IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS 1

A Multiple-Kernel Fuzzy C-Means Algorithm forImage Segmentation

Long Chen, C. L. Philip Chen, Fellow, IEEE, and Mingzhu Lu, Student Member, IEEE

Abstract—In this paper, a generalized multiple-kernel fuzzyC-means (FCM) (MKFCM) methodology is introduced as a frame-work for image-segmentation problems. In the framework, asidefrom the fact that the composite kernels are used in the kernelFCM (KFCM), a linear combination of multiple kernels is pro-posed and the updating rules for the linear coefficients of thecomposite kernel are derived as well. The proposed MKFCMalgorithm provides us a new flexible vehicle to fuse different pixelinformation in image-segmentation problems. That is, differentpixel information represented by different kernels is combinedin the kernel space to produce a new kernel. It is shown thattwo successful enhanced KFCM-based image-segmentation algo-rithms are special cases of MKFCM. Several new segmentation al-gorithms are also derived from the proposed MKFCM framework.Simulations on the segmentation of synthetic and medical imagesdemonstrate the flexibility and advantages of MKFCM-basedapproaches.

Index Terms—Composite kernel, fuzzy C-means (FCM), imagesegmentation, kernel function, multiple kernel.

I. INTRODUCTION

IMAGE segmentation is a central task in many research fieldsincluding computer vision [5] and intelligent image and

video analysis [6]. Its essential goal is to split the pixels ofan image into a set of regions such that the pixels in the sameregion are homogeneous according to some properties and thepixels in different regions are not similar. Clustering, particu-larly fuzzy C-means (FCM)-based clustering and its variants,have been widely used in the task of image segmentation dueto their simplicity and fast convergence [4], [6]–[9], [11], [12],[14], [21]. By carefully selecting input features such as pixelcolor, intensity, texture, or a weighted combination of thesedata, the FCM algorithm can segment images to several re-gions in accordance with resulting clusters. Recently, the FCMand other clustering-based image-segmentation approaches areimproved by including the local spatial information of pixelsin classical clustering procedures [4], [6]–[9], [14], [15], [21],[22]. For example, an additional term about the differencebetween the local spatial information and the cluster centers is

Manuscript received March 5, 2010; revised July 27, 2010 and January 19,2011; accepted February 9, 2011. This work was supported in part by theNational Aeronautics and Space Administration under Grant NNC04GB35Gand in part by The Chinese National Basic Research Program (also called 973Program) under Grant 2011CB302801. The review of this paper was arrangedby Editor A. Gomez Skarmeta.

L. Chen and M. Lu are with the Department of Electrical and ComputerEngineering, The University of Texas, San Antonio, TX 78249-0669 USA(e-mail: gbu922@my.utsa.edu; lwf054@my.utsa.edu).

C. L. P. Chen is with the Faculty of Science and Technology, The Universityof Macau, Macau, China (e-mail: philip.chen@ieee.org).

Digital Object Identifier 10.1109/TSMCB.2011.2124455

attached to the traditional objective functions of FCM algo-rithms [4]. Because of the embedded local spatial information,the new FCM has demonstrated robustness over noises inimages [4], [26], [31].

In addition to the incorporation of local spatial information,the kernelization of FCM has made an important performanceimprovement [3], [11], [19], [23], [26], [30], [31], [36]. Thekernel FCM (KFCM) algorithm is an extension of FCM, whichmaps the original inputs into a much higher dimensional Hilbertspace by some transform function. After this reproduction in thekernel Hilbert space, the data are more easily to be separatedor clustered. Previous research on transformation to the kernelspace has already been studied. Liao et al. [23] have directlyapplied the KFCM in the image-segmentation problems, wherethe input data selected for clustering is the combination ofthe pixel intensity and the local spatial information of a pixelrepresented by the mean or the median of neighboring pixels.Chen and Zhang [31] applied the idea of kernel methods inthe calculation of the distances between the examples and thecluster centers. They compute these distances in the extendedHilbert space, and they have demonstrated that such distancesare more robust to noises. To keep the merit of applying localspatial information, an additional term about the differencebetween the local spatial information and the cluster centers(also computed in the extended Hilbert space) is appended tothe objective function. More kernel methods, the kernelizationof clustering algorithms besides FCM, and their applications inthe problems of image segmentation and classification can befound in [2], [5], [10], [13], [17], [18], [28], [32], [33], [41],and [42].

Recently, developments on kernel methods and their applica-tions have emphasized the need to consider multiple kernels orcomposite kernels instead of a single fixed kernel [2], [25]. Withmultiple kernels, the kernel methods gain more flexibility onkernel selections and also reflect the fact that practical learningproblems often involve data from multiple heterogeneous orhomogeneous sources [1], [2], [17], [20], [24], [25], [27], [29],[41], [42]. Specifically, in image-segmentation problems, theinputs are the properties of image pixels, and they could bederived from different sources. For example, the intensity ofa pixel is directly obtained from the image itself, but somecomplicated texture information is perhaps gained from somewavelet filtering of the image [34]. Multiple-kernel methodsprovide us a great tool to fuse information from differentsources [17]. It is necessary to clarify that, in this paper, weuse the term “multiple kernel” in a wider sense than the oneused in machine learning community. In the machine learningcommunity, “multiple-kernel learning” refers to the learning

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2 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS

using an ensemble of basis kernels (usually a linear ensemble),whose combination is optimized in the learning process. Inthis paper, we focus more on the flexible information fusionby applications of composite kernels constructed by multiplekernels defined in different information channels. The combi-nation of the ensemble kernel can be automatically adjusted inthe learning of multiple-kernel FCM (MKFCM) or it can besettled by trial and error or cross-validation.

In this paper, we propose a general framework of MKFCMmethodology. In the framework, besides the direct applicationsof various composite kernels in the KFCM, a new algorithmthat uses a linear composite of multiple kernels is proposedand the updating rules of the linear coefficients of the com-bined kernel are obtained automatically. When applying theMKFCM framework in image-segmentation problems, thispaper first shows that two successful enhanced KFCM-basedimage-segmentation algorithms, which take advantages of lo-cal spatial information [23], [31], are indeed special cases ofMKFCM. After that, several new variants of MKFCM-basedimage-segmentation algorithms are developed. The proposedMKFCM-based algorithms demonstrate the flexibility in kernelselections and combinations, and therefore, they provide thepotential of significant improvement over traditional methodson image segmentation.

The rest of this paper is organized as follows. In Section II,we review the foundations of KFCM briefly. Based on the prop-erties of kernel functions, the general framework of MKFCM isintroduced. In Section III, two traditional KFCM-based image-segmentation algorithms are proved to be the special casesMKFCM. Moreover, several variants of MKFCM-based image-segmentation algorithms are proposed. The simulation resultsreported in Section IV demonstrate that better segmentationresults are derived from MKFCM-based algorithms. Finally,this paper concludes in Section V.

II. KFCM AND MKFCM

A. Foundations of KFCM

Given a data set X = {x1, . . . ,xn}, where the data pointxj ∈ Ξ ⊆ Rp (j = 1, . . . , n), n is the number of data, and pis the input dimension of a data point, traditional FCM [40]groups X into c clusters by minimizing the weighted sum ofdistances between the data and the cluster centers or prototypesdefined as

Q =c∑

i=1

n∑j=1

umij‖xj − oi‖2. (1)

Here, ‖ · ‖ is the Euclidean distance. uij is the membershipof data xj belonging to cluster i, which is represented by theprototype oi. The constraint on uij is

∑ci=1 uij = 1, and m is

the fuzzification coefficient, which usually takes the value of 2.As an enhancement of classical FCM, the KFCM maps the

data set X from the feature space or the data space Ξ ⊆ Rp intoa much higher dimensional space H (a Hilbert space usuallycalled kernel space) by a transform function ϕ: Ξ → H . In thenew kernel space, the data demonstrate simpler structures orpatterns. According to clustering algorithms, the data in the new

space show clusters that are more spherical and therefore can beclustered more easily by FCM algorithms [3], [30], [36].

Generally, the transform function ϕ is not given out explic-itly, but the kernel function is given and it is defined as k:Ξ × Ξ → R

k(x,y) = 〈ϕ(x), ϕ(y)〉 ∀x,y ∈ Ξ (2)

where 〈, 〉 is the inner product for Hilbert space H . Such kernelfunctions are usually called Mercer kernels or kernel. Givena Mercer kernel k, we know that there is always a transformfunction ϕ: Ξ → H satisfies k(x,y) = 〈ϕ(x), ϕ(y)〉, althoughsometimes, we do not know the specific form of ϕ. Widelyused Mercer kernels include the Gaussian kernel k(x,y) =exp(−‖x − y‖2/r2) and the polynomial kernel k(x,y) = (x ·y + d)2. They are both defined over Rn × Rn. Clearly, dueto the fact that we only know the kernel functions, we needto solve the clustering problems in the kernel space by onlyusing kernel functions, i.e., the inner product of the transformfunction ϕ. Usually this is called “kernel trick” [35].

There are two types of KFCM. If the prototypes oi areconstructed in the kernel space, this type of KFCM is referredas KFCM-K (with K standing for the kernel space) [36]. Theobjective function of KFCM-K is

Q =c∑

i=1

n∑j=1

umij ‖ϕ(xj) − oi‖2 . (3)

The learning algorithm of KFCM-K iteratively updates uij as

uij = 1/c∑

h=1

(dϕ2

ij/dϕ2hj

)1/(m−1)(4)

where

dϕ2ij = k(xj ,xj) −

2∑n

h=1 umihk(xh,xj)∑n

h=1 umih

+∑n

h=1

∑nl=1 um

ihumil k(xh,xl)

(∑n

h=1 umih)2

. (5)

More details about the derivation of (4) and (5) can be referredto [36].

Another type of KFCM confines that the prototypes in thekernel space are actually mapped from the original data spaceor the feature space. That is, the objective function is defined as

Q =c∑

i=1

n∑j=1

umij ‖ϕ(xj) − ϕ(oi)‖2 . (6)

This type of KFCM is referred as KFCM-F (with F standingfor feature space/data space) [36].

Usually, only the Gaussian kernel k(x,y) = exp(−‖x −y‖2/r2) is applied in KFCM-F, and because k(x,x) = 1 forGaussian kernel

‖ϕ(xj) − ϕ(oi)‖2 = 〈ϕ(xj), ϕ(xj)〉 + 〈ϕ(oi), ϕ(oi)〉− 2 〈ϕ(xj), ϕ(oi)〉

=k(xj ,xj) + k(oi,oi) − 2k(xj ,oi)= 2 (1 − k(xj ,oi)) . (7)

CHEN et al.: MULTIPLE-KERNEL FUZZY C-MEANS ALGORITHM FOR IMAGE SEGMENTATION 3

The objective function in (6) is then reformulated as [26]

Q =c∑

i=1

n∑j=1

umij (1 − k(xj ,oi)) . (8)

Here, 1 − k(xj ,oi) can be considered as a robust distance mea-surement derived in the kernel space [26]. For these KFCM-F-applying Gaussian kernels [36], we iteratively update theprototypes and memberships as

uij =(1 − k(xj ,oi))

−1/m−1∑cl=1 (1 − k(xj ,ol))

−1/m−1(9)

oi =∑n

l=1 umil k(xl,oi)xl∑n

l=1 umil k(xl,oi)

. (10)

B. MKFCM

Before the introduction of the MKFCM, we first list somenecessary Mercer kernels’ properties in the following [35].

Theorem 1: Let k1 and k2 be kernels over Ξ × Ξ, Ξ ⊆ Rp,and k3 be a kernel over Rp × Rp. Let function ψ: Ξ → Rp

1) k(x,y) = k1(x,y) + k2(x,y) is a kernel.2) k(x,y) = αk1(x,y) is a kernel, when α > 0.3) k(x,y) = k1(x,y)k2(x,y) is a kernel.4) k(x,y) = k3(ψ(x), ψ(y)) is a kernel.The proof of these properties can be referred to [35].The general framework of MKFCM aims to minimize the

same objective function as the single fixed KFCM, i.e.,

Q =c∑

i=1

n∑j=1

umij ‖ϕcom(xj) − oi‖2 (11)

or

Q =c∑

i=1

n∑j=1

umij ‖ϕcom(xj) − ϕcom(oi)‖2 . (12)

Comparing (3) and (6) to (11) and (12), the only differencebetween them is that the transform function ϕ in (3) and(6) is changed to ϕcom, which is derived from a compos-ite kernel kcom(x,y) = 〈ϕcom(x), ϕcom(y)〉. The compositekernel kcom is defined as a combination of multiple kernelsusing properties introduced in Theorem 1. For example, twosimple composite kernels are kcom = k1 + αk2 and kcom =k1k2. Given that k1 and k2 are Mercer kernels, based onproperties 1), 2), and 3) in Theorem 1, the composite kernelkcom is a Mercer kernel as well. In other words, we canalways find some transformation ϕcom such that kcom(x,y) =〈ϕcom(x), ϕcom(y)〉∀x, y ∈ Ξ.

For MKFCM, it still updates uij according to (4) and (5) or(9) and (10). The difference is that the kernel function k in theseequations is replaced by the combined kernel kcom. Similar toKFCM, if MKFCM assumes the prototypes in the kernel space[using objective function (11)], this type of MKFCM is referredas MKFCM-K; if MKFCM confines that the prototypes aremapped from feature space or data space [using objective func-tion (12)], such a type of MKFCM is referred as MKFCM-F.

When the number of parameters in the combined kernelis small, the parameters can be adjusted by trial and error.For instance, the parameter α in the kcom = k1 + αk2 canbe selected by testing a group α in a predefined range orset. While the number of parameters in the combined kernelis large, the more feasible method is automatically adjustingthese parameters in the learning algorithms. For example, inmachine learning community, a widely used composite kernel isthe linear combination of several kernels, i.e., kcom = w1k1 +w2k2 + . . . + wlkl. Some learning algorithms that adjust theweights wi automatically in typical kernel learning methodslike multiple-kernel regressions and classifications [20], [25]have been studied. Here, we propose a similar algorithm forMKFCM using linearly combined kernels.

To increase the number of selections for kernel functions, alinearly combined kernel function is applied in MKFCM. Thenew composite kernel kL is defined as

kL = wb1k1 + wb

2k2 + · · · + wbl kl (13)

where b > 1 is a coefficient similar to the fuzzy coefficient min (1) and (3). The regulation on weights, w1, w2, . . . , wl, is∑l

i=1 wi = 1.The objective function of the MKFCM with the linearly

combined kernel is still the weighted sum of distances betweenthe data and prototypes in the kernel space

Q =c∑

i=1

n∑j=1

umij ‖ϕL(xj) − oi‖2 (14)

where ϕL is the transformation derived from the linearly com-bined kernel kL(xi,xj) = 〈ϕL(x), ϕL(y)〉 [kL is defined in(13)]. Just the same as in the KFCM [(4) and (5)], the learningrule for membership values is

uij = 1/

c∑k=1

(dϕ2

ij/dϕ2kj

)1/(m−1)(15)

where

dϕ2ij = kL(xj ,xj) −

2∑n

h=1 umihkL(xh,xj)∑n

h=1 umih

+∑n

h=1

∑nl=1 um

ihumil kL(xh,xl)

(∑n

h=1 umih)2

. (16)

Introducing the Lagrange term of the constraint of weightswi(i = 1, . . . , l) into the objective function, we have

Q =c∑

i=1

n∑j=1

umij ‖ϕL(xj) − oi‖2 + η

(1 −

l∑i=1

wi

). (17)

Taking the derivative of Q over wi and setting the resultsto zero, we obtain the updating rule of the weights wi(i =1, . . . , l)

∂Q

∂wi= 0 (i = 1, . . . , l). ⇒ wi = 1/

n∑h=1

(Qi/Qh)1/(b−1)

(18)

4 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS

where

Qh =c∑

i=1

n∑j=1

umij ‖ϕh(xj) − oi‖2 (h = 1, . . . , l). (19)

Here, ϕh is the transform function defined by kh (h = 1, . . . , l)in (13) and

‖ϕh(xj) − oi‖2 = kh(xj ,xj) −2∑n

l=1 umil kh(xl,xj)∑n

l=1 umil

+

∑ng=1

∑nl=1 um

igumil kh(xg,xl)(∑n

g=1 umig

)2 . (20)

The algorithm introduced here is named as LMKFCM(standing for Linear combined MKFCM). LMKFCM can lin-early combine more than two kernels and automatically adjustthe weights of each kernel in the optimization procedure. Bystudying the objective function (14) applied in this LMKFCM,we know that the prototypes are directly defined in the kernelspace; therefore, this LMKFCM is indeed LMKFCM-K (withK standing for the kernel space). Here, we shorten LMKFCM-K as LMKFCM. In feature space, it is very difficult to derive alearning algorithm for feature-space LMKFCM (or LMKFCM-F in short) because the linear combination of basic kernels isnot a Gaussian kernel.

III. MKFCM-BASED IMAGE SEGMENTATION

The application of multiple or composite kernels in theFKCM has its advantages. In addition to the flexibility inselecting kernel functions, it also offers a new approach tocombine different information from multiple heterogeneousor homogeneous sources in the kernel space. Specifically, inimage-segmentation problems, the input data involve propertiesof image pixels sometimes derived from very different sources.For example, as mentioned in Section I, the intensity of apixel is directly gained from the image itself, but the textureinformation of a pixel might be obtained from some waveletfiltering of the image. Therefore, we can define different ker-nel functions purposely for the intensity information and thetexture information separately, and we then combine thesekernel functions and apply the composite kernel in MKFCM(including LMKFCM) to obtain better image-segmentationresults. Examples that are more visible could be found frommultitemporal remote sensing images. The pixel informationin these images inherits from different temporal sensors. As aresult, we can define different kernels for different temperaturechannels and apply the combined kernel in a multiple-kernellearning algorithm.

In this section, we first study some successful enhancedKFCM-based image-segmentation algorithms that considerboth the pixel intensity and the local spatial information. These

algorithms are proved actually the special cases of MKFCM-based methods, which mingle a kernel for the spectral informa-tion and a kernel for the local spatial information. After that,several new variants of MKFCM-based image-segmentationalgorithms are developed. These new variants demonstrate theflexibility of MKFCM in kernel selections and combinationsfor image-segmentation problems and offer the potentials ofimprovement in segmentation results.

At first, we formulate a proposition that is useful in thefollowing.

Proposition 1: For a data point x = [x1, x2, . . . , xn] ∈Rp+q, we also define it as x = [xp,xq], where xp ∈ Rp con-tains p dimensions of the data point x and xq ∈ Rq containsthe remaining q dimensions. If kp: Rp × Rp → R is a kernelover Rp × Rp, then the function k: Rp+q × Rp+q → R, suchthat k(x,y) = kp(xp,yp), is also a kernel over Rp+q × Rp+q.

Indeed, setting k3 = kp and ψ: Rp+q → Rp such thatψ (x) = xp, we obtain the conclusion of Proposition 1 bydirectly applying Theorem 1 [property 4)].

Because the Gaussian kernel k(xp,yp) = exp(−‖xp −yp‖2/r2) and the polynomial kernel k(xp,yp) = (xp · yp +d)2 are typical kernels defined on Rp × Rp, based on Proposi-tion 1, we know the Gaussian function defined as k: Rp+q ×Rp+q → R, such that k(x,y) = exp(−‖xp − yp‖2/r2) andthe polynomial function k(x,y) = (xp · yp + d)2 are both ker-nel functions over Rp+q × Rp+q. Without loss of generality, wecall these two functions the Gaussian kernel and the polynomialkernel as well. We can use such kind of kernels for differentinformation embedded in different subdimensions of the inputdata.

A. Two Enhanced KFCM Algorithms as MKFCM

In order to combine the local spatial information of pixelsinto the classical clustering-based image-segmentation algo-rithms, Liao et al. [23] select input data xj (j = 1, 2, . . . , n) asxj = [xj , xj ] ∈ R2 and directly apply the KFCM-K on theseinput data. Here, xj is the intensity of pixel j and xj is thefiltered intensity of pixel j, which represents the local spatialinformation. In [23], xj is the mean or the median filteredintensity defined in a 3 × 3 window centered at pixel j. Wedenote this algorithm as DKFCM (here, D stands for directapplication of KFCM). Specifically, DKFCM_meanf is used todenote the DKFCM applying the mean filtered intensities as thespatial information and DKFCM_medianf is used for DKFCMwith the median filtered intensities. In DKFCM, the kernelfunction is the Gaussian kernel [23] and the applied learningrules are the same as (4) and (5).

We now prove that DKFCM is a special case of MKFCM.Case 1: DKFCM is a special case of MKFCM-K with

kcom = k1k2 applied on input data xj = [xj , xj ] ∈ R2 (j =1, 2, . . . , n). Here, k1 is the Gaussian kernel for pixel in-tensity k1(xi,xj) := exp(−|xi − xj |2/r2) and k2 is anotherGaussian kernel for local spatial information k2(xi,xj) :=exp(−|xi − xj |2/r2), in which xj is the intensity of pixel jand xj is the local spatial information represented by the filteredintensity of pixel j.

CHEN et al.: MULTIPLE-KERNEL FUZZY C-MEANS ALGORITHM FOR IMAGE SEGMENTATION 5

The kernel function k(xi,xj) is used in DKFCM, wherexj = [xj , xj ] and k is the Gaussian kernel. By its definition

k(xi,xj) = exp

(−‖xi − xj‖

r2

2)

= exp

(−‖[xi, xi] − [xj , xj ]‖2

r2

)

= exp(−|xi − xj |2 + |xi − xj |2

r2

)

= exp(−|xi − xj |2

r2

)exp

(−|xi − xj |2

r2

)

= k1(xi,xj)k2(xi,xj)

= kcom(xi,xj). (21)

That is, the DKFCM uses the same kernel function as MKFCM-K. Considering that both DKFCM and MKFCM-K use theupdating rules (4) and (5), we know that DKFCM is a specialcase of MKFCM-K that uses the composite kernel kcom =k1k2.

Chen and Zhang [31] enhance the Gaussian-kernel-basedKFCM-F by adding a local information term in the objectivefunction, i.e., the new objective function becomes

Q=c∑

i=1

n∑j=1

umij (1−k(xj , oi))+α

c∑i=1

n∑j=1

umij (1−k(xj , oi))

(22)

where xj is the intensity of pixel j. In the new objectivefunction, the additional term is the weighted sum of differ-ences between the filtered intensity xj (the local spatial in-formation) and the clustering prototypes. The differences arealso measured using the kernel-induced distances. Such kindof enhanced KFCM-based algorithm is denoted as AKFCM(with A standing for additional term). Like DKFCM_meanfand DKFCM_medianf, we use AKFCM_meanf to represent theAKFCM applying the mean filtered intensities as the local spa-tial information, and AKFCM_medianf denotes the AKFCMusing the median filtered intensities. In AKFCM, the kernelfunction is the Gaussian kernel [31].

Next, we prove that AKFCM [31] is also a special case ofMKFCM.

Case 2: AKFCM is a special case of MFKCM-F withkcom = k1 + αk2 on the input data xj = [xj , xj ] (j =1, 2, . . . , n), where k1, k2, xj , and xj are the same as the onesdefined in Case 1.

In AKFCM, the goal is to minimize the following objectivefunction:

Q1 =c∑

i=1

n∑j=1

umij (1−k(xj , oi))+α

c∑i=1

n∑j=1

umij (1−k(xj , oi))

=c∑

i=1

n∑j=1

umij (1+α−(k(xj , oi)+αk(xj , oi)))

=c∑

i=1

n∑j=1

umij

(1+α−

(exp

(−|xj−oi|2

r2

)

+α exp(−|xj−oi|2

r2

))). (23)

On the other hand, if kcom = k1 + αk2 is the compositekernel for MKFCM-F, the objective function of the MKFC is

Q=c∑

i=1

n∑j=1

umij‖ϕcom(xj)−ϕcom(oi)‖2

=c∑

i=1

n∑j=1

umij (〈ϕcom(xj), ϕcom(xj)〉

+〈ϕcom(oi), ϕcom(oi)〉−2 〈ϕcom(xj), ϕcom(oi)〉)

=c∑

i=1

n∑j=1

umij (kcom(xj ,xj)+kcom(oi,oi)−2kcom(xj ,oi))

=c∑

i=1

n∑j=1

umij (k1(xj ,xj)+αk2(xj ,xj)+k1(oi,oi)

+αk2(oi,oi)−2 (k1(xj ,oi)+αk2(xj ,oi)))

(by kcom = k1+αk2)

=2c∑

i=1

n∑j=1

umij (1+α−(k1(xj ,oi)+α · k2(xj ,oi)))

(by definitions of k1, k2)

=2c∑

i=1

n∑j=1

umij

(1+α−

(exp

(−|xj−oi|2

r2

)

+α exp

(−|xj−oi|2

r2

)))

=2Q1. (24)

Comparing (23) to (24), we know that AKFCM is actuallythe MKFCM-F using kcom = k1 + αk2 (minimizations of Q1

and 2Q1 are the same problem). In other words, AKFCM is aspecial case of MKFCM-F.

B. Variants of MKFCM-BasedImage-Segmentation Algorithms

As shown in the previous section, AKFCM is a specialcase of MKFCM-F with kcom = k1 + αk2 (k1 is the kernelfor intensity, and k2 is the kernel for local spatial infor-mation). Therefore, similar to MKFCM-F, AKFCM confinesthe prototypes as the points mapped from the original dataspace or the feature space. This limits the search space ofthe prototypes. The natural choice to fix this shortcoming isapplying MKFCM-K, which searches prototypes in the totalkernel space, directly to image-segmentation problems. Indeed,we have demonstrated that DKFCM is an MKFCM-K with

6 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS

a composite kernel kcom = k1k2. More variants of MKFCM-K-based image-segmentation algorithms can be proposed. Forinstance, we propose that the first variant of MKFCM-K thatuses the composite kernel

kcom = k1 + αk2 (25)

on the input data is xj = [xj , xj ] ∈ R2 (j = 1, 2, . . . , n), inwhich xj ∈ R is the intensity of pixel j and xj ∈ R is thefiltered intensity of pixel j that stands for the local spatialinformation. Identical to DKFCM, in the composite kernel, k1

is the Gaussian kernel for pixel intensities, i.e., k1(xi,xj) =exp(−|xi − xj |2/r2), and k2 is the Gaussian kernel forthe local spatial information, i.e., k2(xi,xj) = exp(−|xi −xj |2/r2). As a variant of the general MKFCM-K introducedin Section II, this algorithm still updates uij following the rules(4) and (5), in which the kernel function k is replaced by kcom.

It is worth pointing out that k1 or k2 in the first variantof MKFCM-K-based image segmentation can be changed toany other Mercer kernel function for the information related toimage pixels. This empowers the flexibility to the segmentationalgorithm in kernel function selections and combinations.

For example, a composite kernel that joins different shapedkernels can be defined as

kcom = k1 + αk2 (26)

where k1 is still the Gaussian kernel for pixel intensitiesk1(xi,xj) = exp(−|xi − xj |2/r2), but k2 is a polynomialkernel for the spatial information k2(xi,xj) = (xixj + d)2,where xj is the filtered intensity of pixel j. We denotethis MKFCM-K-based algorithm as the second variant ofMKFCM-K.

Aside from the flexibility of selecting different shaped ker-nel functions for the intensity and the spatial information,MKFCM-K allows us to apply kernel functions for other infor-mation derived from the image. Take the texture information asthe example; we can set the input data xj as xj = [xj , xj , sj ] ∈R3, in which xj ∈ R is the intensity of pixel j. The two-tuple[xj , sj ] ∈ R2 is a simple descriptor of the texture informationat pixel xj[37], where xj is the filtered intensity of pixel jand sj is the standard variance of the intensities of the pixelsin the neighborhood of pixel j. Then, we define the combinedkernel as

kcom = k1 + αk2 (27)

where k1 is the Gaussian kernel for pixel intensities and k2 isthe Gaussian kernel for the texture information k2 (xi,xj) =exp(−‖[xi, si] − [xj , sj ]‖2/r2). This algorithm is denoted asthe third variant of MKFCM-K.

Just like what we did in AKFCM and DKFCM, the notations“_meanf” and “_medianf” can be attached to the namesof algorithms to refer different applied spatial information.Therefore, MKFCM-K_meanf is designated to variants ofMKFCM-K using the mean filtered intensities as the localspatial information, and the MKFCM-K_medianf is used whenthe median filtered intensities are selected as the local spatialinformation in MKFCM-K.

Fig. 1. Segmentation performance based on different alpha values.

To increase the information diversity of an image, LMK-FCM can be applied in image-segmentation problems as well.Specifically, we define different kernels for different imageinformation, linearly ensemble them into a new kernel, andthen apply equations (15)(16) and (18)–(20) to update themembership values and weighting coefficients.

For example, the input image data xj is set to be xj =[xj , xj , sj ] ∈ R3, the same as the third variant of MKFCM-K.Then, the composite kernel is designed as

kL = wb1k1 + wb

2k2 + wb3k3 (28)

where k1 is the Gaussian kernel for pixel intensitiesk1(xi,xj) = exp(−|xi − xj |2/r2), k2 is the Gaussian kernelfor spatial information k2(xi,xj) = exp(−|xi − xj |2/r2), andk3 is the Gaussian kernel for texture information k3 (xi,xj) =exp(−‖[xi, si] − [xj , sj ]‖2/r2). The next section will applythis composite kernel for LMKFCM in simulations. For sim-plicity, we also use LMKFCM to refer the specific segmentationalgorithm utilizing the composite kernel defined in (28).

IV. SIMULATION RESULTS

In this section, we compare the KFCM-based and thenewly proposed MKFCM-based (including LMKFCM) image-segmentation algorithms on several synthetic and medicalimages. Because the performance of FCM-type algorithmsdepends on the initialization, this paper does the initialization100 times and chooses the one with the best objective functionvalue. This increases the reliability of comparison results ac-quired in the simulations.

A. Example 1: Synthetic Noised Two-Cluster Image

The synthetic image is similar to the one used in [4] and[31]. It is in the size of 64 × 64 pixels. It contains two clusterswith two intensity values taken as 128 and 0. Different noises,including “Gaussian noise” and “salt and pepper noise,” areadded to the synthetic image. The AKFCM-, DKFCM-, andMKFCM-K-based algorithms are tested on the noised images,and their performance of segmentation is measured as

Segmentation accuracy

=number of correctly classified pixels

total number of pixels. (29)

CHEN et al.: MULTIPLE-KERNEL FUZZY C-MEANS ALGORITHM FOR IMAGE SEGMENTATION 7

Fig. 2. 5% Gaussian-noised synthetic image segmented by different methods. (a) Noised image. (b) Segmentation result of DKFCM_meanf. (c) Segmentationresult of AKFCM_ meanf. (d) Segmentation result of MKFCM-K_meanf.

Fig. 3. 10% salt-and-pepper-noised synthetic image segmented by different methods. (a) Noised image. (b) Segmentation result of DKFCM_medianf.(c) Segmentation result of AKFCM_medianf. (d) Segmentation result of MKFCM-K_medianf.

TABLE ISEGMENTATION ACCURACIES OF DIFFERENT METHODS ON NOISED IMAGES

The MKFCM-K-based image-segmentation algorithm ap-plied here is the first variant of MKFCM-K introduced inSection III. The kernel functions used in AKFCM, DKFCM,and the first variant of MKFCM-K are all Gaussian kernelsas k(x,y) = exp(−‖x − y‖2/r2), and the parameter r is ofgreat importance to the performance of these kernel methods.As suggested in [31], we take r = 150. In AKFCM and the firstvariant of MKFCM-K, there is a parameter α that balances theimportance of different kernels, more specifically, balances theimportance of pixel intensities and the local spatial informa-tion. Fig. 1 shows the segmentation performance of differentmethods based on different α values. The testing image is the10% Gaussian-noised synthetic image. From Fig. 1, it is easyto draw the conclusion that the larger α is, the better it is for aheavy noised image. In other words, for a heavy noised image,the local spatial information is of greater importance. In thisexample, we select α = 3.8 as suggested in [31].

Fig. 2 shows the 5% Gaussian-noised synthetic image and thesegmentation results of AKFCM_meanf, DKFCM_meanf, andMKFCM-K_meanf. Here, the local spatial information used indifferent algorithms is the mean filtered intensity of the 3 × 3window around the considered pixel because the mean filter isa good tool to smooth the Gaussian noise.

Fig. 3 shows the 10% salt-and-pepper-noised image and thesegmentation results of AKFCM_medianf, DKFCM_medianf,and MKFCM-K_medianf. For the salt-and-pepper-noised im-age, the selected local spatial information is the median filteredvalue of the 3 × 3 window around the considered pixel. Because

the median filter has better results than the mean filter for salt-and-pepper noise reduction, it is used in this salt-and-pepper-noised image.

The segmentation accuracies (SAs) of the three methods fordifferent noised images are listed in Table I. From Figs. 2 and3 and Table I, it is clear that AKFCM- and MKFCM-K-basedmethods that combine the intensity information and the spatialinformation in the kernel spaces have much better performancethan the DKFCM method, in which the corresponding pieces ofinformation are combined in the data space.

B. Example 2: Synthetic Two-Texture Image

To demonstrate the flexibility and the advantages ofMKFCM, a two-texture image is tested in this simulation. Theimage is as shown in Fig. 4(a), in which the left half of theimage and the right half are of great difference because the lefthalf is coarse and the right half is smooth, i.e., their texturesare visibly different. Traditional enhanced KFCM-based algo-rithms like DKFCM and AKFCM cannot deal with this kindof image very well [as shown in Fig. 4(b)–(e)] because theyonly consider the local spatial information. While consideringthe problem in the MKFCM framework, we can simply applythe combined kernel like the one in (27) (the third variant ofMKFCM-K in Section III), where k1 is the Gaussian kernelfor the intensities but k2 is the Gaussian kernel for the textureinformation. Fig. 4(f) shows the segmentation result of thethird variant of MKFCM-K. Due to the consideration of the

8 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS

Fig. 4. Two-textured image segmented by different methods. SA is theabbreviation of segmentation accuracy. (a) Two-textured image. (b) Seg-mentation result of AKFCM_meanf (SA = 0.716). (c) Segmentation resultof AKFCM_medianf (SA = 0.748). (d) Segmentation result of DKFCM_meanf (SA = 0.715). (e) Segmentation result of DKFCM_ medianf (SA =0.747). (f) Segmentation result of MKFCM-K third variant (SA = 0.753).(g) Segmentation result of LMKFCM (SA = 0.853). (h) MKFCM-K, kcom =k1k2k3 (SA = 0.723). (i) Segmentation result of MKFCM-K, kcom = k1 +k2 + k3 (SA = 0.730). (j) Segmentation result of KFCM, single intensitykernel (SA = 0.720). (k) Segmentation result of KFCM, single spatial kernel(SA = 0.709). (l) Segmentation result KFCM, texture kernel (SA = 0.763).

texture information, it has a better result than DKFCM andAKFCM. To further improve the performance of segmentation,we test LMKFCM that linearly combines three kernels as (28),i.e., the first two kernels are the kernels for intensities andthe local spatial information. The third kernel is the texturekernel. Fig. 4(g) shows that the LMKFCM achieves a much

better segmentation result. The corresponding SAs of differentmethods are shown in Fig. 4. The values of SA clearly illustratethat the proposed variants of MKFCM-K and LMKFCM havebetter performances than those of AKFCM and DKFCM. Theresulting weights of the three kernels in LMKFCM are 0.3967,0.3032, and 0.2999, respectively.

To show the advantage of automatically updating the weightsof different kernels in LMKFCM, the results of segmentationby applying a single kernel, or applying a fixed compositekernel kcom = k1 + k2 + k3 or kcom = k1k2k3, are shown inFig. 4(h)–(l). The definitions for k1, k2, and k3 are the same asthe ones in LMKFCM. The SAs of different algorithms demon-strate that the automatic updating of the weights of differentkernels in LMKFCM achieves better segmentation results thanMKFCM using a fixed composite kernel or a single kernel.

C. Example 3: Medical Images

The first medical image in this simulation is the magnetic res-onance (MR) image. The image and its reference segmentationsare obtained from [38]. They are T1-weighted MR phantomwith slice thickness of 1 mm, 3% noise, and no intensityinhomogeneity. The image will be segmented into three clusterscorresponding to White Matters (WMs), Gray Matters (GMs),and Cerebrospinal Fluid (CSF). The SA of algorithm i on classj is calculated as

Sij =Aij ∩ Arefj

Aij ∪ Arefj(30)

where Aij stands for the set of pixels belonging to class j thatare found by algorithm i and Arefj stands for the set of pixelsbelonging to class j that is in the reference segmented image.

After applying the AKFCM_meanf, DKFCM_meanf, andMKFCM-K_meanf (the first variant of MKFCM-K) methodson the MR image [Fig. 5(a)], the segmentation results are shownin Fig. 5(b)–(d). We use the LMKFCM algorithm with threekernels on the MR image as well. The three kernels are the sameas the ones in the two-texture image example. Fig. 5(e) showsthe segmentation result of LMKFCM. The resulting weightsfor the three kernels are 0.1213, 0.5505, and 0.3281. The localspatial information used in this example is the mean of theintensities because there is no salt-and-pepper noise in thisstudied image.

Due to the flexibility of kernel types in MKFCM-K, we caneasily change the second kernel in the first variant of MKFCM-K from the Gaussian kernel for the local spatial information intoa polynomial one as (16) (the second variant of MKFCM-K).For simplicity, we name it as MKFCM-K_poly. The MKFCM-K_poly’s segmentation results are shown in Fig. 5(f). In Fig. 5,the first column is the image as a whole, the second column isthe segmented CSF, the third column is the segmented GM, andthe last column is the segmented WM.

Table II lists the SAs of different methods, in which S1means the SA for the cluster of CSF, S2 is for the clusterof GM, and S3 is for the cluster of WM. In this simulation,because the noise rate of the image is low, the balance rateα is selected as a relatively small number 0.8, which is alsothe suggestion of [31]. From Fig. 5, it is hard to determine

CHEN et al.: MULTIPLE-KERNEL FUZZY C-MEANS ALGORITHM FOR IMAGE SEGMENTATION 9

Fig. 5. Segmentation results of different methods on an MR image. (a) MR image and its correct segmentation. From left to right are the integrated MRimage, the CSF, the GM, and the WM. (b) Segmentation results of AKFCM_meanf. (c) Segmentation results of DKFMC_meanf. (d) Segmentation results ofMKFCM-K_meanf (first variant). (e) Segmentation results of MKFCM-K_poly. (f) LMKFCM.

TABLE IISEGMENTATION ACCURACIES OF DIFFERENT METHODS ON THE MR IMAGE. S1: SEGMENTATION ACCURACY FOR THE CLUSTER OF CSF, S2:

SEGMENTATION ACCURACY FOR THE CLUSTER OF GM, AND S3: SEGMENTATION ACCURACY FOR THE CLUSTER OF WM

10 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS

which method is better. However, from Table II, the newmultiple-kernel methods proposed in this paper, MKFMC-K_poly and LMKFCM, have better segmentation performancesthan those of other methods. MKFCM-K_poly’s good re-sults highlight the potential advantages of applying differ-ent shaped kernel functions for different pieces of imageinformation.

The second medical image shown in Fig. 6(a) is a 128 ×128 positron emission tomography (PET) image of a dog’s lung[39]. We do not have the reference segmentations of this image.The segmentation results obtained by the AKFCM_meanf,DKFCM_meanf, MKFCM-K_meanf (first variant), MKFCM-K_poly, and LMKFCM are shown in Fig. 6(b)–(f). Theseapplied algorithms and their settings are the same as the onesdefined for the previous MR image. As shown in Fig. 6,the proposed MKFCM_poly algorithm derives more homoge-nous regions, which clearly outperforms the competitors.AKFCM_meanf is worse than MKFCM_poly (the misclassi-fication in the circle is noticeable), but it is better than otheralgorithms. Unlike the results of the MR image in Fig. 5,the LMKFCM here does not demonstrate a better result. Thisis because the texture information in the PET image is notsignificant.

D. Discussions and Analysis

The simulations in this section do not intend to prove thatthe MKFCM-based (including LMKFCM) image-segmentationalgorithms are inherently better than other KFCM-based image-segmentation methods. They are used to demonstrate theMKFCM’s significant flexibility in kernel selections and com-binations and the great potential of this flexibility could bringto image-segmentation problems. Under the framework ofMKFCM, changing the Gaussian kernel for local spatial infor-mation in the first variant of MKFCM-K to a polynomial kernelis straightforward, and the corresponded learning algorithmis not changed. By doing so, the segmentation results areimproved, as studied in Example 3. Owing to the frameworkof MKFCM, we can easily fuse the texture information intosegmentation algorithms by just adding a kernel designed forthe texture information in the composite kernel. As in the MRimage-segmentation and two-texture image-segmentation prob-lems, simply adding a Gaussian kernel function of the texturedescriptor in the composite kernel of MKFCM or LMKFCMleads to better segmentation results.

To sum up, the merit of MKFCM-based image-segmentationalgorithms is the flexibility in selections and combinations ofthe kernel functions in different shapes and for different piecesof information. After combining the different kernels in thekernel space (building the composite kernel), there is no need tochange the computation procedures of MFKCM or LMFKCM.This is another advantage to reflect and fuse the image informa-tion from multiple heterogeneous or homogeneous sources.

V. CONCLUSION

In this paper, an MKFCM methodology has been proposedand applied as the general framework for image-segmentation

Fig. 6. Segmentation result of different methods on a PET of dog’s lungimage. (a) PET of dog’s lung. (b) Segmentation result of AKFCM_meanf.(c) Segmentation results of DKFCM_meanf. (d) Segmentation results ofMKFCM_meanf (first variant). (e) Segmentation results of MKFCM_poly.(f) Segmentation results of LMKFCM.

problems, where the kernel function is composited by mul-tiple kernels. These kernels are selected for different piecesof information or properties of image pixels. Aside from theapplications of fixed composite kernels, a new method thatuses a linear combination of multiple kernels is proposed, andthe updating rules of the linear coefficients of the compositekernel are derived. Two traditional spatial KFCM-based image-segmentation algorithms proved the special cases of MKFCM-based image-segmentation methods. Moreover, several newimage-segmentation approaches, derived under the frameworkof MKFCM, are also proposed in this paper.

Considering the image-segmentation problems under theMKFCM framework, the proposed algorithms provide a sig-nificant flexibility in selecting and combining different kernelfunctions. More importantly, a new information fusion methodis obtained, where the information of the image from multipleheterogeneous or homogeneous data sources is combined inthe kernel space. Simulations on several synthetic and medicalimages show the flexibility and the advantages of MKFCM inimage-segmentation problems.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers fortheir valuable comments.

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Long Chen received the B.S. degree in informationsciences from Peking University, Beijing, China, in2000, the M.S.E. degree from the Institute of Au-tomation, Chinese Academy of Sciences, Beijing, in2003, the M.S. degree in computer engineering fromthe University of Alberta, Edmonton, AB, Canada, in2005, and the Ph.D. degree in electrical engineeringfrom The University of Texas, San Antonio, in 2010.

He is currently a Postdoctoral Fellow at TheUniversity of Texas, San Antonio. His current re-search interests include computational intelligence,

Bayesian methods, and other machine learning techniques and their applica-tions.

Dr. Chen has been working in the publications area matters for severalIEEE conferences. He was the Publications Cochair of the IEEE InternationalConference on Systems, Man, and Cybernetics in 2009.

12 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS

C. L. Philip Chen (F’07) received the M.S. degreefrom the University of Michigan, Ann Arbor, in1985, and the Ph.D. degree from Purdue University,West Lafayette, IN, in 1988.

He is currently the Dean and Chair Professor of theFaculty of Science and Technology, The Universityof Macau, Macau, China. He was a Professor and theChair of the Department of Electrical and ComputerEngineering and the Associate Dean for Researchand Graduate Studies of the College of Engineering,The University of Texas, San Antonio. He was a

Visiting Research Scientist at the Materials Directorate, U.S. Air Force WrightLaboratory, OH. He was also a Senior Research Fellow sponsored by the U.S.National Research Council and a Research Faculty Fellow at the NationalAeronautics and Space Administration (NASA) Glenn Research Center forseveral years. Over the last 20 years, his research projects have been sup-ported, continuously and consistently, by the U.S. National Science Foundation,NASA, U.S. Air Force Office Scientific Research, U.S. Air Force, and Officeof Naval Research. His current research interests include theoretic developmentin computational intelligence, intelligent systems, robotics and manufacturingautomation, networking, diagnosis and prognosis, and life prediction and life-extending control.

Dr. Chen has been involved in IEEE professional service for 20 years. Heis the President-Elect and Vice President on Conferences and Meetings of theIEEE Systems, Man, and Cybernetics Society (SMCS), where he has been theVice President of the IEEE SMCS Tech Activities on Systems Science andEngineering, a founding Cochair of three IEEE SMCS technical committees,a founding Cochair of two SMCS chapters, an Associate Editor of the IEEETRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS (SMC)—PART

C: APPLICATIONS AND REVIEWS and the IEEE SYSTEMS JOURNAL, theGeneral Chair of the IEEE SMC 2009, and the General Cochair of the IEEE2007 Secure System Integration and Reliability. He is a member of the TauBeta Pi and Eta Kappa Nu honorary societies. He has been the Founding FacultyAdvisor of an IEEE Computer Society Student Chapter and the Faculty Advisorof the Tau Beta Pi Engineering Honor Society at UTSA.

Mingzhu Lu (S’06) received the M.S. degree incomputer science from Hebei University, Baoding,China, in 2007. She is currently working towardthe Ph.D. degree in the Department of Electricaland Computer Engineering, The University of Texas,San Antonio.

She has been a Reviewer for the InternationalJournal of Machine Learning and Cybernetics, etc.Her current research interests include machine learn-ing, pattern recognition, data mining, Bayesian meth-ods, intelligent systems, and their applications.

Ms. Lu is a member of the Tau Beta Pi and Eta Kappa Nu Engineeringhonor societies. She serves as the Corresponding Secretary of the Tau BetaPi National Engineering Honor Society—currently Texas Mu Chapter. Shehas been a Reviewer for the IEEE TRANSACTIONS ON SYSTEMS, MAN,AND CYBERNETICS—PART B: CYBERNETICS. She was awarded the NSFtravel grant for Women in Machine Learning 2010 (Vancouver, BC, Canada)and CRA-W Graduate Cohort Workshop 2010 (Bellevue WA, USA) and 2011(Boston MA, USA). Moreover, she has served as a volunteer for several IEEEconferences, the Grace Hopper Celebration, and many campus and communityactivities.