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- 1. 522Image Segmentation Using Kernel Fuzzy C-MeansClustering on Level Set Method on Noisy ImagesG.Raghotham ReddylAssoc. Professor, Dept. ofECE, KITS, Warangal-IS,grr_ece@yahoo.comK.Ramudu 2Asst.Professor, Dept. ofECE, KITS, Warangal-lS,ramudukama@gmail.comAbstract- In this paper, kernel fuzzy c-means (KFCM) was used togenerate an initial contour curve which overcomes leaking at theboundary during the curve propagation. Firstly, KFCM algorithmcomputes the fuzzy membership values for each pixel. On the basisof KFCM the edge indicator function was redefined. Using theedge indicator function the segmentation of medical images whichare added with salt and pepper noise was performed to extract theregions of interest for further processing. The results of the aboveprocess of segmentation showed a considerable improvement in theevolution of the level set function.Keywords: Image segmentation; KFCM; level set method; images.I. INTRODUCTIONImage segmentation is plays an important role in the field ofimage understanding, image analysis, pattern identification. Theforemost essential goal of the segmentation process is topartition an image into regions that are homogeneous (uniform)with respect to one or more self characteristics and features.Clustering has long been a popular approach to untested patternrecognition. The fuzzy c-means (FCM)[I] algorithm, as a typicalclustering algorithm, has been utilized in a wide range ofengineering and scientific disciplines such as medical imaging,bioinformatics, pattern recognition, and data mining. Given adata X = {Xi•••••••••• Xn }C R P , the original FCM algorithmpartitions X into c fuzzy subsets by minimizing the followingobjective function( ) � � 2(1)J m U, V = £... .£... u i; Ilxi-V iIII-I k-IWhere c is the number of cluster and selected as a specifiedvalue in the paper, n the number of data points, Uk the memberof xk in class i, satisfyingL:-I Uik m the quantitycontrolling clustering fuzziness and v is set of control clustercenters or a prototypes (Vi E RP) . The function Jm isminimized by the famous alternate iterative algorithm. Since theoriginal FCM uses the squared-norm to measure inner productwith an appropriate kernel function, one similarity betweenprototypes and data points, it can only be effective in clusteringSyed Zaheeruddin3Lecturer, Department ofECE, KITS, Warangal-lS,sdzaheeruddin@gmail.comDr.R.Rameshwar Rao 4Professor, Department of ECE,OUCE, OU, Hyderabad, A.Prameshwarrao@hotmail.comspherical clusters and many algorithms are resulting from theFCM in order to cluster more general dataset. Most of thosealgorithms are realized by replacing the squared-norm in Eq (1)the object function of FCM with other similarity trial (metric) [1-2]. In this paper, a kernel-based fuzzy c-means algorithm(KFCM) is projected. KFCM adopt a new kernel-induced metricin the data space to restore the original Euclidean norm metric inFCM. By replacing the inner product with an appropriate kernelfunction, one can absolutely perform a nonlinear mapping to ahigh dimensional feature space without increasing the number ofparameters.The level set method is [4-7] based on geometric deformablemodel, which translate the problem of evolution 2-D (3-D) closecurve(surface) into the evolution of level set function in thespace with higher dimension to obtain the advantage inmanaging the topology changing of the shape. The level setmethod has had great success in computer graphics and vision.Also, it has been widely used in medical imaging forsegmentation and shape recovery [8-9]. However, there aresome insufficiencies in traditional level set method.Firstly, as using the local marginal information of the image,it is difficult to obtain a perfect result when theres a fuzzy ordiscrete boundary in the region, and the leaking problem hadinescapably appeared; Secondly, solving the partial differentialequation of the level set function requires numerical processingat each point of the image domain which is a time consumingprocess; Finally, if the initial evolution contour is given at will,the iteration time would increase greatly, too large or too smallcontour will cause the convergence of evolution curve to thecontour of object incorrectly. Therefore, some modification hasbeen proposed to improve the speed function of curve evolution[10-12]. In the paper, based on the new variational level setmethod, the edge indicator function was weighted to improvethe ability of detecting fuzzy boundaries of the object. At thesame time, the KFCM algorithm [13-14] was applied to obtainthe appropriate initial contour of evolution curve, so as to get theaccurate contour of object and reduce the evolution time.978-1-4244-9799-7/11/$26.00 ©2011 IEEE
- 2. II. KERNEL FUZZY C-MEANS CLUSTERING (KFCM):Define a nonlinear map as f/J:x � f/J(x)e F ,wherex e X.X denotes the data space and F is the transformedfeature space with higher even infinite dimensions. KFCMminimized the following objective function:J m (U, V)=t. .f u/;11�(x/)- �(v/ f (2.2)i-I k-IWhere11¢(x/)_¢(v/�12 = K(Xk,Xk)+ K(vlv/)- 2K(xk,v/) (2.3)Where K (x,Y)=�(x y �(y) is an inner product of thekernel function. If we adopt the Gaussian function as a kernelfunction, K(x,y)= exp(- llx -Yl12 / 2a2 } thenK(x,x)=1. according to Eq. (2.3), Eq. (2.2) can be rewrittenasJm(u,v)=2f·i:ui;(1-k(xk,Vi)� (2.4)i-l k-IMinimizing Eq. (2.4) under the constraint of, uik,m > 1. Wehavej=1nLUikK(xk,Vi )XkV. =..:.;k...;=I=---_____I nLU�K(XkVi)k=1(2.5)(2.6)Here we now utilize the Gaussian kernel function for straightforwardness. If we use additional kernel functions, there will becorresponding modifications in Eq. (2.5) and (2.6). In fact,Eq.(2.3) can be analyzed as kernel-induced new metric in thedata space, which is defined as the followingd(x,y)�IIf/J(x)- f/J(y�I=�2(1-K(x,y)) (2.7)And it can be proven thatd(x,y)is defmed in Eq. (2.7) is ametric in the original space in case that K(x,y)takes as theGaussian kernel function. According to Eq. (6), the data pointxk is capable with an additional weight K(Xk,Vi), whichmeasures the similarity between xk and Vi and when xk is anoutlier i.e., xk is far from the other data points,then K(xk,Vi )will be very small, so the weighted sum of datapoints shall be more strong. The full explanation of KFCMalgorithm is as follows:KFCMAlgorithm:Step 1: Select initial class prototype{Vi};=1 .Step 2: Update all memberships Uikwith Eq. 2.5.Step 3: Obtain the prototype of clusters in the forms of weightedaverage with Eq. (2.6).Step 4: Repeat step 2-3 till termination. The termination criterionis II�ew -VOid II:::; &.Where 11.11 is the Euclidean norm. V is the vector of clustercenters & is a small number that can be set by user (here&=0.01).III. THE MODIFICATION TO THE LEVEL SET METHODThe level set method was invented by Osher and Sethian [4]to hold the topology changes of curves. A simple representationis that when a surface intersects with the zero plane to give thecurve when this surface changes, and the curve changesaccording with the surface changes. The heart of the level setmethod is the implicit representation of the interface. To get anequation describing varying of the curve or the front with time,we started with the zero level set function at the front as follows:f/J(x,y,t)=O, if (x,y)e 1 (3.1)Then computed its derivative which is also equal to zeroaf/J +af/J.ax+af/J.ay=0(3.2)at ax at ay atConverting the terms to the dot product form of the gradientvector and the x and y derivatives vector, we goaf/J +(af/J.ax).(af/J.ay)=0(3.3)at ax at ay atMultiplying and dividing byV� and taking the other part to beF the equation was gotten as follows:af/J +FIVf/JI=oat(3.4)According to literature [9]11], an energy function was defmed:E(f/J)=f.1Eint(f/J)+Eex,(f/J) (3.5)Where Eext(f/J) was called the external energy, andEint(f/J) wascalled the internal energy. These energy functions wererepresented as:Eint(f/J)= J-!(Vf/J-1Ydxdy (3.6)02Eext(f/J)=ALg(f/J)+vAg(f/J) (3.7)Lg = Jg8(f/J�V f/J�dy (3.8)o523
- 3. (3.9)g=1 + IVG<7 * II(3.10)Where Lg (¢}was the length of zero level curve of ¢; and Agcould be viewed as the weighted area; I was the image and g wasthe edge indicator function. In conventional(traditional) level setmethods, it is numerically necessary to keep the evolving levelset function close to a signed distance function[15][16]. Reinitialization, a technique for occasionally re-initializing thelevel set function to a signed distance function during theevolution, has been extensively used as a numerical remedy formaintaining stable curve evolution and ensuring desirableresults.From the practical viewpoints, the re-initialization processcan be quite convoluted, expensive, and has subtle side effects[17]. In order to overcome the problem, Li et al [9] proposed anew variational level set formulation, which could be easilyimplemented by simple fmite difference scheme, without theneed of re-initialization. The details of the algorithm are in theliterature [9]. However, because only the gradient informationwas imposed in the edge indicator function, Us method has alittle effect on the presence of fuzzy boundaries.In the paper, a innovative method was proposed to modifythe algorithm. The original image was partitioned into some subimages by KFCM. The fuzzy boundary of each sub image wasweighted by a , the edge indicator function was redefmed:g =g+a·g2Where _1g2 -1+ IVG<7 * II I1 Was the image after clustering. The iterative equationof level set functional was:(3.11)(r,-") = p[M -d{I��)1+ M(,)d{gI��)+ vgo(.)(3.12)Taking g =g+a·g2int03.12Il[v" - diV[I::lllM(")iiV[g I::Il¢n+1 =¢n+ i[[ l]+ vgJ(¢)+ a AJ(¢)div g21�:1+vg2(¢)(3.13)524Where a E [0,1] . When processing images of weakboundary or low contrasts, a bigger a was taken; otherwise, asmaller a was taken.IV. THE GENERATION OF INITIAL CONTOUR CURVEOn the basis of KFCM clustering [3] in image segmentation,the over segmentation usually exists. In this paper, the result ofKFCM was used as initial contour curve, and the automatedinitialization of twist model was finished.For all the pixels in each cluster i.e. white matter, if 4neighborhoods included the heterogeneous pixel, the pixel wasregarded as candidate boundary point. Some pixels, such asnoise points, might be included in the candidate boundarypoints. So the algorithm of curve tracing [18] was proposed. Theexterior boundary of the cluster was tracked in the candidateboundary points. Finally, the closed curve was obtained. Thecandidate boundary points, whose Euclidean distances to theorigin coordinates were shortest, were chosen as initiation pointsof curve tracing. The steps of image segmentation with adaptedlevel set method were as follows:Stepl. Set the number of clusters, then the original image wasprocessed with KFCM, and calculate theg2•Step2. Choose one cluster, evaluate the inside area with - p andthe outside area with + p , p is a plus constant. The boundary ofthe area is set to O. The region of interest is defmed initialcontour.Step3. Minimize the overall energy functional with 3.13 formula.V. EXPERIMENTAL RESULTS OF PERFECT LOCKED TEST IMAGEOriginal Image Image with salt & pepper noise Final contourwith KFCMThis test images shows the perfect locked region of interestof particular images as shown in figures above for KFCM andfinal contour with proposal.Final contour with Proposed, 1000 iterations20 40 60 80 100 120 140 160 18020406080100120140160180Final contour with Proposed, 1000 iterations20 40 60 80 100 120 140 160 18020406080100120140160180
- 4. VI. EXPERIMENTAL RESULTSThe segmentation of image takes an important branch in thesurgery navigation and tumor radiotherapy. However, due tomedical imaging characteristics, the low contrast and fuzzyboundary is usually occurred in the images. In the experiment,the samples of images are taken from internet as shown inFigure i. The approximate contour of white matter was got byKFCM algorithm shown in Figure ii. The snooping of regionselse appear as a result of the in excess of segmentation. Theinitial evolution curve was obtained by the automatedinitialization. Because of the improved edge indicator function,the curve regularly evolved to the object boundaries in theprocess of evolution. The result established that the improvedalgorithm can extract the contour of object enhanced.Input image KFCM segmented with final contour withsalt & pepper noise proposed noise" -- . - ..,�. f[[Y" I ..,-J __ ,�/Figure i Figure iiFigure i are the original test images ,,.:. :.:. . ,., ."; .�. I ". .: . � ) ..� P, �. , .Figure iiiFigure ii are the results of KFCM clustering , to extractingthe white matter.Figure iii are the results of final contour with proposed salt &pepper noise method.With the enhanced method, the curve was successfullyevolved to the hollow white matter boundaries, but only to theapproximately white matter boundaries with Lis method. At thesame time, because the curve has been converged to the narrowregion the object boundaries extraction could not beimplemented with Lis method. But the enhanced method solvedthis problem better. On the similar computing proposal, under a3.0GHz Pentium iv PC with 1 GB RAM on board, the averageprocessing time of improved method was 9.6s, and that was30.3s with Lis method. The evolution time was greatly reducedVII. DISCUSSIONSThe need of the re-initialization is completely eliminated bythe proposal of Chunming Li, for pure partial differentialequation driven level set methods, the variational level setmethods. It can be easily implemented by using simple fmitedifference method and is computationally more efficient than thetraditional level set methods. But, in this algorithm, the edgeindicator has little effect on the low contrast image. So it is hardto obtain a perfect result when the region has a fuzzy or discreteboundary. Meanwhile, the initial contour of evolution needs tobe determined by manual, and it has the shortcomings of timeconsuming and user intervention.In this paper, we projected a new method to transform thealgorithm. The original image was partitioned with KFCM, andthe controlled action of the edge indicator function wasincreased. The result of KFCM segmentation was used to obtainthe initial contour of level set method. With the new edgeindicator function, results of image segmentation showed thatthe improved algorithm can exactly extract the correspondingregion of interest. Under the same computing proposal, theaverage time cost was lower. Alternatively the KFCM clusteringis sensitive to noise; some redundant boundaries were appearedin the candidates. Consecutively to solve this problem, thealgorithm of curve tracing was proposed.VIII. CONCLUSIONSIn this paper, we proposed a kernel-induced new metric toreplace the Euclidean norm in fuzzy c-means algorithm in theoriginal space and then derived the alternative kernel-basedfuzzy c-means algorithm. The results of this paper confirmedthat the mixture of KFCM with the level set methods could beused for the segmentation of low contrast images and medicalimages. The method has the advantages of no reinitialization,automation, and reducing the number of iterations. The validityof new algorithm was verified in the process of exacting detailsof images. In the future research, noise was added in imagesprior information on the object boundary extraction with levelset method, such as boundary, shape, and size, would be furtheranalyzed. At the same time, the performance of imagesegmentation algorithms would be improved by modernizationof classic velocity of level set method.525
- 5. IX. REFERENCES[1] J.C.Bezdek ,Pattern Recognition with Fuzzy Objective FunctionAlgorthims,Plenum Press ,New York,1981.[2] K.L.Wu,M.S.Yang,Alternative c-means clustering algorthims,PatternRecognition voI.35,pp.2267-2278,2002.[3] L.Zhang,W.D.zhou,L.C.Jiao.Kernel clustering algorthim, Chinese J.Computers,voI25(6),pp.587-590,2002(in chinese).[4] Osher S, Sethian J.A,Fronts propagating with curvature dependent speed:algorthims based on the Hamilton-Jacobi formulation.Journal ofComputational Physics,1988,pp.12-49.[5] Malladi,R,Sethain,J., and Vemuri,B., Shape modelling with frontpropagation: A level set approach.IEEE Trans.Pattern Anal.Mach.Intell,1995, pp.158-175.[6] Staib,L., Zeng,X., Schultz,R., and Duncan,J., Shape constraints indeformable models.Handbook of MedicalImaging,Bankman,l.,ed.,2000,pp.147-157[7] Leventon,M., Faugeraus,O., Grimson,W., and Wells,W.,Level set basedsegmentation with intensity and curvature priors.Workshop onMathematical Methods in Biomedical Image AnalysisProceedings,2000,pp.4-11.[8] Paragios , Deriche R,Geodesic active contours and level sets for thedetection and tracking of moving objects. IEEE Transaction on patternAnalsis and Machine In telligence,2000,pp.266-280.[9] Vese L A,Chan T F, A multiphase level set frame wor for imagesegmentation using the mumford and shah model. International Journal ofComputer Vision,2002,pp.271-293.[10] Shi Yanggang,Karl W C, Real-time tracking usin g level set, IEEEComputer Society Conference on Computer Vision and PatternRecognition,2005,pp,34-42[11] Li Chunming, xu Chengyang,Gui Changfeng,et al, Level set evolutionwithout re-initialization:a new varitional formulation.IEEE ComputerSociety Conference on Computer Vision and patternRecognition,2005,pp.430-436.[12] Sethain, l, Level set Methods and Fast Marching Methods.CambridgeUniversity Press 1999.[13] Dunn, lC., A fuzzy relative of the ISODATA process and its use indetecting compact well-separated Clusters.J.Cybern., 1973,pp.32-57.[14] Bezedek,J., A convergence thheorem for the fuzzy ISODATA clusteringalgorthims. IEEE Trans.Pattern Anal.Mach.Intell., 1980,pp 78-82.[15] S.Osher and R.Fedkiw, level set methods and Dynamic implicitsurfaces,Sp[ringer,2002,pp.112-113.[16] D.Peng,B?Merrimam,S.Osher,H.zhao, and M.Kang, A PDE- based fastlocal level set method, J.Comp.Phys,1996,pp.410-438.[17] J.Gomes and O.Faugeras,Reconciling distance functions and Level SetsJ.Visiual Communic. And Imag. Representation, 2000, pp.209-223.[18] Mcinerney T, Terzopouls D,Deformable models in medical image analysis:a survey.Medical Analysis,1996,pp.9l-108.[19] Dao-Qiang Zhang, Song-CanChen,Clustering in completed data usigKernel-based fuzzy c-means algorthim. Neural Processing Letters Volume18,Issue 3(december 2003) Pages: 155-162,Year of Publication:2003.526

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