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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. 57 http://sites.google.com/site/ijcsis/ 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 ) 58 http://sites.google.com/site/ijcsis/ 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 59 http://sites.google.com/site/ijcsis/ 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 60 http://sites.google.com/site/ijcsis/ 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 61 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 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 62 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 9, No. 6, June 2011 [15] D.E. Keyes, A. Sameh and V.V. Krishnan, “Parallel Numerical 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. 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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. 64 http://sites.google.com/site/ijcsis/ ISSN 1947-5500

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