Fuzzy k c-means clustering algorithm for medical image

Journal of Information Engineering and Applications www.iiste.org
ISSN 2224-5782 (print) ISSN 2225-0506 (online)
Vol 2, No.6, 2012

Fuzzy k-c-means Clustering Algorithm for Medical Image
Segmentation
Ajala Funmilola A*, Oke O.A, Adedeji T.O, Alade O.M, Adewusi E.A

Department of Computer Science and Engineering, LAUTECH Ogbomoso, Oyo state, Nigeria.

*E-mail of the corresponding author: funfaith2003@yahoo.co.uk.
Abstract

Medical image segmentation is an initiative with tremendous usefulness. Biomedical and anatomical information are
made easy to obtain as a result of success achieved in automating image segmentation. More research and work on it
has enhanced more effectiveness as far as the subject is concerned. Several methods are employed for medical image
segmentation such as Clustering methods, Thresholding method, Classifier, Region Growing, Deformable Model,
Markov Random Model etc. This work has mainly focused attention on Clustering methods, specifically k-means
and fuzzy c-means clustering algorithms. These algorithms were combined together to come up with another method
called fuzzy k-c-means clustering algorithm, which has a better result in terms of time utilization. The algorithms
have been implemented and tested with Magnetic Resonance Image (MRI) images of Human brain. Results have
been analyzed and recorded. Some other methods were reviewed and advantages and disadvantages have been stated
as unique to each. Terms which have to do with image segmentation have been defined along side with other
clustering methods.

Keywords: Clustering algorithms, Fuzzy c-means, K-means, Segmentation.

1 Introduction

Diagnostic imaging is an invaluable tool in medicine today. Magnetic Resonance Imaging (MRI),
Computed Tomography, Digital Mammography, and other imaging modalities provide effective means for
non-invasively mapping the an atomy of a subject. These technologies have greatly increased knowledge of normal
and diseased anatomy for medical research and serves as a critical component in diagnosis and treatment planning.
(Dzung et. al).

Computer algorithms for the delineation of anatomical structures and other regions of interest are a key components
assisting and automating specific radiological tasks. These algorithms are otherwise known as image segmentation
algorithms. They are of great importance in biomedical imaging applications like tissue volume quantification,
diagnosis, localization pathology, study of anatomical structures, treatment planning, partial volume correction of
functional imaging data and computer integrated surgery.

2. Related work

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Numerous methods are available in medical image segmentation. These methods are chosen based on the specific
applications and imaging modalities. Imaging artifacts such as noise, partial volume effects, and motion can also
have significant consequences on the performance of segmentation algorithms. Some of these methods with their
idiosyncrasies were described below

Thresholding Method

Thresholding is the most basic of the medical image segmentation techniques. It is based on separating
pixels in different classes depending on their gray level. Thresholding approaches segment scalar images by creating
a binary partitioning of the image intensities. A thresholding procedure attempts to determine an intensity value,
called the threshold, which separates the desired classes. The segmentation is then achieved by grouping all pixels
with intensity greater than the threshold into one class, and all other pixels into another class. Determination of more
than one threshold value is a process called multi-thresholding. Its main limitations are that in its simplest form only
two classes are generated and it can not be applied to multi-channel images. In addition, thresholding typically does
not take into account the spatial characteristics of an image. This causes it to be sensitive to noise and intensity in
homogeneities, which can occur in magnetic resonance images.

Classifiers
Classifier methods are used in pattern recognition they seek to partition a feature space derived from the
image using data with known labels. A feature space is the range space of any function of the image, with the most
common feature space being the image intensities themselves.
Classifiers are known as supervised methods since they require training data that are manually segmented and then
used as references for automatically segmenting new data.

Markov Random Field Models
Markov random field (MRF) is not a method but a statistical model that can be used within segmentation
methods. MRFs are often incorporated into clustering segmentation algorithms such as the K -means algorithm under
a Bayesian prior model (Pham and et al, 1998). The segmentation is then obtained by maximizing “a posteriori”
probability of the segmentation given the image data using iterative methods such as iterated conditional modes or
simulated annealing. A difficulty associated with MRF models is proper selection of the parameters controlling the
strength of spatial interactions. Too high a setting can result in an excessively smooth segmentation and a loss of
important structural details.

Artificial Neural Networks
Artificial neural networks (ANNs) are massively parallel networks of processing elements or nodes that simulate
biological learning. Each node in an ANN is capable of performing elementary computations. Learning is achieved
through the adaptation of weights assigned to the connections between nodes.
Because of the many interconnections used in a neural network, spatial information can easily be incorporated into
its classification procedures. Although ANNs are inherently parallel, their processing is usually simulated on a
standard serial computer, thus reducing its potential computational advantage (Pham and et al, 1998).

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Atlas-Guided Approaches
Atlas-guided approach uses standard atlas or template is available. This it does by bringing together
information about the anatomy that requires segmenting. This atlas is then used as a reference frame for segmenting
new images. Conceptually, atlas-guided approaches are similar to classifiers except they are implemented in the
spatial domain of the image rather than in a feature space (Dzung L. Pham and et al, 1998).

Deformable Models
Deformable models are model-based techniques which are used for delineating region boundaries by the use
of closed parametric curves or surfaces. This curves or surfaces are deformed under the influence of internal or
external forces. Deformable Models are physically motivated techniques. Delineation of an object boundary in an
image is done by placing a closed curve or surface near the desired boundary then an iterative relaxation process is
allowed to be undergone. Internal forces are computed from within the curve or surface to keep it smooth throughout
the deformation. External forces are usually derived from the image to drive the curve or surface towards the desired
feature of interest (Pham and et al, 1998).

Clustering Analysis

Cluster analysis or clustering is the assignment of a set of observations into subsets (called clusters) so that
observations in the same cluster are similar in some sense (Wikipedia, 2009). Clustering is a method of unsupervised
learning, and a common technique for statistical data analysis used in many fields, including machine learning, data
mining, pattern recognition, image analysis, information retrieval, and bioinformatics. Clustering algorithms and the
classifier method are likely in function but clustering does not use training data instead they iterate between
segmenting the image and characterizing the properties of each class. Consequently they are otherwise termed
unsupervised methods. In a sense, clustering methods train themselves using the available data (Dzung L. Pham and
et al, 1998).
Three commonly used clustering algorithms are the K-means, the fuzzy C-means algorithm, and the
expectation-maximization (EM) algorithm. The K-means clustering algorithm clusters data by iteratively computing
a mean intensity for each class and segmenting the image by classifying each pixel in the class with the closest mean
(Dzung L. Pham and et al, 1998).

Fuzzy C-Means Clustering

Because of the advantages of magnetic resonance imaging (MRI) over other diagnostic imaging, the majority of
researches in medical image segmentation pertain to its use for MR images, and there are a lot of methods available
for MR image segmentation. Among them, fuzzy segmentation methods are of considerable benefits, because they
could retain much more information from the original image than hard segmentation methods. In particular, the
fuzzy C-means (FCM) algorithm, assign pixels to fuzzy clusters without labels. Unlike the hard clustering methods
otherwise known as k-means clustering which force pixels to belong exclusively to one class, FCM allows pixels to
belong to multiple clusters with varying degrees of membership. Because of the additional flexibility, The Fuzzy

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C-means clustering algorithm (FCM) is a soft segmentation method that has been used extensively for segmentation
of MR images applications recently. However, its main disadvantages include its computational complexity and the
fact that the performance degrades significantly with increased noise (NG and et al, 2006).
Fuzzy c-means (FCM) is a method of clustering which allows one piece of data to belong to two or more clusters. In
other word, each point has a degree of belonging to clusters, as in fuzzy logic, rather than belonging completely to one
cluster. Thus, points on the edge of a cluster may be in the cluster to a lesser degree than points in the center of cluster.
Fuzzy c-means has been a very important tool for image processing in clustering objects in an image. In the 70's,
mathematicians introduced the spatial term into the FCM algorithm to improve the accuracy of clustering under noise.
(Wikipedia 2009)

K-Means Clustering

K-means (MacQueen, 1967) is one of the simplest unsupervised learning algorithms that solve the well known
clustering problem. K-means clustering algorithm is a simple clustering method with low computational complexity
as compared to FCM. The clusters produced by K-means clustering do not overlap.
The procedure follows a simple and easy way to classify a given data set through a certain number of clusters (assume
k clusters) fixed a priori. The main idea is to define k centroids, one for each cluster. These centroids should be placed
in a cunning way because of different location causes different result. So, the better choice is to place them as much as
possible far away from each other. The next step is to take each point belonging to a given data set and associate it to
the nearest centroid. When no point is pending, the first step is completed and an early grouping is done. At this point
we need to re-calculate k new centroids as barycenters of the clusters resulting from the previous step. After these k
new centroids, a new binding has to be done between the same data set points and the nearest new centroid. A loop has
been generated. As a result of this loop we may notice that the k centroids change their location step by step until no
more changes are done. In other words centroids do not move any more.

K-means clustering algorithm is an unsupervised method. It is used because it is simple and has relatively low
computational complexity. In addition, it is suitable for biomedical image segmentation as the number of clusters (K)
is usually known for images of particular regions of human anatomy. For example a MR image of the head generally
consists of regions representing the bone, soft tissue, fat and background. Since the regions are 4 in number then K
will be 4. Finally, this algorithm aims at minimizing an objective function, in this case a squared error function. The
objective function

‫ ܬ‬ൌ ∑௞ ∑௡ ||‫ݔ‬௜ ሺ௝ሻ െ ܿ௝ ||ଶ
௝ୀଵ ௜ୀଵ

where ||‫ݔ‬௜ ሺ௝ሻ െ ܿ௝ ||ଶ is a chosen distance measure between a data point ‫ݔ‬௜ ሺ௝ሻ and the cluster centre ܿ௝ , is an indicator
of the distance of the n data points from their respective cluster centres. K-means is a simple algorithm that has been
adapted to many problem domains. It is a good candidate for extension to work with fuzzy feature vectors.

3 METHODOLOGY

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For both clustering methods chosen in this project algorithms and flowcharts have been provided for the proper
implementation. These algorithms have been further combined to formulate another called fuzzy k-c-means algorithm.
The clustering methods have been compared on the bases of the time it takes each to segment a given image, the
number of iteration, and as well as how accurate the result is.

Fuzzy C-Means Algorithm and Flowchart

Fuzzy c-means algorithm allows data to belong to two or more clusters with different membership coefficient. Fuzzy
C-Means clustering is an iterative process. First, the initial fuzzy partition matrix is generated and the initial fuzzy
cluster centers are calculated. In each step of the iteration, the cluster centers and the membership grade point are
updated and the objective function is minimized to find the best location for the clusters. The process stops when the
maximum number of iterations is reached, or when the objective function improvement between two consecutive
iterations is less than the minimum amount of improvement specified.

Moreover the update in the iteration is done using the membership degree as well as the centre of the cluster that is the
two parameter change as the steps are being repeated until a set point called the threshold is reached or the process
stops when the maximum number of iterations is reached, or when the objective function improvement between two
consecutive iterations is less than the minimum amount of improvement specified. In addition a fuzziness coefficient
‘m’ is chosen which may be any real number greater than 1.

The algorithm comprises of the following steps:

1. Read the image into the Matlab environment

2. Try to identify the number of iteration it might possibly do within a given period of time.

3. Get the size of the image.

4. Calculate the distance possible size using repeating structure.

5. Concatenate the given dimension for the image size

6. Repeat the matrix to generate large data items in carrying out possibly distance calculation.
7. Begin Iterations by identifying large component of data vis - a - vis the value of the pixel.

8. Stop Iteration when possible identification elapses.

9. Generate the time taken to segment.

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K-Means Algorithm and Flowchart

The k-means clustering also known as hard c-means clustering provides an algorithm used for partitioning a set of N
vectors into C groups. The algorithm computes the cluster centers (centroids) for each group. This algorithm
minimizes a dissimilarity function. The image to work with is first imputed into the MATLAB work area with the
use of the function called imread. This is followed by the calculation of the colour space by the use of the L*b*a*
colour space derived from the CIE XYZ tri-stimulus values. The L*a*b* space consists of a luminosity layer 'L*',
chromaticity-layer 'a*' indicating where colour falls along the red-green axis, and chromaticity-layer 'b*' indicating
where the colour falls along the blue-yellow axis. All of the colour information is in the 'a*' and 'b*' layers. You can
measure the difference between two colours using the Euclidean distance metric.

Classification of the colours generated in a*b* space is also a very important part of the implementation, k-means
clustering makes this possible. Clustering is a way to separate groups of objects. K-means clustering treats each object
as having a location in space. It finds partitions such that objects within each cluster are as close to each other as
possible, and as far from objects in other clusters as possible. K-means clustering requires that you specify the number

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Vol 2, No.6, 2012

of clusters to be partitioned and a distance metric to quantify how close two objects are to each other. Moreover, an
image is made up of pixels. These pixels are labeled using the result from k-means. For every object in an input,
K-means returns an index corresponding to a cluster. The cluster center output from K-means will be used later in the
demo. Label every pixel in the image with its cluster index. Finally, the images generated through the segmentation of
the original image are created for analysis.

The algorithm has the following steps:

1. Read the image into the MATLAB environment using the imread function
2. Convert the image to L*a*b* colour space using make form and apply form
3. Classify the Colours in 'a*b*' Space Using K-Means Clustering
4. Label every pixel in the Image using the results from K –means

5. Create Images that Segment the H&E Image by colour using clusters.

Fuzzy K-C-Mean Algorithm and Flowchart

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Vol 2, No.6, 2012

In Fuzzy K-C-Means the interest is on making the number of iterations equal to that of the fuzzy c means, and
still get an optimum result. This implies that irrespective of the lower number of iteration, we will still get an
accurate result.

The algorithm has the following steps:

1. Read the image into the Matlab environment
2. Try to identify the number of iteration it might possibly do within a given period of time
3. Reduce number of iteration with distance check
4. Get the size of the image
5. Calculate the distance possible size using repeating structure
6. Concatenate the given dimension for the image size
7. Repeat the matrix to generate large data items in carrying out possibly distance calculation
8. Reduce repeating when possible distance has been attained
9. Iterations begin by identifying large component of data vis a vis the value of the pixel
10. Iteration stops when possible identification elapses
11. Time is generated.

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4. Results and Discussion

The implemented clustering methods have been done in MATLAB. Three images acquired through
Magnetic Resonance Imaging (MRI) were used for comparing the performances of the three methods. The machine
on which they are tested is made up of the following: Pentium (R) M, Processor speed of 1400 MHz, 512 MB of
RAM. The following are the benchmarks used to compare:
• The mode of operation
• The time taken
• The accuracy

Mode of Operation
K-means demands that the user specifies the number of clusters before the segmentation commences. As a result, the
number of clusters is predetermined. The k-means method considered here is operating based on colours contained
by the image. The number of clusters specified by the user must correspond to the number of colour. It is not
necessary to have the pre-knowledge of the number of colours contained by the image because there is provision
made for re-inputting the number of clusters. Maximum number of possible colours provided for is 9 since most
images may have as much as 5-6 colours. It is possible to have an image whose colours are more than this range,
hence the provision for more colours. As soon as k-means gets to the end of the clusters specified it stops.

Fuzzy C-Means converts a coloured image into grey scale before commencing the segmentation. That is it segments
using grey scale. If the image inputted is a non-coloured it will still segment it unlike the k-means which only
segments a coloured image. Usually, Fuzzy C-means iterates based on the number of clusters it comes across on the
image being considered. Unlike K-means, the fuzzy c-means will return the number of clusters after the
segmentation has been done. Therefore the number clusters is approximately the number of iterations.

Fuzzy K-C-Means is a method generated from both fuzzy c-means and k-means but it carries more of fuzzy
c-means properties than that of k-means. Fuzzy k-c-means works on grey scale images like fuzzy c-means, generates
the same number of iterations as in fuzzy c-means.

Time Taken to segment
Based on the tested images k-means appears to be faster than fuzzy c-means while in some cases fuzzy c-means also
appears to be faster than k-means. Whereas both fuzzy c-means and k-means are competing in terms of time, fuzzy
k-c-means has been programmed to generate the same number of iteration with fuzzy c-means with a faster operation
time. That is fuzzy k-c-means is faster than both fuzzy c-means and k-means. The conflict in time between fuzzy
c-means and k-means is assumed to account from the properties of the image under consideration, the efficiency of
the machine on which the methods are tested.

Accuracy

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In terms of accuracy, the number iteration is put into consideration. The more the iterations the more the accuracy.
The iteration that k-means can perform depend largely on the number of colours contained by an image which
make its iterative ability limited unlike that of fuzzy c-means and fuzzy k-c-means which segment based on the
number of iterations or clusters contained in an image. Consequent to this, k-means is less accurate than the other
two methods.

Segmentation results on MRI brain using the Methods

K-means, Fuzzy c-means and Fuzzy k-c-means have been used in segmenting three MRI images in order to compare
the results in each case.

(a) (b) (c)
(d)
Figure 4 (a) Image I, segmentation results; (b) K-Means (c) Fuzzy C-Means (d) Fuzzy K-C Means

Table 1: Comparison of segmentation results on image I

METHODS TIME TAKEN (s) NUMBER OF CLUSTERS NUMBER OF ITERATION
K-MEANS 31.67 9 9
FUZZY C-MEANS 32.45 13 13
FUZZY K-C-MEANS 30.09 13 13

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Table 2: Comparison of segmentation results on image II

METHODS TIME TAKEN (s) NUMBER OF CLUSTERS NUMBER OF ITERATIONS

K-MEANS 31.23 9 9

(a) (b) (c) (d)
Figure 6 (a) Image III, segmentation results; (b) K-Means (c) Fuzzy C-Means (d) Fuzzy K-C Means

Table 3: Comparison of segmentation results on image II

METHODS TIME TAKEN (IN SECONDS) NUMBER OF NUMBER OF ITERATIONS
CLUSTERS
K-MEANS 38.44 5 5

It is pertinent to note in this work that;

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i. for k-means the user is expected to input the number of clusters before segmentation and this is equal to
the number of iterations.
ii. for fuzzy c-means number of clusters is generated iteratively during segmentation and this is equal to
the number of clusters.
iii. for fuzzy k-c-means number of iterations is equal to number of clusters.
The method with the highest iteration value and segments within the shortest period of time takes the more accuracy.
In this case fuzzy k-c-means and fuzzy c-means should have been considered but with clear observation fuzzy
c-means is slower than fuzzy k-c-means therefore fuzzy k-c-means takes the highest accuracy.

5. Conclusion
Medical image segmentation is a case study that is fascinating and very important as well. Fuzzy C-Means,
K-Means and Fuzzy K-C-Means clustering algorithms have been considered so far they have been seen effective in
the image segmentation. They are easy to use unlike some other methods in existence. Time, accuracy, and iterations
have been the major focus here. But there are still limitations that like k-means segmenting with predetermined
number of clusters Fuzzy C-means generating an overlapping results and not being able to segment coloured images
until they are converted into grey scale. Fuzzy K-C-Means also operates almost like Fuzzy C-Means.

6. References

Ng, H.P. Ong, S.H, Foong, K.W.C, Nowinski, W.L. (2005): “An improved watershed algorithm for medical image
segmentation”, Proceedings 12th International Conference on Biomedical Engineering.

MacQueen (1967): “Some methods for classification and analysis of multivariate observations”, Proceedings 5th
Berkeley Symposium on Mathematical Statistics and Probability, pp. 281-297.

Dzung L. Pham, Chenyang Xu, Jerry L. Prince (2010) “ A survey of current methods in medical image segmentation,
Journal of image processing

Heyer, Kruglyak, Yooseph (1999) “Quality Threshold Clustering”

32

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Fuzzy k c-means clustering algorithm for medical image

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