Geometrically Guided Fuzzy C-means Clustering for Multivariate Image Segmentation

J.C. Noordam W.H.A.M. van den Broek L.M.C. Buydens

Institute (ATO), dep. P&CS, Agrotechnological Research Agrotechnological Research Lab. for Anal. Chem, Institute (ATO), dep. P&CS,

P.O. Box 17,6700 AA P.O. Box 17,6700 AA Toernooiveld 1, Wageningen, the Netherlands Wageningen, the Netherlands 6525 ED Nijmegen J. C .Noordam@ ato. wag-ur.nl W.H. A.M.vandenBroek the Netherlands

University of Nijmegen,

@ato.wag-ur.nl L .Buydens@ sci .kun .nl

Abstract

Fuzzy C-means (FCM) clustering is an unsupervised clustering technique and is often used for the unsupervised segmentation of multivariate images. The segmentation of the image in meaningful regions with FCM is based on spec- tral information only. The geometrical relationship between neighbouring pi.yels is not used. In this paper, a semi- supervised FCM technique is used to add geometrical in- formation during clustering. The local neighbourhood of each pixel determines the condition of each pixel, which guides the clustering process. Segmentation experiments with the Geometrically Guided FCM (GG-FCM) show im- proved segmentation above traditional FCM such as more homogeneous regions and less spurious pixels.

1. Introduction

The use of Fuzzy C-Means clustering to segment a mul- tivariate image in meaningful regions has been reported in literature [4, 21. It is known as an unsupervised fuzzy clus- tering technique and uses the measurement data only in or- der to reveal the underlying structure of the data and seg- ment the image in regions with similar spectral properties. When FCM is applied as a segmentation technique in im- age processing, the relationship between pixels in the spatial domain is completely ignored. The partitioning of the mea- surement space depends on the spectral information only. As multivariate imaging offers possibilities to differentiate between both objects of similar spectra and different spatial correlations, FCM can never utilise this property. Adding spatial information during the spectral clustering has ad- vantages above a spectral segmentation procedure followed by a spatial filter, as the spatial filter cannot always correct

segmentation errors. Furthermore, when two overlapping clusters in the spectral domain correspond to two differ- ent objects in the spatial domain, usage of a-priori spatial information can improve the separation of these two over- lapping clusters [3]. In this paper, a modification of the unsupervised fuzzy clustering technique [SI, is utilised to guide the clustering process by adding a-priori geometri- cal information in order to improve the final segmentation results. During each iteration step of the FCM, a condi- tion for each pixel is updated. This condition is based on the membership values of neighbouring pixels in the spatial domain. Thus, the Geometrically Guided FCM (GG-FCM) swaps between the spectral domain and the spatial domain during the clustering process. The principle of geometrical guided FCM presented in this paper uses merely local spa- tial neighbourhood and is therefore considered as a first step to more sophisticated algorithms to search for a specific ge- ometric shape during clustering.

2. Fuzzy C-Means Clustering with partial su- pervision

In this paragraph, the standard Fuzzy C-means clustering and the modified version of FCM with partial supervision are considered.

2.1. Fuzzy C-Means Clustering

Given a set of n data patterns, X = XI, ..., x,, the FCM algorithm minimises the weighted within group sum of squared error objective function J ( U , V) [ 11:

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where xk is the k-th p -dimensional data vector, v i is the prototype of the centre of cluster i, u i k is the degree of membership of x k in the i-th cluster, m is a weighting ex- ponent on each fuzzy membership, d ( x k , v i ) is a distance measure between object xk and cluster centre v i , n is the number of objects and c is the number of clusters. A solu- tion of the objective function J (U, V) can be obtained via an iterative process where the degrees of membership U i k

and the cluster centres v i are updated via:

with the constraints : c n

(3) i=l k = l

2.2. Fuzzy C-Means Clustering with partial super- vision

Clustering is usually seen as an unsupervised routine where no information about the underlying structure of the patterns is known. In cases where clustering is used and some labelled patterns are available, it might be advanta- geous to use these labelled pattems to influence the cluster- ing process. In literature [5], this idea is used and resulted in a modified objective function:

n r

k = l i = l n c

k=l i=l

Here, (Y is a scaling factor to maintain a balance between the supervised and unsupervised data. It is suggested that this scaling factor is proportional to the ratio unlabeled/labelled data. The variable f i k in the second term represents the membership of the labelled pattern IC to cluster i. Variable bk is a Boolean variable to distinguish between labelled and unlabeled pattems. If the Boolean variable bk is zero, the objective function retums to the standard objective function for FCM. This is identical when f i k is set to zero for all clusters i. The update procedure for the partition matrix U is now changed into:

In this paper, the term condition is used to indicate the vari- able f i k instead of membership value, to avoid confusion with the membership values u i k ,

3. Geometrical Guided FCM (GG-FCM)

In literature [5], the values of bk and f i k are fixed and are manually set beforehand. In the method presented in this paper, the values of f i k are allowed to change and are up- dated during the clustering process. As stated in the previ- ous paragraph, the Boolean variable bk indicates if labelled patterns are available. If this is not the case, the Boolean is set to zero and the update equation ( 5 ) returns to the stan- dard update equation (2) for FCM. However, if the condi- tion f i k is set to zero for all clusters, the result is similar and equation 5 returns also the standard update equation (2) for FCM (in both situations, the scaling factor (Y is left over). In the algorithm presented in this paper, the Boolean variable bk is set to 1 for all objects and the condition fik: is used to guide the clustering process. This means that if the condi- tion f i k is set to zero for cluster i, the partition update equa- tion reduces to the update equation of standard FCM (equa- tion 2). This approach avoids the use of an extra condition to determine which objects should be considered as labelled and which objects not. Now, all objects are considered as labelled and only the condition f i k determines the enhance- ment or weakening of object IC for cluster i. High values of f i k will enhance the membership of the pixel for class i and low values will weaken the membership for class i.

3.1. Fuzzy neighbourhood

During the clustering procedure, the membership values of surrounding neighbouring pixels determine the value of condition f i k of each pixel. The value of the condition is a measure of similarity for a pixel compared to surround- ing neighbours. The value of the condition is low when the surrounding pixels have similar membership values and the condition is high when surrounding pixels have devi- ate membership values. As a result, membership values of spurious pixels in the spatial domain can be influenced in- directly when their neighbours have different membership values. The number of rows in the partition matrix U is equal to the number-ofrows times the number-of-columns of the original image. Each column in the partition matrix can be rearranged to an image, which is called a partition image. For each pixel in partition image i, the mean mem- bership deviation (Am) compared to the memberships of neighbouring pixels is determined:

where Amrc,i is the mean membership deviation for the pixel at position (T , e) of partition image i, i is the current cluster, W is a neighbourhood window with (odd) size s, ur1 c~ ,i is the degree of membership of the neighbour pixel

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at position (r' , L) in the window W of partition image i, 'zL,,,~ is the degree of membership of the centre pixel in the window of partition image i. It is clear that for a homo- geneous region the average membership deviation will be zero. To determine for which cluster the current pixel must be enhanced, the membership values covered by the neigh- bourhood window are added for each cluster i. The cluster with the highest sum is considered as the cluster the centre pixel belongs to. For this particular pixel, the condition fik is enhanced with the mean membership deviation Amrc,i. The condition f i k is weakened with the mean membership Amrc,i for all remaining clusters.

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measurement space with traditional FCM. The plot shows two well separated clusters. However, the segmented im- age in figure 1 (middle) shows that foreground objects are contaminated with background pixels and vice versa. Due to the added noise, background pixels have shifted to the foreground cluster and foreground pixels have shifted to the background cluster. As the traditional FCM uses no in- formation from the spatial domain, .this result is to be ex- pected. Figure l(right) shows the segmented image and

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4. Experiments

90 + Foregroundclass Background class

0 Cluster Cen1roid

To show the improvements of the geometrical guided FCM compared to traditional FCM, experiments have been carried out on synthetic and real world images. The syn- thetic image demonstrates the principle of the geometri- cally guided clustering and the real-world image demon- strates that the algorithm performs well on real images. Af- ter the clustering procedure, the fuzzy images are converted to crisp images by applying the maximum membership pro- cedure. This means that an object is classified to the cluster with highest membership.

4.1. Experiment on synthetic image

The first experiment is carried out on a synthetic image consisting of two squares of similar colour (R=150, G=50, B=50) on a background (R=125, G=75, B=SO). The amount of foreground pixels is equal to the amount of background pixels. The image is contaminated with Gaussian noise (p = 0, c = 10) to simulate cluster overlap. A 3x3 win-

Figure 1. RGB-image with Gaussian noise (left), FCM segmented image (middle) and GG-FCM segmented image (right)

dow contains the a-priori information of single distributed pixels. The segmentation results are shown in figures 1 and 2. The plot in figure 2 shows the partitioning of the

-*** * - * Backgroundclass + Foreground class 0 Cluslercemr~id

*

20 40 60 80 too 120 80

Green

Figure 2. FCM result: the Red versus Green plot. The cross hair represents the centre of a cluster

Red versus Green plot lor Synthefic image ,POY

Figure 3. GG-FCM result: the Red versus Green plot. The cross hair represents the centre of a cluster

figure 3 shows the plot of GG-FCM. The improvement of the segmentation due to the extra spatial information is ob- vious, the two foreground squares and the background are almost perfectly segmented. The plot in figure 3 shows

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that foreground pixels appear in the background cluster and vice versa. This cluster overlap is a result of the added a- priori spatial information which directs spurious pixels to the other class.

4.2. Experiment on real world image

A second experiment is carried out on a classical image in pattem recognition, referred to as the peppers image. To

Figure 6. Peppers image segmented by GG- FCM.

are merged with the class of the surrounding pixels. The segmented regions are more homogeneous and spurious lit- tle blobs are removed from the segmented image.

5. Conclusions and further research

Figure 4. Original peppers image.

illustrate the improvement, a sub-image is considered (Fig- ure 4). For the initialisation of the FCM routines, 5 classes are initiated. No noise is added to the image. A 3x3 win- dow is selected during the GG-FCM clustering. The im-

Figure 5. Peppers image segmented by FCM.

age shown in figure 5 is segmented by the standard FCM. It shows that some of segmented regions are enveloped by spurious edges. Most of these spurious envelopes are sev- eral pixels wide. The spurious edges are removed in the image segmented by the GG-FCM, shown in figure 6. The extra spatial information sees to it that spurious edge pixels

In this paper, a technique to guide the clustering process of the FCM based on geometrical information is presented. The use of Geometrically Guided FCM as an image seg- mentation process clearly shows improvements above clus- tering with traditional FCM. The addition of a-priori infor- mation from the spatial domain makes it possible to inter- vene in the clustering process and guide the clustering. The segmented images show more homogenous regions in com- parison with standard FCM, which uses no spatial informa- tion. Currently, a local neighbourhood window is used to determine the condition, but the addition of more sophisti- cated techniques to search for a specific shape in the image will be the topic for further research.

References

[ 11 J. Bezdek. Pattern recognition with fuzzy objective functions. Plenum Press, New York, 198 1.

[2] J. Bezdek, L. Hall, and L. Clarke. Review of mr image segmentation techniques using pattem recognition. Medical Physics, 20(4), 1993.

[3] J. C. Noordam and W. H. A. M. van den Broek. Multivariate image segmentation based on geometrically guided fuzzy c- means clustering. Submitted for publication, 2000.

[4] S. H. Park, I. D. Yun, and S. U. Lee. Color image segmenta- tion based on 3-d clustering: Morphological approach. Pat- tern Recognition, 21(8):1061-1076, 1998.

[SI W. Pedrycz and J. Waletzky. Fuzzy clustering with partial supervision. IEEE transactions 011 systems, man and cyber- netics, 27(5):787-795, 1997.

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