Project No.49

Image Segmentation using

Advanced Fuzzy c-means Algorithm

B.Tech Final Year Project Report Submitted as requirement

for award of degree of

BACHELOR OF TECHNOLOGY

in

Electrical Engineering

Submitted By:

J Koteswar Rao Ankit Agarawal

Guided By:

Dr. R.P.MAHESHWARI

DEPARTMENT OF ELECTRICAL ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY ROORKEE

ROORKEE-247667(INDIA)

MAY,2012

Acknowledgement

We express our most sincere and heartfelt gratitude to Dr. R.P. Maheshwari for helping us through every step of the project. Sparing time for us even through his extremely busy schedule, his guidance was essential for the successful completion of this project. We would also like to thank the research scholar, Mr. Subramaniam, for helping us and sharing his knowledge and experience in the field of Digital Image Processing. We have gained good experience and knowledge in the field of Digital Image Processing while doing this project and we are grateful to everyone who has helped us in our endeavour. Lastly we would like to thank God and our parents for their blessings and motivation throughout this project.

ABSTRACT

K-means algorithm is still extensively used in various fields like image segmentation, data

mining, etc. for clustering data and retrieving information. To enhance its quality of output many

new concepts and algorithms have been developed like fuzzy c-means (fcm) which introduced

the concept of partial membership. Since then huge amount of research has been done on fcm

and many of its variants have been developed like Bias-corrected fuzzy c-means (BCFCM),

some kernel versions of FCM with spatial constraints, such as KFCM_S1 and KFCM_S2, to

solve those drawbacks of BCFCM. Then, a Gaussian kernel-based fuzzy c-means algorithm

(GKFCM) with a spatial bias correction was proposed to decrease the heavy effect of

parameters. In our project, we have implemented all these algorithms for image segmentation

which have been proved to give very good results and created a GUI. Later above our pre-stated

objectives, we developed an algorithm which improves them by automatically choosing the

number of clusters, better selection of initial cluster centres and tremendously increasing the

speed to enable their use in real-time video segmentation.

Segmentation of flowers

CHAPTER 1 INTRODUCTION

Due to ever increasing amount of visual data and need to retrieve the useful information out of it,

it has become very necessary to develop better image segmentation tools and methods which

enable retrieving desired information and that too fast. Huge amount of research has been and is

being done in this field and various algorithms have been developed. Clustering is one such

technique which helps not only in image segmentation but is also useful in various other fields

like data mining, pattern recognition, etc. K-means is the most common clustering algorithm

which is widely used due to its speed, simplicity and good output. It has been improved over the

years and many of its variants have been developed like fuzzy c-means, Bias corrected fuzzy c-

means, etc. which give better results and remove many drawbacks of K-means.

In our project we first implemented K-means algorithm for image segmentation to understand

the effect of clustering on an image and its segmentation results. We discovered that it suffers

from many drawbacks like non-convergence and is susceptible to effects from noise. Hence, we

implemented Fuzzy c-means algorithm which gives partial membership of each cluster to each

pixel and then its variant which was proposed to yield better results as it took into account the

effect of neighbouring pixels to decrease the effect of noise(BCFCM). Since it is

computationally time taking and lacks enough robustness to noise and outliers, some kernel

versions of FCM with spatial constraints, such as KFCM_S1 and KFCM_S2, were proposed to

solve those drawbacks of BCFCM. However, KFCM_S1 and KFCM_S2 are heavily affected by

their parameters. So,we implemented a Gaussian kernel-based fuzzy c-means algorithm

(GKFCM) with a spatial bias correction which was proposed to be more efficient and robust.

While implementing them, we noticed that we can improve the results of K-means algorithm by

hitting on its basic inputs and we realized research is being done in that direction as well. Inputs

to a K-means algorithm are number of clusters and the initial cluster centres. We noticed that

initial clusters were in general randomly chosen and the number of clusters was derived by

applying k-means a number of times and then comparing the results. So, we came up with the

idea that we can apply a moving average filter on the histogram of an image and take the number

of peaks as the number of clusters and those peaks as initial cluster centres. We found that this

way the problem of non-convergence was removed and the results were better as the initial

cluster centres were found to be close to final cluster centres. This way we also avoided the

input, number of clusters, from the user. Then, we happened to notice that we can apply the k-

means algorithm on the histogram of the image rather than the image itself which increased the

speed of the algorithm since the iterations became very fast. We incorporated an additional

feature which automatically varies the input to moving average filter and named it auto-pilot,

which depending on the application itself decides the value of n for moving average filter. As the

algorithm became very fast and didn’t require any input from the user we realized that we can

even try real-time video segmentation which is very difficult due to limitation of time.

We have developed a GUI(Graphic User Interface) to display all the algorithms implemented,

modifications which we are proposing, etc. so that the user can easily look through our work and

use it as well. We have some interesting features like report generation which makes a report of

the segmented results of an image in word file, live video segmentation, etc.

In the report starting with introduction to Digital Image Segmentation we have explained various

clustering algorithms implemented by us, then highlighted a few drawbacks of K-means and

Fuzzy c-means which we have removed by our proposed algorithm which is explained in the

next chapter along with auto-pilot and real-time video segmentation. At last we have explained

the features of GUI developed.

Segmentation using proposed algorithm (tumor got completely separated)

Chapter 2 Digital Image Segmentation

2.1 What is Digital Image Segmentation?

When acquiring an image, it might be interesting to know what parts of the image do belong to

each other. It could be a satellite image, where we wish to quantify and locate different types of

vegetation, or a medical MRI image where the doctors are interested to know how much of each

tissue type is present, and where it is located. Image segmentation techniques offer a method to

perform these tasks, and thus, can be regarded as the process of dividing an image into

groups of pixels which from a preset property are connected to each other. The goal of

segmentation is to simplify and/or change the representation of an image into something that is

more meaningful and easier to analyze.

Image segmentation is typically used to locate objects and boundaries (lines, curves, etc.) in

images. More precisely, image segmentation is the process of assigning a label to every pixel in

an image such that pixels with the same label share certain common visual characteristics. These

characteristics or computed property may be, such as color, intensity or texture. Commonly this

means that pixels with almost the same intensity values are grouped together, or pixels with the

same colour code. There are techniques for finding for instance objects with closed contours,

convex objects and the boundaries of an object, etc.

Segmentation is a fundamental task of image processing because of its many areas

of application.

Medical Imaging

Locate tumors and other pathologies

Measure tissue volumes

Computer-guided surgery

Diagnosis

Treatment planning

Study of anatomical structure

Locate objects in satellite images (roads, forests, etc.)

Face Recognition

Iris Recognition

Fingerprint Recognition

Traffic Control Systems

Brake Light Detection

Machine Vision

Segmentation techniques have not been widely applied, partly because they are time consuming

and partly because there are no overall techniques that are suitable for all different types of

images. All intensity based segmentation techniques are sensible to the situation that different

objects have almost equal intensities. This will often lead to misclassification (wrongly classified

pixels) if the objects from the view of the human eye should not belong to the same class.

[In general all the segmentation techniques have to be based on intensity since the only data

available from an image is an array of numbers which signify the intensity. Then these numbers

are used in segmentation in various formats to represent just the intensity based(for gray level

images) or colour based(which uses three or more intensity matrices) or any kind of a pattern like

a texture(which again may use more than one parameter), etc. Here in our project we have

concentrated only on gray level images.]

2.2 Different ways of Image Segmentation

Image segmentation can proceed on three different ways,

1. Manually

2. Automatically

3. Semi-automatically

2.2.1 Manual Segmentation

The pixels belonging to the same intensity range could manually be pointed out, but clearly this

is a very time consuming method if the image is large. A better choice would be to mark the

contours of the objects. This could be done discrete from the keyboard, giving high accuracy, but

low speed, or it could be done with the mouse with higher speed but less accuracy. The manual

techniques all have in common the amount of time spent in tracing the objects, and human

resources are expensive. Tracing algorithms can also make use of geometrical figures like

ellipses to approximate the boundaries of the objects. This has been done a lot for medical

purposes, but the approximations may not be very good.

2.2.2 Automatic Segmentation

Fully automatic segmentation is difficult to implement due to the high complexity and variation

of images. Most algorithms need some a priori information to carry out the segmentation, and for

a method to be automatic, this a priori information must be available to the computer. The

needed a priori information could for instance be noise level and probability of the objects

having a special distribution. In computer vision it is required that computer is able to do

automatic segmentation and then take the required actions according to the segmentation results.

In our project we have made an attempt to go closer to automatic segmentation by requiring less

and less input from the user.

2.2.3 Semi-automatic Segmentation

Semiautomatic segmentation combines the benefits of both manual and automatic segmentation.

By giving some initial information about the structures, we can proceed with automatic methods.

Thresholding If the distribution of intensities is known, thresholding divides the image into two

regions, separated by a manually chosen threshold value ‘a’ as follows:

if B(i; j) ¸ a; B(i; j) = 1 (object)

else B(i; j) = 0 (background)

for all i; j over the image B. This can be repeated for each region, dividing

them by the threshold value, which results in four regions etc. However, a successful

segmentation requires that some properties of the image is known beforehand. This

method has the drawback of including separated regions which correctly lie within

the limits specified, but regionally do not belong to the selected region. These pixels

could for instance appear from noise. The simplest way of choosing the threshold

value would be a fixed value, for instance the mean value of the image.

A better choice would be a histogram derived threshold. This method includes some

knowledge of the distribution of the image, and will result in less misclassification.

Isodata algorithm is an iterative process for finding the threshold value. First segment the image

into two regions according to a temporary chosen threshold value. Then calculate the mean value

of the image corresponding to the two segmented regions. Calculate a new threshold value from

thresholdnew = mean(meanregion1 + meanregion2)

and repeat until the threshold value does not change any more. Finally choose this value for the

threshold segmentation.

Various other thresholding methods are used like adaptive thresholding, Otsu algorithm,

Minimum error algorithm, Minimum entropy method, etc.

Boundary tracking Edge-finding or edge-detection by gradients is the method of selecting a boundary manually, and

automatically follow this gradient until returning to the same point. Returning to the same point

can be a major problem of this method. Boundary tracking will wrongly include all interior holes

in the region, and will meet problems if the gradient specifying the boundary is varying or is very

small. A way to overcome this problem is first to calculate the gradient and then apply a

threshold segmentation. This will exclude some wrongly included pixels compared to the

threshold method only.

Zero-crossing based procedure is a method based on the Laplacian. Assume the boundaries of an

object has the property that the Laplacian will change sign across them. Assume the boundary is

blurred, and the gradient will have a shape like in Figure 2.2. The Laplacian will change sign just

around the assumed edge for position = 0.

For noisy images the noise will produce large second derivatives around zero crossings, and the

zero-crossing based procedure needs a smoothing filter to produce satisfactory results.

Various edge detection tools are used and huge development have been made in this like various

masks have been developed e.g. Robert operator, Sobel’s operator, Prewitt Operator, etc.

Neighbouring methods

Region growing is a statistical method for segmentation. We group the pixels into different

regions from the principles of proximity and homogeneity. The principle of proximity splits the

image into smaller regions and merges them again such that all pixels have neighbours similar to

themselves. This means that the intensities of the neighbours all lie within a certain threshold.

The principle of homogeneity compares the statistical probability of different combinations of

regions to belong to the same class. This requires some a priori information about the image we

are trying to segment. Various statistical measures like standard deviation, skewness, kurtosis

etc. are used for determining whether the regions are similar or different. But here also the input

for these measures to differentiate has to be given by the user based on a priori information.

Mathematical Morhphology

Mathematical morphology (MM) is a theory and technique for the analysis and processing of

geometrical structures. MM is most commonly applied to digital images, but it can be employed

as well on graphs, surface meshes, solids, and many other spatial structures.

Topological and geometrical continuous-space concepts such as

size, shape, convexity, connectivity, and geodesic distance were introduced by MM on both

continuous and discrete spaces. MM is also the foundation of morphological image processing,

which consists of a set of operators that transform images according to the above

characterizations.

Besides extending the main concepts (such as dilation, erosion, etc...) to functions, this

generalization yielded new operators, such as morphological gradients, and

the Watershed (MM's main segmentation approach). The watershed transformation considers the

gradient magnitude of an image as a topographic surface. Pixels having the highest gradient

magnitude intensities (GMIs) correspond to watershed lines, which represent the region

boundaries. Water placed on any pixel enclosed by a common watershed line flows downhill to a

common local intensity minimum (LIM). Pixels draining to a common minimum form a catch

basin, which represents a segment.

Clustering Methods Clustering is the process of dividing data elements into classes or clusters so that items in the

same class are as similar as possible, and items in different classes are as dissimilar as possible.

Depending on the nature of the data and the purpose for which clustering is being used, different

measures of similarity may be used to place items into classes, where the similarity measure

controls how the clusters are formed. Some examples of measures that can be used as in

clustering include distance, connectivity, and intensity. Hence, clustering has been used as a very

useful tool in data mining, image segmentation, pattern recognition, information retrieval, etc.

Many clustering algorithms have been developed like K-means, Fuzzy c-means, etc. With

moderate processing time they give very good result and are very commonly used.

We’ll be discussing about clustering algorithms at length in the next chapter and discuss about

various clustering algorithms evolved and their drawbacks. Later in the report we’ll mention the

efforts made by us in trying to remove those drawbacks and getting better results.

Chapter 3 CLUSTERING METHODS

3.1 Introduction

Cluster analysis or clustering is the task of assigning a set of objects into groups

(called clusters) so that the objects in the same cluster are more similar (in some sense or

another) to each other than to those in other clusters.

Clustering is widely used in image analysis, pattern recognition, and many other fields,

including data mining, statistical analysis, machine learning, information retrieval

and bioinformatics. Its wide use can be attributed to the fact that they are very fast and simple

and give good results for unsupervised approaches.

In intensity based image segmentation for example clustering algorithms finds its use because an

object generally has same intensity, pattern or texture etc. for a particular orientation. Hence to

segment a particular object out of an image what we require to know is its intensity. Similarly

different objects can be segmented based on their intensity in the image. So by the process of

clustering we identify various groups of intensities which may refer to different objects and thus

segment the image into its constituting parts separately.

Cluster analysis itself is not one specific algorithm, but the general task to be solved. It can be

achieved by various algorithms that differ significantly in their notion of what constitutes a

cluster and how to efficiently find them. Popular notions of clusters include groups with

low distances among the cluster members, dense areas of the data space, intervals or particular

statistical distributions. Clustering can therefore be formulated as a multi-objective

optimization problem. The appropriate clustering algorithm and parameter settings (including

values such as the distance function to use, a density threshold or the number of expected

clusters) depend on the individual data set and intended use of the results. Cluster analysis as

such is not an automatic task, but an iterative process of knowledge discovery or interactive

multi-objective optimization that involves trial and failure. It will often be necessary to modify

preprocessing and parameters until the result achieves the desired properties.

3.2 CLUSTERS AND CLUSTERING

The notion of a "cluster" varies between algorithms and is one of the many decisions to take

when choosing the appropriate algorithm for a particular problem. At first the terminology of a

cluster seems obvious: a group of data objects. However, the clusters found by different

algorithms vary significantly in their properties, and understanding these "cluster models" is key

to understanding the differences between the various algorithms. There may be various cluster

models but we’ll be confining our discussion to centroid based models like K-means algorithm

and its variants in which a cluster is represented by a single mean vector.

A "clustering" is essentially a set of such clusters, usually containing all objects in the data set.

Additionally, it may specify the relationship of the clusters to each other, for example a hierarchy

of clusters embedded in each other. Clusterings can be roughly distinguished in:

hard clustering: each object belongs to a cluster or not

soft clustering (also: fuzzy clustering): each object belongs to each cluster to a certain

degree

In the next few sections of the chapter we would discuss various clustering algorithms which we

have implemented in our project.

3.3 K-Means Algorithm

K-means (MacQueen) is one of the simplest unsupervised learning algorithms that solve the well

known clustering problem. It is very popular because of its ability to cluster a kind of huge data,

and also outliers, quickly and efficiently. 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. Every pixel has to belong to only and

only one cluster. These centroids shoud be placed in a cunning way because 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 groupage is

done. At this point we need to re-calculate k new centroids for the clusters resulting from the

previous step. After we have 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.

Finally, this algorithm aims at minimizing an objective function, in this case a squared error

function. The objective function

Where,

is a chosen distance measure between a data point and the cluster

centre , is an indicator of the distance of the n data points from their respective cluster centres.

K-means has proved to be very useful algorithm for image segmentation being both simple and

fast is sometimes used as a pre-segmentation for other segmentation processes.

Place K points in

space one for each

cluster randomly.

Assign each pixel to

the cluster whose

centroid or centre is

closest

Calculate new

centres as the

centroid of the group

cj(new)-cj(old)<Є

Cluster centres have

been obtained

yes

No

figure: Flowchart for K-means algorithm

3.4 Fuzzy C-Means (FCM) Algorithm

Zadeh proposed fuzzy sets that introduced the idea of partial memberships described by

membership functions, it has been successfully applied in various areas. Especially, fuzzy sets

could allow membership functions to all clusters in a data set so that it is very suitable for cluster

analysis i.e. each pixel can have membership to all the clusters but the membership would be

defined. In contrast to K-means where each pixel is a member of only one cluster Ruspini first

proposed fuzzy c-partitions as a fuzzy approach to clustering. Later, the fuzzy c-means (FCM)

algorithms with a weighting exponent m = 2 proposed by Dunn , and then generalized by Bezdek

with m > 1 became popular.

Let X = {x1, . . .,xn} be a data set in an s-dimensional Euclidean space R. Let c be a positive

integer greater than one. A partition of X into c parts can be presented by a mutually disjoint set

X1, . . .,Xc such that X1 U….U Xc = X, or equivalently by the indicator functions µ1, . . .,µc

such that µij = µi(xj) = 1 if xj Є Xi and µij = 0 if xj ∉ Xi for i = 1,. . ., c and j = 1,. . .,n. The set

of indicator functions {µ1, . . ., µc} is known as a hard c-partition of clustering X into c clusters.

Ruspini first considered an extension to allow µij = -µi(xj) to be the membership functions of

fuzzy sets µi on X assuming values in the interval [0,1] such that ∑ µ for all xj in X.

In this fuzzy extension, {µ1, . . ., µc} is called a fuzzy c-partition of X.

The fuzzy c-partitions of X can be represented in a matrix form as follows:

µ = {µij}cxn Є Mfcn where Mfcn is a partition matrix with

Mfcn = {µ=[µij]cn | ∀ i, ∀j µij ≥0, ∑ µ , n> ∑ µ

> 0 }

Dunn first embedded the fuzzy c-partitions into K-means and then proposed the fuzzy c-means

(FCM) objective function with

∑∑µ

where µij Є Mfcn and {a1, . . .,ac} denote the cluster centres of the data set X. Bezdek extended

the weighting exponent m = 2 to any m > 1 with the FCM objective function Jm as

µ ∑∑ µ

Thus, the FCM algorithm is iterated through the necessary conditions for minimizing Jm(µ,a)

with the following update equations:

∑ µ

∑ µ

, i=1,2,…c (1)

µ ∑ , i=1,2,….c (2)

j=1,2,….n

Based on a sequence of execution for stage s using stage (s -1) according to the update Eqs. (1)

and (2), the FCM can be described as follows:

FCM procedure

Input:

(1) X = { ,…. }, Є R, the data set

(2) c, 2 ≤ c ≤ n, the number of clusters

(3) Ɛ > 0, the stopping criterion of algorithm

(4) (

) the initials of cluster centers.

OUTPUT: a = { }, the final cluster centers

Algorithm:

Step 1: Let s = 1

Step 2: Compute µ using (2).

Step 3: Update the cluster centers with µ using (1).

Step 4: Compare to in a convenient matrix norm ||.||.

IF || || < Ɛ, STOP and OUTPUT.

ELSE s = s + 1 and return to step 2.

Fuzzy clustering has been widely studied and applied in a variety of areas, such as numerical

taxonomy, image processing, pattern recognition, medicine, economics, ecology, marketing,

artificial intelligence, data mining, engineering systems, and gene expression. In fuzzy

clustering, the fuzzy c-means (FCM) algorithm plays an important role.

3.5 Bias-corrected Fuzzy C-Means (BCFCM)

Although the FCM algorithm is the best known, it has several drawbacks. For example, the

points in the data set are supposed to be equally important, the number of points in the clusters is

almost equal, nearly all points do not have a membership value of one, and the outliers always

affect the clustering results. To overcome these drawbacks, many generalized FCM algorithms

have been proposed in the literature. Among them, Ahmed et al. (2002) first modified the FCM

algorithm as a bias-corrected FCM(BCFCM) with a spatial neighbourhood regularization term

by regularizing the following FCM objective function .

where Nj stands for the set of neighbours that exist in a window around xj and is the

cardinality of Nj. The effect of neighbouring terms is controlled by the parameter a. Thus, the

BCFCM algorithm is iterated through the necessary conditions for minimizing with the

following update equations:

However, Chen and Zhang (2004) pointed out a shortcoming of the BCFCM update equations

that computing the term of neighbourhoods Nj may take too much time than FCM. They then

proposed a modified objective function as

where is the sample mean within the window around xj where can be computed in advance.

The idea is that the term

Thus, the modified BCFCM algorithm is iterated through the necessary conditions for

minimizing with the following update equations:

3.6 Kernel based Fuzzy C-means (KFCM)

Chen and Zhang (2004) then replaced the Euclidean distance || || with a Gaussian kernel-

induced distance 1- K( =1 - exp(- / ). They gave the kernel version

of

as

The necessary conditions for minimizing (µ,a)are the following update equations:

where K(x,y) =exp(- / ). Note that different kernels can be chosen by replacing the

Euclidean distance . for different purposes. However, a Gaussian kernel is suitable

for clustering in which it can actually induce the necessary conditions (7) and (8). They then

proposed the KFCM_S1 as follows:

KFCM_S1 procedure

Input:

(1) X = { ,…. }, Є R, the data set

(2) c, 2 ≤ c ≤ n,, the number of clusters

(3) Ɛ > 0, the stopping criterion of algorithm

(4) and , the values of parameters

(5) (

) the initials of cluster centres.

Output: a = { }, the final cluster centres

Algorithm:

Step 1: Let s = 1

Step 2: Compute µ using (8).

Step 3: Update the cluster centres with and µ using (7).

Step 4: IF || || < Ɛ, STOP and OUTPUT.

ELSE s = s + 1 and return to step 2.

If is taken as the median of the neighbours within the window around , then the

algorithm is called KFCM_S2 by Chen and Zhang(2004) where they compare the KFCM_S1

and KFCM_S2 with FCM and BCFCM. However, the parameter in BCFCM, KFCM_S1 and

KFCM_S2 heavily affects the final clustering results. For estimating the parameter and

learning the parameter , we propose a generalized type of BCFCM, KFCM_S1 and KFCM_S2

where the parameters and can be automatically estimated and learned from the data in the

next section.

3.7 Gaussian kernel-based FCM (GKFCM)

We know that Chen and Zhang (2004) considered a kernel version of FCM by replacing the

Euclidean distance || || with the kernel substitution as

( )

where / is a nonlinear map from the data space into the feature space with its corresponding

kernel K. They specially assumed K(x,y) = 1 and then proposed the kernel-type objective

function with

µ ∑∑µ

Thus, the update equations for the necessary conditions for minimizing µ are as follows:

∑ µ

( )

∑ µ

( ) i = 1,2,…,c (9)

µ

( )

∑ ( )

, i = 1,2,…c; (10)

j=1,….,n

We point out that the necessary conditions for minimizing µ are update Eqs. (9) and (10)

only when the kernel function K is chosen to be the Gaussian function with K( ) = exp(-

).

µ ∑∑µ

∑∑µ

Where, K(x,y) =exp(- / ) is taken. Thus, the necessary conditions for minimizing

µ are Eqs. (7) and (8) as shown in Section 2.We mention that the parameter is used to

control the effect of the neighbors for adjusting the spatial bias correction term. In fact, the

parameter heavily affects the clustering results of BCFCM, KFCM_S1 and KFCM_S2.

Intuitively, it would be better if we can adjust each spatial bias correction term separately for

each cluster i. That is, the overall parameter is better replaced with that is correlated to each

cluster i. In this sense, we will consider the following modified objective function with

µ ∑ ∑ µ

∑ ∑

µ

(11)

where K(x,y) =exp(- / ).

Based on the concept of machine learning with a learning capability to improve the performance

of clustering results, an exponential-type distance is bounded and monotone increasing. Hence it

can be robust to noise and outliers when = 1- exp(- ) is used to

replace the Euclidean distance . However, there are parameters and

in the

proposed objective function . Since is presented as a dispersion, we use the sample

variance to estimate with

∑

with ∑ (12)

On the other hand, because the parameter controls the effect of the neighbouring term for each

cluster i, a ratio of the two distance-based influence terms, and

, can be used as a learning scheme for . This is because the term

presents the separation of data set for the cluster i. If the value of is larger,

then the cluster i will be more isolated so that the value of should be larger. However, the

scale of for each data set may depend on the total separation of the data set in

which it can be measured by the term . Thus, the ratio of the two terms

and can be well used as a standardized separation strength of

cluster i. Moreover, we also consider these influencing factors simultaneously in replacing the

Euclidean distance with the exponential-type distance 1- K(x,y) = 1-exp(- / ).

Therefore, the parameter is estimated as follows:

, i = 1,2,…c (13)

Thus, the update equations for the necessary conditions of minimizing µ are

∑ µ

( ) ( )

∑ µ

( ) , i = 1,…c (14)

µ

( ( )) ( )

∑ ( ( )) ( )

, i = 1,…,c; j = 1,…,n (15)

where K(x,y) = exp(- / ).

We see that the KFCM objective function µ is a special case of

µ when =

for all i. Thus, we have the GKFCM as follows:

GKFCM procedure

Input:

(1) X = { ,…. }, Є R, the data set

(2) c, 2 ≤ c ≤ n, the number of clusters

(3) Ɛ > 0, the stopping criterion of algorithm

(4) (

) the initials of cluster centres.

Output: a = { }, the final cluster centers

Algorithm:

Step 1: Let s = 1and estimate using (12).

Step 2: Compute

using (13).

Step 3: Compute µ and

using (15).

Step 4: Update the cluster centres with ,

and µ using (14).

Step 5: IF || || < Ɛ, STOP and OUTPUT.

ELSE s = s + 1 and return to step 2.

If in Eqs. (14) and (15) is considered as the sample mean within the window around , then

we call the algorithm as GKFCM1. If in Eqs. (14) and (15) is replaced with the median within

the window around , then the algorithm is called GKFCM2. Note that the parameters and

in BCFCM, KFCM_S1 and KFCM_S2 are priori assigned by users where they heavily affect the

clustering results. But in our proposed GKFCM1 and GKFCM2, the parameter is estimated by

Eq. (12) and the parameter iis updated at each iteration using Eq. (13) based on the data and

learning schemes.

CHAPTER 4 Drawbacks of K-means and FCM Algorithms

4.1 Introduction

The K-means and FCM algorithms are having many drawbacks due to which their results are not

good and hence the algorithms which have k-means as their starting point or are from its

variations have to suffer from various drawbacks. These are drawbacks are:

1. No. of clusters has to be a human input

2. May not converge (K-means)

3. Different initial cluster centres may yield different final cluster centres hence sometimes

the result may be completely erroneous

4. Slow (FCM)

In the next few sections we discuss these drawbacks along with some methods which were

proposed to modify and improve it. They also have suffered from the drawbacks of their own so

we have suggested a new method in the next chapter and its certain features have been discussed

along with the drawbacks of previously proposed algorithms.

4.2 Number of clusters

The first issue is that the number of clusters k needs to be determined in advance as an input to

clustering algorithms. In a real data set, k is usually unknown. In practice, different values

of k are tried, and cluster validation techniques are used to measure the clustering results and

determine the best value of k, see, for instance, [9]. In [10], Li et al. presented an agglomerative

fuzzy k-means clustering algorithm for numerical data, an extension to the standard fuzzy k-

means algorithm by introducing a penalty term to the objective function. The algorithm can

determine the number of clusters by analyzing the penalty factor. For categorical data, a bottom-

up hierarchical algorithm ACE was proposed in [11], which uses entropy as an index function to

capture the candidates for the number of clusters.

All these methods are time consuming and computation expensive to apply. For example, [9] for

instance requires whole K-means algorithm to be applied multiple times to gain the no. of

clusters to divide the image. In [10] the algorithm needs to randomly select a subset from data set

as initial cluster centers, which results in an uncertainty and then suffers with the problem as

stated in the next drawback. [11] is very time consuming and is dependent on the size of the

image because if the size of the image is very large, the ACE algorithm is not efficient due to its

computational burden of O(n2log2n) with n being the number of data points.

Hence, we would like to propose a method to find out the number of clusters for clustering of an

image for the purpose of segmentation which is lot simpler, fast and gives good results. First, the

histogram of the image is developed and on the histogram of the image a moving average filter is

applied whose size can be decided by the user depending upon the image. Then this moving

average filter will do the smoothing of the filter and give rise to some smooth peaks and valleys.

In general, it has been noticed that the intensity of an object in an image is clustered in one

particular region which forms a peak and the peaks of two different objects (or the object from

the background) are separated by a valley. So, this feature of a digital image is made use of to

determine the number of clusters. It is proposed that for intensity based segmentation each peak

will refer to a separate segment of the image. Hence, the number of peaks is our required number

of clusters. As such it can be said that this can easily be known by just viewing the histogram of

an image. But, for the purpose of computer vision and machine learning it is important that

computer itself identifies the number of clusters. This particular method aids computer in

identifying the number of cluster on its own. Though it can be said that the size of the moving

average filter is to be an input which cannot be done by computer, but later in the project we’ve

tried to sort that problem as well.

4.3 INITIAL cluster centres

Through the iterative partitioning, k-means algorithm minimizes the sum of distance from each

data to its clusters. However, one of the biggest drawbacks of the k-means algorithm is, it is

very sensitive to the designated initial starting points as cluster centers because they have direct

impact on final cluster centres. K-Means does not guarantee unique clustering as we get different

results with randomly chosen initial cluster centres. The final cluster centroids may not be the

optimal ones as the algorithm can converge into local optimal solutions. An empty cluster can be

obtained if no points are allocated to the cluster during the assignment step. Thus, computer

vision and machine learning practitioners find it difficult to rely on the results thus obtained.

Therefore, it is quite important for k-means to have good initial cluster centers.

Several methods proposed to solve the cluster initialization for k-means algorithm.

Since clusters are separated groups in a feature space, it is desirable to select initial centers which

are well separated. So the image is first divided into k parts on the basis of its intensity and then

the centre of each part is taken to be the initial cluster centres. But, the clustering of the data can

be in any region and need not be spread throughout the range and it is dangerous to select

outliers as initial centers, since they are away from normal samples. This will not only take time

for reaching to the final cluster centres there may be the problem of convergence as is discussed

in next section as well.

For the initial cluster center, Jain and Dubes (1988) applied the k-means with several times by

randomly selected initial values and selected the average of these final cluster centers.

Bradley and Fayyad (1998) proposed the refinement algorithm that builds a set of small random

sub-samples of the data, then clusters data in each sub-samples by k-means. All centroids of all

sub-samples are then clustered together by k-means using the k-centroids of each sub-sample as

initial centers. The centers of the final clusters that give minimum clustering error are to be used

as the initial centers for clustering the original set of data using k-means algorithm.

Penã et al. (1999) presented a comparative study for different initialization methods for the K-

means algorithm. The result of their experiments illustrate that the random initialization method

outperforms the rest of the compared methods as they make the K-means more effective and

more independent on initial clustering and on instance order.

This clearly showed that the methods developed till then were not giving effective but

the need to find better initial cluster was there in order to get better final cluster centres

and also to avoid any problem of convergence which may occur due to inappropriate

selection of initial cluster centres.

Hence, more research was being done to find better initial cluster which are closer to the

final cluster centres which would make the results better and along with that would be

speedy since convergence will be faster and number of iterations required will be less.

Likas et al. (2003) proposed the global k-means algorithm which is an incremental approach to

clustering which dynamically adds one cluster center at a time through a deterministic global

search procedure consisting of N (with N being the size of the dataset) executions of the k-means

algorithm from suitable initial positions. But it is a very long and time consuming process as

error finding method is run about (K-1)*N times in this.

Khan and Ahmad (2004) proposed Cluster Center Initialization Algorithm (CCIA) to solve

cluster initialization problem. CCIA is based on two observations, which some patterns are very

similar to each other. It initiates with calculating mean and standard deviation for data attributes,

and then separates the data with normal curve into certain partition. CCIA uses k-means and

density-based multi scale data condensation to observe the similarity of data patterns before

finding out the final initial clusters. The experiment results of the CCIA performed the

effectiveness and robustness this method to solve the several clustering problems.

Deelers and Auwatanamongkol (2007) proposed an algorithm to compute initial cluster centers

for k-means algorithm. They partitioned the data set in a cell using a cutting plane that divides

cell in two smaller cells. The plane is perpendicular to the data axis with the highest variance and

is designed to reduce the sum squared errors of the two cells as much as possible, while at the

same time keep the two cells far apart as possible. Also they partitioned the cells one at a time

until the number of cells equals to the predefined number of clusters, k. In their method the

centers of the k cells become the initial cluster centers for k-means algorithm.

Presently research is being done to make use of many attributes of the data set to find initial

cluster centres but this is again very time consuming and computation expensive process for just

finding the initial cluster centres as all the other methods mentioned above though they might

give very good result but if the data set is large and the processing time available for segmenting

images (or in that case any other kind of data) is less i.e. there is huge amount of images or video

data which has to be segmented then they would not fulfill the requirement.

Our proposed algorithm calculates the initial cluster centers that are quite close to the desired

cluster centers. The peaks which are obtained by using a moving average filter on the histogram

of the image are taken to be our initial cluster centres and they may not be the modes of the

histogram. This method is really fast in comparison to other methods as explained above in

finding the initial cluster centres (and is almost completed by the end of the process to find the

number of clusters) and better as well.

Firstly, it is not based on any kind of random choice for initial cluster centres which may lead to

the problem of non-convergence or may give erratic results(it will be very difficult to get unique

final cluster centres) for different initial centres which are chosen randomly and we can’t even

have any kind of control over them.

Secondly, since the initial cluster centres are found to be close to the final cluster centres the

whole clustering algorithm becomes fast as the number of iterations required to converge will

decrease whereas if the initial cluster centres are equally spaced throughout the intensity range

and the data is confined only to one section of intensity range, then unnecessarily more iterations

will be required and the cluster centres may also get trapped in local clustering of data.

Thirdly, it is independent of the size of the image. Only while making the histogram we require

going through all the pixels of the image otherwise all the calculations can be conducted on the

histogram itself which saves a lot of computation time [and is kind of uniform for different

image sizes].

Fourthly, it can be used on histogram and doesn’t require any other attribute to find the initial

cluster centres though there may be compromise with quality. The ones which use many

attributes to find the initial cluster centres are very calculation intensive and would make the

algorithm slow.

4.4 Convergence

There is a problem of convergence which may occur in K-means algorithm if care is not taken

and when random selection of initial cluster centres is done then since there is no control over

the process the algorithm may not even converge to final cluster centres. This may happen if the

initial cluster is put in a region where there are no data points. So when the new cluster centres

will be calculated in the next step then since there are no data points in that cluster the

denominator will become zero, hence there will be the problem of convergence.

A way was devised to avoid this problem by assigning the initial cluster centres between the

minimum and the maximum values of the data points. But still this problem may arise if cluster

centre drops in a region where there are no points pertaining to the data i.e. data need not be

spread throughout uniformly. So, this problem is also resolved by choosing correct initial cluster

centres and having some control over it.

4.5 Speed

Speed is always a factor which determines the applicability of a particular method. Simple K-

means algorithm applied on images is very slow because it is iteratively applied on the image

and as the size of the image increases its time of evaluating cluster centres will increase

proportionately as the computer will have to peruse all the pixels again. Though K-means is still

very fast in comparison to other methods for clustering as the computation is less. But it can be

made faster if it is applied on the histogram of the image rather than the image pixels.

Hence, as we have mentioned earlier we can work on the histogram of the image. The initial

cluster centre rather than being a pixel and iteratively measuring the distance of intensity of each

pixel from the intensity of the cluster centre pixel what can be done is we can take the intensities

of the peaks as the initial cluster centres and then cluster the intensities on the basis of their

nearness to respective cluster centre. Then using their frequency in the image as their weight and

calculate the new cluster centres. This way the number of data points is reduced from any

number depending on the size of the image to 255(if 8-bit representation). So the iterations

become faster and hence the same result can be obtained in lesser time.

CHAPTER 5 PROPOSED ALGORITHMS

5.1 optimum cluster centers and optimum number of clusters

Clustering algorithms like k-means and fuzzy c-means , although appears very simple, we get

long time or short time to converge depending upon initial cluster centers chosen and sometimes

it so happens that it doesn’t converge at all. For computer vision, we need a reliable algorithm

producing fast and sure shot convergence.

Number of clusters and initial cluster centers are human input in k-means algorithm. So, there is

a (n+1) degrees of freedom to get a particular segmentation, which causes inconsistency in

results, and there is known charge against k-means algorithm that it doesn’t have spatial

coherence.

Keeping these two things in mind, We propose a new concept of finding initial cluster centers as

a function of size of filter used to filter the histogram. Using this concept, degree of freedom

reduces to one i.e. size of the filter instead of number of clusters and it accounts for spatial

coherence too. Convergence becomes faster as initial cluster centers are somewhat near final

cluster centers.

We propose two methods of finding such optimum cluster centers using filtered histogram of the

image.

METHOD-1

Step 1: Apply moving average filter of size ‘t’ on histogram and obtain Histogram-1

Step 2: Apply moving average filter of size ‘round(t/2)’ on Histogram-1 and obtain Histogram-2

Step 3: Intensities having local maxima in the Histogram-2 can be taken as optimum initial

cluster centers. Since Histogram-2 is a filtered histogram, there are only very few local maximas

METHOD-2

Step 1: Apply moving average filter of size ‘t’ on histogram and obtain Histogram-1

Step 2: Apply moving average filter of size ‘round(t/2)’ on Histogram-1 and obtain Histogram-2

Step 3: Intensities having minima in the histogram can be taken as separating criteria for dividing

the original histograms into sub-histograms.

Step 4: Within those sub histograms, intensities having maximum frequency can be taken as

optimum initial cluster centers.

Although, choosing initial cluster centers at equal distance has been in practice so far. But by

using this method, we account for spatial coherency and speed increases very significantly as the

initial cluster centers obtained are somewhat near final cluster centers.

It is observed that greater the size of filter, lesser is the number of clusters obtained to start with.

And there is a high degree of consistency in final cluster centers obtained by same number of

cluster within certain range of size of filter used.

Whole idea is that input is size of the filter not number of clusters we want to divide the image

into, forcibly.

5.2 Proposed Algorithms for clustering

5.2.1 K-means using histogram

Step 1: Divide histogram into sub histograms by criteria of mid-point between cluster centers.

Step 2: perform k-means on histogram using these cluster centers to obtain new cluster centers

by finding the weighted average of intensities within the corresponding histogram. Weights are

calculated by dividing corresponding frequency of intensity with sum of all frequencies within

the sub histogram. This weighted average becomes the new cluster center

Step 3: Check if error between consecutive iterations is less than maximum error allowed

If Yes, final cluster centers are obtained and intensities within corresponding histogram belong to

corresponding cluster

Otherwise, Go to step 1

5.2.2 Fuzzy c-means using histogram

Equation for calculating membership remains same but while calculating average membership

term is multiplied with corresponding frequency of that intensity.

5.2.3 FCM +k-means

In FCM, while segmenting finally maximum an intensity falls in the cluster in which it has

maximum intensity.Finding maximum membership consumes lot of time. This can be easily

avoided by dividing in the way we divide in k-means (an intensity value belongs to the cluster if

it falls in sub-histogram of that particular cluster)

5.2.4 kf-means

K-means is fast whereas fuzzy c-means ensures sure shot convergence, therefore we propose a

method of combining these two to find the final cluster center in optimum time and with sure

shot convergence. We carry on iterations as we do in k-means, recalling the (sum of sub-

histogram) criteria discussed above can be used to call a single iteration of fuzzy c-means to

place the cluster centers in such a manner that convergence will happen as the criteria doesn’t

sense the problem of convergence once FCM is called.

5.3 Effect of using initial cluster centers proposed by two theorems

Original Image

Normal Initial cluster centers: 32 96 159 223

Final cluster centers: 19 77 137 237

Time taken: 1.59 secs

Optimum initial cluster centers by theorem 1(t=22): 33 78 123 255

Final cluster centers: 18 75 135 236

Time Taken: 0.445 secs

Optimum initial cluster centers by theorem 1(t=22): 14 65 111 255

Final cluster centers: 16 71 132 235

Time Taken:0.703 secs

Original Histogram

0

500

1000

1500

2000

2500

0 50 100 150 200 250

Filtered Histogram

One may ask what about time taken for finding these initial cluster centers. But finding normal

cluster centers also takes time as range of image need to be calculated as if by chance cluster

center outside the range is chosen, algorithm doesn’t converge.

0 50 100 150 200 250 3000

200

400

600

800

1000

1200

1400

1600

1800

2000

5.4 Addressing problem of convergence in clustering algorithms

In spite of choosing optimum cluster centers, we cannot ensure convergence because of inherent

nature of the image. One of the reason might be discontinuous histogram. So we propose a

solution of using histogram based k-means algorithm. By which speed increases drastically. We

can have a constant check in every iteration, if algorithm can converge for given number of

cluster centers. Especially in computer vision, predefined number of clusters based on training

but still there are problems of convergence encountered.

We propose a flexible number of clusters method based on histogram.

Use optimum initial cluster centers obtained from METHOD-1.

Step 1: Divide histogram into sub histograms by criteria of mid-point between cluster centers.

Step 2: Check if sum of all frequencies in the sub histogram is greater than 0

If yes, continue.

Otherwise, delete this cluster center and merge the left part of sub histogram to left cluster center

and right part of sub histogram to right cluster center.

Original image with segmented edge Segmented image with segmented edge

Step 3: perform k-means on histogram using these cluster centers to obtain new cluster centers

by finding the weighted average of intensities within the corresponding histogram. Weights are

calculated by dividing corresponding frequency of intensity with sum of all frequencies within

the sub histogram.

Step 4: Check if difference between consecutive iteration is less than criterion predefined.

If yes, obtained cluster centers are final cluster centers.

Otherwise, go to step1.

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 50 100 150 200 250

Now if we use normal initial cluster centers=32 96 159 223, algorithm doesn’t converge.

Because 3rd

initial cluster center (159) can’t have any pixel in his cluster.

if we use cluster centers obtained by theorem 1=44 88 125 237

Final cluster centers=14 59 104 215

Time taken=0.0911 secs

if we use cluster centers obtained by theorem 2=14 75 200 255

Final cluster centers=23 91 201 250

Time taken=0.09 secs

So, using these cluster centers, algorithm converges.

In spite of using these cluster centers we cannot ensure sure shot convergence , so reduction in

number of clusters remains as only solution as discussed above when it is encountered that

algorithm cannot converge using sub-histogram criteria

Main reason behind using histogram instead of image is to have some criteria for checking if

convergence is possible and take described action.

Thus the advantages of the proposed algorithms are:

1. It automatically finds the number of clusters for segmentation which used to be a

human input, thus, heading to automatic segmentation.

2. The initial cluster centres are obtained in a simple and refined way which removes the

problems which are created due to random selection of initial cluster centres like non-

convergence, improper clustering results, etc. , and is faster than other algorithms

which use many attributes to calculate initial cluster center though the result may be a

bit poor.

3. The problem of convergence has also been removed.

4. The speed has been increased manifold as is shown in next section containing results

because of various reasons:

a. Use of histogram for iterations rather than applying it on the image itself.

b. Number of clusters is chosen faster.

c. Initial cluster centres are also chosen very fast and they are found to be near

final cluster centre which decreases the overall number of iterations.

But there is a drawback in this method of segmentation that it is susceptible to noise because

noise can bring heavy shift in the final cluster centres and thus degrade the segmentation

result. The effect of noise can be removed by using a filter before segmenting the image to

get better result.

CHAPTER 6 Auto-segmentation/auto-pilot

Introduction

In a step to go further to automatic segmentation, to be more useful for computer vision and

machine learning, we made an effort wherein we don’t even require giving the input to the

moving average filter used in our proposed algorithm. Thus, we have been able to advance

further in our research to make the algorithm more user-friendly and faster.

The program itself applies the moving average filter on the histogram of the image for different

values of ‘t’ in steps of 2. This way a number of filtered outputs will be obtained and the number

of peaks obtained from the histogram may keep on decreasing due to more and more smoothing

of the histogram. And hence the number of clusters or the value of k may also keep on changing

according to our algorithm. The program will then see what are the values of k which is/are

stable, i.e. when the size of the moving average filter is varied which value of k occur

repetitively. Then, only for those values of k the segmentation is done and the results are shown.

This way we have been able to avoid any kind of human intervention and made the program

easier and faster. Presently, if a biomedical practitioners or some other professional interested in

segmentation uses K-means algorithm, he/she would have to try various values of k and then see

which value is suitable for him/her or is giving good results, which can be tedious, but here

he/she can just give one input to the program i.e. the image to be segmented, then the program

itself will find the stable values of k and show the segmented image only for those values of k.

Auto pilot

Auto pilot is a concept of finding all stable segmentations of an image.

Initialize t=3, s=maximum clusters you want, sm=minimum clusters you want

Step 1: increment ‘t’ in step of 2

Step 2: find initial cluster centers,

Step 3: if number of clusters<’s’. s=number of clusters. Segment the image

Otherwise, go to step 1

Step 4: s<=sm, autopilot has finished its job,break

Otherwise, go to step 1

Inverse auto pilot

Initialize t=80,sm=minimum clusters you want, s=maximum clusters you want

Step 1: decrement ‘t’ in step of 1

Step 2: find initial cluster centers,

Step 3: if number of clusters>=’sm’. sm=number of clusters. Segment the image

Otherwise, go to step 1

Step 4: sm>=s, autopilot has finished its job,break.

Otherwise, go to step 1

Original image

This image has been segmented into 4,3,2 clusters by auto pilot as shown below

clusters=4

clusters=3

clusters=2

CHAPTER 7

Fast Video Segmentation with automatic

change of number of clusters

Real-time Video-segmentation

Introduction

A video is made up of many images called frames which are linked together and these frames are

displayed at such a high speed that it appears to us as if it is one single component(one video).

But, it is not so, hence, video segmentation is similar to image segmentation only major

difference being that we segment each frame of the video in order to segment it. But various

other aspects come into picture while segmenting a video since it is not random images which

are to be segmented they are all linked in a manner.

Several video segmentation algorithms have been proposed. They can be classified into three

types:

1. edge information based video segmentation,

2. image segmentation based video segmentation and

3. change detection based video segmentation.

1. Edge information based algorithms, first apply edge detector algorithms to find edge

information of each frame and then keep tracking these edges. A morphology motion

filter is also applied to find edges belonging to foreground objects. Next, a filling

technique can connect edge information to generate final object masks. This method can

deal with both still camera and moving camera situations; however, the computation load

is very large.

2. Image segmentation based algorithms first apply image segmentation algorithms, such

as watershed transform and colour segmentation on each frame to separate a frame into

many homogeneous regions. By combining motion information derived with motion

estimation, or frame difference, regions with motion vectors different from the global

motion are merged as foreground regions. These algorithms often can give segmentation

results with accurate boundaries, but the computation load for image segmentation and

motion information calculation is also high, and the region merging process often has

many parameters to set. Both these two kinds of algorithms are too complex to be

integrated into a real-time system.

3. Change detection based segmentation algorithms, threshold the frame difference to

form change detection mask. Then the change detection masks are further processed to

generate final object masks. The processing speed is high, but it is often not robust. The

segmentation results are suffered from the uncovered background situations; still object

situations, light changing, shadow, and noise. The robustness can be promoted by a lot of

post-processing algorithms; however, complex post-processing will make the efficiency

of less computation lost. The threshold of change detection is very critical and cannot be

automatically decided. These reasons make this kind of algorithms not practical for real

applications.

All these algorithms have their own specific areas of excellence like image segmentation type

algorithms can give very good segmentation results but are too heavy to be used for real-time

applications, and change detection algorithms being light are often used for real-time

applications though a compromise on quality of segmentation is to be made.

Real time video segmentation is a very difficult task since the segmentation of the frame has to

be done very fast i.e. before the next frame arrives. Though the speed of arrival of frames can be

varied but still it has to be fast enough so that it doesn’t appear as a collection of images but a

video. In general, a speed of 25frames/second is found to be sufficient but that too requires the

segmentation algorithm to be really fast.

Proposed real-time video segmentation algorithm

We noticed that since our proposed algorithm is very fast in comparison to regular K-means

algorithm we thought that we can go one more step forward and try real-time video

segmentation. Our method would qualify as the second category of segmentation i.e. image

segmentation based algorithm, wherein we segment each frame of the video separately till the

next frame arrives. These types of algorithms are famous for their segmentation quality and poor

time efficiency, but our proposed method has not only improved the quality but also has

increased the speed large enough to be used for real-time purposes which earlier was not possible

for other image segmentation type algorithms.

By using moving average filter to find the number of clusters we have been able to break the

video into variable number of clusters for each frame i.e. each frame will have its own number of

clusters. If general K-means algorithm is applied to segment a video (not even real time) in

which the number of clusters is given as an input, it cannot be given for each frame as it will be

very difficult, so K will be given for a complete video. Hence, the program will try to break

every frame into equal number of clusters. For example, if the value of k is given as 4 then the

program will try to break each frame into 4 clusters even if in a frame there is only one intensity

value. This will give rise to the problem of non-convergence since the rest of the three clusters

won’t be able to find final cluster centres and it will give error. But by our method if any of the

frame has many objects of different intensities then it will be appropriately broken into many

clusters and if there is only one intensity then it will have only one cluster.

The additional feature of auto-pilot which has been incorporated so that the program itself

decides the value of n for moving average filter to give the number of clusters brings it closer to

being unsupervised. We just need to give the input that how much detail do we require from the

video i.e. If we require good amount of detail then we can keep number of clusters to be the

maximum value of n which comes out to be stable or if we require only crude discrimination

then we can have minimum stable value of n as number of clusters.

In video segmentation, clustering has been used only for video summary, but not for

segmentation of video frame itself, because of large computing time. By using autopilot as

discussed, segmentation of video is possible with automatic change in number of clusters when

there is introduction of new object in the image. This automatic change becomes a crucial point

of consideration in computer vision.

This can be established by using the auto pilot by choosing proper limits of clusters and breaking

the autopilot at first possible segmentation itself.

This fastest video segmentation algorithm as speed can go even up to 100 frames per second

based on limits of clusters whereas speed of previous algorithms is limited to only 25 frames per

second.

In applications like where we are concerned about only watching the video segmentation we can

just use color map to give the effect of segmentation without actually getting into the image and

cause increased computation time. This makes algorithm time independent of size of frame.

Uses,

We can also use this algorithm for lossy video compression in internet video chating or

live streaming etc.

In algorithms having to segment large database of images. This can save lot of time.

Frames of same video with change in number of clusters by introduction of object in the

video

CHAPTER 8 GRAPHIC USER INTERFACE

Layout of GUI

Divided into 7 panels:

1. Loader Panel

2. Error and Selection Panel

3. Viewer Panel

4. Interaction Panel

5. Initialization Panel

6. Segmentation Panel

7. Results Panel

Loader Panel

Used for loading the image in the software

Image name should be written in ‘file name.file format’

Image is loaded as soon as Load Image button is pressed

Software works only for 2 dimensional images. Therefore Image should be first

converted into 2 dimension and then it can be loaded

Error & Selection Panel

Used for error selection & Theorem selection

Error is the minimum error allowed between consecutive iteration to stop the iterations. It

acts as breaking criteria

10 Algorithms have been loaded in the software. User can select any one of them for

segmenting the image

First 6 Algorithms are already existing and last 4 Algorithms are proposed ones

Existing Algorithms:

1. K-means

2. Fuzzy-cmean

3. Fcm for image with salt and pepper noise

4. Fcm for image with Gaussian noise

5. GKFCM

6. Modified GKFCM

Proposed Algorithms

1. K-means using histogram

2. Fuzzy c-means using histogram

3. F+k means

4. Kf-means

Viewer Panel

Axes 11 for showing image with initial cluster centers(to give an idea about

segmentation)

Axes 6 for showing original image

Initialization panel

Axes 1 is used for showing original histogram and filtered histogram

Slider is used for giving size of moving average filter as user input

Popup menu to select algorithm for finding initial cluster centers

Provision for starting with normal cluster centers as ‘Normal’

KHT and FHT for speeding up the segmentation of images that can use only image based

algorithms like image with noise

Crop and Zoom In buttons for cropping the images and finding the pixel values

respectively.

Interaction Panel

Used for snapshot

Live segmentation demo(for computer vision )

Chating demo

User input: No. of frames

Segmentation Panel

Display and editing of initial cluster centers

Starting Segmentation

Choosing color map and color of edge

For choosing variables controlling neighborhood effect

Choosing Auto Pilots:

1. Auto pilot using k-means

2. Auto Pilot using fuzzy c-means

Results Panel

Display of final cluster centers.

Display of time taken for segmentation algorithm.

Display of number of iterations carried out.

CHAPTER 9 Experimental Results

Original Image

Original Histogram

Filtered Histogram

Comparision between speeds for initial cluster centers found with different theorems

(Algorithm used is simple k-means).

Normal Initial cluster centers for number of clusters as 3= [24 73 122]

Final cluster centers= [19 47 91], Time=0.3773 secs, iterations=12

Initial cluster centers using theorem 1 for ‘t’ as 15= [19 39 99]

Final cluster centers= [18 45 90], Time=0.1466 secs, iterations=4

Initial cluster centers using theorem 2 for ‘t’ as 15= [0 27 86]

Final cluster centers= [13 38 88], Time=0.2331 secs, iterations=7

Observe the reduction in no. of iterations required to get the result.

Since, we have found fastest results with theorem-1, Segmentation results in that case are being shown:

Comparision between results obtained by different algorithms(using initial cluster centers

of theorem -1):

Initial Cluster centers:[34 94 128]

k-means:

Fuzzy c-means:

k-means using proposed method:

Fcm using propsed method:

F+k means:

kf-means

Segmentations results obtained by kf-means:

Segmentation of noisy images:

(i)Salt & Pepper noise:

BCFCM_S2 is used

Initial Cluster Centers: [32 78 123 255]

‘t’=28

‘af’=10

‘sig’=20

cluster1 cluster2

cluster3 cluster4

Original Image with segmented edge Segmented Image with segmented edge

Edge obtained by matlab Edge obtained by propesed method

(ii)Gaussian Noise

Algorithm used=BCFCM_S1

Initial cluster centers=[57 77 255]

‘t’=41

‘af’=60

‘sig’=20

cluster1 cluster2 cluster3

Original Image with segmented edge Segmented Image with segmented edge

GKFCM:

Edge obtained by matlab Edge obtained by propesed method

Cropping

Sometimes detail required in the mage may be very small portion of entire image. In that case cropping

is necessary to obtain useful results so that the detail plays a significant role in the segmentation

Cropped image

To be able to see the tumor clearly

Tumor has got separated in cluster 5.

cluster1 cluster2 cluster3

cluster4 cluster5

CHAPTER 10 ACCOMPLISHMENTS, FUTURE RESEARCH AND APPLICATIONS

Accomplishments:

1. We have been able to implement six different algorithms for clustering K-means, Fuzzy

C-means, BCFCM, KFCM_S1, KFCM_S2 and GKFCM.

2. We have been able to make modifications to already existing algorithms to remove the

problem of number of clusters, initial cluster centres and non-convergence.

3. Four new methods have been proposed by us.

4. By speeding-up of the algorithms we have been able to do real-time image segmentation.

Future research or development:

1. Better selection of number of clusters can be done, though auto-pilot gives decent results

and is fast but it can be further improved depending upon the applications.

2. The algorithm is much faster than required to do 25frames/second so the work can be

done on improving the quality of the segmented video.

3. A further research can be done on choosing better membership function and relationship

between pixel and its neighbours that removes more noise gives better segmentation

results.

Applications:

1. For many segmentation algorithm K-means is pre-processing step on which other

segmentation tools are used so our method can speed-up segmentation and quality of

output as well , and can speed-up other processes requiring clustering like data mining,

information retrieval, etc.

2. By using the auto-pilot option the K-means algorithm has been made more unsupervised

and can be useful in computer vision and machine learning.

3. Real-time video segmentation can be useful in video-surveillance where data size is large

and time to survey it is less or in on-line chatting when the speed of transmission is poor

then the segmented image can be compressed more for fast transmission and retrieval.

References

1. Digital Image Processing using MATLAB, by Gonzalez, Woods and Eddins, 2009

2. Miin-Shen Yang, Hsu-Shen Tsai, 2008, ‘A Gaussian kernel-based fuzzy c-means

algorithm with a spatial bias correction’

3. Erlend Hodneland ,July, 2003, ‘Segmentation of Digital Image’

4. Ursula Gonzales-Baron, Francis Butler, March 2005, ‘A comparison of seven

thresholding techniques with the k-means clustering algorithm for measurement of bread-

crumb features by digital image analysis’

5. D.A. Clausi, 2002, K-means Iterative Fisher (KIF) unsupervised clustering algorithm

applied to image texture segmentation

6. Likas, Nikos Vlassis, Jakob J. Verbeek, 2003, The global k-means clustering algorithm

7. Sadullah Sakallioglu, Murat Erisoglu, Nazif Calis ,2011, ‘A new algorithm for initial

cluster centers in k-means algorithm 2011’

8. Liang Bai, Jiye Liang, Chuangyin Dang, 2011, ‘An initialization method to

simultaneously find initial cluster centers and the number of clusters for clustering

categorical data’

9. Fuyuan Cao, Jiye Liang, Guang Jiang, 2009, ‘An initialization method for the K-Means

algorithm using neighborhood model’

10. A.K. Jain, R.C. Dubes, Algorithms for Clustering Data

11. K.K. Chen, L. Liu, 2008, ‘Best K: critical clustering structures in categorical datasets’

12. J.J. Li, M.K. Ng, Y.M. Cheng, Z.H. Huang, 2008, ‘Agglomerative fuzzy k-means

clustering algorithm with selection of number of clusters’

13. J.M Peña, J.A Lozano, P Larrañaga, 1999, An empirical comparison of four initialization

methods for the K-means algorithm

14. Yasira Beevi and Dr. S. Natarajan, Dec 2009, An efficient Video Segmentation

Algorithm with Real time Adaptive Threshold Technique

Accomplishments:

5. We have been able to implement six different algorithms for clustering

Future research or development:

4. Better selection of number of clusters can be done, though auto-pilot gives decent results

and is fast but it can be further improved depending upon the applications.

5. The algorithm is much faster than required to do 25frames/second so the work can be

done on improving the quality of the segmented video.

6. A further research can be done on choosing better membership function and relationship

between pixel and its neighbours that removes more noise gives better segmentation

results.

Applications:

4. For many segmentation algorithm K-means is pre-processing step on which other

segmentation tools are used so our method can speed-up segmentation and quality of

output as well , and can speed-up other processes requiring clustering like data mining,

information retrieval, etc.

5. By using the auto-pilot option the K-means algorithm has been made more unsupervised

and can be useful in computer vision and machine learning.

6. Real-time video segmentation can be useful in video-surveillance where data size is large

and time to survey it is less or in on-line chatting when the speed of transmission is poor

then the segmented image can be compressed more for fast transmission and retrieval.