1120583119901

119894119896(119909119896 minus 120573119896)

sum119873

119896=1120583119901

119894119896

(16)





119878119896119895 =

S119904 119896119895 times 119878119892 119896119895 119895 = 119896

0 119895 = 119896

(17)


119878119904 119896119895 = exp(minusmax (1003816100381610038161003816

1003816119901119895 minus 119901119896

10038161003816100381610038161003816

10038161003816100381610038161003816119902119895 minus 119902119896

10038161003816100381610038161003816)

120582119904

) (18)


119878119892 119896119895 = exp(minus

10038171003817100381710038171003817119909119896 minus 119909119895

10038171003817100381710038171003817

2

1205821198921205902

119892 119896

) (19)


120590119892 119896 = sqrt(sum119895isin119873119896

10038171003817100381710038171003817119909119895 minus 119909119896

10038171003817100381710038171003817

2

119873119877

) (20)



119869 =

119888

sum

119894=1

119902

sum

119896=1

120574119896120583119901

119894119896(120585119896 minus V119894)

2 (21)


119896=1120574119896 = 119873 and


120585119896 =

sum119895isin119873119896

119878119896119895119909119895

sum119895isin119873119896

119878119896119895

(22)


120583119894119896 =


sum119888

119895=1((120585119896 minus V119895)


)

V119894 =sum119902

119896=1120574119896120583119901

119894119896120585119896

sum119902

119896=1120574119896120583119901

119894119896

(23)



119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119909119896 V119894))

+

119888

sum

119894=1

119873

sum

119896=1

120578119894120583119901

119894119896(1 minus 119870 (119896 V119894))

(24)


119896=1(119909119896 minus 119909

2120578119894)

with119909 = sum119873

119895=1(119909119896119873) and 120578119894 = min1198941015840 =119894(1minus119870(V

1015840

119894 V119894))max119895(1minus

119870(V119895 119909))



120583119894119896

=


sum119888

119895=1((1 minus 119870 (119909119896 V119895)) + 120578119894 (1 minus 119870 ( V119895)))


V119894 =sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) 119909119896 + 120578119894119870(119896 V119894) 119896)

sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) + 120578119894119870(119896 V119894))

(25)




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

+

120572

119873119877

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896( sum

119910119903isin119873119896

1003817100381710038171003817119910119903 minus V119894

1003817100381710038171003817

2)

+ 120573

119888

sum

119894=1

119873

sum

119896=1

(120583119894119896 minus 1205831015840

119894119896)

119901 1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

(26)



1205831015840

119894119896=

1205831015840

119894119896 if max (120583119905+1

119894119896) gt 119870

0 otherwise(27)


120583119894119896 =

1 + 120573sum119888

119895=1((1205831015840

1198941198961198892

119894119896minus 1205831015840

1198941198951198892

119894119896) ((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119894119896)))

sum119888

119895=1(((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)) ((1 + 120573) 119889

2

119895119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)))

V119894 =sum119873

119896=11205832

119894119896(119909119896 + (120572119873119877)sum119909

119903isin119873119896

119909119903) + 120573sum119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)

2

119909119896

(1 + 120572)sum119873

119896=11205832

119894119896+ 120573sum

119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)2

(28)





119869 =

119888

sum

119894=1

119873

sum

119896=1

[(1 minus 120572) 120583119901

119894119896(119910119896 minus 119887119896 minus V119894)

2

+ 120572 sum

119903isin119873119896

120583119901

119894119896120596119896119903 (119910119903 minus 119887119903 minus V119894)

2] +

119873

sum

119896=1

120572119896

119888

sum

119894=1

120583119894119896 (1


119894119896)

(29)




120583119894119896 = (

119888

sum

119895=1

(

(1 minus 120572) 119889119896119894 + 120572sum119903isin119873119896

120596119896119903119889119903119894 minus 120572119896

(1 minus 120572) 119889119896119895 + 120572sum119903isin119873119896

120596119896119903119889119903119895 minus 120572119896

))

minus1

V119894

=

sum119873

119896=1120583119901


119896

120596119896119903 (119910119903 minus 119887119903)

sum119873

119896=1120583119901

119894119896

119887119896 =

119872

sum

119896=1

120596119896119892119896 (119896) = 120596119879119866 (119896)

(30)


119879 and119866(119896) = (1198921(119896) 1198922(119896) 119892119872(119896))

119879 and here 120596119896 isin R 119896 =


int

Ω

119892119894 (119909) 119892119895 (119909) = 120575119894119895 (31)

where 120575119894119895 = 0 119894 = 119895 120575119894119895 = 1 119894 = 119895 120596 = 119860minus1119861 119860 =

sum119896isinΩ

119866(119896)119866(119896)119879sum119888

119894=1120583119901

119894119896 and 119861 = sum

119896isinΩ119866(119896)sum

119888

119894=1120583119901

119894119896(119910119896 minus




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+ (1 +

119888

sum

119894=1

120583119901

119894119896) (32)



120583lowast

119894119896=

119888

sum

119895=1

((

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+

10038171003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119895

10038171003817100381710038171003817

2

+

)

1(119901minus1)

)

minus1

Vlowast119894=

sum119873

119896=1120583119901

119894119896(119910119896 minus 120596119896120573119896)

sum119873

119896=1120583119901

119894119896

120573lowast

119896=

1

120596119896

(119910119896 minus

sum119888

119894=1120583119901

119894119896V119894

sum119888

119894=1120583119901

119894119896

)

(33)



119869FCM 119902119887 =119888

sum

119894=1

sum

119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897(1 minus V119894)

2

(34)



(119905)

119897= 119899 And



120583119894119897 =


sum119888

119895=1(1 minus V119895)


V119894 =sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897119897

sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897

(35)


119888

119894=1120583119901

119894119897V119894sum119888

119894=1120583119901



119896

(36)

with 119897119896 = 119910119896 minus 119887119905minus1

119896





SA = (


)

times 100

(37)







119869 (1198781 1198782) =

1198781 cap 1198782

1198781 cup 1198782

(38)


120581 (1198781 1198782) =

210038161003816100381610038161198781 cap 1198782

1003816100381610038161003816

10038161003816100381610038161198781

1003816100381610038161003816+10038161003816100381610038161198782

1003816100381610038161003816

(39)



119878 (119894) =

119887 (119894) minus 119886 (119894)

max 119887 (119894) 119886 (119894) (40)









10 Conclusion



Table1Com

paris

onof

thea

lgorith

msu

sedforsegmentatio

nforb

rain

MRim

ages

with

intensity

inho

mogeneitycorrectio

nin

term

sofadvantagesa

ndlim

itatio

ns

Algorith

mAd

vantages

Limitatio

nsQuantitativ

eevaluation

Bias

correctedFC

M(BCF

CM)

[14]

BCFC

Mcompensatefor

theg

ray(in

tensity

)levels

inho


gof

apixel

tobe

influ

encedby

thelabels

inits

immediate

neighb

ourhoo

d

BCFC

Mcompu

testhe

neighb

ourhoo

dterm

ineach

iteratio

nste

pwhich

isvery

compu

tatio

nally

time-consum

ingandlackse

noug

hrobu

stnessto

noise

andou

tliersa

ndisdifficultto

cluste

rno

n-Eu

clidean

structure

data

Segm

entatio

naccuracy

(SA)

Spatially


FCM

(FCM

S1FCM

S2and

KFCM

S1K

FCM

S2)[21]

Clustersthen

on-Euclid

eanstr

ucturesindata

Robu

stnesso

fthe

originalclu

sterin

galgorithm

sto

noise

andou

tliersw

hileretaining

compu

tatio

nalsim

plicity

KFCM

S1andKF

CMS2

areh

eavilyaffectedby

theirp

aram

eters

Segm

entatio

naccuracy

andsim

ilaritymeasure

Improved

FCM

(IFC

M)[26]

IFCM

perfo

rmsb


nalF

CMSensitive

toalgorithm

icparameter

values

election

Und

ersegm

entatio

n(U

nS)oversegm

entatio

n(O


n(In

S)and

comparis

onscore

Fastgeneralized

FCM

(FGFC

M)

[68]

FGFC

Mlessensthe

disadvantageso

ffuzzy

119888-m

eans

(FCM

)algorith

msw

ithspatial

constraints(FC

MS)

andatthes

ametim

eenhances

thec

luste

ringperfo

rmanceTh

eintro

ductionof

localspatia

linformationto

the

correspo

ndingob

jectivefun

ctions

enhances

their

insensitivenessto

noise

Lack

enou

ghrobu

stnesstono

iseandou

tliers

especiallyin

absenceo

fprio

rkno

wledgeo

fthe

noise

Thetim

eofsegmentin

gan

imageis

depend

ento

ntheimages

izeandhencethe

larger

thes

izeo

fthe

imagethem

orethe

segm

entatio

ntim

e

Segm

entatio

naccuracy

andsim

ilaritymeasure

GaussianKe

rnelized

FCM

(GKF

CM)[27]

Itisag

eneralized

type

ofFC

MB

CFCM

KF

CMS1and

KFCM

S2algorithm

sand

ismore

efficientand

robu

stwith

good

parameter

learning

schemeincomparis

onwith

FCM

andits

varia

nt

Forthe

case

ofGaussianno

iseof

10level

KFCM

Shasb

etterresultthanGKF

CMSegm

entatio

naccuracy

Mod

ified

FCM

classificatio

nmetho

dusingmultiscaleFu

zzy

119888-m

eans

(MsFCM

)[44

]

Them

etho

disrobu

stforn

oise

andlowcontrast

MRim

ages

becauseo

fitsmultiscalediffu

sion

filterin

gscheme

Thethresho

ldvalueo

fparam

eter119896used

inthe

algorithm

may

need

tobe

adjuste

dwhenthis

metho

disappliedto

otherimages

with

adifferent

noise

level

Dicec

oefficientand

overlap

ratio

Mod

ified

fastFC

M(M

FCM)for

brainMRim

ages

segm

entatio

n[15]

Intensity

inho

mogeneitynoiseand

partial

volume(PV

)effectsa

retakeninto

accoun

tfor

images

egmentatio

nAu

tomated

metho

dto

determ

inethe

initialvalues

ofthec

entro

idsa

ndadaptiv

emetho

dto

incorporatethe

localspatia

lcontinuityto

overcomethe

noise

effectiv

elyand

preventthe

edge

from

blurrin

g

Quantitativ

eaccuracydecreasesw

henthen

oise

levelinbrainMRim

ages

increasesLarger

radius

ofneighb

ourhoo

dleadstothelosso

ftextureso

CSFcann

otbe

segm

entedaccurately

Jaccardsim

ilarityandDicec

oefficient

Mod

ified

robu

stFC

Malgorithm

with

weightedbias

estim

ation

(MRF

CM-w

BE)[19]

Initializes

thec

entro

idsu

singdist-

max


toredu

cethen

umbero

fiteratio

ns

Thea

lgorith

mwas

teste

don

T1-T2weighted

images

andsim

ulated

brainim

ages

corrup

tedby

Gaussianno

iseon

lySilhou

ettevalue



Competing Interests


References

















































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




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












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



4 FCM Clustering


119873


119869FCM = (119880119881) =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2 (4)

where 119881 = V119894119888



2=


119873

119896=1120583119894119896 lt 119873 1 le 119894 le 119888

and sum119888119894=1



120583119894119896 =

1

sum119888

119895=1(1198892

1198941198961198892

119895119896)

1119901

V119894 =sum119873

119896=1120583119901

119894119896119909119896

sum119873

119896=1120583119901

119894119896

(5)







(a)

(b)



119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119910119896 minus 120573119896 minus V119894

1003817100381710038171003817

2

+

120572

119873119877

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896( sum

119910119903isin119873119896

1003817100381710038171003817119910119903 minus 120573119903 minus V119894

1003817100381710038171003817

2)

(6)



120583lowast

119894119896

=

1

sum119888

119895=1((119863119894119896 + (120572119873119877) 120574119894) (119863119895119896 + (120572119873119877) 120574119895))

1(119901minus1)

Vlowast119894=

sum119873

119896=1120583119901

119894119896((119910119896 minus 120573119896) + (120572119873119877)sum119910

119903isin119873119896

(119910119903 minus 120573119896))

(1 + 120572)sum119873

119896=1120583119901

119894119896

120573lowast

119896= 119910119896 minus

sum119888

119894=1120583119901

119894119896V119894

sum119888

119894=1120583119901

119894119896

(7)


119910119903isin119873119896

119910119903 minus






119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2+ 120572

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1003817100381710038171003817119896 minus V119894

1003817100381710038171003817

2) (8)


119903isin119873119896

119909119903minusV1198942 can


119909119903 minus 1198962+ 119896 minus V119894


120583119894119896 =

(1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2+ 120572

1003817100381710038171003817119896 minus V119894

1003817100381710038171003817

2)


sum119888

119895=1(

10038171003817100381710038171003817119909119896 minus V119895

10038171003817100381710038171003817

2

+ 120572

10038171003817100381710038171003817119896 minus V119895

10038171003817100381710038171003817

2

)


V119894 =sum119873

119896=1120583119901

119894119896(119909119896 + 119896)

(1 + 120572)sum119873

119896=1120583119901

119894119896

(9)



21205902) and


119869119872 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119909119896 V119894))

+ 120572

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119896 V119894))

(10)


120583119894119896

=


sum119888

119895=1((1 minus 119870 (119909119896 V119895)) + 120572 (1 minus 119870 (119896 V119895)))


V119894 =sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) 119909119896 + 120572119870 (119896 V119894) 119896)

sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) + 120572119870 (119896 V119894))

(11)





119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896119889 (119909119896 V119894) (12)


119889 (119909119896 V119894) =1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2(1 minus 120582119867119894119896 minus 120585119865119894119896)

(13)


119867119894119896 =

sum119878

119895=1120583119894119895119892119896119895

sum119878

119895=1119892119896119895

119865119894119896 =

sum119878

119895=11205832

1198941198951199022

119896119895

sum119878

119895=11199022

119896119895

(14)


120573119896 = 119909119896 minus 119901119896 minus

sum119873

119896=1(119909119896 minus 119901119896)

sum119888

119894=1120583119901

1198941198961198722sum119873

119896=1(sum119888

119894=1120583119901

1198941198961198722)

(15)

where 119901119896 = sum119888

119894=1120583119901

1198941198961198722V119894sum

119888

119894=1120583119901

1198941198961198722 and 119872 = 1 minus 120582119867119894119896 minus


120583119894119896 =

1

sum119888

119895=1(119889 (119909119896 V119894) 119889 (119909119896 V119895))

2(119901minus1)

V119894 =sum119873

119896=1120583119901

119894119896(119909119896 minus 120573119896)

sum119873

119896=1120583119901

119894119896

(16)





119878119896119895 =

S119904 119896119895 times 119878119892 119896119895 119895 = 119896

0 119895 = 119896

(17)


119878119904 119896119895 = exp(minusmax (1003816100381610038161003816

1003816119901119895 minus 119901119896

10038161003816100381610038161003816

10038161003816100381610038161003816119902119895 minus 119902119896

10038161003816100381610038161003816)

120582119904

) (18)


119878119892 119896119895 = exp(minus

10038171003817100381710038171003817119909119896 minus 119909119895

10038171003817100381710038171003817

2

1205821198921205902

119892 119896

) (19)


120590119892 119896 = sqrt(sum119895isin119873119896

10038171003817100381710038171003817119909119895 minus 119909119896

10038171003817100381710038171003817

2

119873119877

) (20)



119869 =

119888

sum

119894=1

119902

sum

119896=1

120574119896120583119901

119894119896(120585119896 minus V119894)

2 (21)


119896=1120574119896 = 119873 and


120585119896 =

sum119895isin119873119896

119878119896119895119909119895

sum119895isin119873119896

119878119896119895

(22)


120583119894119896 =


sum119888

119895=1((120585119896 minus V119895)


)

V119894 =sum119902

119896=1120574119896120583119901

119894119896120585119896

sum119902

119896=1120574119896120583119901

119894119896

(23)



119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119909119896 V119894))

+

119888

sum

119894=1

119873

sum

119896=1

120578119894120583119901

119894119896(1 minus 119870 (119896 V119894))

(24)


119896=1(119909119896 minus 119909

2120578119894)

with119909 = sum119873

119895=1(119909119896119873) and 120578119894 = min1198941015840 =119894(1minus119870(V

1015840

119894 V119894))max119895(1minus

119870(V119895 119909))



120583119894119896

=


sum119888

119895=1((1 minus 119870 (119909119896 V119895)) + 120578119894 (1 minus 119870 ( V119895)))


V119894 =sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) 119909119896 + 120578119894119870(119896 V119894) 119896)

sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) + 120578119894119870(119896 V119894))

(25)




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

+

120572

119873119877

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896( sum

119910119903isin119873119896

1003817100381710038171003817119910119903 minus V119894

1003817100381710038171003817

2)

+ 120573

119888

sum

119894=1

119873

sum

119896=1

(120583119894119896 minus 1205831015840

119894119896)

119901 1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

(26)



1205831015840

119894119896=

1205831015840

119894119896 if max (120583119905+1

119894119896) gt 119870

0 otherwise(27)


120583119894119896 =

1 + 120573sum119888

119895=1((1205831015840

1198941198961198892

119894119896minus 1205831015840

1198941198951198892

119894119896) ((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119894119896)))

sum119888

119895=1(((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)) ((1 + 120573) 119889

2

119895119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)))

V119894 =sum119873

119896=11205832

119894119896(119909119896 + (120572119873119877)sum119909

119903isin119873119896

119909119903) + 120573sum119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)

2

119909119896

(1 + 120572)sum119873

119896=11205832

119894119896+ 120573sum

119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)2

(28)





119869 =

119888

sum

119894=1

119873

sum

119896=1

[(1 minus 120572) 120583119901

119894119896(119910119896 minus 119887119896 minus V119894)

2

+ 120572 sum

119903isin119873119896

120583119901

119894119896120596119896119903 (119910119903 minus 119887119903 minus V119894)

2] +

119873

sum

119896=1

120572119896

119888

sum

119894=1

120583119894119896 (1


119894119896)

(29)




120583119894119896 = (

119888

sum

119895=1

(

(1 minus 120572) 119889119896119894 + 120572sum119903isin119873119896

120596119896119903119889119903119894 minus 120572119896

(1 minus 120572) 119889119896119895 + 120572sum119903isin119873119896

120596119896119903119889119903119895 minus 120572119896

))

minus1

V119894

=

sum119873

119896=1120583119901


119896

120596119896119903 (119910119903 minus 119887119903)

sum119873

119896=1120583119901

119894119896

119887119896 =

119872

sum

119896=1

120596119896119892119896 (119896) = 120596119879119866 (119896)

(30)


119879 and119866(119896) = (1198921(119896) 1198922(119896) 119892119872(119896))

119879 and here 120596119896 isin R 119896 =


int

Ω

119892119894 (119909) 119892119895 (119909) = 120575119894119895 (31)

where 120575119894119895 = 0 119894 = 119895 120575119894119895 = 1 119894 = 119895 120596 = 119860minus1119861 119860 =

sum119896isinΩ

119866(119896)119866(119896)119879sum119888

119894=1120583119901

119894119896 and 119861 = sum

119896isinΩ119866(119896)sum

119888

119894=1120583119901

119894119896(119910119896 minus




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+ (1 +

119888

sum

119894=1

120583119901

119894119896) (32)



120583lowast

119894119896=

119888

sum

119895=1

((

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+

10038171003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119895

10038171003817100381710038171003817

2

+

)

1(119901minus1)

)

minus1

Vlowast119894=

sum119873

119896=1120583119901

119894119896(119910119896 minus 120596119896120573119896)

sum119873

119896=1120583119901

119894119896

120573lowast

119896=

1

120596119896

(119910119896 minus

sum119888

119894=1120583119901

119894119896V119894

sum119888

119894=1120583119901

119894119896

)

(33)



119869FCM 119902119887 =119888

sum

119894=1

sum

119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897(1 minus V119894)

2

(34)



(119905)

119897= 119899 And



120583119894119897 =


sum119888

119895=1(1 minus V119895)


V119894 =sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897119897

sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897

(35)


119888

119894=1120583119901

119894119897V119894sum119888

119894=1120583119901



119896

(36)

with 119897119896 = 119910119896 minus 119887119905minus1

119896





SA = (


)

times 100

(37)







119869 (1198781 1198782) =

1198781 cap 1198782

1198781 cup 1198782

(38)


120581 (1198781 1198782) =

210038161003816100381610038161198781 cap 1198782

1003816100381610038161003816

10038161003816100381610038161198781

1003816100381610038161003816+10038161003816100381610038161198782

1003816100381610038161003816

(39)



119878 (119894) =

119887 (119894) minus 119886 (119894)

max 119887 (119894) 119886 (119894) (40)









10 Conclusion



Table1Com

paris

onof

thea

lgorith

msu

sedforsegmentatio

nforb

rain

MRim

ages

with

intensity

inho

mogeneitycorrectio

nin

term

sofadvantagesa

ndlim

itatio

ns

Algorith

mAd

vantages

Limitatio

nsQuantitativ

eevaluation

Bias

correctedFC

M(BCF

CM)

[14]

BCFC

Mcompensatefor

theg

ray(in

tensity

)levels

inho


gof

apixel

tobe

influ

encedby

thelabels

inits

immediate

neighb

ourhoo

d

BCFC

Mcompu

testhe

neighb

ourhoo

dterm

ineach

iteratio

nste

pwhich

isvery

compu

tatio

nally

time-consum

ingandlackse

noug

hrobu

stnessto

noise

andou

tliersa

ndisdifficultto

cluste

rno

n-Eu

clidean

structure

data

Segm

entatio

naccuracy

(SA)

Spatially


FCM

(FCM

S1FCM

S2and

KFCM

S1K

FCM

S2)[21]

Clustersthen

on-Euclid

eanstr

ucturesindata

Robu

stnesso

fthe

originalclu

sterin

galgorithm

sto

noise

andou

tliersw

hileretaining

compu

tatio

nalsim

plicity

KFCM

S1andKF

CMS2

areh

eavilyaffectedby

theirp

aram

eters

Segm

entatio

naccuracy

andsim

ilaritymeasure

Improved

FCM

(IFC

M)[26]

IFCM

perfo

rmsb


nalF

CMSensitive

toalgorithm

icparameter

values

election

Und

ersegm

entatio

n(U

nS)oversegm

entatio

n(O


n(In

S)and

comparis

onscore

Fastgeneralized

FCM

(FGFC

M)

[68]

FGFC

Mlessensthe

disadvantageso

ffuzzy

119888-m

eans

(FCM

)algorith

msw

ithspatial

constraints(FC

MS)

andatthes

ametim

eenhances

thec

luste

ringperfo

rmanceTh

eintro

ductionof

localspatia

linformationto

the

correspo

ndingob

jectivefun

ctions

enhances

their

insensitivenessto

noise

Lack

enou

ghrobu

stnesstono

iseandou

tliers

especiallyin

absenceo

fprio

rkno

wledgeo

fthe

noise

Thetim

eofsegmentin

gan

imageis

depend

ento

ntheimages

izeandhencethe

larger

thes

izeo

fthe

imagethem

orethe

segm

entatio

ntim

e

Segm

entatio

naccuracy

andsim

ilaritymeasure

GaussianKe

rnelized

FCM

(GKF

CM)[27]

Itisag

eneralized

type

ofFC

MB

CFCM

KF

CMS1and

KFCM

S2algorithm

sand

ismore

efficientand

robu

stwith

good

parameter

learning

schemeincomparis

onwith

FCM

andits

varia

nt

Forthe

case

ofGaussianno

iseof

10level

KFCM

Shasb

etterresultthanGKF

CMSegm

entatio

naccuracy

Mod

ified

FCM

classificatio

nmetho

dusingmultiscaleFu

zzy

119888-m

eans

(MsFCM

)[44

]

Them

etho

disrobu

stforn

oise

andlowcontrast

MRim

ages

becauseo

fitsmultiscalediffu

sion

filterin

gscheme

Thethresho

ldvalueo

fparam

eter119896used

inthe

algorithm

may

need

tobe

adjuste

dwhenthis

metho

disappliedto

otherimages

with

adifferent

noise

level

Dicec

oefficientand

overlap

ratio

Mod

ified

fastFC

M(M

FCM)for

brainMRim

ages

segm

entatio

n[15]

Intensity

inho

mogeneitynoiseand

partial

volume(PV

)effectsa

retakeninto

accoun

tfor

images

egmentatio

nAu

tomated

metho

dto

determ

inethe

initialvalues

ofthec

entro

idsa

ndadaptiv

emetho

dto

incorporatethe

localspatia

lcontinuityto

overcomethe

noise

effectiv

elyand

preventthe

edge

from

blurrin

g

Quantitativ

eaccuracydecreasesw

henthen

oise

levelinbrainMRim

ages

increasesLarger

radius

ofneighb

ourhoo

dleadstothelosso

ftextureso

CSFcann

otbe

segm

entedaccurately

Jaccardsim

ilarityandDicec

oefficient

Mod

ified

robu

stFC

Malgorithm

with

weightedbias

estim

ation

(MRF

CM-w

BE)[19]

Initializes

thec

entro

idsu

singdist-

max


toredu

cethen

umbero

fiteratio

ns

Thea

lgorith

mwas

teste

don

T1-T2weighted

images

andsim

ulated

brainim

ages

corrup

tedby

Gaussianno

iseon

lySilhou

ettevalue



Competing Interests


References

















































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




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



4 FCM Clustering


119873


119869FCM = (119880119881) =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2 (4)

where 119881 = V119894119888



2=


119873

119896=1120583119894119896 lt 119873 1 le 119894 le 119888

and sum119888119894=1



120583119894119896 =

1

sum119888

119895=1(1198892

1198941198961198892

119895119896)

1119901

V119894 =sum119873

119896=1120583119901

119894119896119909119896

sum119873

119896=1120583119901

119894119896

(5)







(a)

(b)



119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119910119896 minus 120573119896 minus V119894

1003817100381710038171003817

2

+

120572

119873119877

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896( sum

119910119903isin119873119896

1003817100381710038171003817119910119903 minus 120573119903 minus V119894

1003817100381710038171003817

2)

(6)



120583lowast

119894119896

=

1

sum119888

119895=1((119863119894119896 + (120572119873119877) 120574119894) (119863119895119896 + (120572119873119877) 120574119895))

1(119901minus1)

Vlowast119894=

sum119873

119896=1120583119901

119894119896((119910119896 minus 120573119896) + (120572119873119877)sum119910

119903isin119873119896

(119910119903 minus 120573119896))

(1 + 120572)sum119873

119896=1120583119901

119894119896

120573lowast

119896= 119910119896 minus

sum119888

119894=1120583119901

119894119896V119894

sum119888

119894=1120583119901

119894119896

(7)


119910119903isin119873119896

119910119903 minus






119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2+ 120572

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1003817100381710038171003817119896 minus V119894

1003817100381710038171003817

2) (8)


119903isin119873119896

119909119903minusV1198942 can


119909119903 minus 1198962+ 119896 minus V119894


120583119894119896 =

(1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2+ 120572

1003817100381710038171003817119896 minus V119894

1003817100381710038171003817

2)


sum119888

119895=1(

10038171003817100381710038171003817119909119896 minus V119895

10038171003817100381710038171003817

2

+ 120572

10038171003817100381710038171003817119896 minus V119895

10038171003817100381710038171003817

2

)


V119894 =sum119873

119896=1120583119901

119894119896(119909119896 + 119896)

(1 + 120572)sum119873

119896=1120583119901

119894119896

(9)



21205902) and


119869119872 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119909119896 V119894))

+ 120572

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119896 V119894))

(10)


120583119894119896

=


sum119888

119895=1((1 minus 119870 (119909119896 V119895)) + 120572 (1 minus 119870 (119896 V119895)))


V119894 =sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) 119909119896 + 120572119870 (119896 V119894) 119896)

sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) + 120572119870 (119896 V119894))

(11)





119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896119889 (119909119896 V119894) (12)


119889 (119909119896 V119894) =1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2(1 minus 120582119867119894119896 minus 120585119865119894119896)

(13)


119867119894119896 =

sum119878

119895=1120583119894119895119892119896119895

sum119878

119895=1119892119896119895

119865119894119896 =

sum119878

119895=11205832

1198941198951199022

119896119895

sum119878

119895=11199022

119896119895

(14)


120573119896 = 119909119896 minus 119901119896 minus

sum119873

119896=1(119909119896 minus 119901119896)

sum119888

119894=1120583119901

1198941198961198722sum119873

119896=1(sum119888

119894=1120583119901

1198941198961198722)

(15)

where 119901119896 = sum119888

119894=1120583119901

1198941198961198722V119894sum

119888

119894=1120583119901

1198941198961198722 and 119872 = 1 minus 120582119867119894119896 minus


120583119894119896 =

1

sum119888

119895=1(119889 (119909119896 V119894) 119889 (119909119896 V119895))

2(119901minus1)

V119894 =sum119873

119896=1120583119901

119894119896(119909119896 minus 120573119896)

sum119873

119896=1120583119901

119894119896

(16)





119878119896119895 =

S119904 119896119895 times 119878119892 119896119895 119895 = 119896

0 119895 = 119896

(17)


119878119904 119896119895 = exp(minusmax (1003816100381610038161003816

1003816119901119895 minus 119901119896

10038161003816100381610038161003816

10038161003816100381610038161003816119902119895 minus 119902119896

10038161003816100381610038161003816)

120582119904

) (18)


119878119892 119896119895 = exp(minus

10038171003817100381710038171003817119909119896 minus 119909119895

10038171003817100381710038171003817

2

1205821198921205902

119892 119896

) (19)


120590119892 119896 = sqrt(sum119895isin119873119896

10038171003817100381710038171003817119909119895 minus 119909119896

10038171003817100381710038171003817

2

119873119877

) (20)



119869 =

119888

sum

119894=1

119902

sum

119896=1

120574119896120583119901

119894119896(120585119896 minus V119894)

2 (21)


119896=1120574119896 = 119873 and


120585119896 =

sum119895isin119873119896

119878119896119895119909119895

sum119895isin119873119896

119878119896119895

(22)


120583119894119896 =


sum119888

119895=1((120585119896 minus V119895)


)

V119894 =sum119902

119896=1120574119896120583119901

119894119896120585119896

sum119902

119896=1120574119896120583119901

119894119896

(23)



119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119909119896 V119894))

+

119888

sum

119894=1

119873

sum

119896=1

120578119894120583119901

119894119896(1 minus 119870 (119896 V119894))

(24)


119896=1(119909119896 minus 119909

2120578119894)

with119909 = sum119873

119895=1(119909119896119873) and 120578119894 = min1198941015840 =119894(1minus119870(V

1015840

119894 V119894))max119895(1minus

119870(V119895 119909))



120583119894119896

=


sum119888

119895=1((1 minus 119870 (119909119896 V119895)) + 120578119894 (1 minus 119870 ( V119895)))


V119894 =sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) 119909119896 + 120578119894119870(119896 V119894) 119896)

sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) + 120578119894119870(119896 V119894))

(25)




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

+

120572

119873119877

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896( sum

119910119903isin119873119896

1003817100381710038171003817119910119903 minus V119894

1003817100381710038171003817

2)

+ 120573

119888

sum

119894=1

119873

sum

119896=1

(120583119894119896 minus 1205831015840

119894119896)

119901 1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

(26)



1205831015840

119894119896=

1205831015840

119894119896 if max (120583119905+1

119894119896) gt 119870

0 otherwise(27)


120583119894119896 =

1 + 120573sum119888

119895=1((1205831015840

1198941198961198892

119894119896minus 1205831015840

1198941198951198892

119894119896) ((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119894119896)))

sum119888

119895=1(((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)) ((1 + 120573) 119889

2

119895119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)))

V119894 =sum119873

119896=11205832

119894119896(119909119896 + (120572119873119877)sum119909

119903isin119873119896

119909119903) + 120573sum119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)

2

119909119896

(1 + 120572)sum119873

119896=11205832

119894119896+ 120573sum

119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)2

(28)





119869 =

119888

sum

119894=1

119873

sum

119896=1

[(1 minus 120572) 120583119901

119894119896(119910119896 minus 119887119896 minus V119894)

2

+ 120572 sum

119903isin119873119896

120583119901

119894119896120596119896119903 (119910119903 minus 119887119903 minus V119894)

2] +

119873

sum

119896=1

120572119896

119888

sum

119894=1

120583119894119896 (1


119894119896)

(29)




120583119894119896 = (

119888

sum

119895=1

(

(1 minus 120572) 119889119896119894 + 120572sum119903isin119873119896

120596119896119903119889119903119894 minus 120572119896

(1 minus 120572) 119889119896119895 + 120572sum119903isin119873119896

120596119896119903119889119903119895 minus 120572119896

))

minus1

V119894

=

sum119873

119896=1120583119901


119896

120596119896119903 (119910119903 minus 119887119903)

sum119873

119896=1120583119901

119894119896

119887119896 =

119872

sum

119896=1

120596119896119892119896 (119896) = 120596119879119866 (119896)

(30)


119879 and119866(119896) = (1198921(119896) 1198922(119896) 119892119872(119896))

119879 and here 120596119896 isin R 119896 =


int

Ω

119892119894 (119909) 119892119895 (119909) = 120575119894119895 (31)

where 120575119894119895 = 0 119894 = 119895 120575119894119895 = 1 119894 = 119895 120596 = 119860minus1119861 119860 =

sum119896isinΩ

119866(119896)119866(119896)119879sum119888

119894=1120583119901

119894119896 and 119861 = sum

119896isinΩ119866(119896)sum

119888

119894=1120583119901

119894119896(119910119896 minus




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+ (1 +

119888

sum

119894=1

120583119901

119894119896) (32)



120583lowast

119894119896=

119888

sum

119895=1

((

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+

10038171003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119895

10038171003817100381710038171003817

2

+

)

1(119901minus1)

)

minus1

Vlowast119894=

sum119873

119896=1120583119901

119894119896(119910119896 minus 120596119896120573119896)

sum119873

119896=1120583119901

119894119896

120573lowast

119896=

1

120596119896

(119910119896 minus

sum119888

119894=1120583119901

119894119896V119894

sum119888

119894=1120583119901

119894119896

)

(33)



119869FCM 119902119887 =119888

sum

119894=1

sum

119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897(1 minus V119894)

2

(34)



(119905)

119897= 119899 And



120583119894119897 =


sum119888

119895=1(1 minus V119895)


V119894 =sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897119897

sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897

(35)


119888

119894=1120583119901

119894119897V119894sum119888

119894=1120583119901



119896

(36)

with 119897119896 = 119910119896 minus 119887119905minus1

119896





SA = (


)

times 100

(37)







119869 (1198781 1198782) =

1198781 cap 1198782

1198781 cup 1198782

(38)


120581 (1198781 1198782) =

210038161003816100381610038161198781 cap 1198782

1003816100381610038161003816

10038161003816100381610038161198781

1003816100381610038161003816+10038161003816100381610038161198782

1003816100381610038161003816

(39)



119878 (119894) =

119887 (119894) minus 119886 (119894)

max 119887 (119894) 119886 (119894) (40)









10 Conclusion



Table1Com

paris

onof

thea

lgorith

msu

sedforsegmentatio

nforb

rain

MRim

ages

with

intensity

inho

mogeneitycorrectio

nin

term

sofadvantagesa

ndlim

itatio

ns

Algorith

mAd

vantages

Limitatio

nsQuantitativ

eevaluation

Bias

correctedFC

M(BCF

CM)

[14]

BCFC

Mcompensatefor

theg

ray(in

tensity

)levels

inho


gof

apixel

tobe

influ

encedby

thelabels

inits

immediate

neighb

ourhoo

d

BCFC

Mcompu

testhe

neighb

ourhoo

dterm

ineach

iteratio

nste

pwhich

isvery

compu

tatio

nally

time-consum

ingandlackse

noug

hrobu

stnessto

noise

andou

tliersa

ndisdifficultto

cluste

rno

n-Eu

clidean

structure

data

Segm

entatio

naccuracy

(SA)

Spatially


FCM

(FCM

S1FCM

S2and

KFCM

S1K

FCM

S2)[21]

Clustersthen

on-Euclid

eanstr

ucturesindata

Robu

stnesso

fthe

originalclu

sterin

galgorithm

sto

noise

andou

tliersw

hileretaining

compu

tatio

nalsim

plicity

KFCM

S1andKF

CMS2

areh

eavilyaffectedby

theirp

aram

eters

Segm

entatio

naccuracy

andsim

ilaritymeasure

Improved

FCM

(IFC

M)[26]

IFCM

perfo

rmsb


nalF

CMSensitive

toalgorithm

icparameter

values

election

Und

ersegm

entatio

n(U

nS)oversegm

entatio

n(O


n(In

S)and

comparis

onscore

Fastgeneralized

FCM

(FGFC

M)

[68]

FGFC

Mlessensthe

disadvantageso

ffuzzy

119888-m

eans

(FCM

)algorith

msw

ithspatial

constraints(FC

MS)

andatthes

ametim

eenhances

thec

luste

ringperfo

rmanceTh

eintro

ductionof

localspatia

linformationto

the

correspo

ndingob

jectivefun

ctions

enhances

their

insensitivenessto

noise

Lack

enou

ghrobu

stnesstono

iseandou

tliers

especiallyin

absenceo

fprio

rkno

wledgeo

fthe

noise

Thetim

eofsegmentin

gan

imageis

depend

ento

ntheimages

izeandhencethe

larger

thes

izeo

fthe

imagethem

orethe

segm

entatio

ntim

e

Segm

entatio

naccuracy

andsim

ilaritymeasure

GaussianKe

rnelized

FCM

(GKF

CM)[27]

Itisag

eneralized

type

ofFC

MB

CFCM

KF

CMS1and

KFCM

S2algorithm

sand

ismore

efficientand

robu

stwith

good

parameter

learning

schemeincomparis

onwith

FCM

andits

varia

nt

Forthe

case

ofGaussianno

iseof

10level

KFCM

Shasb

etterresultthanGKF

CMSegm

entatio

naccuracy

Mod

ified

FCM

classificatio

nmetho

dusingmultiscaleFu

zzy

119888-m

eans

(MsFCM

)[44

]

Them

etho

disrobu

stforn

oise

andlowcontrast

MRim

ages

becauseo

fitsmultiscalediffu

sion

filterin

gscheme

Thethresho

ldvalueo

fparam

eter119896used

inthe

algorithm

may

need

tobe

adjuste

dwhenthis

metho

disappliedto

otherimages

with

adifferent

noise

level

Dicec

oefficientand

overlap

ratio

Mod

ified

fastFC

M(M

FCM)for

brainMRim

ages

segm

entatio

n[15]

Intensity

inho

mogeneitynoiseand

partial

volume(PV

)effectsa

retakeninto

accoun

tfor

images

egmentatio

nAu

tomated

metho

dto

determ

inethe

initialvalues

ofthec

entro

idsa

ndadaptiv

emetho

dto

incorporatethe

localspatia

lcontinuityto

overcomethe

noise

effectiv

elyand

preventthe

edge

from

blurrin

g

Quantitativ

eaccuracydecreasesw

henthen

oise

levelinbrainMRim

ages

increasesLarger

radius

ofneighb

ourhoo

dleadstothelosso

ftextureso

CSFcann

otbe

segm

entedaccurately

Jaccardsim

ilarityandDicec

oefficient

Mod

ified

robu

stFC

Malgorithm

with

weightedbias

estim

ation

(MRF

CM-w

BE)[19]

Initializes

thec

entro

idsu

singdist-

max


toredu

cethen

umbero

fiteratio

ns

Thea

lgorith

mwas

teste

don

T1-T2weighted

images

andsim

ulated

brainim

ages

corrup

tedby

Gaussianno

iseon

lySilhou

ettevalue



Competing Interests


References

















































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




(a)

(b)



119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119910119896 minus 120573119896 minus V119894

1003817100381710038171003817

2

+

120572

119873119877

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896( sum

119910119903isin119873119896

1003817100381710038171003817119910119903 minus 120573119903 minus V119894

1003817100381710038171003817

2)

(6)



120583lowast

119894119896

=

1

sum119888

119895=1((119863119894119896 + (120572119873119877) 120574119894) (119863119895119896 + (120572119873119877) 120574119895))

1(119901minus1)

Vlowast119894=

sum119873

119896=1120583119901

119894119896((119910119896 minus 120573119896) + (120572119873119877)sum119910

119903isin119873119896

(119910119903 minus 120573119896))

(1 + 120572)sum119873

119896=1120583119901

119894119896

120573lowast

119896= 119910119896 minus

sum119888

119894=1120583119901

119894119896V119894

sum119888

119894=1120583119901

119894119896

(7)


119910119903isin119873119896

119910119903 minus






119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2+ 120572

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1003817100381710038171003817119896 minus V119894

1003817100381710038171003817

2) (8)


119903isin119873119896

119909119903minusV1198942 can


119909119903 minus 1198962+ 119896 minus V119894


120583119894119896 =

(1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2+ 120572

1003817100381710038171003817119896 minus V119894

1003817100381710038171003817

2)


sum119888

119895=1(

10038171003817100381710038171003817119909119896 minus V119895

10038171003817100381710038171003817

2

+ 120572

10038171003817100381710038171003817119896 minus V119895

10038171003817100381710038171003817

2

)


V119894 =sum119873

119896=1120583119901

119894119896(119909119896 + 119896)

(1 + 120572)sum119873

119896=1120583119901

119894119896

(9)



21205902) and


119869119872 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119909119896 V119894))

+ 120572

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119896 V119894))

(10)


120583119894119896

=


sum119888

119895=1((1 minus 119870 (119909119896 V119895)) + 120572 (1 minus 119870 (119896 V119895)))


V119894 =sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) 119909119896 + 120572119870 (119896 V119894) 119896)

sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) + 120572119870 (119896 V119894))

(11)





119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896119889 (119909119896 V119894) (12)


119889 (119909119896 V119894) =1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2(1 minus 120582119867119894119896 minus 120585119865119894119896)

(13)


119867119894119896 =

sum119878

119895=1120583119894119895119892119896119895

sum119878

119895=1119892119896119895

119865119894119896 =

sum119878

119895=11205832

1198941198951199022

119896119895

sum119878

119895=11199022

119896119895

(14)


120573119896 = 119909119896 minus 119901119896 minus

sum119873

119896=1(119909119896 minus 119901119896)

sum119888

119894=1120583119901

1198941198961198722sum119873

119896=1(sum119888

119894=1120583119901

1198941198961198722)

(15)

where 119901119896 = sum119888

119894=1120583119901

1198941198961198722V119894sum

119888

119894=1120583119901

1198941198961198722 and 119872 = 1 minus 120582119867119894119896 minus


120583119894119896 =

1

sum119888

119895=1(119889 (119909119896 V119894) 119889 (119909119896 V119895))

2(119901minus1)

V119894 =sum119873

119896=1120583119901

119894119896(119909119896 minus 120573119896)

sum119873

119896=1120583119901

119894119896

(16)





119878119896119895 =

S119904 119896119895 times 119878119892 119896119895 119895 = 119896

0 119895 = 119896

(17)


119878119904 119896119895 = exp(minusmax (1003816100381610038161003816

1003816119901119895 minus 119901119896

10038161003816100381610038161003816

10038161003816100381610038161003816119902119895 minus 119902119896

10038161003816100381610038161003816)

120582119904

) (18)


119878119892 119896119895 = exp(minus

10038171003817100381710038171003817119909119896 minus 119909119895

10038171003817100381710038171003817

2

1205821198921205902

119892 119896

) (19)


120590119892 119896 = sqrt(sum119895isin119873119896

10038171003817100381710038171003817119909119895 minus 119909119896

10038171003817100381710038171003817

2

119873119877

) (20)



119869 =

119888

sum

119894=1

119902

sum

119896=1

120574119896120583119901

119894119896(120585119896 minus V119894)

2 (21)


119896=1120574119896 = 119873 and


120585119896 =

sum119895isin119873119896

119878119896119895119909119895

sum119895isin119873119896

119878119896119895

(22)


120583119894119896 =


sum119888

119895=1((120585119896 minus V119895)


)

V119894 =sum119902

119896=1120574119896120583119901

119894119896120585119896

sum119902

119896=1120574119896120583119901

119894119896

(23)



119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119909119896 V119894))

+

119888

sum

119894=1

119873

sum

119896=1

120578119894120583119901

119894119896(1 minus 119870 (119896 V119894))

(24)


119896=1(119909119896 minus 119909

2120578119894)

with119909 = sum119873

119895=1(119909119896119873) and 120578119894 = min1198941015840 =119894(1minus119870(V

1015840

119894 V119894))max119895(1minus

119870(V119895 119909))



120583119894119896

=


sum119888

119895=1((1 minus 119870 (119909119896 V119895)) + 120578119894 (1 minus 119870 ( V119895)))


V119894 =sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) 119909119896 + 120578119894119870(119896 V119894) 119896)

sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) + 120578119894119870(119896 V119894))

(25)




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

+

120572

119873119877

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896( sum

119910119903isin119873119896

1003817100381710038171003817119910119903 minus V119894

1003817100381710038171003817

2)

+ 120573

119888

sum

119894=1

119873

sum

119896=1

(120583119894119896 minus 1205831015840

119894119896)

119901 1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

(26)



1205831015840

119894119896=

1205831015840

119894119896 if max (120583119905+1

119894119896) gt 119870

0 otherwise(27)


120583119894119896 =

1 + 120573sum119888

119895=1((1205831015840

1198941198961198892

119894119896minus 1205831015840

1198941198951198892

119894119896) ((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119894119896)))

sum119888

119895=1(((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)) ((1 + 120573) 119889

2

119895119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)))

V119894 =sum119873

119896=11205832

119894119896(119909119896 + (120572119873119877)sum119909

119903isin119873119896

119909119903) + 120573sum119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)

2

119909119896

(1 + 120572)sum119873

119896=11205832

119894119896+ 120573sum

119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)2

(28)





119869 =

119888

sum

119894=1

119873

sum

119896=1

[(1 minus 120572) 120583119901

119894119896(119910119896 minus 119887119896 minus V119894)

2

+ 120572 sum

119903isin119873119896

120583119901

119894119896120596119896119903 (119910119903 minus 119887119903 minus V119894)

2] +

119873

sum

119896=1

120572119896

119888

sum

119894=1

120583119894119896 (1


119894119896)

(29)




120583119894119896 = (

119888

sum

119895=1

(

(1 minus 120572) 119889119896119894 + 120572sum119903isin119873119896

120596119896119903119889119903119894 minus 120572119896

(1 minus 120572) 119889119896119895 + 120572sum119903isin119873119896

120596119896119903119889119903119895 minus 120572119896

))

minus1

V119894

=

sum119873

119896=1120583119901


119896

120596119896119903 (119910119903 minus 119887119903)

sum119873

119896=1120583119901

119894119896

119887119896 =

119872

sum

119896=1

120596119896119892119896 (119896) = 120596119879119866 (119896)

(30)


119879 and119866(119896) = (1198921(119896) 1198922(119896) 119892119872(119896))

119879 and here 120596119896 isin R 119896 =


int

Ω

119892119894 (119909) 119892119895 (119909) = 120575119894119895 (31)

where 120575119894119895 = 0 119894 = 119895 120575119894119895 = 1 119894 = 119895 120596 = 119860minus1119861 119860 =

sum119896isinΩ

119866(119896)119866(119896)119879sum119888

119894=1120583119901

119894119896 and 119861 = sum

119896isinΩ119866(119896)sum

119888

119894=1120583119901

119894119896(119910119896 minus




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+ (1 +

119888

sum

119894=1

120583119901

119894119896) (32)



120583lowast

119894119896=

119888

sum

119895=1

((

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+

10038171003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119895

10038171003817100381710038171003817

2

+

)

1(119901minus1)

)

minus1

Vlowast119894=

sum119873

119896=1120583119901

119894119896(119910119896 minus 120596119896120573119896)

sum119873

119896=1120583119901

119894119896

120573lowast

119896=

1

120596119896

(119910119896 minus

sum119888

119894=1120583119901

119894119896V119894

sum119888

119894=1120583119901

119894119896

)

(33)



119869FCM 119902119887 =119888

sum

119894=1

sum

119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897(1 minus V119894)

2

(34)



(119905)

119897= 119899 And



120583119894119897 =


sum119888

119895=1(1 minus V119895)


V119894 =sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897119897

sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897

(35)


119888

119894=1120583119901

119894119897V119894sum119888

119894=1120583119901



119896

(36)

with 119897119896 = 119910119896 minus 119887119905minus1

119896





SA = (


)

times 100

(37)







119869 (1198781 1198782) =

1198781 cap 1198782

1198781 cup 1198782

(38)


120581 (1198781 1198782) =

210038161003816100381610038161198781 cap 1198782

1003816100381610038161003816

10038161003816100381610038161198781

1003816100381610038161003816+10038161003816100381610038161198782

1003816100381610038161003816

(39)



119878 (119894) =

119887 (119894) minus 119886 (119894)

max 119887 (119894) 119886 (119894) (40)









10 Conclusion



Table1Com

paris

onof

thea

lgorith

msu

sedforsegmentatio

nforb

rain

MRim

ages

with

intensity

inho

mogeneitycorrectio

nin

term

sofadvantagesa

ndlim

itatio

ns

Algorith

mAd

vantages

Limitatio

nsQuantitativ

eevaluation

Bias

correctedFC

M(BCF

CM)

[14]

BCFC

Mcompensatefor

theg

ray(in

tensity

)levels

inho


gof

apixel

tobe

influ

encedby

thelabels

inits

immediate

neighb

ourhoo

d

BCFC

Mcompu

testhe

neighb

ourhoo

dterm

ineach

iteratio

nste

pwhich

isvery

compu

tatio

nally

time-consum

ingandlackse

noug

hrobu

stnessto

noise

andou

tliersa

ndisdifficultto

cluste

rno

n-Eu

clidean

structure

data

Segm

entatio

naccuracy

(SA)

Spatially


FCM

(FCM

S1FCM

S2and

KFCM

S1K

FCM

S2)[21]

Clustersthen

on-Euclid

eanstr

ucturesindata

Robu

stnesso

fthe

originalclu

sterin

galgorithm

sto

noise

andou

tliersw

hileretaining

compu

tatio

nalsim

plicity

KFCM

S1andKF

CMS2

areh

eavilyaffectedby

theirp

aram

eters

Segm

entatio

naccuracy

andsim

ilaritymeasure

Improved

FCM

(IFC

M)[26]

IFCM

perfo

rmsb


nalF

CMSensitive

toalgorithm

icparameter

values

election

Und

ersegm

entatio

n(U

nS)oversegm

entatio

n(O


n(In

S)and

comparis

onscore

Fastgeneralized

FCM

(FGFC

M)

[68]

FGFC

Mlessensthe

disadvantageso

ffuzzy

119888-m

eans

(FCM

)algorith

msw

ithspatial

constraints(FC

MS)

andatthes

ametim

eenhances

thec

luste

ringperfo

rmanceTh

eintro

ductionof

localspatia

linformationto

the

correspo

ndingob

jectivefun

ctions

enhances

their

insensitivenessto

noise

Lack

enou

ghrobu

stnesstono

iseandou

tliers

especiallyin

absenceo

fprio

rkno

wledgeo

fthe

noise

Thetim

eofsegmentin

gan

imageis

depend

ento

ntheimages

izeandhencethe

larger

thes

izeo

fthe

imagethem

orethe

segm

entatio

ntim

e

Segm

entatio

naccuracy

andsim

ilaritymeasure

GaussianKe

rnelized

FCM

(GKF

CM)[27]

Itisag

eneralized

type

ofFC

MB

CFCM

KF

CMS1and

KFCM

S2algorithm

sand

ismore

efficientand

robu

stwith

good

parameter

learning

schemeincomparis

onwith

FCM

andits

varia

nt

Forthe

case

ofGaussianno

iseof

10level

KFCM

Shasb

etterresultthanGKF

CMSegm

entatio

naccuracy

Mod

ified

FCM

classificatio

nmetho

dusingmultiscaleFu

zzy

119888-m

eans

(MsFCM

)[44

]

Them

etho

disrobu

stforn

oise

andlowcontrast

MRim

ages

becauseo

fitsmultiscalediffu

sion

filterin

gscheme

Thethresho

ldvalueo

fparam

eter119896used

inthe

algorithm

may

need

tobe

adjuste

dwhenthis

metho

disappliedto

otherimages

with

adifferent

noise

level

Dicec

oefficientand

overlap

ratio

Mod

ified

fastFC

M(M

FCM)for

brainMRim

ages

segm

entatio

n[15]

Intensity

inho

mogeneitynoiseand

partial

volume(PV

)effectsa

retakeninto

accoun

tfor

images

egmentatio

nAu

tomated

metho

dto

determ

inethe

initialvalues

ofthec

entro

idsa

ndadaptiv

emetho

dto

incorporatethe

localspatia

lcontinuityto

overcomethe

noise

effectiv

elyand

preventthe

edge

from

blurrin

g

Quantitativ

eaccuracydecreasesw

henthen

oise

levelinbrainMRim

ages

increasesLarger

radius

ofneighb

ourhoo

dleadstothelosso

ftextureso

CSFcann

otbe

segm

entedaccurately

Jaccardsim

ilarityandDicec

oefficient

Mod

ified

robu

stFC

Malgorithm

with

weightedbias

estim

ation

(MRF

CM-w

BE)[19]

Initializes

thec

entro

idsu

singdist-

max


toredu

cethen

umbero

fiteratio

ns

Thea

lgorith

mwas

teste

don

T1-T2weighted

images

andsim

ulated

brainim

ages

corrup

tedby

Gaussianno

iseon

lySilhou

ettevalue



Competing Interests


References

















































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2+ 120572

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1003817100381710038171003817119896 minus V119894

1003817100381710038171003817

2) (8)


119903isin119873119896

119909119903minusV1198942 can


119909119903 minus 1198962+ 119896 minus V119894


120583119894119896 =

(1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2+ 120572

1003817100381710038171003817119896 minus V119894

1003817100381710038171003817

2)


sum119888

119895=1(

10038171003817100381710038171003817119909119896 minus V119895

10038171003817100381710038171003817

2

+ 120572

10038171003817100381710038171003817119896 minus V119895

10038171003817100381710038171003817

2

)


V119894 =sum119873

119896=1120583119901

119894119896(119909119896 + 119896)

(1 + 120572)sum119873

119896=1120583119901

119894119896

(9)



21205902) and


119869119872 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119909119896 V119894))

+ 120572

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119896 V119894))

(10)


120583119894119896

=


sum119888

119895=1((1 minus 119870 (119909119896 V119895)) + 120572 (1 minus 119870 (119896 V119895)))


V119894 =sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) 119909119896 + 120572119870 (119896 V119894) 119896)

sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) + 120572119870 (119896 V119894))

(11)





119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896119889 (119909119896 V119894) (12)


119889 (119909119896 V119894) =1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2(1 minus 120582119867119894119896 minus 120585119865119894119896)

(13)


119867119894119896 =

sum119878

119895=1120583119894119895119892119896119895

sum119878

119895=1119892119896119895

119865119894119896 =

sum119878

119895=11205832

1198941198951199022

119896119895

sum119878

119895=11199022

119896119895

(14)


120573119896 = 119909119896 minus 119901119896 minus

sum119873

119896=1(119909119896 minus 119901119896)

sum119888

119894=1120583119901

1198941198961198722sum119873

119896=1(sum119888

119894=1120583119901

1198941198961198722)

(15)

where 119901119896 = sum119888

119894=1120583119901

1198941198961198722V119894sum

119888

119894=1120583119901

1198941198961198722 and 119872 = 1 minus 120582119867119894119896 minus


120583119894119896 =

1

sum119888

119895=1(119889 (119909119896 V119894) 119889 (119909119896 V119895))

2(119901minus1)

V119894 =sum119873

119896=1120583119901

119894119896(119909119896 minus 120573119896)

sum119873

119896=1120583119901

119894119896

(16)





119878119896119895 =

S119904 119896119895 times 119878119892 119896119895 119895 = 119896

0 119895 = 119896

(17)


119878119904 119896119895 = exp(minusmax (1003816100381610038161003816

1003816119901119895 minus 119901119896

10038161003816100381610038161003816

10038161003816100381610038161003816119902119895 minus 119902119896

10038161003816100381610038161003816)

120582119904

) (18)


119878119892 119896119895 = exp(minus

10038171003817100381710038171003817119909119896 minus 119909119895

10038171003817100381710038171003817

2

1205821198921205902

119892 119896

) (19)


120590119892 119896 = sqrt(sum119895isin119873119896

10038171003817100381710038171003817119909119895 minus 119909119896

10038171003817100381710038171003817

2

119873119877

) (20)



119869 =

119888

sum

119894=1

119902

sum

119896=1

120574119896120583119901

119894119896(120585119896 minus V119894)

2 (21)


119896=1120574119896 = 119873 and


120585119896 =

sum119895isin119873119896

119878119896119895119909119895

sum119895isin119873119896

119878119896119895

(22)


120583119894119896 =


sum119888

119895=1((120585119896 minus V119895)


)

V119894 =sum119902

119896=1120574119896120583119901

119894119896120585119896

sum119902

119896=1120574119896120583119901

119894119896

(23)



119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119909119896 V119894))

+

119888

sum

119894=1

119873

sum

119896=1

120578119894120583119901

119894119896(1 minus 119870 (119896 V119894))

(24)


119896=1(119909119896 minus 119909

2120578119894)

with119909 = sum119873

119895=1(119909119896119873) and 120578119894 = min1198941015840 =119894(1minus119870(V

1015840

119894 V119894))max119895(1minus

119870(V119895 119909))



120583119894119896

=


sum119888

119895=1((1 minus 119870 (119909119896 V119895)) + 120578119894 (1 minus 119870 ( V119895)))


V119894 =sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) 119909119896 + 120578119894119870(119896 V119894) 119896)

sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) + 120578119894119870(119896 V119894))

(25)




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

+

120572

119873119877

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896( sum

119910119903isin119873119896

1003817100381710038171003817119910119903 minus V119894

1003817100381710038171003817

2)

+ 120573

119888

sum

119894=1

119873

sum

119896=1

(120583119894119896 minus 1205831015840

119894119896)

119901 1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

(26)



1205831015840

119894119896=

1205831015840

119894119896 if max (120583119905+1

119894119896) gt 119870

0 otherwise(27)


120583119894119896 =

1 + 120573sum119888

119895=1((1205831015840

1198941198961198892

119894119896minus 1205831015840

1198941198951198892

119894119896) ((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119894119896)))

sum119888

119895=1(((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)) ((1 + 120573) 119889

2

119895119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)))

V119894 =sum119873

119896=11205832

119894119896(119909119896 + (120572119873119877)sum119909

119903isin119873119896

119909119903) + 120573sum119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)

2

119909119896

(1 + 120572)sum119873

119896=11205832

119894119896+ 120573sum

119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)2

(28)





119869 =

119888

sum

119894=1

119873

sum

119896=1

[(1 minus 120572) 120583119901

119894119896(119910119896 minus 119887119896 minus V119894)

2

+ 120572 sum

119903isin119873119896

120583119901

119894119896120596119896119903 (119910119903 minus 119887119903 minus V119894)

2] +

119873

sum

119896=1

120572119896

119888

sum

119894=1

120583119894119896 (1


119894119896)

(29)




120583119894119896 = (

119888

sum

119895=1

(

(1 minus 120572) 119889119896119894 + 120572sum119903isin119873119896

120596119896119903119889119903119894 minus 120572119896

(1 minus 120572) 119889119896119895 + 120572sum119903isin119873119896

120596119896119903119889119903119895 minus 120572119896

))

minus1

V119894

=

sum119873

119896=1120583119901


119896

120596119896119903 (119910119903 minus 119887119903)

sum119873

119896=1120583119901

119894119896

119887119896 =

119872

sum

119896=1

120596119896119892119896 (119896) = 120596119879119866 (119896)

(30)


119879 and119866(119896) = (1198921(119896) 1198922(119896) 119892119872(119896))

119879 and here 120596119896 isin R 119896 =


int

Ω

119892119894 (119909) 119892119895 (119909) = 120575119894119895 (31)

where 120575119894119895 = 0 119894 = 119895 120575119894119895 = 1 119894 = 119895 120596 = 119860minus1119861 119860 =

sum119896isinΩ

119866(119896)119866(119896)119879sum119888

119894=1120583119901

119894119896 and 119861 = sum

119896isinΩ119866(119896)sum

119888

119894=1120583119901

119894119896(119910119896 minus




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+ (1 +

119888

sum

119894=1

120583119901

119894119896) (32)



120583lowast

119894119896=

119888

sum

119895=1

((

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+

10038171003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119895

10038171003817100381710038171003817

2

+

)

1(119901minus1)

)

minus1

Vlowast119894=

sum119873

119896=1120583119901

119894119896(119910119896 minus 120596119896120573119896)

sum119873

119896=1120583119901

119894119896

120573lowast

119896=

1

120596119896

(119910119896 minus

sum119888

119894=1120583119901

119894119896V119894

sum119888

119894=1120583119901

119894119896

)

(33)



119869FCM 119902119887 =119888

sum

119894=1

sum

119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897(1 minus V119894)

2

(34)



(119905)

119897= 119899 And



120583119894119897 =


sum119888

119895=1(1 minus V119895)


V119894 =sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897119897

sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897

(35)


119888

119894=1120583119901

119894119897V119894sum119888

119894=1120583119901



119896

(36)

with 119897119896 = 119910119896 minus 119887119905minus1

119896





SA = (


)

times 100

(37)







119869 (1198781 1198782) =

1198781 cap 1198782

1198781 cup 1198782

(38)


120581 (1198781 1198782) =

210038161003816100381610038161198781 cap 1198782

1003816100381610038161003816

10038161003816100381610038161198781

1003816100381610038161003816+10038161003816100381610038161198782

1003816100381610038161003816

(39)



119878 (119894) =

119887 (119894) minus 119886 (119894)

max 119887 (119894) 119886 (119894) (40)









10 Conclusion



Table1Com

paris

onof

thea

lgorith

msu

sedforsegmentatio

nforb

rain

MRim

ages

with

intensity

inho

mogeneitycorrectio

nin

term

sofadvantagesa

ndlim

itatio

ns

Algorith

mAd

vantages

Limitatio

nsQuantitativ

eevaluation

Bias

correctedFC

M(BCF

CM)

[14]

BCFC

Mcompensatefor

theg

ray(in

tensity

)levels

inho


gof

apixel

tobe

influ

encedby

thelabels

inits

immediate

neighb

ourhoo

d

BCFC

Mcompu

testhe

neighb

ourhoo

dterm

ineach

iteratio

nste

pwhich

isvery

compu

tatio

nally

time-consum

ingandlackse

noug

hrobu

stnessto

noise

andou

tliersa

ndisdifficultto

cluste

rno

n-Eu

clidean

structure

data

Segm

entatio

naccuracy

(SA)

Spatially


FCM

(FCM

S1FCM

S2and

KFCM

S1K

FCM

S2)[21]

Clustersthen

on-Euclid

eanstr

ucturesindata

Robu

stnesso

fthe

originalclu

sterin

galgorithm

sto

noise

andou

tliersw

hileretaining

compu

tatio

nalsim

plicity

KFCM

S1andKF

CMS2

areh

eavilyaffectedby

theirp

aram

eters

Segm

entatio

naccuracy

andsim

ilaritymeasure

Improved

FCM

(IFC

M)[26]

IFCM

perfo

rmsb


nalF

CMSensitive

toalgorithm

icparameter

values

election

Und

ersegm

entatio

n(U

nS)oversegm

entatio

n(O


n(In

S)and

comparis

onscore

Fastgeneralized

FCM

(FGFC

M)

[68]

FGFC

Mlessensthe

disadvantageso

ffuzzy

119888-m

eans

(FCM

)algorith

msw

ithspatial

constraints(FC

MS)

andatthes

ametim

eenhances

thec

luste

ringperfo

rmanceTh

eintro

ductionof

localspatia

linformationto

the

correspo

ndingob

jectivefun

ctions

enhances

their

insensitivenessto

noise

Lack

enou

ghrobu

stnesstono

iseandou

tliers

especiallyin

absenceo

fprio

rkno

wledgeo

fthe

noise

Thetim

eofsegmentin

gan

imageis

depend

ento

ntheimages

izeandhencethe

larger

thes

izeo

fthe

imagethem

orethe

segm

entatio

ntim

e

Segm

entatio

naccuracy

andsim

ilaritymeasure

GaussianKe

rnelized

FCM

(GKF

CM)[27]

Itisag

eneralized

type

ofFC

MB

CFCM

KF

CMS1and

KFCM

S2algorithm

sand

ismore

efficientand

robu

stwith

good

parameter

learning

schemeincomparis

onwith

FCM

andits

varia

nt

Forthe

case

ofGaussianno

iseof

10level

KFCM

Shasb

etterresultthanGKF

CMSegm

entatio

naccuracy

Mod

ified

FCM

classificatio

nmetho

dusingmultiscaleFu

zzy

119888-m

eans

(MsFCM

)[44

]

Them

etho

disrobu

stforn

oise

andlowcontrast

MRim

ages

becauseo

fitsmultiscalediffu

sion

filterin

gscheme

Thethresho

ldvalueo

fparam

eter119896used

inthe

algorithm

may

need

tobe

adjuste

dwhenthis

metho

disappliedto

otherimages

with

adifferent

noise

level

Dicec

oefficientand

overlap

ratio

Mod

ified

fastFC

M(M

FCM)for

brainMRim

ages

segm

entatio

n[15]

Intensity

inho

mogeneitynoiseand

partial

volume(PV

)effectsa

retakeninto

accoun

tfor

images

egmentatio

nAu

tomated

metho

dto

determ

inethe

initialvalues

ofthec

entro

idsa

ndadaptiv

emetho

dto

incorporatethe

localspatia

lcontinuityto

overcomethe

noise

effectiv

elyand

preventthe

edge

from

blurrin

g

Quantitativ

eaccuracydecreasesw

henthen

oise

levelinbrainMRim

ages

increasesLarger

radius

ofneighb

ourhoo

dleadstothelosso

ftextureso

CSFcann

otbe

segm

entedaccurately

Jaccardsim

ilarityandDicec

oefficient

Mod

ified

robu

stFC

Malgorithm

with

weightedbias

estim

ation

(MRF

CM-w

BE)[19]

Initializes

thec

entro

idsu

singdist-

max


toredu

cethen

umbero

fiteratio

ns

Thea

lgorith

mwas

teste

don

T1-T2weighted

images

andsim

ulated

brainim

ages

corrup

tedby

Gaussianno

iseon

lySilhou

ettevalue



Competing Interests


References

















































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






119878119896119895 =

S119904 119896119895 times 119878119892 119896119895 119895 = 119896

0 119895 = 119896

(17)


119878119904 119896119895 = exp(minusmax (1003816100381610038161003816

1003816119901119895 minus 119901119896

10038161003816100381610038161003816

10038161003816100381610038161003816119902119895 minus 119902119896

10038161003816100381610038161003816)

120582119904

) (18)


119878119892 119896119895 = exp(minus

10038171003817100381710038171003817119909119896 minus 119909119895

10038171003817100381710038171003817

2

1205821198921205902

119892 119896

) (19)


120590119892 119896 = sqrt(sum119895isin119873119896

10038171003817100381710038171003817119909119895 minus 119909119896

10038171003817100381710038171003817

2

119873119877

) (20)



119869 =

119888

sum

119894=1

119902

sum

119896=1

120574119896120583119901

119894119896(120585119896 minus V119894)

2 (21)


119896=1120574119896 = 119873 and


120585119896 =

sum119895isin119873119896

119878119896119895119909119895

sum119895isin119873119896

119878119896119895

(22)


120583119894119896 =


sum119888

119895=1((120585119896 minus V119895)


)

V119894 =sum119902

119896=1120574119896120583119901

119894119896120585119896

sum119902

119896=1120574119896120583119901

119894119896

(23)



119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896(1 minus 119870 (119909119896 V119894))

+

119888

sum

119894=1

119873

sum

119896=1

120578119894120583119901

119894119896(1 minus 119870 (119896 V119894))

(24)


119896=1(119909119896 minus 119909

2120578119894)

with119909 = sum119873

119895=1(119909119896119873) and 120578119894 = min1198941015840 =119894(1minus119870(V

1015840

119894 V119894))max119895(1minus

119870(V119895 119909))



120583119894119896

=


sum119888

119895=1((1 minus 119870 (119909119896 V119895)) + 120578119894 (1 minus 119870 ( V119895)))


V119894 =sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) 119909119896 + 120578119894119870(119896 V119894) 119896)

sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) + 120578119894119870(119896 V119894))

(25)




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

+

120572

119873119877

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896( sum

119910119903isin119873119896

1003817100381710038171003817119910119903 minus V119894

1003817100381710038171003817

2)

+ 120573

119888

sum

119894=1

119873

sum

119896=1

(120583119894119896 minus 1205831015840

119894119896)

119901 1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

(26)



1205831015840

119894119896=

1205831015840

119894119896 if max (120583119905+1

119894119896) gt 119870

0 otherwise(27)


120583119894119896 =

1 + 120573sum119888

119895=1((1205831015840

1198941198961198892

119894119896minus 1205831015840

1198941198951198892

119894119896) ((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119894119896)))

sum119888

119895=1(((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)) ((1 + 120573) 119889

2

119895119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)))

V119894 =sum119873

119896=11205832

119894119896(119909119896 + (120572119873119877)sum119909

119903isin119873119896

119909119903) + 120573sum119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)

2

119909119896

(1 + 120572)sum119873

119896=11205832

119894119896+ 120573sum

119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)2

(28)





119869 =

119888

sum

119894=1

119873

sum

119896=1

[(1 minus 120572) 120583119901

119894119896(119910119896 minus 119887119896 minus V119894)

2

+ 120572 sum

119903isin119873119896

120583119901

119894119896120596119896119903 (119910119903 minus 119887119903 minus V119894)

2] +

119873

sum

119896=1

120572119896

119888

sum

119894=1

120583119894119896 (1


119894119896)

(29)




120583119894119896 = (

119888

sum

119895=1

(

(1 minus 120572) 119889119896119894 + 120572sum119903isin119873119896

120596119896119903119889119903119894 minus 120572119896

(1 minus 120572) 119889119896119895 + 120572sum119903isin119873119896

120596119896119903119889119903119895 minus 120572119896

))

minus1

V119894

=

sum119873

119896=1120583119901


119896

120596119896119903 (119910119903 minus 119887119903)

sum119873

119896=1120583119901

119894119896

119887119896 =

119872

sum

119896=1

120596119896119892119896 (119896) = 120596119879119866 (119896)

(30)


119879 and119866(119896) = (1198921(119896) 1198922(119896) 119892119872(119896))

119879 and here 120596119896 isin R 119896 =


int

Ω

119892119894 (119909) 119892119895 (119909) = 120575119894119895 (31)

where 120575119894119895 = 0 119894 = 119895 120575119894119895 = 1 119894 = 119895 120596 = 119860minus1119861 119860 =

sum119896isinΩ

119866(119896)119866(119896)119879sum119888

119894=1120583119901

119894119896 and 119861 = sum

119896isinΩ119866(119896)sum

119888

119894=1120583119901

119894119896(119910119896 minus




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+ (1 +

119888

sum

119894=1

120583119901

119894119896) (32)



120583lowast

119894119896=

119888

sum

119895=1

((

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+

10038171003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119895

10038171003817100381710038171003817

2

+

)

1(119901minus1)

)

minus1

Vlowast119894=

sum119873

119896=1120583119901

119894119896(119910119896 minus 120596119896120573119896)

sum119873

119896=1120583119901

119894119896

120573lowast

119896=

1

120596119896

(119910119896 minus

sum119888

119894=1120583119901

119894119896V119894

sum119888

119894=1120583119901

119894119896

)

(33)



119869FCM 119902119887 =119888

sum

119894=1

sum

119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897(1 minus V119894)

2

(34)



(119905)

119897= 119899 And



120583119894119897 =


sum119888

119895=1(1 minus V119895)


V119894 =sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897119897

sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897

(35)


119888

119894=1120583119901

119894119897V119894sum119888

119894=1120583119901



119896

(36)

with 119897119896 = 119910119896 minus 119887119905minus1

119896





SA = (


)

times 100

(37)







119869 (1198781 1198782) =

1198781 cap 1198782

1198781 cup 1198782

(38)


120581 (1198781 1198782) =

210038161003816100381610038161198781 cap 1198782

1003816100381610038161003816

10038161003816100381610038161198781

1003816100381610038161003816+10038161003816100381610038161198782

1003816100381610038161003816

(39)



119878 (119894) =

119887 (119894) minus 119886 (119894)

max 119887 (119894) 119886 (119894) (40)









10 Conclusion



Table1Com

paris

onof

thea

lgorith

msu

sedforsegmentatio

nforb

rain

MRim

ages

with

intensity

inho

mogeneitycorrectio

nin

term

sofadvantagesa

ndlim

itatio

ns

Algorith

mAd

vantages

Limitatio

nsQuantitativ

eevaluation

Bias

correctedFC

M(BCF

CM)

[14]

BCFC

Mcompensatefor

theg

ray(in

tensity

)levels

inho


gof

apixel

tobe

influ

encedby

thelabels

inits

immediate

neighb

ourhoo

d

BCFC

Mcompu

testhe

neighb

ourhoo

dterm

ineach

iteratio

nste

pwhich

isvery

compu

tatio

nally

time-consum

ingandlackse

noug

hrobu

stnessto

noise

andou

tliersa

ndisdifficultto

cluste

rno

n-Eu

clidean

structure

data

Segm

entatio

naccuracy

(SA)

Spatially


FCM

(FCM

S1FCM

S2and

KFCM

S1K

FCM

S2)[21]

Clustersthen

on-Euclid

eanstr

ucturesindata

Robu

stnesso

fthe

originalclu

sterin

galgorithm

sto

noise

andou

tliersw

hileretaining

compu

tatio

nalsim

plicity

KFCM

S1andKF

CMS2

areh

eavilyaffectedby

theirp

aram

eters

Segm

entatio

naccuracy

andsim

ilaritymeasure

Improved

FCM

(IFC

M)[26]

IFCM

perfo

rmsb


nalF

CMSensitive

toalgorithm

icparameter

values

election

Und

ersegm

entatio

n(U

nS)oversegm

entatio

n(O


n(In

S)and

comparis

onscore

Fastgeneralized

FCM

(FGFC

M)

[68]

FGFC

Mlessensthe

disadvantageso

ffuzzy

119888-m

eans

(FCM

)algorith

msw

ithspatial

constraints(FC

MS)

andatthes

ametim

eenhances

thec

luste

ringperfo

rmanceTh

eintro

ductionof

localspatia

linformationto

the

correspo

ndingob

jectivefun

ctions

enhances

their

insensitivenessto

noise

Lack

enou

ghrobu

stnesstono

iseandou

tliers

especiallyin

absenceo

fprio

rkno

wledgeo

fthe

noise

Thetim

eofsegmentin

gan

imageis

depend

ento

ntheimages

izeandhencethe

larger

thes

izeo

fthe

imagethem

orethe

segm

entatio

ntim

e

Segm

entatio

naccuracy

andsim

ilaritymeasure

GaussianKe

rnelized

FCM

(GKF

CM)[27]

Itisag

eneralized

type

ofFC

MB

CFCM

KF

CMS1and

KFCM

S2algorithm

sand

ismore

efficientand

robu

stwith

good

parameter

learning

schemeincomparis

onwith

FCM

andits

varia

nt

Forthe

case

ofGaussianno

iseof

10level

KFCM

Shasb

etterresultthanGKF

CMSegm

entatio

naccuracy

Mod

ified

FCM

classificatio

nmetho

dusingmultiscaleFu

zzy

119888-m

eans

(MsFCM

)[44

]

Them

etho

disrobu

stforn

oise

andlowcontrast

MRim

ages

becauseo

fitsmultiscalediffu

sion

filterin

gscheme

Thethresho

ldvalueo

fparam

eter119896used

inthe

algorithm

may

need

tobe

adjuste

dwhenthis

metho

disappliedto

otherimages

with

adifferent

noise

level

Dicec

oefficientand

overlap

ratio

Mod

ified

fastFC

M(M

FCM)for

brainMRim

ages

segm

entatio

n[15]

Intensity

inho

mogeneitynoiseand

partial

volume(PV

)effectsa

retakeninto

accoun

tfor

images

egmentatio

nAu

tomated

metho

dto

determ

inethe

initialvalues

ofthec

entro

idsa

ndadaptiv

emetho

dto

incorporatethe

localspatia

lcontinuityto

overcomethe

noise

effectiv

elyand

preventthe

edge

from

blurrin

g

Quantitativ

eaccuracydecreasesw

henthen

oise

levelinbrainMRim

ages

increasesLarger

radius

ofneighb

ourhoo

dleadstothelosso

ftextureso

CSFcann

otbe

segm

entedaccurately

Jaccardsim

ilarityandDicec

oefficient

Mod

ified

robu

stFC

Malgorithm

with

weightedbias

estim

ation

(MRF

CM-w

BE)[19]

Initializes

thec

entro

idsu

singdist-

max


toredu

cethen

umbero

fiteratio

ns

Thea

lgorith

mwas

teste

don

T1-T2weighted

images

andsim

ulated

brainim

ages

corrup

tedby

Gaussianno

iseon

lySilhou

ettevalue



Competing Interests


References

















































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





120583119894119896

=


sum119888

119895=1((1 minus 119870 (119909119896 V119895)) + 120578119894 (1 minus 119870 ( V119895)))


V119894 =sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) 119909119896 + 120578119894119870(119896 V119894) 119896)

sum119873

119896=1120583119901

119894119896(119870 (119909119896 V119894) + 120578119894119870(119896 V119894))

(25)




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

+

120572

119873119877

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896( sum

119910119903isin119873119896

1003817100381710038171003817119910119903 minus V119894

1003817100381710038171003817

2)

+ 120573

119888

sum

119894=1

119873

sum

119896=1

(120583119894119896 minus 1205831015840

119894119896)

119901 1003817100381710038171003817119909119896 minus V119894

1003817100381710038171003817

2

(26)



1205831015840

119894119896=

1205831015840

119894119896 if max (120583119905+1

119894119896) gt 119870

0 otherwise(27)


120583119894119896 =

1 + 120573sum119888

119895=1((1205831015840

1198941198961198892

119894119896minus 1205831015840

1198941198951198892

119894119896) ((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119894119896)))

sum119888

119895=1(((1 + 120573) 119889

2

119894119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)) ((1 + 120573) 119889

2

119895119896+ (120572119873119877) (sum119909

119903isin119873119896

1198892

119895119896)))

V119894 =sum119873

119896=11205832

119894119896(119909119896 + (120572119873119877)sum119909

119903isin119873119896

119909119903) + 120573sum119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)

2

119909119896

(1 + 120572)sum119873

119896=11205832

119894119896+ 120573sum

119873

119896=1(120583119894119896 minus 120583

1015840

119894119896)2

(28)





119869 =

119888

sum

119894=1

119873

sum

119896=1

[(1 minus 120572) 120583119901

119894119896(119910119896 minus 119887119896 minus V119894)

2

+ 120572 sum

119903isin119873119896

120583119901

119894119896120596119896119903 (119910119903 minus 119887119903 minus V119894)

2] +

119873

sum

119896=1

120572119896

119888

sum

119894=1

120583119894119896 (1


119894119896)

(29)




120583119894119896 = (

119888

sum

119895=1

(

(1 minus 120572) 119889119896119894 + 120572sum119903isin119873119896

120596119896119903119889119903119894 minus 120572119896

(1 minus 120572) 119889119896119895 + 120572sum119903isin119873119896

120596119896119903119889119903119895 minus 120572119896

))

minus1

V119894

=

sum119873

119896=1120583119901


119896

120596119896119903 (119910119903 minus 119887119903)

sum119873

119896=1120583119901

119894119896

119887119896 =

119872

sum

119896=1

120596119896119892119896 (119896) = 120596119879119866 (119896)

(30)


119879 and119866(119896) = (1198921(119896) 1198922(119896) 119892119872(119896))

119879 and here 120596119896 isin R 119896 =


int

Ω

119892119894 (119909) 119892119895 (119909) = 120575119894119895 (31)

where 120575119894119895 = 0 119894 = 119895 120575119894119895 = 1 119894 = 119895 120596 = 119860minus1119861 119860 =

sum119896isinΩ

119866(119896)119866(119896)119879sum119888

119894=1120583119901

119894119896 and 119861 = sum

119896isinΩ119866(119896)sum

119888

119894=1120583119901

119894119896(119910119896 minus




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+ (1 +

119888

sum

119894=1

120583119901

119894119896) (32)



120583lowast

119894119896=

119888

sum

119895=1

((

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+

10038171003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119895

10038171003817100381710038171003817

2

+

)

1(119901minus1)

)

minus1

Vlowast119894=

sum119873

119896=1120583119901

119894119896(119910119896 minus 120596119896120573119896)

sum119873

119896=1120583119901

119894119896

120573lowast

119896=

1

120596119896

(119910119896 minus

sum119888

119894=1120583119901

119894119896V119894

sum119888

119894=1120583119901

119894119896

)

(33)



119869FCM 119902119887 =119888

sum

119894=1

sum

119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897(1 minus V119894)

2

(34)



(119905)

119897= 119899 And



120583119894119897 =


sum119888

119895=1(1 minus V119895)


V119894 =sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897119897

sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897

(35)


119888

119894=1120583119901

119894119897V119894sum119888

119894=1120583119901



119896

(36)

with 119897119896 = 119910119896 minus 119887119905minus1

119896





SA = (


)

times 100

(37)







119869 (1198781 1198782) =

1198781 cap 1198782

1198781 cup 1198782

(38)


120581 (1198781 1198782) =

210038161003816100381610038161198781 cap 1198782

1003816100381610038161003816

10038161003816100381610038161198781

1003816100381610038161003816+10038161003816100381610038161198782

1003816100381610038161003816

(39)



119878 (119894) =

119887 (119894) minus 119886 (119894)

max 119887 (119894) 119886 (119894) (40)









10 Conclusion



Table1Com

paris

onof

thea

lgorith

msu

sedforsegmentatio

nforb

rain

MRim

ages

with

intensity

inho

mogeneitycorrectio

nin

term

sofadvantagesa

ndlim

itatio

ns

Algorith

mAd

vantages

Limitatio

nsQuantitativ

eevaluation

Bias

correctedFC

M(BCF

CM)

[14]

BCFC

Mcompensatefor

theg

ray(in

tensity

)levels

inho


gof

apixel

tobe

influ

encedby

thelabels

inits

immediate

neighb

ourhoo

d

BCFC

Mcompu

testhe

neighb

ourhoo

dterm

ineach

iteratio

nste

pwhich

isvery

compu

tatio

nally

time-consum

ingandlackse

noug

hrobu

stnessto

noise

andou

tliersa

ndisdifficultto

cluste

rno

n-Eu

clidean

structure

data

Segm

entatio

naccuracy

(SA)

Spatially


FCM

(FCM

S1FCM

S2and

KFCM

S1K

FCM

S2)[21]

Clustersthen

on-Euclid

eanstr

ucturesindata

Robu

stnesso

fthe

originalclu

sterin

galgorithm

sto

noise

andou

tliersw

hileretaining

compu

tatio

nalsim

plicity

KFCM

S1andKF

CMS2

areh

eavilyaffectedby

theirp

aram

eters

Segm

entatio

naccuracy

andsim

ilaritymeasure

Improved

FCM

(IFC

M)[26]

IFCM

perfo

rmsb


nalF

CMSensitive

toalgorithm

icparameter

values

election

Und

ersegm

entatio

n(U

nS)oversegm

entatio

n(O


n(In

S)and

comparis

onscore

Fastgeneralized

FCM

(FGFC

M)

[68]

FGFC

Mlessensthe

disadvantageso

ffuzzy

119888-m

eans

(FCM

)algorith

msw

ithspatial

constraints(FC

MS)

andatthes

ametim

eenhances

thec

luste

ringperfo

rmanceTh

eintro

ductionof

localspatia

linformationto

the

correspo

ndingob

jectivefun

ctions

enhances

their

insensitivenessto

noise

Lack

enou

ghrobu

stnesstono

iseandou

tliers

especiallyin

absenceo

fprio

rkno

wledgeo

fthe

noise

Thetim

eofsegmentin

gan

imageis

depend

ento

ntheimages

izeandhencethe

larger

thes

izeo

fthe

imagethem

orethe

segm

entatio

ntim

e

Segm

entatio

naccuracy

andsim

ilaritymeasure

GaussianKe

rnelized

FCM

(GKF

CM)[27]

Itisag

eneralized

type

ofFC

MB

CFCM

KF

CMS1and

KFCM

S2algorithm

sand

ismore

efficientand

robu

stwith

good

parameter

learning

schemeincomparis

onwith

FCM

andits

varia

nt

Forthe

case

ofGaussianno

iseof

10level

KFCM

Shasb

etterresultthanGKF

CMSegm

entatio

naccuracy

Mod

ified

FCM

classificatio

nmetho

dusingmultiscaleFu

zzy

119888-m

eans

(MsFCM

)[44

]

Them

etho

disrobu

stforn

oise

andlowcontrast

MRim

ages

becauseo

fitsmultiscalediffu

sion

filterin

gscheme

Thethresho

ldvalueo

fparam

eter119896used

inthe

algorithm

may

need

tobe

adjuste

dwhenthis

metho

disappliedto

otherimages

with

adifferent

noise

level

Dicec

oefficientand

overlap

ratio

Mod

ified

fastFC

M(M

FCM)for

brainMRim

ages

segm

entatio

n[15]

Intensity

inho

mogeneitynoiseand

partial

volume(PV

)effectsa

retakeninto

accoun

tfor

images

egmentatio

nAu

tomated

metho

dto

determ

inethe

initialvalues

ofthec

entro

idsa

ndadaptiv

emetho

dto

incorporatethe

localspatia

lcontinuityto

overcomethe

noise

effectiv

elyand

preventthe

edge

from

blurrin

g

Quantitativ

eaccuracydecreasesw

henthen

oise

levelinbrainMRim

ages

increasesLarger

radius

ofneighb

ourhoo

dleadstothelosso

ftextureso

CSFcann

otbe

segm

entedaccurately

Jaccardsim

ilarityandDicec

oefficient

Mod

ified

robu

stFC

Malgorithm

with

weightedbias

estim

ation

(MRF

CM-w

BE)[19]

Initializes

thec

entro

idsu

singdist-

max


toredu

cethen

umbero

fiteratio

ns

Thea

lgorith

mwas

teste

don

T1-T2weighted

images

andsim

ulated

brainim

ages

corrup

tedby

Gaussianno

iseon

lySilhou

ettevalue



Competing Interests


References

















































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





120583119894119896 = (

119888

sum

119895=1

(

(1 minus 120572) 119889119896119894 + 120572sum119903isin119873119896

120596119896119903119889119903119894 minus 120572119896

(1 minus 120572) 119889119896119895 + 120572sum119903isin119873119896

120596119896119903119889119903119895 minus 120572119896

))

minus1

V119894

=

sum119873

119896=1120583119901


119896

120596119896119903 (119910119903 minus 119887119903)

sum119873

119896=1120583119901

119894119896

119887119896 =

119872

sum

119896=1

120596119896119892119896 (119896) = 120596119879119866 (119896)

(30)


119879 and119866(119896) = (1198921(119896) 1198922(119896) 119892119872(119896))

119879 and here 120596119896 isin R 119896 =


int

Ω

119892119894 (119909) 119892119895 (119909) = 120575119894119895 (31)

where 120575119894119895 = 0 119894 = 119895 120575119894119895 = 1 119894 = 119895 120596 = 119860minus1119861 119860 =

sum119896isinΩ

119866(119896)119866(119896)119879sum119888

119894=1120583119901

119894119896 and 119861 = sum

119896isinΩ119866(119896)sum

119888

119894=1120583119901

119894119896(119910119896 minus




119869 =

119888

sum

119894=1

119873

sum

119896=1

120583119901

119894119896

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+ (1 +

119888

sum

119894=1

120583119901

119894119896) (32)



120583lowast

119894119896=

119888

sum

119895=1

((

1003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119894

1003817100381710038171003817

2+

10038171003817100381710038171003817119910119896 minus 120596119896120573119896 minus V119895

10038171003817100381710038171003817

2

+

)

1(119901minus1)

)

minus1

Vlowast119894=

sum119873

119896=1120583119901

119894119896(119910119896 minus 120596119896120573119896)

sum119873

119896=1120583119901

119894119896

120573lowast

119896=

1

120596119896

(119910119896 minus

sum119888

119894=1120583119901

119894119896V119894

sum119888

119894=1120583119901

119894119896

)

(33)



119869FCM 119902119887 =119888

sum

119894=1

sum

119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897(1 minus V119894)

2

(34)



(119905)

119897= 119899 And



120583119894119897 =


sum119888

119895=1(1 minus V119895)


V119894 =sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897119897

sum119897isinΩ(119905)

ℎ119905

119897120583119901

119894119897

(35)


119888

119894=1120583119901

119894119897V119894sum119888

119894=1120583119901



119896

(36)

with 119897119896 = 119910119896 minus 119887119905minus1

119896





SA = (


)

times 100

(37)







119869 (1198781 1198782) =

1198781 cap 1198782

1198781 cup 1198782

(38)


120581 (1198781 1198782) =

210038161003816100381610038161198781 cap 1198782

1003816100381610038161003816

10038161003816100381610038161198781

1003816100381610038161003816+10038161003816100381610038161198782

1003816100381610038161003816

(39)



119878 (119894) =

119887 (119894) minus 119886 (119894)

max 119887 (119894) 119886 (119894) (40)









10 Conclusion



Table1Com

paris

onof

thea

lgorith

msu

sedforsegmentatio

nforb

rain

MRim

ages

with

intensity

inho

mogeneitycorrectio

nin

term

sofadvantagesa

ndlim

itatio

ns

Algorith

mAd

vantages

Limitatio

nsQuantitativ

eevaluation

Bias

correctedFC

M(BCF

CM)

[14]

BCFC

Mcompensatefor

theg

ray(in

tensity

)levels

inho


gof

apixel

tobe

influ

encedby

thelabels

inits

immediate

neighb

ourhoo

d

BCFC

Mcompu

testhe

neighb

ourhoo

dterm

ineach

iteratio

nste

pwhich

isvery

compu

tatio

nally

time-consum

ingandlackse

noug

hrobu

stnessto

noise

andou

tliersa

ndisdifficultto

cluste

rno

n-Eu

clidean

structure

data

Segm

entatio

naccuracy

(SA)

Spatially


FCM

(FCM

S1FCM

S2and

KFCM

S1K

FCM

S2)[21]

Clustersthen

on-Euclid

eanstr

ucturesindata

Robu

stnesso

fthe

originalclu

sterin

galgorithm

sto

noise

andou

tliersw

hileretaining

compu

tatio

nalsim

plicity

KFCM

S1andKF

CMS2

areh

eavilyaffectedby

theirp

aram

eters

Segm

entatio

naccuracy

andsim

ilaritymeasure

Improved

FCM

(IFC

M)[26]

IFCM

perfo

rmsb


nalF

CMSensitive

toalgorithm

icparameter

values

election

Und

ersegm

entatio

n(U

nS)oversegm

entatio

n(O


n(In

S)and

comparis

onscore

Fastgeneralized

FCM

(FGFC

M)

[68]

FGFC

Mlessensthe

disadvantageso

ffuzzy

119888-m

eans

(FCM

)algorith

msw

ithspatial

constraints(FC

MS)

andatthes

ametim

eenhances

thec

luste

ringperfo

rmanceTh

eintro

ductionof

localspatia

linformationto

the

correspo

ndingob

jectivefun

ctions

enhances

their

insensitivenessto

noise

Lack

enou

ghrobu

stnesstono

iseandou

tliers

especiallyin

absenceo

fprio

rkno

wledgeo

fthe

noise

Thetim

eofsegmentin

gan

imageis

depend

ento

ntheimages

izeandhencethe

larger

thes

izeo

fthe

imagethem

orethe

segm

entatio

ntim

e

Segm

entatio

naccuracy

andsim

ilaritymeasure

GaussianKe

rnelized

FCM

(GKF

CM)[27]

Itisag

eneralized

type

ofFC

MB

CFCM

KF

CMS1and

KFCM

S2algorithm

sand

ismore

efficientand

robu

stwith

good

parameter

learning

schemeincomparis

onwith

FCM

andits

varia

nt

Forthe

case

ofGaussianno

iseof

10level

KFCM

Shasb

etterresultthanGKF

CMSegm

entatio

naccuracy

Mod

ified

FCM

classificatio

nmetho

dusingmultiscaleFu

zzy

119888-m

eans

(MsFCM

)[44

]

Them

etho

disrobu

stforn

oise

andlowcontrast

MRim

ages

becauseo

fitsmultiscalediffu

sion

filterin

gscheme

Thethresho

ldvalueo

fparam

eter119896used

inthe

algorithm

may

need

tobe

adjuste

dwhenthis

metho

disappliedto

otherimages

with

adifferent

noise

level

Dicec

oefficientand

overlap

ratio

Mod

ified

fastFC

M(M

FCM)for

brainMRim

ages

segm

entatio

n[15]

Intensity

inho

mogeneitynoiseand

partial

volume(PV

)effectsa

retakeninto

accoun

tfor

images

egmentatio

nAu

tomated

metho

dto

determ

inethe

initialvalues

ofthec

entro

idsa

ndadaptiv

emetho

dto

incorporatethe

localspatia

lcontinuityto

overcomethe

noise

effectiv

elyand

preventthe

edge

from

blurrin

g

Quantitativ

eaccuracydecreasesw

henthen

oise

levelinbrainMRim

ages

increasesLarger

radius

ofneighb

ourhoo

dleadstothelosso

ftextureso

CSFcann

otbe

segm

entedaccurately

Jaccardsim

ilarityandDicec

oefficient

Mod

ified

robu

stFC

Malgorithm

with

weightedbias

estim

ation

(MRF

CM-w

BE)[19]

Initializes

thec

entro

idsu

singdist-

max


toredu

cethen

umbero

fiteratio

ns

Thea

lgorith

mwas

teste

don

T1-T2weighted

images

andsim

ulated

brainim

ages

corrup

tedby

Gaussianno

iseon

lySilhou

ettevalue



Competing Interests


References

















































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in







SA = (


)

times 100

(37)







119869 (1198781 1198782) =

1198781 cap 1198782

1198781 cup 1198782

(38)


120581 (1198781 1198782) =

210038161003816100381610038161198781 cap 1198782

1003816100381610038161003816

10038161003816100381610038161198781

1003816100381610038161003816+10038161003816100381610038161198782

1003816100381610038161003816

(39)



119878 (119894) =

119887 (119894) minus 119886 (119894)

max 119887 (119894) 119886 (119894) (40)









10 Conclusion



Table1Com

paris

onof

thea

lgorith

msu

sedforsegmentatio

nforb

rain

MRim

ages

with

intensity

inho

mogeneitycorrectio

nin

term

sofadvantagesa

ndlim

itatio

ns

Algorith

mAd

vantages

Limitatio

nsQuantitativ

eevaluation

Bias

correctedFC

M(BCF

CM)

[14]

BCFC

Mcompensatefor

theg

ray(in

tensity

)levels

inho


gof

apixel

tobe

influ

encedby

thelabels

inits

immediate

neighb

ourhoo

d

BCFC

Mcompu

testhe

neighb

ourhoo

dterm

ineach

iteratio

nste

pwhich

isvery

compu

tatio

nally

time-consum

ingandlackse

noug

hrobu

stnessto

noise

andou

tliersa

ndisdifficultto

cluste

rno

n-Eu

clidean

structure

data

Segm

entatio

naccuracy

(SA)

Spatially


FCM

(FCM

S1FCM

S2and

KFCM

S1K

FCM

S2)[21]

Clustersthen

on-Euclid

eanstr

ucturesindata

Robu

stnesso

fthe

originalclu

sterin

galgorithm

sto

noise

andou

tliersw

hileretaining

compu

tatio

nalsim

plicity

KFCM

S1andKF

CMS2

areh

eavilyaffectedby

theirp

aram

eters

Segm

entatio

naccuracy

andsim

ilaritymeasure

Improved

FCM

(IFC

M)[26]

IFCM

perfo

rmsb


nalF

CMSensitive

toalgorithm

icparameter

values

election

Und

ersegm

entatio

n(U

nS)oversegm

entatio

n(O


n(In

S)and

comparis

onscore

Fastgeneralized

FCM

(FGFC

M)

[68]

FGFC

Mlessensthe

disadvantageso

ffuzzy

119888-m

eans

(FCM

)algorith

msw

ithspatial

constraints(FC

MS)

andatthes

ametim

eenhances

thec

luste

ringperfo

rmanceTh

eintro

ductionof

localspatia

linformationto

the

correspo

ndingob

jectivefun

ctions

enhances

their

insensitivenessto

noise

Lack

enou

ghrobu

stnesstono

iseandou

tliers

especiallyin

absenceo

fprio

rkno

wledgeo

fthe

noise

Thetim

eofsegmentin

gan

imageis

depend

ento

ntheimages

izeandhencethe

larger

thes

izeo

fthe

imagethem

orethe

segm

entatio

ntim

e

Segm

entatio

naccuracy

andsim

ilaritymeasure

GaussianKe

rnelized

FCM

(GKF

CM)[27]

Itisag

eneralized

type

ofFC

MB

CFCM

KF

CMS1and

KFCM

S2algorithm

sand

ismore

efficientand

robu

stwith

good

parameter

learning

schemeincomparis

onwith

FCM

andits

varia

nt

Forthe

case

ofGaussianno

iseof

10level

KFCM

Shasb

etterresultthanGKF

CMSegm

entatio

naccuracy

Mod

ified

FCM

classificatio

nmetho

dusingmultiscaleFu

zzy

119888-m

eans

(MsFCM

)[44

]

Them

etho

disrobu

stforn

oise

andlowcontrast

MRim

ages

becauseo

fitsmultiscalediffu

sion

filterin

gscheme

Thethresho

ldvalueo

fparam

eter119896used

inthe

algorithm

may

need

tobe

adjuste

dwhenthis

metho

disappliedto

otherimages

with

adifferent

noise

level

Dicec

oefficientand

overlap

ratio

Mod

ified

fastFC

M(M

FCM)for

brainMRim

ages

segm

entatio

n[15]

Intensity

inho

mogeneitynoiseand

partial

volume(PV

)effectsa

retakeninto

accoun

tfor

images

egmentatio

nAu

tomated

metho

dto

determ

inethe

initialvalues

ofthec

entro

idsa

ndadaptiv

emetho

dto

incorporatethe

localspatia

lcontinuityto

overcomethe

noise

effectiv

elyand

preventthe

edge

from

blurrin

g

Quantitativ

eaccuracydecreasesw

henthen

oise

levelinbrainMRim

ages

increasesLarger

radius

ofneighb

ourhoo

dleadstothelosso

ftextureso

CSFcann

otbe

segm

entedaccurately

Jaccardsim

ilarityandDicec

oefficient

Mod

ified

robu

stFC

Malgorithm

with

weightedbias

estim

ation

(MRF

CM-w

BE)[19]

Initializes

thec

entro

idsu

singdist-

max


toredu

cethen

umbero

fiteratio

ns

Thea

lgorith

mwas

teste

don

T1-T2weighted

images

andsim

ulated

brainim

ages

corrup

tedby

Gaussianno

iseon

lySilhou

ettevalue



Competing Interests


References

















































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Table1Com

paris

onof

thea

lgorith

msu

sedforsegmentatio

nforb

rain

MRim

ages

with

intensity

inho

mogeneitycorrectio

nin

term

sofadvantagesa

ndlim

itatio

ns

Algorith

mAd

vantages

Limitatio

nsQuantitativ

eevaluation

Bias

correctedFC

M(BCF

CM)

[14]

BCFC

Mcompensatefor

theg

ray(in

tensity

)levels

inho


gof

apixel

tobe

influ

encedby

thelabels

inits

immediate

neighb

ourhoo

d

BCFC

Mcompu

testhe

neighb

ourhoo

dterm

ineach

iteratio

nste

pwhich

isvery

compu

tatio

nally

time-consum

ingandlackse

noug

hrobu

stnessto

noise

andou

tliersa

ndisdifficultto

cluste

rno

n-Eu

clidean

structure

data

Segm

entatio

naccuracy

(SA)

Spatially


FCM

(FCM

S1FCM

S2and

KFCM

S1K

FCM

S2)[21]

Clustersthen

on-Euclid

eanstr

ucturesindata

Robu

stnesso

fthe

originalclu

sterin

galgorithm

sto

noise

andou

tliersw

hileretaining

compu

tatio

nalsim

plicity

KFCM

S1andKF

CMS2

areh

eavilyaffectedby

theirp

aram

eters

Segm

entatio

naccuracy

andsim

ilaritymeasure

Improved

FCM

(IFC

M)[26]

IFCM

perfo

rmsb


nalF

CMSensitive

toalgorithm

icparameter

values

election

Und

ersegm

entatio

n(U

nS)oversegm

entatio

n(O


n(In

S)and

comparis

onscore

Fastgeneralized

FCM

(FGFC

M)

[68]

FGFC

Mlessensthe

disadvantageso

ffuzzy

119888-m

eans

(FCM

)algorith

msw

ithspatial

constraints(FC

MS)

andatthes

ametim

eenhances

thec

luste

ringperfo

rmanceTh

eintro

ductionof

localspatia

linformationto

the

correspo

ndingob

jectivefun

ctions

enhances

their

insensitivenessto

noise

Lack

enou

ghrobu

stnesstono

iseandou

tliers

especiallyin

absenceo

fprio

rkno

wledgeo

fthe

noise

Thetim

eofsegmentin

gan

imageis

depend

ento

ntheimages

izeandhencethe

larger

thes

izeo

fthe

imagethem

orethe

segm

entatio

ntim

e

Segm

entatio

naccuracy

andsim

ilaritymeasure

GaussianKe

rnelized

FCM

(GKF

CM)[27]

Itisag

eneralized

type

ofFC

MB

CFCM

KF

CMS1and

KFCM

S2algorithm

sand

ismore

efficientand

robu

stwith

good

parameter

learning

schemeincomparis

onwith

FCM

andits

varia

nt

Forthe

case

ofGaussianno

iseof

10level

KFCM

Shasb

etterresultthanGKF

CMSegm

entatio

naccuracy

Mod

ified

FCM

classificatio

nmetho

dusingmultiscaleFu

zzy

119888-m

eans

(MsFCM

)[44

]

Them

etho

disrobu

stforn

oise

andlowcontrast

MRim

ages

becauseo

fitsmultiscalediffu

sion

filterin

gscheme

Thethresho

ldvalueo

fparam

eter119896used

inthe

algorithm

may

need

tobe

adjuste

dwhenthis

metho

disappliedto

otherimages

with

adifferent

noise

level

Dicec

oefficientand

overlap

ratio

Mod

ified

fastFC

M(M

FCM)for

brainMRim

ages

segm

entatio

n[15]

Intensity

inho

mogeneitynoiseand

partial

volume(PV

)effectsa

retakeninto

accoun

tfor

images

egmentatio

nAu

tomated

metho

dto

determ

inethe

initialvalues

ofthec

entro

idsa

ndadaptiv

emetho

dto

incorporatethe

localspatia

lcontinuityto

overcomethe

noise

effectiv

elyand

preventthe

edge

from

blurrin

g

Quantitativ

eaccuracydecreasesw

henthen

oise

levelinbrainMRim

ages

increasesLarger

radius

ofneighb

ourhoo

dleadstothelosso

ftextureso

CSFcann

otbe

segm

entedaccurately

Jaccardsim

ilarityandDicec

oefficient

Mod

ified

robu

stFC

Malgorithm

with

weightedbias

estim

ation

(MRF

CM-w

BE)[19]

Initializes

thec

entro

idsu

singdist-

max


toredu

cethen

umbero

fiteratio

ns

Thea

lgorith

mwas

teste

don

T1-T2weighted

images

andsim

ulated

brainim

ages

corrup

tedby

Gaussianno

iseon

lySilhou

ettevalue



Competing Interests


References

















































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





Competing Interests


References

















































































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





















































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Review Article FCM Clustering Algorithms for Segmentation...

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Transcript of Review Article FCM Clustering Algorithms for Segmentation...