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*                                                             Abd Rahni Mt Piah
                  Universiti Sains Malaysia                                                       Universiti Sains Malaysia
              School of Mathematical Sciences                                                  School of Mathematical Sciences
                  11800 USM Pulau Pinang                                                          11800 USM Pulau Pinang
                         MALAYSIA                                                                        MALAYSIA
           E-mail: ssenthilkumar1974@yahoo.co.in                                                 E-mail: arahni@cs.usm.my
                   ssenthilkumar@usm.my


Abstract— This paper concentrates on employing different                           developed by Butcher [8-10] to solve many computational
parallel RK(5,6) techniques for hole-filing via unique                             problems. Evans and Sanugi [11] developed parallel
characteristics of improved fuzzy cellular neural network                          integration techniques of Runge-Kutta for step by step solution
(IFCNN) simulation to improve the performance of an                                of ordinary differential equations to obtain results.
image or handwritten character recognition. Results are                            Ponalagusamy and Ponammal [12-14] developed new parallel
presented according to the range of template selected for                          fifth order algorithm to solve robot arm model, time varying
simulation.                                                                        network for first order initial value problems and new
                                                                                   generalised plasticity equation for compressible powder
    Keywords- Parallel 5-order 6-stage numerical integration                       metallurgy materials with results on stability region for test
techniques, Improved fuzzy cellular neural network, Hole filing,                   equation. Keyes et al. [15] provided a survey towards
Simulation, Ordinary differential equations.                                       applications requiring memories and processing rates of large-
                                                                                   scale parallelism, leading algorithmicist applications of
                          I.     INTRODUCTION                                      parallel numerical algorithms. Further, focused on practical
                                                                                   medium-granularity parallelism, approachable through
    Parallel computing techniques are used to carry out                            traditional programming languages. Gear [16] gave the
computations simultaneously, operating on the principle that                       potentiality behavior for parallelism in solving real time
large problems are often can be divided into smaller ones,                         problems using ordinary differential equations. A survey of
which can then be solved concurrently. It is a simultaneous                        potential for parallelism in Runge-Kutta techniques and
process of multiple computing resources to solve a                                 parallel numerical techniques for initial value problems for
computational problem easily and quickly. In real time it is                       ordinary differential equations are demonstrated by Norsett
practically believed by researchers that a possible way of                         and Jackson [17] and Jackson [18]. Using fourth order explicit
solving many significant computationally intensive problems                        Runge-Kutta method, a parallel mesh chopping algorithm for a
in science and engineering is by employing parallel algorithms                     class of initial value problem is illustrated by Katti and
effectively.                                                                       Srivastava [19]. Harrer et al. [20] introduced explicit Euler,
                                                                                   predictor-corrector and fourth-order Runge-Kutta algorithms
    From the literature, it is observed that most of the real time                 for simulating cellular neural networks. The RK-Butcher
problems are solved by adapting Runge-Kutta (RK) methods                           algorithm has been introduced by Bader [21, 22] for finding
which in turn are applied to compute numerical solutions for                       truncation error estimates, intrinsic accuracies and early
various problems, which are modeled in terms of initial value                      detection of stiffness in coupled differential equations that
problems as in Alexander and Coyle [3], Evans [4], Hung [5],                       arises in theoretical chemistry problems. Senthilkumar and
Shampine and Watts [6] and Shampine and Gordon [7].                                Piah [23] implemented parallel Runge-Kutta arithmetic mean
Shampine and Watts [6] developed mathematical codes for                            algorithm to obtain a solution to a system of second order
Runge-Kutta fourth order method to solve many numerical                            robot arm. In this paper a new attempt has been made to
problems. Runge-Kutta formula of fifth order has been                              employ parallel RK(5,6) algorithm for hole filing problem
This research work is carried out by the first author under a post doctoral        under IFCNN environment. Oliveira [24] introduced a popular
fellow scheme at the School of Mathematical Sciences, Universiti Sains             sequential RK-Gill algorithm to evaluate effectiveness factor
Malaysia, 11800 USM Pulau Pinang, MALAYSIA.                                        of immobilized enzymes.
*Corresponding Author.




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                                                                                                              ISSN 1947-5500
                                                            (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                Vol. 9, No. 6, June 2011

    Computing value is easy in case of implementing VLSI                sufficiently utilized. FCNN is a locally connected network
CNN chips, thereby making real-time operations possible.                [37] and the output of a neuron is connected to the inputs of
Roska [28] and Roska et al. [29] have presented the first               every neuron/cell in its r × r neighborhood, and similarly the
widely used simulation system which allows simulation of a              inputs of a neuron are only connected to the outputs of every
large class of CNN and is especially suited for image                   neuron in its r × r neighborhood. It is apparent that feedback
processing applications. It also includes signal processing,            (not recurrent) connections are presented in detail. The
pattern recognition and solving ordinary and partial                    architecture of IFCNN is shown in Figure 1.
differential equations, as in Gonzalez et al. [30]. The existing
RK-Butcher fifth order method hole filing problem has been
studied 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 fourth
order 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, fuzzy
cellular 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 CNNs
introduced 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 local
interconnection 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 cell
into its state equation, resulting in enhancing the boundary
integrity. Laiho et al. [38] proposed template design for CNNs                                                Figure 1. Architecture of IFCNN
with 1-bit weights.
                                                                        The state equation of IFCNN is given by,
    This paper is ordered as follows. A brief introduction on               dxij           −1
improved fuzzy cellular non-linear network is presented in              c             =       xij + ∑                       A(i, j; k , l ) ykl +
section 2. Section 3 deals with the performance of hole-filler                dt           Rx      c ( k ,l )∈N r ( i , j )
template design and simulation results. Section 4 discusses
parallel RK(5,6) numerical integration techniques. Finally,                      ∑
                                                                         c ( k ,l )∈N r ( i , j )
                                                                                                    B(i, j; k , l )ukl
concluding remarks is presented in section 5.
                                                                        + I ij +                    ∧
                                                                                                    %             ( Af min (i, j; k , l ) + ykl ) +
              II.   A BRIEF OVERVIEW OF IFCNN                                          c ( k ,l )∈N r ( i , j )

                                                                                 ∨
                                                                                 %                 ( Af max (i, j; k , l ) + ykl ) +                               (1)
                                                                        c ( k ,l )∈N r ( i , j )
    The capability of the conventional cellular neural network
to solve different kinds of image processing problems and the           +             ∧
                                                                                      %                ( B f min (i, j; k , l )ukl ) +
                                                                            c ( k ,l )∈N r ( i , j )
capability of fuzzy logic to cope with uncertainty in images
are the inherent features of FCNN [37]. Moreover, it also has                    ∨
                                                                                 %                 ( B f max (i, j; k , l )ukl ) +
                                                                        c ( k ,l )∈N r ( i , j )
inbuilt connections with mathematical morphology. The
unique characteristic of IFCNN is incorporating novel fuzzy                       ∧
                                                                                  %                ( Ff min (i, j; k , l ) xkl ) +
status with feed-forward and feedback templates in FCNN                 c ( k ,l )∈N r ( i , j )

such that the useful information beyond the region can be                        ∨
                                                                                 %                 ( Ff max (i, j; k , l ) xkl )
                                                                        c ( k ,l )∈N r ( i , j )




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                                                                                                                            ISSN 1947-5500
                                                                    (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                        Vol. 9, No. 6, June 2011

and the input equation of Cij is given by,                                     considered in the above template to congregate the IFCNN’s
                                                                               symmetric requirements.
uij = Eij ≥ 0,                                                      (2)
                                                                                     III.         A BRIEF SKETCH ON HOLE-FILLER AND SIMULATION
1 ≤ i ≤ M; 1 ≤ j ≤ N.                                                                                                                     RESULTS
                                                                               In a bipolar image, all the holes are filled and remains
                                                                               unaltered outside the holes, in case of hole filing IFCNN
the output equation of Cij is given by,
                                                                               simulation [46-50]. Allow R x = 1, C = 1 and take +1 to
                                1⎡
yij = f ( xij ) =                  xij + 1 − xij − 1 ⎤ ,            (3)        represent the black pixel and –1 for the white pixel. If the
                                2⎣                   ⎦
                                                                                                                                                  { }
                                                                               bipolar image is input with U = u ij into IFCNN and images
1 ≤ i ≤ M; 1 ≤ j ≤ N.                                                          having holes are enclosed by the black pixels, then initial state
                                                                               values are set to be xij (0) = 1 . The output values are obtained
The constraints /conditions are given by
                                                                               as y ij (0) = 1,1 ≤ i ≤ M ,1 ≤ j ≤ N from equation (1).

Afmax(i,j;k,l) = Afmin(k,l;i,j);                                               Consider the templates A, B and independent current source I
                                                                               as
Afmax(i,j;k,l) = Afmax(k,l;i,j);

Ffmax(i,j;k,l) = Ffmin(k,l;i,j);                                                   ⎡0 a 0⎤
                                                                               A = ⎢a b a ⎥ ,
                                                                                   ⎢      ⎥                                              a > 0, b > 0
Ffmax(i,j;k,l) = Ffmax(k,l;i,j);
                                                                                   ⎢0 a 0⎥
                                                                                   ⎣      ⎦
1 ≤ i ≤ M; 1 ≤ j ≤ N.                                               (4)
                                                                                                                                                                                 (6)
 xij (0) ≤ 1 ;1 ≤ i ≤ M; 1 ≤ j ≤ N.                                                 ⎡0 0 0 ⎤
                                                                                B = ⎢0 4 0 ⎥ ,
                                                                                    ⎢      ⎥                                               I = -1
u ij (0) ≤ 1 ;1 ≤ i ≤ M; 1 ≤ j ≤ N.                                                 ⎢0 0 0 ⎥
                                                                                    ⎣      ⎦

 A(i, j; k , l ) = A(k , l ; i, j )
                                                                               where the template parameters a and b are to be determined. In
                                                        ~       ~              order to make the outer edge cells become the inner ones,
From the above Eqs. (1) - (4), ∧ , ∨ , Nr(i,j), and A are                      normally auxiliary cells are added along the outer boundary of
identical as in FCNN. Comparing (4) with FCNN, the only                        the image and their state values are set to be zeros by circuit
one discrepancy between the equation is the novel fuzzy                        realization resulting in zero output values. The state equation
status.                                                                        (1) can then be rewritten as

                ~                                                              dxij
   (         ∧
       ckl ∈N r ( i , j )
                            ( Ff min (i, j; k , l ) + xkl ) +
                                                                                 dt
                                                                                         = − xij +                      ∧
                                                                                                                        %
                                                                                                              c ( k ,l )∈N r ( i , j )
                                                                                                                                         ( Af min (i, j; k , l ) + ykl ) +
                                                                                                                                                                                 (7)
            ~                                                                           ∨
                                                                                        %                 ( Af max (i, j; k , l ) + ykl ) + 4uij (t ) − I .
          ∨
   ckl ∈N r ( i , j )
                        ( Ff max (i, j; k , l ) + xkl ) +           (5)        c ( k ,l )∈N r ( i , j )



   xkl ))                                                                      For instance, here the cells C(i+1,j), C(i-1,j), C(i,j+1) and
                                                                               C(i,j-1) are non-diagonal cells. Designing of hole-filler
is adhered to Eq. (1), which obviously reflects the required                   template [31] and its various sub-problems are discussed using
information where Ffmin(i,j;k,l) and Ffmax(i,j;k,l) indicates the              CNN simulations [46-50]. Figures 2 and 3 show the hole filing
connected weights between cell Cij and Ckl respectively.                       of an image (before and after) by employing a parallel
Hence, the complete template determines the connection                         RK(5,6) type-III technique. The settling time Ts and
between cell and its neighbors, consists of (2r × 1) and (2r ×                 computation time Tc for different step sizes are considered for
1) matrices A, B, Ffmin and Ffmax. The symmetric matrices are                  the purpose of comparison. The settling time Ts is the time




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                                                                                                                                          ISSN 1947-5500
                                                                           (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                               Vol. 9, No. 6, June 2011


from start of computation until the last cell leaves the interval                   which is based on a specified limit (e.g., |dx/dt|< 0.01). The
[-1.0, 1.0]                                                                         computation time Tc is the time taken for settling the network
                                                                                    and adjusting the cell for proper position once the network is
                                                                                    settled. 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 shown
                                                                                    in figure 4 for the simulation.
                                                                                        IV.   PARALLEL RUNGE-KUTTA FIFTH ORDER TECHNIQUES: A
                                                                                                          BRIEF OVERVIEW

                                                                                    A. Parallel Runge-Kutta 5-Order 6-Stage Type-I Technique
             Figure 2(a). Original image and hole filed image


                                                                                    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 yj and y j , j = 1, 2,3,....m such
                                                                                                                            &
                                                                                    that
                                                                                                       7      32     2    32    7
                                                                                    y n +1 = y n + [      k1 + k 3 + k 4 + k 5 + k 6 ]
            Figure 2(b). Original image and hole filed image
                                                                                                       90     90    90    90    90
                                                                                                                                                       (8)

   Figure 2. Hole filing before and after adapting type-III parallel RK(5,6)        Thus, the corresponding parallel Runge-Kutta 5-order 6-stage
                                  technique                                         type –I technique of Butcher array represents


                                                                                    0

                                                                                    2          2
                                                                                    5          5

                                                                                    1          11        5
                                                                                    4          64        64
  Figure 3. Hole filing before and after employing type-III parallel RK(5,6)
                                  technique                                         1           3         5
                                                                                    2          16        16

                                                                                    3          9         − 27   3           9
                                                                                    4          32         32    4          16

                                                                                               −9        35                − 12       8
                                                                                    1                            0
                                                                                               28        28                 7         7


                                                                                               7                32          2         32         7
                                                                                                          0
                      Figure 4. Range of the template
                                                                                               90               90         90         90         90



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                                                                                                                ISSN 1947-5500
                                                                     (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                         Vol. 9, No. 6, June 2011

The 6 stage 5th order algorithm with 5 parallel and 2                               0
processors, by selecting a43 = 0 to evaluate k3 and k4
simultaneously is given by                                                          1          1
                                                                                    3          3
.   k1ij = Δtf ( xij (t n )) ,
                                                                                    2          4          6
                        2 Δt      2                                                   .
    k = Δtf ( xij (tn +
      ij
      2                      ) ) + k1ij ,                                           5          25         25
                         5        5
                                                                                    1          1                    15
                        Δt    11      5 ij                                                               -3
    k3 = Δtf ( xij (tn + ) ) + k1ij +
     ij                                        *
                                         k2 = k3 ij ,                               2          4                     4
                        4     64      64
                                                                                    2          6          − 90       50         8
                       Δt     3       5 ij                                          3          81          81        81         81
    k = Δtf ( xij (tn + ) ) + k1ij +
      ij
      4
                                              *
                                        k2 = k4 ij ,
                       4     16      16
                                                                                    4          −6         36          10        8
                 Δt    9       275 ij 3 ij 9 ij                                     5          75         75          75        75
k = Δtf ( xij (tn ) ) + k1ij −
      ij
      5                           k 2 + k3 + k 4
                 2     32      32      4    16
   *ij
= k5 ,
                                                                                               23                   125              − 812           125
                                                                                                            0                   0
                       3   9      35 ij 12 ij 8 ij                                            192                   192               192            192
k6 = Δtf ( xij (tn + Δt ) − k1ij + k2 − k4 + k5
  ij

                       4 28       28     7    7
     *ij
= k6 .                                          (9)                             Therefore, the final integration is a weighted sum of four
                                                                                calculated derivatives per time step which is given by
Therefore, the final integration is a weighted sum of the five
calculated derivatives which is given as                                                                23      125       81      125
                                                                                    y n +1 = y n + h[      k1 +     k3 −     k5 +     k6 ] .
                                                                                                        90      192      192      192
                            Δt
    tn+1
                                                                                                                                                           (12)
     ∫     f ( x(t ))dt =      [7k1ij + 32k3ij + 12k4 + 32 k5 + 7 k6 ].
                                                    ij      ij     ij

    tn
                            90                                                  The 6 stage 5th order algorithm with 5 parallel and 2
                                                                    (10)        processors by selecting a65 = 0 to evaluate k5 and k6
                                                                                simultaneously is given by

B. Parallel Runge-Kutta 5-Order 6-Stage Type-II Technique                       .   k1ij = Δtf ( xij (t n )) ,

A parallel Runge-Kutta 5-order 6-stage type-II technique [12-                                                 Δt      1
14] is also one of the simplest method used to solve ordinary                       k 2 = Δtf ( xij (t n +
                                                                                      ij
                                                                                                                 ) ) + k1ij ,
differential equations. It is an explicit formula which adapts                                                3       3
the Taylor’s series expansion in order to obtain the
approximation. A parallel Runge-Kutta 5-order 6-stage type-II                                                 2 Δt      4       6 ij
                                                                                    k 3 = Δtf ( xij (t n +
                                                                                      ij
                                                                                                                   ) ) + k1ij +    k2 ,
technique determines yj and y j , j = 1,2,3,....m such that
                               &                                                                               3        25      25
                        23      125       81      125
    yn +1 = yn + h[        k1 +     k3 −     k5 +     k6 ].                                        Δt k1ij           15k 3
                                                                                                                         ij
                        90      192      192      192                           k = Δtf (t n + ) +
                                                                                        ij
                                                                                        4                   − 3k 2 +
                                                                                                                 ij

                                                                    (11)                            2    4             4
                                                                                                         2     6 ij 90 ij 50 ij 8 ij
Thus, the corresponding parallel Runge-Kutta 5-order 6-stage                    k 5 = Δtf ( xij (t n + Δt ) + k1 − k 2 − k 3 + k 4
                                                                                  ij

technique of type-II Butcher array represents
                                                                                                         3 81          81   81  81
                                                                                     *ij
                                                                                = k5




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                                                                                                                    ISSN 1947-5500
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               4 Δt      6
k6 = Δtf ( xij (tn +
 ij
                    ) ) − k1ij +                                           Therefore, the final integration is a weighted sum of five
                5        75         *ij
                                 = k6 .                       (12)         calculated derivatives per time step which is given by
36 ij 10 ij 8 ij
   k2 +    k3 + k 4
75      75     75                                                                           17       250
                                                                               yn +1 = yn + h[  k1 −     k3 +
                                                                                           306       153
Therefore, the final integration is a weighted sum of the five                                                                                         (15)
                                                                               442      8192       31
calculated derivatives which is given by                                           k4 +      k5 +     k6 ].
tn+1
                         Δt                                                    255      9945      234
 ∫     f ( x(t ))dt =       [23k1ij + 125k3 − 81k5ij + 125k6 ].
                                          ij               ij

 tn
                        192
                                                              (13)         The 6 stage 5th order algorithm with 5 parallel and 2
                                                                           processors by selecting a54 = 0 to evaluate k5 and k4
C. Parallel Runge-Kutta 5-Order 6-Stage Type-III Technique                 simultaneously is given by

A parallel Runge-Kutta 5-order 6-stage type-III technique [12-             .   k1ij = Δtf ( xij (t n )) ,
14] is another simple method used to solve ordinary
differential equations. It is also an explicit formula which
adapts the Taylor’s series expansion for an approximation. A                                              Δt      1
                                                                               k 2 = Δtf ( xij (t n +
                                                                                 ij
                                                                                                             ) ) + k1ij
parallel Runge-Kutta 5-order 6-stage type-III technique                                                   5       5
determines yj and y j , j = 1, 2,3,....m such that
                   &
             17       250                                                                                 2 Δt       39 ij    5 ij
yn +1 = yn + h[  k1 −     k3 +                                                 k 3 = Δtf ( xij (t n +
                                                                                 ij
                                                                                                               )) +     k1 +    k2 ,
            306       153                                                                                  5        160      32
                                                              (14)
442      8192       31
    k4 +      k5 +     k6 ].                                                               Δt k1ij 5k 2
                                                                                                      ij
                                                                                                           2k 3
                                                                                                              ij

255      9945      234                                                         k = Δtf (t n ) +
                                                                                 ij
                                                                                 4                −      +          *ij
                                                                                                                 = k4
                                                                                           2    24 24       3
Thus, the corresponding parallel Runge-Kutta 5-order 6-stage
technique of type-III Butcher array represents                                                                3    1       3 ij 1 ij
                                                                               k 5 = Δtf ( xij (t n + Δt
                                                                                 ij
                                                                                                                ) + k1ij − k 2 − k 3 = k5 ij
                                                                                                                                        *

                                                                                                             16 8         16    4
0
                                                                                                9 ij
                                                                                                  k1 +
                                                                               k6 = Δtf ( xij (tn + Δt ) ) −
                                                                                ij

1            1                                                                                 14                                                      (16)
5            5                                                             15 ij 8 ij 12 ij 8 ij
                                                                              k 2 + k 3 − k 4 + k5 .
                                                                           14      7     7     7
2            39           5
  .                                                                        Therefore, the final integration is a weighted sum of the five
5           160          32                                                calculated derivatives which is given by

1            1          −5       2                                             tn+1
                                                                                                            17 k1ij 250k3ij

2            24         24       3                                              ∫
                                                                               tn
                                                                                      f ( x(t ))dt = Δt[−
                                                                                                             306
                                                                                                                   −
                                                                                                                     153
                                                                                                                            +
                                                                                                                                                       (17)
 3           1          −3       1                                             442k   8192k
                                                                                        ij
                                                                                              31k    ij         ij

16           8          16       4                                                  +   4
                                                                                            +     ]. 5          6
                                                                                255    9945   234
             −9         15        8      12       8
 1
             14         14        7       7       7                                                 V.      CONCLUDING REMARKS

             − 17            − 250     442      8192     31
                  0                                                        In this paper, hole filing problem is addressed under IFCNN
             306              153      255      9945    234                model 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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                                                                                                                ISSN 1947-5500
                                                                       (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                           Vol. 9, No. 6, June 2011

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filled and the outside image remains unaffected, that is, the                              Algorithm”, Kluwer Academic Publishers, 1997.
edges of the images are preserved and are intact. The                               [16]   G.W. Gear, “The potential for parallelism in ordinary differential
templates of the cellular neural network are not unique and                                equations”, Technical Report UIUCDCS-R-86-1246, Computer
this is important in its implementation. The significance of this                          Science Department, University of Illinois, Urbana, IL, 1986.
                                                                                    [17]   K.R. Jackson and S.P. Norsett, “The potential for parallelism in
work is to improve the performance of handwritten character                                Runge-Kutta methods: Part-I: RK formulas in standard form”,
recognition because in many language scripts, numerals and in                              SIAM Journal on Numerical Analysis, Vol. 32, pp. 49-82, 1995.
images etc., there are many holes and the CNN described                             [18]    K.R. Jackson, “A survey of parallel numerical methods for initial
above can be used in addition to the connected component                                   value problems for ordinary differential equations”, IEEE
                                                                                           Transactions on Magnetics, Vol. 27, pp. 3792-3797, 1991.
detector. It is also noticed that IFCNN preserves the boundary                      [19]   C.P. Katti and D.K. Srivastava, “On a parallel mesh chopping
integrity.                                                                                 algorithm for fourth order explicit Runge-Kutta method”, Applied
                                                                                           Mathematics and Computation, Vol. 143, pp. 563-570, 2003.
                                                                                    [20]   H. Harrer, A. Schuler and E. Amelunxen, “Comparison of different
                         ACKNOWLEDGMENT                                                    numerical integrations for simulating cellular neural networks”, In
                                                                                           CNNA-90 Proceedings of IEEE International Workshop on
                                                                                           Cellular Neural Networks and their Applications, pp. 151-159,
    The first author would like to extend his sincere gratitude                            1990.
                                                                                    [21]    M. Bader, “A comparative study of new truncation error estimates
to Universiti Sains Malaysia for supporting this work under its                            and intrinsic accuracies of some higher order Runge-Kutta
post doctoral fellowship scheme. Much of this work was                                     algorithms”, Computers & Chemistry, Vol. 11, pp.121-124, 1987.
carried out during his stay at Universiti Sains Malaysia in                         [22]   M. Bader, “A new technique for the early detection of stiffness in
                                                                                           coupled differential equations and application to standard Runge-
2011. He wishes to acknowledge Universiti Sains Malaysia’s                                 Kutta algorithms”, Theoretical Chemistry Accounts, Vol. 99, pp.
financial support.                                                                         215-219, 1988.
                                                                                    [23]   S. Senthilkumar and A.R.M. Piah, “Solution to a system of second
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         Journal of Computer, Mathematical Sciences and Applications,                      Zealand Two-Stream International Conference on Artificial Neural
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         Sciences, Vol. 33, pp. 397-409, 2008.                                             and Their Applications to Cognitive and Decision Processes”,
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                                                                               63                                http://sites.google.com/site/ijcsis/
                                                                                                                 ISSN 1947-5500
                                                                     (IJCSIS) International Journal of Computer Science and Information Security,
                                                                                                                         Vol. 9, No. 6, June 2011

[38]   M. Laiho, A. Paasio, J. Flak and K. A. I. Halonen, “Template                                             Senthilkumar was born in Neyveli Township,
       design for cellular nonlinear networks with 1-bit weights”, IEEE                                         Cuddalore District, Tamilnadu, India on 18th July
       Transactions on Circuits and Systems-I: Regular Papers, Vol. 55,                                         1974. He received his B.Sc in Mathematics from
       No. 3, pp. 904-913, 2008.                                                                                Madras University in 1994, M.Sc in Mathematics
[39]    T. Yang, L. B. Yang, C. W. Wu, L. O. Chua, “Fuzzy cellular                                              from Bharathidasan University in 1996, M.Phil
       neural networks: Theory”, In Proceedings of IEEE International                                           in Mathematics from Bharathidasan University in
       Workshop on Cellular Neural Networks and Applications, pp.181-                                           1999 and M.Phil in Computer Science &
       186, 1996.                                                                                               Engineering from Bharathiar University in 2000.
[40]   T. Yang, L. B. Yang, “The global stability of fuzzy cellular neural                                      He also has a PGDCA and PGDCH in Computer
       networks”, IEEE Transactions on Circuit and Systems-I, Vol. 43,            Science and Applications and Computer Hardware from Bharathidasan
       pp. 880-883, 1996.                                                         University which he obtained in 1996 and 1997, respectively. He has a
[41]   T. Yang and L.B. Yang, “Fuzzy cellular neural network: A new               doctoral degree in Mathematics and Computer Applications from National
       paradigm for image processing”, International Journal of Circuit           Institute of Technology [REC], Tiruchirappalli, Tamilnadu, India. Currently,
       Theory and Applications, Vol. 25, pp. 469-481, 1997.                       he is a post doctoral fellow at the School of Mathematical Sciences, Universiti
[42]   T. Yang and L.B. Yang, “Application of fuzzy cellular neural               Sains Malaysia, 11800 USM Pulau Pinang, Malaysia. Prior to this
       networks to Euclidean distance transformation”, IEEE                       appointment, he was a lecturer/assistant professor in the Department of
       Transactions on Circuits and Systems-I, CAS-44, pp. 242-246,               Computer Science at Asan Memorial College of Arts and Science, Chennai,
       1997.                                                                      Tamilnadu, India. He has published many good research papers in
[43]   A. Kandel, “Fuzzy Techniques in Pattern Recognition”, John                 international conference proceedings and peer-reviewed/refereed international
       Wiley, New York, 1982.                                                     journals with high impact factor. He has made significant and outstanding
[44]    R.R. Yager and L.A. Zadeh (eds), “An Introduction to Fuzzy
                                                                                  contributions to various activities related to research work. He is also an
       Logic in Intelligent Systems”, Kluwer, Boston, 1992.
                                                                                  associate editor, editorial board member, reviewer and referee for many
[45]   J.A. Nossek, G. Seiler, T. Roska and L.O. Chua, “Cellular neural
                                                                                  scientific international journals. His current research interests include
       networks: Theory and circuit design”, International Journal of
                                                                                  advanced cellular neural networks, advanced digital image processing,
       Circuit Theory and Applications, Vol. 20, pp. 533-553, 1992.
                                                                                  advanced numerical analysis and methods, advanced simulation and
[46]   G. F. Dalla Betta, S. Graffi, M. Kovacs and G. Masetti, “CMOS
                                                                                  computing and other related areas.
       implementation of an analogy programmed cellular neural
       network”, IEEE Transactions on Circuits and Systems-Part–II,
       Vol. 40, pp. 206–214, 1993.                                                Abd Rahni Mt Piah was born in Baling, Kedah Malaysia on 8th May 1956. He
[47]   C.L. Yin, J.L. Wan, H. Lin and W.K. Chen, “Brief                                                       received his B.A. (Cum Laude) in Mathematics
       Communication: The cloning template design of a cellular neural                                        from Knox College, Illinois, USA in 1979. He
       network”, Journal of the Franklin Institute, Vol. 336, pp. 903-909,                                    received his M.Sc in Mathematics from
       1999.                                                                                                  Universiti Sains Malaysia in 1986. He obtained
[48]    L. O. Chua and P. Thiran, “An analytic method for designing                                           his Ph.D in Approximation Theory from the
       simple cellular neural networks”, IEEE Transactions on                                                 University of Dundee, Scotland UK in 1993. He
       Circuitsand Systems-I, Vol. 38, pp. 1332-1341, 1991.                                                   has been an academic staff member of the School
[49]    T. Matsumoto, L.O. Chua and R. Furukawa, “CNN cloning                                                 of Mathematical Sciences; Universiti Sains
       template: hole filler”, IEEE Transactions on Circuits and Systems,                                     Malaysia since 1981 and at present is an
       Vol. 37, pp. 635-638, 1990.                                                Associate Professor. He was a program chairman and deputy dean in the
[50]   K. Murugesan and P. Elango, “CNN based hole filler template                School of Mathematical Sciences, Universiti Sains Malaysia for many years.
       design using numerical integration technique”, LNCS 4668, pp.              He has published various research papers in refereed national and international
       490-500, 2007.                                                             conference proceedings and journals. His current research areas include
                                                                                  Computer Aided Geometric Design (CAGD), Medical Imaging, Numerical
                                                                                  Analysis and Techniques and other related areas.




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                                                                                                                   ISSN 1947-5500