Hole Filing IFCNN Simulation by Parallel RK(5,6) Techniques

8/6/2019 Hole Filing IFCNN Simulation by Parallel RK(5,6) Techniques

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(IJCSIS) International Journal of Computer Science and Information Security,

Vol. 9, No. 6, June 2011

Hole Filing IFCNN Simulation by Parallel RK(5,6)

Techniques(Hole Filing by Parallel RK(5,6))

Sukumar Senthilkumar*

Universiti Sains Malaysia

School of Mathematical Sciences11800 USM Pulau Pinang

MALAYSIA

E-mail: [email protected]

[email protected]

Abd Rahni Mt Piah

Universiti Sains Malaysia

School of Mathematical Sciences11800 USM Pulau Pinang

MALAYSIA

E-mail: [email protected]

Abstract — This paper concentrates on employing different

parallel RK(5,6) techniques for hole-filing via unique

characteristics of improved fuzzy cellular neural network(IFCNN) simulation to improve the performance of an

image or handwritten character recognition. Results are

presented according to the range of template selected for

simulation.

Keywords- Parallel 5-order 6-stage numerical integration

techniques, Improved fuzzy cellular neural network, Hole filing,

Simulation, Ordinary differential equations.

I. I NTRODUCTION

Parallel computing techniques are used to carry out

computations simultaneously, operating on the principle that

large problems are often can be divided into smaller ones,

which can then be solved concurrently. It is a simultaneous

process of multiple computing resources to solve a

computational problem easily and quickly. In real time it is practically believed by researchers that a possible way of

solving many significant computationally intensive problems

in science and engineering is by employing parallel algorithmseffectively.

From the literature, it is observed that most of the real time

problems are solved by adapting Runge-Kutta (RK) methods

which in turn are applied to compute numerical solutions for various problems, which are modeled in terms of initial value

problems as in Alexander and Coyle [3], Evans [4], Hung [5],Shampine and Watts [6] and Shampine and Gordon [7].Shampine and Watts [6] developed mathematical codes for

Runge-Kutta fourth order method to solve many numerical

problems. Runge-Kutta formula of fifth order has been

developed by Butcher [8-10] to solve many computational

problems. Evans and Sanugi [11] developed parallel

integration techniques of Runge-Kutta for step by step solutionof ordinary differential equations to obtain results.

Ponalagusamy and Ponammal [12-14] developed new parallel

fifth order algorithm to solve robot arm model, time varying

network for first order initial value problems and new

generalised plasticity equation for compressible powder metallurgy materials with results on stability region for test

equation. Keyes et al. [15] provided a survey towards

applications requiring memories and processing rates of large-scale parallelism, leading algorithmicist applications of

parallel numerical algorithms. Further, focused on practical

medium-granularity parallelism, approachable through

traditional programming languages. Gear [16] gave the

potentiality behavior for parallelism in solving real time problems using ordinary differential equations. A survey of potential for parallelism in Runge-Kutta techniques and

parallel numerical techniques for initial value problems for

ordinary differential equations are demonstrated by Norsettand Jackson [17] and Jackson [18]. Using fourth order explicit

Runge-Kutta method, a parallel mesh chopping algorithm for a

class of initial value problem is illustrated by Katti and

Srivastava [19]. Harrer et al. [20] introduced explicit Euler,

predictor-corrector and fourth-order Runge-Kutta algorithmsfor simulating cellular neural networks. The RK-Butcher

algorithm has been introduced by Bader [21, 22] for finding

truncation error estimates, intrinsic accuracies and earlydetection of stiffness in coupled differential equations that

arises in theoretical chemistry problems. Senthilkumar and

Piah [23] implemented parallel Runge-Kutta arithmetic mean

algorithm to obtain a solution to a system of second order

robot arm. In this paper a new attempt has been made toemploy parallel RK(5,6) algorithm for hole filing problem

under IFCNN environment. Oliveira [24] introduced a popular

sequential RK-Gill algorithm to evaluate effectiveness factor

of immobilized enzymes.

This research work is carried out by the first author under a post doctoral

fellow scheme at the School of Mathematical Sciences, Universiti Sains

Malaysia, 11800 USM Pulau Pinang, MALAYSIA.

*Corresponding Author.

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Vol. 9, No. 6, June 2011

Computing value is easy in case of implementing VLSICNN chips, thereby making real-time operations possible.

Roska [28] and Roska et al. [29] have presented the first

widely used simulation system which allows simulation of alarge class of CNN and is especially suited for image

processing applications. It also includes signal processing,

pattern recognition and solving ordinary and partial

differential equations, as in Gonzalez et al. [30]. The existingRK-Butcher fifth order method hole filing problem has beenstudied by Murugesh and Badri [32] via CNN simulation

model. Similarly, hole filing problem has been analyzed by

Murugesan and Elango [50] by means of existing RK fourthorder method under CNN simulation. Dalla Betta et al. [46]

implemented CMOS implementation of an analogy

programmed cellular neural network. Anguita et al. [31]

discussed in detail about parameter configurations for hole

extraction in cellular neural networks.

Zadeh [35] and Zadeh et al. [36] introduced the concept of

fuzzy sets (FSs) theory. Different notions of higher-order FSs

have been proposed by different researchers. Recently, fuzzycellular neural network (CNN) model [43-45] has attracted a

great deal of interest among researchers from different

disciplines. A locally interconnected, regularly repeated,

analogue (continuous- or discrete-time) circuits with a one-or-

two-or three-dimensional grid architecture called CNNsintroduced by Chua and Yang [25-26] and Chua [27]. Each

cell (neuron) in CNN is a non-linear dynamic system coupled

only to its nearest neighbors. Because of this localinterconnection property, CNNs have been considered

specifically suitable for very-large-scale integration

implementations. Shitong et al. [37] proposed improved fuzzy

cellular networks to incorporate the novel fuzzy status

containing the useful information beyond a white blood cellinto its state equation, resulting in enhancing the boundary

integrity. Laiho et al. [38] proposed template design for CNNs

with 1-bit weights.

This paper is ordered as follows. A brief introduction on

improved fuzzy cellular non-linear network is presented in

section 2. Section 3 deals with the performance of hole-filler

template design and simulation results. Section 4 discusses

parallel RK(5,6) numerical integration techniques. Finally,

concluding remarks is presented in section 5.

II. A BRIEF OVERVIEW OF IFCNN

The capability of the conventional cellular neural network

to solve different kinds of image processing problems and the

capability of fuzzy logic to cope with uncertainty in imagesare the inherent features of FCNN [37]. Moreover, it also has

inbuilt connections with mathematical morphology. The

unique characteristic of IFCNN is incorporating novel fuzzystatus with feed-forward and feedback templates in FCNN

such that the useful information beyond the region can be

sufficiently utilized. FCNN is a locally connected network [37] and the output of a neuron is connected to the inputs of

every neuron/cell in its r × r neighborhood, and similarly the

inputs of a neuron are only connected to the outputs of every

neuron in its r × r neighborhood. It is apparent that feedback

(not recurrent) connections are presented in detail. Thearchitecture of IFCNN is shown in Figure 1.

Figure 1. Architecture of IFCNN

The state equation of IFCNN is given by,

( , ) ( , )

( , ) ( , )

min( , ) ( , )

max( , ) ( , )

min( , ) ( , )

( , )

1( , ; , )

( , ; , )

( ( , ; , ) )

( ( , ; , ) )

( ( , ; , ) )

r

r

r

r

r

r

ij

ij kl

c k l N i j x

kl

c k l N i j

ij f klc k l N i j

f klc k l N i j

f klc k l N i j

c k l N

dxc x A i j k l y

dt R

B i j k l u

I A i j k l y

A i j k l y

B i j k l u

∈

∈

∈

∈

∈

∈

−= + +

+ + ∧ + +

∨ + +

+ ∧ +

∨

∑

∑

%

%

%

%max

( , )

min( , ) ( , )

max( , ) ( , )

( ( , ; , ) )

( ( , ; , ) )

( ( , ; , ) )

r

r

f kli j

f klc k l N i j

f klc k l N i j

B i j k l u

F i j k l x

F i j k l x

∈

∈

+

∧ +

∨

%

%

(1)

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Vol. 9, No. 6, June 2011

and the input equation of Cij is given by,

0,ij iju E = ≥ (2)

1 ≤ i ≤ M ; 1 ≤ j ≤ N .

the output equation of C ij is given by,

1( ) 1 1 ,

2ij ij ij ij y f x x x⎡ ⎤= = + − −⎣ ⎦ (3)

1 ≤ i ≤ M ; 1 ≤ j ≤ N .

The constraints /conditions are given by

Afmax(i, j;k ,l) = Afmin(k ,l;i, j);

Afmax(i, j;k ,l) = Af max(k ,l;i, j);

F fmax(i, j;k ,l) = F fmin(k ,l;i, j);

F fmax(i, j;k ,l) = F f max(k ,l;i, j);

1 ≤ i ≤ M ; 1 ≤ j ≤ N . (4)

1)0( ≤ij x ;1 ≤ i ≤ M ; 1 ≤ j ≤ N .

1)0( ≤iju ;1 ≤ i ≤ M ; 1 ≤ j ≤ N .

),;,(),;,( jilk Alk ji A =

From the above Eqs. (1) - (4),~

∧ ,~

∨ , N r (i, j), and A areidentical as in FCNN. Comparing (4) with FCNN, the only

one discrepancy between the equation is the novel fuzzy

status.

~

min( , )

~

max

( , )

( ( ( , ; , ) )

( ( , ; , ) )

))

kl r

kl r

f klc N i j

f kl

c N i j

kl

F i j k l x

F i j k l x

x

∈

∈

+ +

+ +

∧

∨ (5)

is adhered to Eq. (1), which obviously reflects the required

information where F fmin(i, j;k ,l) and F fmax(i, j;k ,l) indicates theconnected weights between cell C ij and C kl respectively.

Hence, the complete template determines the connection

between cell and its neighbors, consists of (2r × 1) and (2r ×1) matrices A, B, F fmin and F fmax. The symmetric matrices are

considered in the above template to congregate the IFCNN’ssymmetric requirements.

III. A BRIEF SKETCH ON HOLE-FILLER AND SIMULATION

RESULTS

In a bipolar image, all the holes are filled and remains

unaltered outside the holes, in case of hole filing IFCNN

simulation [46-50]. Allow 1,1 == C R x and take +1 to

represent the black pixel and –1 for the white pixel. If the

bipolar image is input with ijuU = into IFCNN and images

having holes are enclosed by the black pixels, then initial state

values are set to be 1)0( =ij x . The output values are obtained

as N j M i yij ≤≤≤≤= 1,1,1)0( from equation (1).

Consider the templates A, B and independent current source I

as

,

00

00

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

a

aba

a

A a > 0, b > 0

(6)

,

000

040

000

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

= B I = -1

where the template parameters a and b are to be determined. In

order to make the outer edge cells become the inner ones,

normally auxiliary cells are added along the outer boundary of

the image and their state values are set to be zeros by circuit

realization resulting in zero output values. The state equation

(1) can then be rewritten as

min( , ) ( , )

max( , ) ( , )

( ( , ; , ) )

( ( , ; , ) ) 4 ( ) .

r

r

ij

ij f klc k l N i j

f kl ijc k l N i j

dx x A i j k l y

dt

A i j k l y u t I

∈

∈

= − + ∧ + +

∨ + + −

%

%

(7)

For instance, here the cells C (i+1, j), C (i-1, j), C (i, j+1) andC (i, j-1) are non-diagonal cells. Designing of hole-filler template [31] and its various sub-problems are discussed using

CNN simulations [46-50]. Figures 2 and 3 show the hole filing

of an image (before and after) by employing a parallel

RK(5,6) type-III technique. The settling time Ts andcomputation time Tc for different step sizes are considered for

the purpose of comparison. The settling time Ts is the time


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Vol. 9, No. 6, June 2011

Figure 2(a). Original image and hole filed image

Figure 2(b). Original image and hole filed image

from start of computation until the last cell leaves the interval

[-1.0, 1.0]

Figure 2. Hole filing before and after adapting type-III parallel RK(5,6)technique

Figure 3. Hole filing before and after employing type-III parallel RK(5,6)

technique

Figure 4. Range of the template

which is based on a specified limit (e.g., |d x/dt |< 0.01). The

computation time Tc is the time taken for settling the network

and adjusting the cell for proper position once the network issettled. The simulation shows the desired output for every

neuron/cell. Specifically, note that +1 and -1 indicate the black

and white pixels, respectively. The marked selected template

parameters a and b are restricted to the shaded area, as shownin figure 4 for the simulation.

IV. PARALLEL R UNGE-K UTTA FIFTH ORDER TECHNIQUES: A

BRIEF OVERVIEW

A. Parallel Runge-Kutta 5-Order 6-Stage Type-I Technique

A parallel Runge-Kutta 5-order 6-stage type-I technique [12-14] is one of the simplest method used to solve ordinary

differential equations. It is an explicit formula which adapts

the Taylor’s series expansion in order to obtain the

approximation. A parallel Runge-Kutta 5-order 6-stage type-I

technique is used to determine y j and , 1, 2, 3,.... j y j m=&such

that

]90

7

90

32

90

2

90

32

90

7[ 654311 k k k k k y y nn +++++=+

(8)

Thus, the corresponding parallel Runge-Kutta 5-order 6-stage

type –I technique of Butcher array represents

0

5

2

5

2

4

1

64

11

64

5

2

1

16

3

16

5

4

3

32

9

32

27−

4

3

16

9

128

9−

28

350

7

12−

7

8

90

70

90

32

90

2

90

32

90

7


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Vol. 9, No. 6, June 2011

The 6 stage 5th order algorithm with 5 parallel and 2 processors, by selecting a43 = 0 to evaluate k 3 and k 4

simultaneously is given by

. ))((1 nij

ijt xtf k Δ= ,

2 12 2( ( ) )5 5

ij ij

ij nt k tf x t k Δ= Δ + + ,

3 1 2

11 5( ( ) )

4 64 64

ij ij ij

ij n

t k tf x t k k

Δ= Δ + + + =

*

3

ijk ,

4 1 2

3 5( ( ) )

4 16 16

ij ij ij

ij n

t k tf x t k k

Δ= Δ + + + =

*

4

ijk ,

5 1 2 3 4

9 275 3 9( ( ) )

2 32 32 4 16

ij ij ij ij ij

ij n

t k tf x t k k k k

Δ= Δ + − + +

=*

5

ijk ,

6 1 2 4 5

3 9 35 12 8( ( )

4 28 28 7 7

ij ij ij ij ij

ij nk tf x t t k k k k = Δ + Δ − + − +

=*

6 .ijk (9)

Therefore, the final integration is a weighted sum of the five

calculated derivatives which is given as

1

1 3 4 5 6( ( )) [7 32 12 32 7 ].

90

n

n

t

ij ij ij ij ij

t

t f x t dt k k k k k

+ Δ= + + + +∫

(10)

B. Parallel Runge-Kutta 5-Order 6-Stage Type-II Technique

A parallel Runge-Kutta 5-order 6-stage type-II technique [12-

14] is also one of the simplest method used to solve ordinarydifferential equations. It is an explicit formula which adapts

the Taylor’s series expansion in order to obtain the

approximation. A parallel Runge-Kutta 5-order 6-stage type-II

technique determines y j and m j y j ,....3,2,1, =& such that

1 1 3 5 6

23 125 81 125[ ].90 192 192 192

n n y y h k k k k + = + + − +

(11)


technique of type-II Butcher array represents

0

3

1

3

1

5

2

. 25

4

25

6

2

1

4

1-3

4

15

3

2

81

6

81

90−

81

50

81

8

5

4

75

6−

75

36

75

10

75

8

192

230

192

1250

192

812−

192

125

Therefore, the final integration is a weighted sum of four calculated derivatives per time step which is given by

]192

125

192

81

192

125

90

23[ 65311 k k k k h y y nn +−++=+ .

(12)

The 6 stage 5th order algorithm with 5 parallel and 2

processors by selecting a65 = 0 to evaluate k 5 and k 6 simultaneously is given by

. ))((1 nij

ijt xtf k Δ= ,

ij

nij

ij k t

t xtf k 123

1))

3(( +

Δ+Δ= ,

ijij

nij

ijk k

t t xtf k 213

25

6

25

4))

3

2(( ++

Δ+Δ= ,

4

153

4)

2( 3

21

4

ij

ij

ij

n

ij k k

k t t tf k +−+

Δ+Δ=

ijijij

nij

ij kk k k t t xtf k 321581

8

81

50

81

90

81

6)

3

2(( +−−+Δ+Δ=

=*

5

ijk


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Vol. 9, No. 6, June 2011

6 1

2 3 4

4 6( ( ) )

5 75

36 10 8

75 75 75

ij ij

ij n

ij ij ij

t k tf x t k

k k k

Δ= Δ + − +

+ +

=*

6 .ijk (12)

Therefore, the final integration is a weighted sum of the five

calculated derivatives which is given by1

1 3 5 6( ( )) [23 125 81 125 ].192

n

n

t

ij ij ij ij

t

t f x t dt k k k k

+ Δ= + − +∫

(13)

C. Parallel Runge-Kutta 5-Order 6-Stage Type-III Technique

A parallel Runge-Kutta 5-order 6-stage type-III technique [12-

14] is another simple method used to solve ordinarydifferential equations. It is also an explicit formula which

adapts the Taylor’s series expansion for an approximation. A

parallel Runge-Kutta 5-order 6-stage type-III technique

determines y j and , 1, 2, 3,.... j y j m=&such that

1 1 3

4 5 6

17 250[306 153

442 8192 31].

255 9945 234

n n y y h k k

k k k

+ = + − +

+ +

(14)


technique of type-III Butcher array represents

0

5

1

5

1

5

2.

160

39

32

5

2

1

24

1

24

5−

3

2

16

3

8

1

16

3−

4

1

114

9−

14

15

7

8

7

12

7

8

306

17−0

153

250−

255

442

9945

8192

234

31

Therefore, the final integration is a weighted sum of fivecalculated derivatives per time step which is given by

1 1 3

4 5 6

17 250[306 153

442 8192 31

].255 9945 234

n n y y h k k

k k k

+ = + − +

+ +

(15)

The 6 stage 5th order algorithm with 5 parallel and 2 processors by selecting a54 = 0 to evaluate k 5 and k 4

simultaneously is given by

. ))((1 nij

ijt xtf k Δ= ,

ij

nij

ij k t

t xtf k 125

1))

5(( +

Δ+Δ=

ijij

nij

ij k k t

t xtf k 21332

5

160

39))

5

2(( ++

Δ+Δ= ,

3

2

24

5

24)

2( 321

4

ijijij

n

ij k k k t t tf k +−+

ΔΔ= =

*

4

ijk

ijijij

nij

ij k k k t t xtf k 32154

1

16

3

8

1)

16

3(( −−+Δ+Δ= =

*

5

ijk

6 1

2 3 4 5

9

( ( ) ) 14

15 8 12 8.

14 7 7 7

ij ij

ij n

ij ij ij ij

k tf x t t k

k k k k

= Δ + Δ − +

+ − +

(16)

Therefore, the final integration is a weighted sum of the fivecalculated derivatives which is given by

1

31

5 64

25017( ( )) [

306 153

8192 31442

].255 9945 234

n

n

t ijij

t

ij ijij

k k f x t dt t

k k k

+

= Δ − − +

+ +

∫(17)

V. CONCLUDING REMARKS

In this paper, hole filing problem is addressed under IFCNNmodel using parallel RK(5,6) techniques and its validity is

illustrated by simulation results. It is observed that the hole is


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Vol. 9, No. 6, June 2011

filled and the outside image remains unaffected, that is, theedges of the images are preserved and are intact. The

templates of the cellular neural network are not unique and

this is important in its implementation. The significance of thiswork is to improve the performance of handwritten character

recognition because in many language scripts, numerals and in

images etc., there are many holes and the CNN described

above can be used in addition to the connected componentdetector. It is also noticed that IFCNN preserves the boundaryintegrity.

ACKNOWLEDGMENT

The first author would like to extend his sincere gratitude

to Universiti Sains Malaysia for supporting this work under its

post doctoral fellowship scheme. Much of this work was

carried out during his stay at Universiti Sains Malaysia in

2011. He wishes to acknowledge Universiti Sains Malaysia’sfinancial support.

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Senthilkumar was born in Neyveli Township,

Cuddalore District, Tamilnadu, India on 18th July

1974. He received his B.Sc in Mathematics from

Madras University in 1994, M.Sc in Mathematics

from Bharathidasan University in 1996, M.Philin Mathematics from Bharathidasan University in

1999 and M.Phil in Computer Science &

Engineering from Bharathiar University in 2000.

He also has a PGDCA and PGDCH in Computer

Science and Applications and Computer Hardware from BharathidasanUniversity which he obtained in 1996 and 1997, respectively. He has a

doctoral degree in Mathematics and Computer Applications from National

Institute of Technology [REC], Tiruchirappalli, Tamilnadu, India. Currently,he is a post doctoral fellow at the School of Mathematical Sciences, Universiti

Sains Malaysia, 11800 USM Pulau Pinang, Malaysia. Prior to this

appointment, he was a lecturer/assistant professor in the Department of Computer Science at Asan Memorial College of Arts and Science, Chennai,

Tamilnadu, India. He has published many good research papers in

international conference proceedings and peer-reviewed/refereed international journals with high impact factor. He has made significant and outstanding

contributions to various activities related to research work. He is also an

associate editor, editorial board member, reviewer and referee for many

scientific international journals. His current research interests includeadvanced cellular neural networks, advanced digital image processing,

advanced numerical analysis and methods, advanced simulation and

computing and other related areas.

Abd Rahni Mt Piah was born in Baling, Kedah Malaysia on 8th May 1956. Hereceived his B.A. (Cum Laude) in Mathematics

from Knox College, Illinois, USA in 1979. He

received his M.Sc in Mathematics from

Universiti Sains Malaysia in 1986. He obtainedhis Ph.D in Approximation Theory from the

University of Dundee, Scotland UK in 1993. Hehas been an academic staff member of the School

of Mathematical Sciences; Universiti Sains

Malaysia since 1981 and at present is an

Associate Professor. He was a program chairman and deputy dean in theSchool of Mathematical Sciences, Universiti Sains Malaysia for many years.

He has published various research papers in refereed national and international

conference proceedings and journals. His current research areas include

Computer Aided Geometric Design (CAGD), Medical Imaging, Numerical

Analysis and Techniques and other related areas.


ISSN 1947-5500

Hole Filing IFCNN Simulation by Parallel RK(5,6) Techniques

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