Model-based fuzzy c-shells clusteringprofdoc.um.ac.ir/articles/a/1022796.pdf · Model-based fuzzy c-shells clustering ... image processing [1–3]. The aim of clustering is ﬁnding

ORIGINAL ARTICLE

Model-based fuzzy c-shells clustering

Hadi Hossein-Abad Mahdipour • Morteza Khademi •

Hadi Yazdi Sadoghi

Received: 20 September 2010 / Accepted: 11 March 2011 / Published online: 9 April 2011

� Springer-Verlag London Limited 2011

Abstract In this paper, a new shell clustering method is

presented to cluster model-based shells in 2-dimensions.

Shells, that each one of them can be expressed by a center

and non-negative radius in each angle (by considering

polar coordinate system), can cluster by using the proposed

model-based fuzzy c-shells (MFCS) clustering method. In

this paper, firstly one of the most famous traditionally

clustering methods, i.e. fuzzy c-spherical shells (FCSS)

clustering method, is extracted from the proposed MFCS

clustering method as a specific state of it. Then, the per-

formance of proposed method is examined in three exam-

ples when it is applied over shells with various shapes in

2-dimensions. Since the resulted systems of equations in

the studied examples cannot be solved directly, the particle

swarm optimization (PSO) algorithm is used to numeri-

cally solve the resulted equations systems. The simulation

results show the acceptable performance of the proposed

MFCS method.

Keywords Model-based fuzzy c-shells (MFCS) �Fuzzy c-spherical shell (FCSS) � Clustering

1 Introduction

Clustering is a significant point of interest for researchers

in the pattern recognition area. Such methods have been

widely applied in various research areas such as taxonomy,

geology, business, engineering systems, medicine, and

image processing [1–3]. The aim of clustering is finding

data structure and also partitioning data sets into groups

with similar individuals. The hard clustering methods

restrict each point of the data set to exactly one cluster [4].

Zadeh [5] has proposed fuzzy sets that can produce partial

membership of belonging, described by a membership

function. Fuzzy clustering has been widely studied as well

as applied in variety of substantive areas. After the Zadeh’s

masterwork, researches have been focused on applying

fuzzy state to crisp cases. In the literature, the fuzzy

c-mean (FCM) clustering algorithm is the most well-known

method [2, 6]. The fuzzy C-shells (FCS) algorithm pro-

posed by Dave has been successful in clustering spherical

shells [7]. This algorithm has been further generalized to

adaptive FCS (AFCS) for the case of elliptical shells [8].

Krishnapuram et al. [9] have developed the fuzzy

C-spherical shells (FCSS) algorithm to reduce the com-

putational costs of FCS by introducing an algebraic (non-

Euclidean) distance measure. In this way, the prototypes

can be calculated directly and therefore, solution of cou-

pled nonlinear equations would not necessarily be as the

FCS case. Specifically, for two-dimensional (2D) cases,

Man and Gath developed the fuzzy C-rings (FCR)

algorithm [10] for clustering ring data, while Gath and

Hoory proposed the fuzzy C-ellipses (FCE) algorithm

[11] for ellipse data. Krishnapuram et al. [12] have

developed the fuzzy C-quadric shells (FCQS) algorithm,

which detects quadrics like circles, ellipses, hyperbolas, or

lines.

H. Hossein-Abad Mahdipour � M. Khademi

Electrical Engineering Department,

Ferdowsi University of Mashhad, Mashhad, Iran

e-mail: [email protected]

M. Khademi


H. Yazdi Sadoghi (&)

Computer Engineering Department,

Ferdowsi University of Mashhad, Mashhad, Iran


123

Neural Comput & Applic (2012) 21 (Suppl 1):S29–S41

DOI 10.1007/s00521-011-0571-0

The clustering algorithms for detecting rectangular

shells have also been presented in the literature, such as

the norm-induced shell prototypes (NISP) algorithm by

Bezdek et al. [13] and the fuzzy C-rectangular shell

(FCRS) algorithm by Hoeppner [14]. Wang [15] has pre-

sented a type of alternating optimization-based probabi-

listic c-shell algorithm for clustering template-based

shapes. Also, Song et al. [16] have presented information

fuzzy C-spherical shells (IFCSS) algorithm that tackles the

intertwined robust fuzzy clustering problems of outlier

detection, prototype initialization, and cluster validity in a

unified framework of information clustering.

In this paper, the conventional FCSS clustering method

is extended to cluster the model-based shells. Shells, with a

center and non-negative radius in each angle (by consid-

ering polar coordinate system), can be clustered by using

the proposed model-based fuzzy c-shells (MFCS) cluster-

ing method. We have applied proposed MFCS over real

numbers in 2-dimensions.

The remainder of this paper is organized as follows. The

conventional FCSS clustering method, applied over real

numbers, is presented in Sect. 2. Section 3 presents the

proposed MFCS clustering method in 2-dimensions. While

the proposed MFCS method is applied over four various

model-based shells, the correspondent relations are

obtained in Sect. 4. Simulation results of MFCS imple-

menting, applied over three examples of Sect. 4, is pre-

sented in Sect. 5. Finally, the paper is concluded in Sect. 6.

2 Preliminaries

In this section, firstly we explain the conventional FCSS.

Furthermore, it will be shown the resulted systems of equa-

tions in this paper cannot be solved directly; therefore, as we

have employed particle swarm optimization (PSO) algo-

rithm as a numeric approach to solve the systems of equa-

tions, in this section the PSO algorithm will be explained too.

2.1 FCSS clustering algorithm

The aim of the FCSS algorithm is minimizing the follow-

ing objective function to find fuzzy memberships uik , and

clusters with centers vi and radiuses ri for i ¼ 1; 2; . . .; c

and k ¼ 1; 2; . . .; n [9]:

JðU;V ;RÞ ¼Xn

k¼1

Xc

i¼1

umi;kd2 xk; vi; rið Þ ð1Þ

d2 xk; vi; rið Þ ¼ xk � vik k2�r2i

� �2

¼ xk � við ÞT xk � við Þ � r2i

� �2; ð2Þ

where X ¼ x1; . . .; xnf g denotes the set of input numbers,

xk ¼ xk;1; . . .; xk;p

� �T2 Rp, k ¼ 1; 2; . . .; n, is the kth input

number, c is the number of clusters, and m [ 1 is the

fuzziness index. The matrix U ¼ ui;k

� �c�n

is called a

constrained fuzzy c partition of X, if the entries of U

satisfy:

Xc

i¼1

ui;k ¼ 1; where ui;k 2 0; 1½ �; for i ¼ 1; . . .; c

and k ¼ 1; . . .; n ð3Þ

where uik is the membership grad of the kth input number to

the ith cluster, V ¼ v1 ; . . .; vcf g is clusters centers set,

vi ¼ vi;1; . . .; vi;p

� �T2 Rp, i ¼ 1; 2; . . .; c, is the center of ith

cluster and R ¼ r1; . . .; rcf g is the set of spherical clusters

radiuses.

Using Lagrange multiplayers, we can minimize

JðU;V ;RÞ subject to (3) and get to updated relations of

unknown parameters uik , vi, and ri as follows:

L V ;R;U; kð Þ ¼Xn

k¼1

Xc

i¼1

umi;k xk � við ÞT xk � við Þ � r2

i

� �2

�Xn

k¼1

kk

Xc

i¼1

ui;k � 1

!; ð4Þ

ui;k ¼Xc

j¼1

d2 xk; vi; rið Þd2 xk; vj; rj

� � ! 1

m�1

0

@

1

A�1

¼Xc

j¼1

xk � við ÞT xk � við Þ � r2i

� �

xk � vj

� �Txk � vj

� �� r2

j

� �

0

@

1

A

2m�1

0

B@

1

CA

�1

;

for i ¼ 1; 2; . . .; c and k ¼ 1; 2; . . .; n

ð5Þ

oL=ovi¼ 0; oL=ori

¼ 0)vi ¼ � 1

2gi;1; gi;2; . . .; gi;p

� �

ri ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivt

ivi � gi;pþ1

p ;

(

for i ¼ 1; 2; . . .; c: ð6Þ

where :

gi ¼ 12H�1

i wi for i¼ 1;2; . . .;c: ð6:1Þ

Hi ¼Pn

k¼1

Umi;kykyT

k for i¼ 1;2; . . .;c: ð6:2Þ

wi ¼Pn

k¼1

Umi;ksk for i¼ 1;2; . . .;c: ð6:3Þ

sk ¼ 2 xTk xk

� �yk for k¼ 1;2; . . .;n: ð6:4Þ

yk ¼ xk;1;xk;2; . . .;xk;p;1� �T

for k¼ 1;2; . . .;n: ð6:5Þ

8>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>:

2.2 Particle swarm optimization

A PSO system [17, 18] starts with the random initialization

of a population (swarm) of individuals (particles) in the

S30 Neural Comput & Applic (2012) 21 (Suppl 1):S29–S41

123

search space and works on the social behavior in the

swarm. The position and the velocity of the lth particle

in the d-dimensional search space can be represented as

Xl ¼ xl;1; . . .; xl;d

� �Tand Sl ¼ sl;1; . . .; sl;d

� �T, respectively.

Each particle has its own best position (pbest) Pl ¼pl;1; . . .; pl;d

� �Tcorresponding to the personal best objective

value obtained so far at time t. The global best particle

(gbest) is denoted by Pg, which represents the best particle

found so far at time t in the entire swarm. The new velocity

of each particle is calculated as follows:

sl;j t þ 1ð Þ ¼ x sl;j tð Þ þ c1r1 pl;j � xl;j tð Þ� �

þ c2r2 pg;j � xl;j tð Þ� �

; j ¼ 1; . . .; d; ð7Þ

where c1 and c2 are acceleration coefficients, xis inertia

factor, and r1 and r2 are two independent random numbers

uniformly distributed in the range of 0; 1½ �. Thus, the

position of each particle is updated in each generation

according to the following equation:

xl;j t þ 1ð Þ ¼ xl;j tð Þ þ sl;j t þ 1ð Þ; j ¼ 1; . . .; d ð8Þ

Generally, the value of each component in Sl(sl;j) can be

clamped to the range �smax; smax½ � to control excessive

roaming of particles outside the search space. Then, the

particles file toward new positions according to (8). This

process is repeated until a user-defined stopping criterion is

reached.

Finally, the procedure of standard PSO can summarized

as follows:

Step 1 Initialize a population of particles with random

positions and velocities, where each particle contains d

variables.

Step 2 Evaluate the objective values of all particles; let

pbest of each particle (Pl) and its objective value equal to

its current position and objective value, respectively; and

let gbest (Pg) and its objective value equal to the position

and objective value of the best initial particle,

respectively.

Step 3 Update the velocity and position of each particle

according to (7) and (8).

Step 4 Evaluate the objective values of all particles.

Step 5 For each particle, compare its current objective

value with the objective value of its pbest. If current value

is better, then update pbest and its objective value with

the current position and objective value.

Step 6 Determine the best particle of the current swarm

with the best objective value. If the objective value is

better than the objective value of gbest, then update gbest

and its objective value with the position and objective

value of the current best particle.

Step 7 If a stopping criterion is met, then output gbest and

its objective value; otherwise go back to Step 3.

3 MFCS clustering

Let we want cluster any model-based shells in 2-dimen-

sions (p ¼ 2). It is supposed that the desired shells have a

center and a non-negative derivable radius in each angle h,

0� h\2p (by considering polar coordinate system). For

example, let the radius has one of the following relations

with respect to angle h:

f hð Þ ¼ 1; 0� h\2p ð9Þ

f hð Þ ¼ffiffiffiffiffi2hp

; 0� h\2p ð10Þ

f hð Þ ¼ h 2p� hð Þ h� pð Þ2; 0� h\2p ð11Þf hð Þ ¼ 1þ sin hð Þ; 0� h\2p ð12Þ

Each one of the (9–12) equations is called as the basic

structure. Demonstration of mentioned basic structures

regarding to angle h, in Cartesian and polar coordinate

systems, are illustrated in Figs. 1, 2, 3, and 4.

In MFCS, the input numbers (xk, k ¼ 1; . . .; n) must be

clustered to c clusters. Suppose Ki set, for i ¼ 1; . . .; c,

represents the input numbers that are clustered as ith cluster

(these numbers will have maximum membership grades to

ith cluster respect to other clusters). In noise-free case, the

members of Ki set are truth in the relation of ith desired

model-based shell where the desired model-based shells

are scaled, rotated, and moved shells of a specific and

known basic structure f hð Þ. Indeed we have:

Ki ¼ xk : xk � vik k2�R2i hkð Þ

� �2

¼ 0

for k ¼ 1; . . .; n and i ¼ 1; . . . ; c

xkf gnk¼1¼

Sc

i¼1

Ki

Ki

TKj ¼ fg; for 1� i\j� c

9>>>>>>>>>>=

>>>>>>>>>>;

ð13Þ

where Ri hkð Þ ¼ rif hi;k � hi;0

� �and hi;k ¼ tan�1 xk;2�vi;2

xk;1�vi;1

� �.

For input numbers (xk, k ¼ 1; . . .; n) and a defined and

known basic structure f hð Þ, the proposed MFCS method

will find centers vi, scales coefficients ri, rotations angles

hi;0 and suitable membership grades of each input number

to clusters (uik ) for i ¼ 1; . . .; c and k ¼ 1; . . .; n.

In the proposed MFCS clustering method, similar to

FCSS method case, distance definition and resulted

Lagrange function (to minimize (1)) are as follows:

Neural Comput & Applic (2012) 21 (Suppl 1):S29–S41 S31

123

d2 xk; vi; ri; hi;0

� �¼ xk � vik k2�R2

i hkð Þ� �2

¼ xk � við ÞT xk � við Þ � r2i f 2 hi;k � hi;0

� �� 2

ð14Þ

L U;V ;R; h; kð Þ ¼Xn

k¼1

Xc

i¼1

umi;kd2 xk; vi; ri; hi;0

� �

þXn

k¼1

kk

Xc

i¼1

ui;k � 1

!ð15Þ

where hi;k ¼ tan�1 xk;2�vi;2

xk;1�vi;1

� �; vi, ri and hi;0 for i ¼ 1; . . .; c

and k ¼ 1; . . .; n are unknown and f hð Þ is a known basic

structure. Similar to conventional FCSS and (5), the

membership value of ui;k can be updated as follows:

ui;k¼Xc

j¼1

d2 xk;vi;ri;hi;0

� �

d2 xk;vj;rj;hj;0

� � ! 1

m�1

0

@

1

A�1

¼Xc

j¼1

xk�við ÞT xk�við Þ�r2i f 2 hi;k�hi;0

� ��

xk�vj

� �Txk�vj

� ��r2

j f 2 hj;k�hj;0

� ��

0@

1A

2m�1

0

B@

1

CA

�1

ð16Þ

where i ¼ 1; 2; . . .; c and k ¼ 1; 2; . . .; n.

Let ti set (ti ¼ vi;1; vi;2; . . .; vi;p; ri; hi;0

� �¼ ti;j

� �pþ2

j¼1,

ti;j ¼ vi;j for j ¼ 1; . . .; p, ti;jþ1 ¼ ri and ti;jþ2 ¼ hi;0)

Fig. 1 f hð Þ ¼ 1 for 0� h\2p,

in Cartesian coordinate system

(left side) and polar coordinate

system (right side)

Fig. 2 f hð Þ ¼ffiffiffiffiffi2hp

for

0� h\2p, in Cartesian

coordinate system (left side) and

polar coordinate system (rightside)


123

includes independent and unknown parameters of the ith

cluster. In 2-dimensions (p ¼ 2), to achieve the updated

relations of ti;j for i ¼ 1; 2; . . .; c and j ¼ 1; . . .; 4, we must

calculate derivation of Lagrange function with respect to

parameter ti;j and put resulting equations equal to zero for

i ¼ 1; 2; . . .; c and j ¼ 1; . . .; 4. For this purpose, we start

from a used procedure in [9] and define qi xkð Þ vector for

i ¼ 1; 2; . . .; c and k ¼ 1; 2; . . .; n; yk; sk vectors for

k ¼ 1; 2; . . .; n; Mk matrix for k ¼ 1; 2; . . .; n, and bk scalar

for k ¼ 1; 2; . . .; n as follows:

qi xkð Þ ¼ �2vi;1;�2vi;2; vTi vi � r2

i f 2 hi;k � hi;0

� �� T

for i ¼ 1; 2; . . .; c and k ¼ 1; 2; . . .; n

yk ¼ xk;1; xk;2; 1� �T

; Mk ¼ ykyTk ; sk ¼ 2 xT

k xk

� �yk;

bk ¼ xTk xk

� �2for k ¼ 1; 2; . . .; n

ð17Þ

According to the above definitions, the algebraic

distance of (14) can be rewritten as follows (see [9]):

Fig. 3 f hð Þ ¼ h 2p� hð Þ h� pð Þ2 for 0� h\2p, in Cartesian coordinate system (left side) and polar coordinate system (right side)

Fig. 4 f hð Þ ¼ 1þ sin hð Þ for

0� h\2p, in Cartesian

coordinate system (left side) and

polar coordinate system (rightside)


123

d2 xk; vi; ri; hi;0

� �¼ qT

i xkð ÞMk qi xkð Þ þ sTk qi xkð Þ þ bk ð18Þ

As all independent and unknown parameters ti;j, in (18),

only are evident in qi xkð Þ phrase, therefore, we can apply

partial derivation to get:

oL

oti;j¼Xn

k¼1

oqTi xkð Þoti;j

:oL

oqi xkð Þ¼ 0; for i ¼ 1; 2; . . .; c and

j ¼ 1; . . .; 4: ð19Þ

If we denote oqTi xkð Þ

oti;j with nti;j xkð Þ and rewrite

oL=oqi xkð Þ by using (15) and (18), the resulted equation

from (19) will be as follows:

nti;j xkð Þ ¼ oqTi xkð Þ

oti;j; for i ¼ 1; 2; . . .; c;

j ¼ 1; . . .; 4 and k ¼ 1; . . .; n: ð20ÞXn

k¼1

umi;k nti;j xkð Þ 2Mk qi xkð Þ þ skð Þ ¼ 0;

for ı ¼ 1; 2; . . .; c and j ¼ 1; . . .; 4: ð21Þ

Therefore, we can get to updated relations of ti;jparameters as follows:

Pn

k¼1

umi;k nti;j xkð Þ 2 Mk qi xkð Þ þ skð Þ ¼ 0

for j¼ 1;2;3;4 and i¼ 1; . . .; c

9=

; )After Solving the

Resulted Equations Systems

ti;1 or vi;1 ¼ � � �ti;2 or vi;2 ¼ � � �ti;3 or ri ¼ � � �ti;4 or hi;0 ¼ � � �

8>>><

>>>:for i¼ 1; . . . ;c:

ð22Þ

where Mk, qi xkð Þ and sk are obtained from (17). However

when qi xkð Þ and nti;j xkð Þ are independent from input

numbers, i.e. qi xkð Þ ¼ qi and nti;j xkð Þ ¼ nti;j , the optimum

qi (~qi, that can be obtained from (21)) and updated relations

of ti;j can be obtained as follows:

nti;j

Xn

k¼1

umi;k 2 Mk qi þ skð Þ ¼ 0 )

nti;j6¼0

qið Þopt:

¼ ~qi ¼ �1

2

Xn

k¼1

umi;k Mk

!�1 Xn

k¼1

umi;k sk

!;

for i ¼ 1; 2; . . .; c: ð23Þ

nti;j qi ¼ nti;j ~qi

for j ¼ 1; 2; 3; 4 and i ¼ 1; . . .; c

)

After Solving the

Resulted Equations Systems

ti;1 or vi;1 ¼ � � �ti;2 or vi;2 ¼ � � �ti;3 or ri ¼ � � �ti;4 or hi;0 ¼ � � �

8>>><

>>>:for i ¼ 1; . . .; c: ð24Þ

For completing the proposed MFCS clustering method,

we obtain nti;j xkð Þ for all ti;j ( ti;j� �4

j¼1¼ vi;1; vi;2; ri; hi;0

� �for

i ¼ 1; 2; . . .; c) according to the definition of qi xkð Þ and

nti;j xkð Þ in (17) and (20) as follows:

nti;j xkð Þ��j¼1¼ nvi;1

xkð Þ

¼ �2; 0; 2vi;1 � 2r2i fi;kf 0i;k

xk;2�vi;2

xk;1�vi;1ð Þ2þ xk;2�vi;2ð Þ2� �


xkð Þ

¼ 0;�2; 2vi;2 þ 2r2i fi;kf 0i;k

xk;1�vi;1

xk;1�vi;1ð Þ2þ xk;2�vi;2ð Þ2� �

nti;j xkð Þ��j¼3¼ nri

xkð Þ ¼ 0; 0;�2rif2i;k

h i

nti;j xkð Þ��j¼4¼ nhi;0

xkð Þ ¼ 0; 0; 2r2i fi;kf 0i;k

h i

8>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>:

;

for i ¼ 1; 2; . . . ; c:

ð25Þ

where fi;k ¼ f hð Þjh¼hi;k�hi;0, f 0i;k ¼

of hð Þoh

��h¼hi;k�hi;0

and hi;k ¼tan�1 xk;2�vi;2

xk;1�vi;1

� �for i ¼ 1; 2; . . .; c and k ¼ 1; 2; . . .; n.

Therefore, for each ri hkð Þ, if qi xkð Þ (resulted from (17))

and nti;j xkð Þ (resulted from (25)) be dependent or indepen-

dent from input numbers (xk), the equations (22) and (24)

are, respectively, usable to create an equations system with

four equations and four unknown parameters vi;1 ; vi;2 ; ri

and hi;0 for i ¼ 1; 2; . . .; c. Since these systems of equations

are related to ri hkð Þ, we create and examine these equations

systems for four examples in the next section.

4 Examples

In this section, we use the proposed MFCS method to

cluster shells in four different examples. In the first

example, we extract the conventional FCSS clustering

method from the proposed MFCS method where (9) is used

as basic structure. In the next examples, we use MFCS to

cluster shells that have basic structures (10–12).

4.1 First example: extracting the conventional FCSS

from the proposed MFCS

We can extract theoretically most of traditional fuzzy

c-shells clustering methods, e.g. spherical, elliptical, and

etc., in 2-dimensions from the proposed MFCS method. The

extracting of these traditional methods is very similar to

extracting of FCSS method that will be obtained in this

subsection. Of course it is obvious; as a result, the perfor-

mance of the MFCS and its simulation results in these cases

is precisely same with correspondent traditional methods.


123

In order to cluster spherical shells, we consider f hð Þ ¼ 1

and Ri hkð Þ ¼ ri; where ri for i ¼ 1; 2; . . .; c are radiuses of

spherical shells. Therefore, it can be seen f 0 hð Þ ¼ 0. In this

case qi xkð Þ ¼ �2vi;1;�2vi;2; vTi vi � r2

i

� �T¼ qi and we can

rewrite (25) as follows:


xkð Þ ¼ �2; 0; 2vi;1

� �¼ nti;j

��j¼1¼ nvi;1


xkð Þ ¼ 0;�2; 2vi;2

� �¼ nti;j

��j¼2¼ nvi;2


xkð Þ ¼ 0; 0;�2ri½ � ¼ nti;j

��j¼3¼ nri


xkð Þ ¼ 0; 0; 0½ � ¼ nti;j

��j¼4¼ nhi;0

8>>>>>>>>>>>><

>>>>>>>>>>>>:

ð26Þ

Since in this case, qi xkð Þ and nti;j xkð Þ are independent

from the input numbers (qi xkð Þ ¼ qi and nti;j xkð Þ ¼ nti;j ), so

we use (24) to achieve systems of equations. Resulted

systems of equations in this case are as follows:

2vi;1 þ vi;1 v2i;1 þ v2

i;2 � r2i

� �¼ �~qi;1 þ vi;1 ~qi;3 ð27Þ

2vi;2 þ vi;2 v2i;1 þ v2

i;2 � r2i

� �¼ �~qi;2 þ vi;2 ~qi;3 ð28Þ

v2i;1 þ v2

i;2 � r2i ¼ ~qi;3 ð29Þ

where ~qi is obtained from (23) and i ¼ 1; . . .; c. To solve

the above equations system, we use (29) to replace the term

~qi;3 in (27) and (28) with the term v2i;1 þ v2

i;2 � r2i . Finally,

we can conclude the updated relations of (6) for this

example (FCSS clustering method ([9]) extracting from

MFCS in 2-dimensions (p ¼ 2)). In this case, we cannot

calculate hi;0 and it can have any value in 0; 2p½ � for

i ¼ 1; 2; . . .; c.

4.2 Second example: clustering of shells with radical-

based structure

In this subsection, we cluster shells that have radical-based

radiuses; i.e. f hð Þ ¼ffiffiffiffiffi2hp

and Ri hkð Þ ¼ ri

ffiffiffiffiffiffiffiffiffi2hi;k

pfor i ¼

1; 2; . . .; c and k ¼ 1; 2; . . .; n. In this case, it is supposed

that hi;0 ¼ 0 for i ¼ 1; 2; . . .; c. Therefore, fi;k and f 0i;k cal-

culate as follows:

fi;k ¼ f hð Þjh¼hi;k�hi;0¼


p

f 0i;k ¼of hð Þoh


¼ 1 ffiffiffiffiffiffiffiffiffi

2hi;k

p

8><

>:ð30Þ

In this case, we can rewrite (25) as follows:


xkð Þ

¼ �2; 0; 2vi;1 � 2r2i

xk;2 � vi;2

� �

xk;1 � vi;1

� �2þ xk;2 � vi;2

� �2

" #


xkð Þ

¼ 0;�2; 2vi;2 þ 2r2i

xk;1 � vi;1

� �

xk;1 � vi;1

� �2þ xk;2 � vi;2

� �2

" #


xkð Þ ¼ 0; 0;�4rihi;k

� �


xkð Þ ¼ 0; 0; 0½ �

8>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>:

ð31Þ

In (31), nti;j xkð Þ are dependent to the input numbers.

Therefore, we use (22) to get the systems of equations as

follows:

Xn

k¼1

umi;k �2; 0; 2vi;1 � 2r2

i

xk;2 � vi;2

� �

xk;1 � vi;1

� �2þ xk;2 � vi;2

� �2

" #

2 Mk qi xkð Þ þ skð Þ ¼ 0 ð32Þ

Xn

k¼1

umi;k 0;�2; 2vi;2 þ 2r2

i

xk;1 � vi;1

� �

xk;1 � vi;1

� �2þ xk;2 � vi;2

� �2

" #

2 Mk qi xkð Þ þ skð Þ ¼ 0 ð33ÞXn

k¼1

umi;k 0; 0;�4rihi;k

� �2 Mk qi xkð Þ þ skð Þ ¼ 0 ð34Þ

Xn

k¼1

umi;k 0; 0; 0½ � 2Mk qi xkð Þ þ skð Þ ¼ 0 ð35Þ

where Mk, qi xkð Þ and sk are obtained from (17) and hi;k ¼tan�1 xk;2�vi;2

xk;1�vi;1

� �and i ¼ 1; . . .; c. It is observed the above

system of equations (32–34) is a nonlinear equations

system respect to unknown parameters vi;1 ; vi;2 and ri

for i ¼ 1; . . .; c; because of appear hi;k ¼ tan�1 xk;2�vi;2

xk;1�vi;1

� �,

��v2

i;1þv2

i;2þ��, r2

i and etc. phrases. Therefore, we cannot solve

above systems of equations directly. The PSO numerical

and iterative approach is employed to solve this equations

system for i ¼ 1; . . .; c.

4.3 Third example: clustering of shells

with polynomial-based structure

In this subsection, we cluster shells that have polynomial-based

radiuses; i.e. f hð Þ ¼ h 2p� hð Þ h� pð Þ2 and Ri hkð Þ ¼ rihi;k

2p� hi;k

� �hi;k � p� �2

for i ¼ 1; 2; . . .; c and k ¼ 1; 2; . . .; n.


123

In this case, it is supposed hi;0 ¼ 0 for i ¼ 1; 2; . . .; c.

Therefore, fi;k and f 0i;k have the following forms:

fi;k ¼ f hð Þjh¼hi;k�hi;0¼ hi;k 2p� hi;k

� �hi;k � p� �2



¼ 2p� hi;k

� �hi;k � p� �2

�hi;k hi;k � p� �2þ2hi;k 2p� hi;k

� �hi;k � p� �

8>><

>>:ð36Þ

In this case, similar to previous case, we can get to

nti;j xkð Þ by using (25) as follows:

Because nti;j xkð Þ dependent on the input numbers, we

use (22) to get the systems of equations as same as

previous example. As the size of resulted equations

systems in this case is very large and furthermore getting

to it is very simple and similar to the previous case,

we avoid expressing of these equations systems in

thiscase.

4.4 Fourth example: clustering of shells

in sinusoidal-based structure

In this section, we cluster shells that have sinusoidal

radiuses; i.e. f hð Þ ¼ 1þ sin hð Þ and Ri hkð Þ ¼ ri 1þðsin hi;k � hi;0

� �Þ for i ¼ 1; 2; . . .; c and k ¼ 1; 2; . . .; n. In

this case, fi;k and f 0i;k have the following forms:

fi;k ¼ f hð Þjh¼hi;k�hi;0¼ 1þ sin hi;k � hi;0

� �



¼ cos hi;k � hi;0

� �

8><

>:ð38Þ

The expressing of (25) in this case is as follows:


xkð Þ

¼ �2; 0; 2vi;1 � 2r2i

1þsin hi;k�hi;0ð Þð Þ cos hi;k�hi;0ð Þ xk;2�vi;2ð Þxk;1�vi;1ð Þ2þ xk;2�vi;2ð Þ2

� �


xkð Þ

¼ 0;�2; 2vi;2 þ 2r2i

1þsin hi;k�hi;0ð Þð Þ cos hi;k�hi;0ð Þ xk;1�vi;1ð Þxk;1�vi;1ð Þ2þ xk;2�vi;2ð Þ2

� �


xkð Þ ¼ 0; 0;�2ri 1þ sin hi;k � hi;0

� �� 2h i


xkð Þ¼ 0; 0; 2r2

i 1þ sin hi;k � hi;0

� �� cos hi;k � hi;0

� ��

8>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>:

ð39Þ

As nti;j xkð Þ are dependent to input numbers, therefore, we

use (22) to get systems of equations as follows:


xkð Þ ¼ �2; 0; 2vi;1 � 2r2i hi;k 2p� hi;k

� �hi;k � p� �2

h

� � � 2p� hi;k

� �hi;k � p� �2�hi;k hi;k � p

� �2þ2hi;k 2p� hi;k

� �hi;k � p� ��

xk;2�vi;2

xk;1�vi;1ð Þ2þ xk;2�vi;2ð Þ2�


xkð Þ ¼ 0;�2; 2vi;2 þ 2r2i hi;k 2p� hi;k

� �hi;k � p� �2

h

� � � 2p� hi;k

� �hi;k � p� �2�hi;k hi;k � p


� �hi;k � p� ��

xk;1�vi;1

xk;1�vi;1ð Þ2þ xk;2�vi;2ð Þ2�


xkð Þ ¼ 0; 0;�2rih2i;k 2p� hi;k

� �2hi;k � p� �4

h i


xkð Þ ¼ 0; 0; 2r2i hi;k 2p� hi;k

� �hi;k � p� �2

h

� � � 2p� hi;k

� �hi;k � p� �2�hi;k hi;k � p


� �hi;k � p� �� i

8>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>:

ð37Þ


123

Fig. 5 Input numbers and resulted shells by MFCS clustering method in case 1

Table 1 Simulation results in

case 1Two Input Class Numbers Resulted Shells by the MFCS

First dimension of centers ( vi;1

� �2i¼1

) 0 5 0.0029 4.9997

Second dimension of centers ( vi;2

� �2i¼1

) 0 5 0.0038 4.9834

Clusters’ scale coefficients ( ri½ �2i¼1) 0.8 1.5 0.7999 1.5000

Clusters’ rotation angles ( hi;0

� �2i¼1

) 0 0 0.0065 0.0001

Fig. 6 Input numbers and

resulted shells by MFCS

clustering method in case 2

Table 2 Simulation results in case 2

Four Input Class Numbers Resulted Shells by MFCS


� �4i¼1

) 3 5 7 4.5 2.9980 4.9981 6.9931 4.5791


� �4i¼1

) 3 5 4 6.5 2.9984 5.0115 4.0482 6.6035

Clusters’ scale coefficients ( ri½ �4i¼1) 1 1.5 1.3 0.9 0.9998 1.5102 1.3268 0.9330


� �4i¼1

) 0 0 0 0 0.0368 0.0025 0.0002 0.0050


123

Xn

k¼1

umi;k

��2; 0; 2vi;1 � 2r2

i

�1þ sin hi;k � hi;0


� �xk;2 � vi;2

� �

xk;1 � vi;1

� �2þ xk;2 � vi;2

� �2

�

� 2 Mk qi xkð Þ þ skð Þ ¼ 0 ð40Þ

Xn

k¼1

umi;k

"0;�2; 2vi;2 þ 2r2

i

�1þ sin hi;k � hi;0


� �xk;1 � vi;1

� �

xk;1 � vi;1

� �2þ xk;2 � vi;2

� �2

#

� 2 Mk qi xkð Þ þ skð Þ ¼ 0 ð41ÞXn

k¼1

umi;k 0; 0;�2ri 1þ sin hi;k � hi;0

� �� 2h i

� 2 Mk qi xkð Þ þ skð Þ ¼ 0 ð42ÞXn

k¼1

umi;k 0; 0; 2r2

i 1þ sin hi;k � hi;0


� ��

� 2 Mk qi xkð Þ þ skð Þ ¼ 0 ð43Þ

where Mk, qi xkð Þ and sk are obtained from (17), hi;k ¼tan�1 xk;2�vi;2

xk;1�vi;1

� �and i ¼ 1; . . .; c. Because, the terms such as

r2i sin tan�1

xk;2�vi;2xk;1�vi;1

� ��hi;0

� �cos tan�1

xk;2�vi;2xk;1�vi;1

� ��hi;0

� �vi;1þ��

v2i;1þv2

i;2þ�� are

appeared, the equations system in this case is nonlinear

with respect to unknown parameters vi;1, vi;2, ri and hi;0 for

i ¼ 1; . . .; c as same as two previous cases.

The performance of MFCS in clustering of well-known

shells, e.g. elliptical and spherical, is precisely same with

correspondent traditional clustering methods (e.g. [8] and

[9] for elliptical and spherical shells, respectively). There-

fore, we do not present the simulation results of MFCS

applying over these shells in this paper. We only examine

the performance of the MFCS method in three last examples

cases, in the next section, i.e. simulation results section.

5 Simulation results

Simulations are performed for three last examples of the

previous section in two cases for each example. We define





Two Input Class Numbers Resulted Shells by MFCS


� �2i¼1

) 0 25 -0.0147 25.0505


� �2i¼1

) 0 25 -0.0002 24.9792



� �2i¼1

) 0 0 0.0000 0.0008


123

the sum squares of equations first term, in correspondent

systems of equations for each case, as an objective function

that must be minimized by the PSO algorithm. For example,

when we solve the problem of Sect. 4.2, the sum squares of

first terms (32–34) equations, for i ¼ 1; . . .; c, is considered

as the PSO-objective function. In Sect. 4.4 case, the sum

squares of first terms (40–43) equations, for i ¼ 1; . . .; c, is

considered as the PSO-objective function and etc. The

resulted shells, by using the MFCS method, are illustrated in

the correspond figure of each case, together with the input

numbers. Furthermore, parameters of input numbers shells

and resulted shells, by using the MFCS method, are pre-

sented in the correspondent table of each case.

Case 1: In this case, MFCS is applied to two separate

shapes with Ri hkð Þ ¼ ri


p. The input numbers and

resulted shells are illustrated in Fig. 5. Table 1 shows

the parameters of input shells numbers and resulted

shells by using the proposed MFCS clustering method.

Case 2: In this case, MFCS is applied to four shapes with

Ri hkð Þ ¼ ri


p, while they are interfering with each

other. The input numbers and resulted shells are

illustrated in Fig. 6. Table 2 shows the parameters of

input shells numbers and resulted shells by using the

proposed MFCS clustering method.


shapes with Ri hkð Þ ¼ rihi;k 2p� hi;k

� �hi;k � p� �2

. The

input numbers and resulted shells are illustrated in

Fig. 7. Table 3 shows the parameters of input shells

numbers and resulted shells by using the proposed

MFCS clustering method.

Case 4: In this case, MFCS is applied to two shapes with

Ri hkð Þ ¼ rihi;k 2p� hi;k

� �hi;k � p� �2

, while they are

interfering with each other. The input numbers and





shapes with Ri hkð Þ ¼ ri 1þ sin hi;k � hi;0

� �� . The input

numbers and resulted shells are illustrated in Fig. 9.

Table 5 shows the parameters of input shells numbers

and resulted shells by using the proposed MFCS

clustering method.

Case 6: In this case, MFCS is applied to three shapes

with Ri hkð Þ ¼ ri 1þ sin hi;k � hi;0

� �� , while they are







� �2i¼1

) 15 8 15.0979 7.892


� �2i¼1

) 2 8 2.0297 7.968

Clusters’ scale coefficients ( ri½ �2i¼1) 1 1.5 1 1.4993


� �2i¼1

) 0 0 0.0009 0.0052


123

interfering with each other. The input numbers and




It is seen in simulation results; the proposed MFCS clus-

tering method can estimate the parameters of model-based

shells and cluster them. While only some numbers, which are

members of desired model-based shells, are given as input




� �2i¼1

) 0 3 0.0000 3.0000


� �2i¼1

) 0 3 0.0000 3.0000



� �2i¼1

) 3.1416 1.5708 3.1416 1.5708








123

numbers of the clustering process. The proposed MFCS

method has an acceptable performance for complicated and

interfering model-based shells (e.g., cases 2 and 6) as well as

simple and non-interfering model-based shells (e.g. cases 1,

3, 4 and 5). Although, the performance of the proposed

method is slightly decreased where more than 2-shells are

clustered and they are interfering with each other.

6 Conclusion

In this paper, the conventional fuzzy c-spherical shells

(FCSS) clustering method was extended to cluster the

model-based shells. It was showed; shells that have a center

and non-negative radius in each angle (by considering

polar coordinate system) can be clustered by the proposed

model-based fuzzy c-shells (MFCS) clustering method.

The resulted systems of equations in the clustering process,

when the MFCS clustering method is used, may be com-

plicated and non-solvable directly depending on basic

structure of model-based shells e.g. three examined

examples in this paper. Therefore, we must use iterative

and numerical methods inevitably to solve the systems of

equations. In this paper, we used the particle swarm opti-

mization (PSO) method to numerically solve the resulted

system of equations. It was showed that the proposed

method is a useful method to cluster shells when they are

complicated and interfering each other as well as they are

simple and separate shells. In future works, the proposed

method can be applied to preprocessed images (i.e. edge

detected images) to specific target detection of them.

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Three Input Class Numbers Resulted Shells by MFCS


� �3i¼1

) 0 1.5 0.5 -0.0481 1.4429 0.5476


� �3i¼1

) 0.8 1.5 2 0.7125 1.5210 1.9492

Clusters’ scale coefficients ( ri½ �3i¼1) 1 1.2 0.8 0.9680 1.1801 0.7746


� �3i¼1

) 3.1416 1.5708 4.1888 3.1956 1.5992 4.1664


123

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