Improved evidential fuzzy c-means method

JIANG Wen,YANG Tian,SHOU Yehang,TANG Yongchuan,and HU Weiwei

School of Electronics and Information,Northwestern Polytechnical University,Xi’an 710072,China

1.Introduction

Recently,image processing is more and more important.In the existing image processing technology,image fusion is an important part in the information fusion,which can greatly reduce the uncertainty and inaccuracy of information.At the same time,image segmentation is an essential task in image processing and computer vision applications[1–4].The purpose of image segmentation is to divide the given images into different regions for subsequent processing based on the claim.Detecting generic object categories is a fundamental issue in computer vision[5].It has to be done despite many hurdles like noise,nonuniform illumination,and uneven contrast in other homogenous re-gion[6].For example,in the field of medicine,the complexity of breast cell histopathology(BCH)images makes reliable segmentation and classification hard[7].With the development of medical imaging technology,magnetic resonance imaging(MRI)has become more and more popular[8].Medical images are indispensable for disease diagnosis.For example,disease identification is conducted by using the YUV quantitative analysis of auto fluorescence bronchoscopy(AFB)images in the target areas[9].Noise,partial volume effect(PVE)and intensity non-uniformity(INU)[10]are three main norms in MRI.The main source of noise is divided into biological and scanning noise.Organizational unevenness and limitations in hardware design are the primary causes of biological and scanning noise.

In practical applications,it is difficult to get the precise information[11].How to deal with the uncertain information conflict information effectively is still an open issue.Information fusion techniques could be used to increase the decisions accuracy by decreasing the imprecision and uncertainty of the information through the use of redundancy.To process the uncertain information and improve the prise of the information,many mathematical methods are proposed such as fuzzy sets theory[12],Dempster-Shafer evidence theory(DS theory)[13–17],rough sets theory[18,19]and Z-numbers[20,21].DS theory[22–24]was introduced by Dempster and then developed by Shafer..DS theory can well describe and process uncertainty information by assigning probability to a set,therefore it is diffusely used in information fusion[25].

In recent years,many kinds of images are studied and used[26–28].In the medical image processing[29,30],to the best of our knowledge,MRI images are applied to accurate image segmentation since MRI is safer and less invasive compared to computed tomography(CT)scans[31].It plays an essential role in the assessment of various disease, brain development and treatment progress.Moreover,image segmentation[32–34]is widely applied to image processing[35].Some methods based on mixed intelligent algorithms are utilized for image segmentation[36–39].Some researchers have applied the DS theory to image segmentation.For example,Liu et al.[40]have applied the DS theory to one image to get a bet-ter image fusion result and some other researchers have combined the DS theory with other algorithms or modified fuzzy c-means(FCM)for image segmentation and image fusion[41–43].

However,we notice that most of the existing methods do not premeditate the conflict information in the multiple sources images and spatial information.Only a few methods are based on spatial information[44–46].However,in these methods,both the conflict information and the effect of the different spatial locations are not considered.In this paper,a method is proposed to obtain a better image segmentation result.Firstly,an average fusion image is obtained by average fusion to realize the processing of the conflict information between the two images,and it also can reduce the effect of the motion noise.Secondly,FCM is applied to the average fusion image to get two kinds of mass functions.One mass function is generated without any spatial information,which can contain rich boundary information.The other mass function is generated by considering the influence of spatial location information of neighborhood pixels.Finally,the final data fusion is achieved by the DS theory,which is used to complete the image segmentation.

The remainder of this paper is organized as follows.The basic concepts are briefly introduced in Section 2.In Section 3,our method is proposed and applied to the simulated images.Fusion results of MRI images segmentation are demonstrated in Section 4,which show the efficiency of the proposed method.Finally,this paper is concluded in Section 5.

2.Preliminaries

2.1 DS theory

In this subsection,the DS theory is briefly introduced.

definition 1If Θ is a frame of discernment(FOD),then a function m:2Θ→ [0,1]is a basic probability assignment(BPA)[14]whenever

The quantity m(A)is the basic probability number of A,and it can be understood as the belief that is exactly given to A.If m(A)＞0,A is a focal element.

definition 2Suppose m1and m2are two BPAs on the same FOD Θ.Then the Dempster’s combination rule[14]is defined as

2.2 Fuzzy clustering by FCM

The FCM algorithm is an unsupervised fuzzy clustering algorithm.The clustering analysis is defined as the process of grouping objects which are similar in some respects[49].Under the condition of a general formulation,the data to be classified can be expressed as an M-dimensional vector X={x1,x2,...,xM}.Assuming that C＞2 is an integer designating the number of clusters,where X will be classified.RC×Mis the set of all real C×M matrices.A fuzzy C-partition of X is represented as,where μik= μi(xk)represents the degree of membership of the element xkin the cluster i.The following constraints can be verified[47,48]:

where U is used to depict the clusters of X,and a partition of X can be obtained by the min of the FCM objective function[50]:

where q∈[1,+∞]decides the fuzzy degree of classification results[51],and V =(v1,v2,...,vC)is the vector of the cluster centers.The approximate optimization of Jqcan be obtained by iteration with the following formulas.

In the FCM algorithm[47,48]:assume?xk-vi?2＞0,1≤i≤C,1≤k≤M.Jqcan be minimized by(U,V),only if

The FCM algorithm is composed by iterations alternating between(5)and(6).The algorithm is restrained to either a saddle point of Jqor a local minimum.

Step 1Give a value as the number of clusters C and the threshold value ε,where 2 ＜ C ＜ L,L is the number of gray levels.

Step 2Initialize μikas the following way:

Step 3Calculate the centroid viby(6).

Step 4Renew the membership degrees

Step 6Defuzzification.

In general,clusters are exact number,and q is often defined as 2.The clustering process is stopped when ε≤10-5.

2.3 Weighted average fusion of images[52]

The processing of conflict information[53]is very important for the fusion of multiple sources images.In this paper,the weighted average fusion of images is used to better decrease the conflict information among the multiple sources images,which can effectively realize the convergence.Taking the two sources images fusion process as an example,and the process of fusion is explained as follows.Multiple sources images smoothing can be done in the same manner. Assuming that sources images are named as A and B respectively,and the size of images is M×N.The image after fusion is named as F.Then,a weighted average fusion process can be represented on the basis of the pixel gray values of two sources images A and B,as follows[52]:

where m=1,2,...,M is the number of rows of pixels in the image,n=1,2,...,N is the number of columns of pixels in the image.ω1and ω2are the weighted coefficients,and ω1+ ω2=1.The weighted coefficients are used to adjust the importance of different images for the final fusion result.If ω1= ω2=0.5,the process is named as the average fusion of images.It shows that the two images have the same importance in the final fusion result.If one is more important than the other,we can give it a higher weight.

3.The proposed method

The general steps of the proposed method are shown in Fig.1.Our proposed method performs the image fusion twice.Firstly,the average fusion is used to get one fusion image,which can realize the decrease of the conflict information between the two images.The effect of the motion noise and uncertainty of the images can be reduced,too.Secondly,the neighborhood information and the different influence of spatial location of neighborhood pixels are taken into consideration to take full use of the spatial information.Finally,the fusion of the two images is completed by the DS theory to achieve the final image segmentation.Then how to generate the two mass functions is presented before the general steps of the proposed method is introduced.

Fig.1 Step chart of the proposed method

3.1 Determination of mass function without neighboring information

One mass function is determined without the neighborhood pixels information to protect boundaries information.The simple hypotheses and the double hypotheses are generated separately by the following process[44].

The generation of BPA is like this:if|μi(l)-μt(l)|＞ δ,then the simple hypotheses BPA is generated as

If|μi(l)- μt(l)|≤ δ[44],the generation of double hypotheses BPA is

Fig.2 Construction of double hypothesis by membership functions without spatial information

Fig.3 The 3×3 neighbors lattice of(x,y)

In the end,the BPA m is obtained by normalizing the power set 2Θso as to satisfy

3.2 Determination of mass function with neighboring information

The other mass function is generated with the neighboring information,which also makes the boundaries blurred.A part of the boundary information will be lost.The central pixels membership function is calculated by weighting those of its neighboring pixels with the Gaussian filtered membership function.

The simple hypotheses and the double hypotheses are generated by the following process.The BPA generation method is like(9)and(10).

3.3 General steps of the proposed method

The proposed method involves the following steps.

Step 1An average fusion image is obtained by average fusion of the two original images,according to(1)where ω1= ω2=0.5.The average fusion is simple and intuitive,which is suitable for real-time processing.It can improve the signal to noise ratio(SNR)of the image and eliminate the pole to some extent.

Step 2The FCM is applied to the average fusion image.

Step 3One mass function can be directly got with FCM,yet it does not include domain pixel points information.Its simple hypotheses and double hypotheses can be obtained according to Section 3.1 and Fig.2.The other mass function with the spatial neighborhood information is obtained by the Gaussian filtered membership functions related to the pixel(x,y).In Fig.3 and(13),it considers not only the effect of pixels,but also the influence of location of the neighborhood pixels.Every pixel is only associated with a few nearest neighbors[54].According to the practical analysis,the nearer the neighborhood pixel is to the center pixel,the bigger influence of the pixel should be,and the greater the weighting coefficient should be used.The simple hypotheses and the double hypotheses can be obtained in Section 3.2 and in Fig.4.

Fig.4 Construction of the two sub-sets hypothesis by triangular membership functions with spatial information

Step 4Data fusion by the DS theory.The Dempster’s combination rule is applied to the sensor data fusion result.

Step 5Make decision and segment image.

3.4 Test and analysis

To demonstrate the accuracy and the effectiveness of our proposed method,Gaussian noise,whose expectation is 0 and the variance is 0.015,is added to two images.The figure of the bright background is used to simulate a strong X-ray acquisition.As shown in Fig.5(a),the upper gray belt is close to the background gray,which is easy to be confused.Fig.5(b)is used to simulate a weak X-ray acquisition.Gray belts of two layers below are difficult to be distinguished in Fig.5(b).Fig.6(a)and Fig.6(b)are segmentation results of Fig.5(a)and Fig.5(b)respectively by FCM.In Fig.6(a)and Fig.6(b),the four regions are not presented evidently.Fig.6(c)and Fig.6(d)are the edge detection results of Fig.6(a)and Fig.6(b)respectively.In Fig.6(c)and Fig.6(d),the detected edges are not clear and accurate.The fusion results by using the method[44]and the proposed method are shown in Fig.7(a)and Fig.7(b)respectively.Fig.7(c)and Fig.7(d)are the edge detection results of Fig.7(a)and Fig.7(b)respectively.In this paper,the Canny edge detector operation[55]is chosen to identify the edges of the image. It has the following advantages.

(i)It can effectively restrain noise and pinpoint the location of the edge.

(ii)It can carry out the measure according to the SNR and product positioning to get the optimal approximation operator.

Fig.5 Simulation images for fusion

Fig.6 FCM segmentation results and edge detection results of the images

Fig.7 DS segmentation results and edge detection results

The purpose of the edge detection is to obtain the local maximum of the image gradient,which is calculated by using the derivative of the Gaussian filter.The Canny’s method uses two thresholds to distinguish strong and weak edges.For weak edges,they are only included in the output when they are attached to a strong one.Thus the Canny detection is not susceptible to noise,and it can detect the weak edges.From the comparison of Fig.7(a)and Fig.7(b),the four regions are clearly presented by the proposed method.From the comparison of Fig.7(c)and Fig.7(d),the edges detected by our method are more clear and accurate.Above all,the sensor data fusion results and the edge detection results with the proposed method are more perfect.

4.Numerical examples of the MRI images and results

In this section,two slices of a human brain are selected to do the experiment.One is the normal brain tissue,and the other one is the abnormal brain tissue with infarction lesion.They are from the same patient.Fig.8(a),Fig.8(b),Fig.8(c)and Fig.8(d)are corresponding to the slice of the normal brain tissue.Fig.8(a)is T2 weighted image,and Fig.8(b)is D weighted image.The conflict information[56–59]is shown obviously between Fig.8(a)and Fig.8(b).The segmentation result by the Zhu et al’s method[44]is shown in Fig.8(c).In Fig.8(c)and Fig.8(e),segmentation results cannot better help the doctor make accurate decision,because the Zhu et al’s method does not consider the process of the conflict information between the two images.In Fig.8(e),Liu et al’s method does not consider the multi-source image information.The sensor data fusion result with the proposed method is shown in Fig.8(d).The shape and size of brain tissues are well segmented in Fig.8(d).From the comparison,the better segmentation performance can be obtained by the proposed method.The fusion result of the normal brain tissue shows the effectiveness of the proposed method.Furthermore,the two methods are used in the other slice of the brain tissue with infarction lesion.Fig.9(a),Fig.9(b),Fig.9(c)and Fig.9(d)are corresponding to the slice of the abnormal brain tissue with infarction lesion.Fig.9(a)is T2 weighted image,and Fig.9(b)is D weighted image.Similarly,there is conflict information between Fig.9(a)and Fig.9(b).In Fig.9(c),Fig.9(d)and Fig.9(e),the presence of four classes is shown by the segmentation result:they are gray matter,white matter,infarction lesion of the brain and cerebrospinal fluid,and background.InFig.9(d),the segmented regions have significant improvement in increasing boundary precision.The homogeneous region enables the brain tissue volumes measured correctly.The location of the lesion is more accurate.The fusion results of the abnormal brain tissue show the validity of the proposed method as well.In both cases,the proposed method,which better decreases the conflict information in multiple sources images to achieve convergence and efficiently uses the information of the spatial location of the neighborhood pixels,and has a significant improvement in the sensor data fusion results by the DS theory to realize better image segmentation results.It can help doctors to make decisions effectively.

Fig.8 Normal brain tissue-MRI images segmentation

Fig.9 Abnormal brain tissue-MRI images segmentation

5.Conclusions

In this paper, a new method, which effectively reduces the conflict information between the two images and efficiently uses the information of the spatial location of the neighborhood pixels, is proposed.One mass function withoutspatial information is generated by applying the FCM algorithm to the image of average fusion to contain rich boundaries information. The other mass function is generated by considering the spatial location information of neighborhood pixels. The final data fusion is completed by the DS theory to realize the image segmentation. Also, our method is robust in decreasing the effect of noise. The simulated images show notable validity in increasing boundary precision and region harmony. At the same time, the experimental results with the MRI images demonstrate that our proposed method is more reasonable and accuracy in handling vague edges and in better solving the conflict information in the multiple sources images to achieve convergence.

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