This paper presents, a design method for the template of the hole-filler used to improve the performance of the character recognition using Numerical integration algorithms. This is done by analyzing the features of the hole-filler template and the dynamic process of CNN and by using popular numerical algorithms to obtain a set of inequalities satisfying its output characteristics as well as the parameter range of the hole-filler template. Some simulation results and comparisons are also presented

Batri, K.

Murugesh, V.

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International Journal of Computers, Communications & ControlVol. II (2007), No. 4, pp. 367-374An Efficient Numerical Integration Algorithm for Cellular NeuralNetwork BasedHole-Filler Template DesignV. Murugesh, Krishnan BatriAbstract: This paper presents, a design method for the template of the hole-fillerused to improve the performance of the character recognition using Numerical inte-gration algorithms. This is done by analyzing the features of the hole-filler templateand the dynamic process of CNN and by using popular numerical algorithms to ob-tain a set of inequalities satisfying its output characteristics as well as the parameterrange of the hole-filler template. Some simulation results and comparisons are alsopresented.Keywords: Cellular Neural Networks; Euler Algorithm; RK-Gill Algorithm; RK-Butcher Algorithm; Ordinary differential equations, Hole-filler.1 IntroductionCellular Neural Networks (CNNs) are analog, time-continuous, nonlinear dynamical systems andformally belong to the class of recurrent neural networks. Since their introduction in 1988 (by Chua andYang [1, 2], it has been the subject of intense research. Initial applications include image processing,signal processing, pattern recognition and solving partial differential equations, etc.Runge-Kutta (RK) methods have become very popular, both as a computational technique as well asa subject of research, which are discussed by Butcher [3, 4]. This method was derived by Runge aroundthe year 1894 and extended by Kutta a few years later. They developed algorithms to solve differentialequations efficiently and these are the equivalent of approximating the exact solutions by matching ‘n’terms of the Taylor series expansion.Butcher [3] derived the best RK pair along with an error estimate and by all statistical measures itappeared as the RK-Butcher algorithm. This RK-Butcher algorithm is nominally considered as sixthorder, since it requires six function evaluations, but in actual practice the “working order” is close to five(fifth order).Bader [4, 5] introduced the RK-Butcher algorithm for finding the truncation error estimates andintrinsic accuracies and the early detection of stiffness in coupled differential equations that arise intheoretical chemistry problems. Recently Devarajan et al [7] used the RK-Butcher algorithm for findingthe numerical solution of an industrial robot arm control problem. Oliveria [8] introduced the popularRK-Gill algorithm for the evaluation of ’effectiveness factor’ of immobilized enzymes.In this paper, we describe the dynamic behavior of CNN in section 2, Hole-filler template design ideasin Section 3, Numerical integration algorithms and its description is shown in Section 4, and simulationresults in Section 5.2 Dynamic Analysis of CNNThe dynamic equation of cell C(i, j) in an M x N cellular neural network is given by Chua and Yang[1, 2].Cdxi j(t)dt=− 1Rxxi j(t)+ ∑C(k,l)∈Nr(i, j)A(i, j;k, l)Ykl(t)+ ∑C(k,l)∈Nr(i, j)B(i, j;k, l)Ukl + I (1)Yi j(t) =[∣∣xi j(t)+1∣∣− ∣∣xi j(t)∣∣−1] ,1≤ i≤M,1≤ j ≤ n (2)Copyright © 2006-2007 by CCC Publications368 V. Murugesh, Krishnan Batriuij xij· · ·yijCR IYijn IUijn IFigure 1: CNN–Cellwherexi j, yi j and ui j are the state voltage, output voltage and input voltage respectively and they arefunctions of time t. Rx is a linear resistance, C is a linear capacitor, and A(i, j;k, l) and B(i, j;k, l) are thetransconductances of the output and input voltages of C(k, l) with respect to C(i, j) called the cloningtemplates of CNN. Nr(i, j) denotes the rth- neighbor of C(i, j) and I is an independent current source.From equation (2) one can see that the output voltage is nonlinear. We can rewrite the cell equation (1)as follows:Cdxi j(t)dt=− f [xi j(t)]+g(t) (3)Wheref [xi j(t)] =1Rxxi j(t) (4)g(t) = ∑C(k,l)∈Nr(i, j)C(k,l)6=C(i, j)A(i, j;k, l)Ykl(t)+ ∑C(k,l)B(i, j;k, l)Ukl + I (5)3 Hole-filler Template DesignThe Hole-Filler is a cellular neural network discussed by Yin et al [9], which fills up all the holes andremains unaltered outside the holes in a bipolar image. Let Rx = 1, C = 1 and let +1 stand for the blackpixel and -1 for the white one. We shall discuss the images having holes enclosed by the black pixels,when the bipolar image is input with U = {ui j} into CNN. The initial state values are set as Xi j(0) = 1.From the equation (2) the output values are Yi j(0) = 1, 1≤ i≤M, 1≤ j ≤ N.Suppose that the template A and B and the independent current source I are given asA =0 a 0a b a0 a 0 , a > 0, b > 0, B =0 0 00 4 00 0 0 , I =−1 (6)Where the template parameters a and b are to be determined. In order to make the outer edge cellsbecome the inner ones, normally auxiliary cells are added along the outer boundary of the image, andtheir state values are set to zeros by circuit realization, resulting in the zero output values. The stateequation (1) can be rewritten asdxi j(t)dt=−xi j(t)+ ∑C(k,l)∈Nr(i, j)A(i, j;k, l)Yi j(t)+4ui j(t)− I (7)For the cell C(i, j), we call the cells C(i+1, j), C(i−1, j), C(i, j+1) and C(i, j−1) to be the non-diagonal cells. Here, several cases are to be considered.An Efficient Numerical Integration Algorithm for Cellular Neural Network BasedHole-Filler Template Design 369Case 1: The input value ui j = +1 for cell C(i, j), signaling the black pixel. Because the initial statevalue of the cell C(i, j) has been set to 1, xi j(0) = 1, and from equation (2) its initial output value is alsoyi j(0) = 1 . According to the hole-filler demands, its eventual output should be yi j(∞) = 1. To obtainthis result we setdxi j(t)dt≥ 0 (8)Substituting this input ui j = 1 and equation (6) into equation (7), we obtaindxi j(t)dt=−xi j(t)+a[y(i−1) j(t)+ y(i+1) j(t)+ yi( j−1)(t)+ yi( j+1)(t)]+byi j(t)+3 (9)Combining equations (8) and (9) and considering the minimum value of xi j(t) = 1 this case yieldsa[y(i−1) j(t)+ y(i+1) j(t)+ yi( j−1)(t)+ yi( j+1)(t)]+byi j(t)+2≥ 0 (10)To facilitate our discussion, two sub cases are distinguished.Sub Case 1: The cell C(i, j) is inside the holes. Since xi j(0) = 1 , from equation (2) its initial outputvalue yi j(0) = 1. Considering equations (8) and (2), yi j(t)≥ 1. According to the hole-filler demands, itsinitial output of non-diagonal black pixels should not be changed inside the holes. The weights of a andb are equal to +4 and +1, respectively.Since A(i, j;k, l)> 1Rx the parameter b is found to be b > 1, or4a+b+2≥ 0, b > 1 (11a)Sub Case 2: The cell C(i, j) is outside the holes. To satisfy equation (10), we need to check only theminimum value on the left-hand side of equation (10). This is true when there are four non-diagonalwhite pixels around the cell C(i, j), where the weight of a in equation (10) is -4. Since yi j(t) ≥ 1, theweight of b is equal to 1. Combining this with b > 1 gives−4a+b+2≥ 0, b > 1 (11b)Case 2: The input value of cell C(i, j) is ui j = 1, signaling the white pixel. Substituting this input valuein equation (7) givesdxi j(t)dt=−xi j(t)+a[y(i−1) j(t)+ y(i+1) j(t)+ yi( j−1)(t)+ yi( j+1)(t)]+byi j(t)−5 (12)Sub Case 1: The cell C(i, j) is inside the holes. Since xi j(0) = 1, from equation (2) its initial outputvalue is yi j(0) = 1. According to the hole-filler demands, the holes should be filled by the black pixels,whereas its initial black pixels remain unaltered:dxi j(t)dt≥ 0 (13)Combining equations (12) and (13) and considering xi j(t)≥ 1 yieldsaby(i−1) j(t)+ y(i+1) j(t)+ yi( j−1)(t)+ yi( j+1)(t)c+byi j(t)−6≥ 0 (14)where we use the minimum value of xi j(t) in equation (12). Since the cell is inside the holes, itsinitial output of non-diagonal black pixels remain unchanged. The weight of a and b are equal to +4 and+1, respectively. Combining this with b > 1 gives4a+b−6≥ 0, b > 1 (15)370 V. Murugesh, Krishnan BatriSub Case 2: The cell C(i, j) is outside the holes. Since xi j(0) = 1, from equation (2) its initial outputvalue is yi j(0) = 1. According to the hole-filler demands, the final output of this cell should be white,and in this case yi j(∞)≤−1.dxi j(t)dt< 0 (16)Combining equations (12) and (16) and considering xi j(t)≤ 1. we getaby(i−1) j(t)+ y(i+1) j(t)+ yi( j−1)(t)+ yi( j+1)(t)c+byi j(t)−6 < 0 (17)where we use the maximum value of xi j(t) in equation (12).Initially yi j(0) = 1. How can the output of cell C(i, j) be changed to -1?. Where does this changebegin?. First we consider the situation where the change begins from the inside of the bipolar image. Ifthe maximum value on the left-hand side in equation (17) is less than zero, equation (17) holds. Inside theimage and outside the holes, the maximum weights of a and b are +4 and +1, respectively. This case wasdescribed by equation (15). In fact, the change of the output of the cell C(i, j) is like a wave propagatingfrom the edges to the inside of the image and it is verified from the simulated result. Therefore, weshould first consider the edge cell C(i, j), i = 1 or M, j = 1 or N. For this the maximum weight of a inequation (17) is +3, which is also the maximum weight of a outside the holes. The maximum weight ofb is +1, occurring at the initial time:3a+b−6 < 0, b > 1 (18)Combining Cases 1 and 2, we obtain3a+b−6 < 0,4a+b−6 ≥ 0, (19)−4a+b+2 ≥ 0.4 Numerical Integration AlgorithmsThe CNN dynamics on a digital architecture requires discretization in time and suitable numericalintegration algorithms. Three of the most widely used Numerical Integration Algorithms are used inRaster CNN Simulation described here. They are the Euler’s Algorithm, RK-Gill Algorithm discussedby Oliveria [8] and the RK-Butcher Algorithm discussed by Badder [5, 6] and Murugesh and Murugesan[10, 11, 12].4.1 Euler AlgorithmEuler’s method is the simplest of all algorithms for solving ODEs. It is explicit formula which usesthe Taylor-series expansion to calculate the approximation.xi j((n+1)τ) = xi j(piτ)+ τ f ′(x(piτ)) (20)4.2 RK-Gill AlgorithmThe RK-Gill algorithm discussed by Oliveria [8] is an explicit method requiring the computation offour derivatives per time step. The increase of the state variable xi j is stored in the constant ki j1 . ThisAn Efficient Numerical Integration Algorithm for Cellular Neural Network BasedHole-Filler Template Design 371result is used in the next iteration for evaluating ki j2 . The same must be done for ki j3 and ki j4 .ki j1 = f′ (xi j (τpi))ki j2 = f′(xi j (τpi)+12ki j1)(21)ki j3 = f′(xi j (τpi)+(1√2− 12)ki j1 +(1− 1√2)ki j2)ki j4 = f′(xi j (τpi)− 1√2ki j2 +(1+1√2)ki j3)The final integration is a weighted sum of the four calculated derivatives:xi j((n+1)τ) = xi j +16[ki j1+(2−√2)ki j2 +(2+√2)ki j3 + ki j4](22)4.3 RK-Butcher AlgorithmThe RK-Butcher algorithm discussed by Badder [5, 6] and Murugesh and Murugesan [10, 11, 12],is an explicit method. It starts with a simple Euler step. The increase of the state variable xi j is stored inthe constant ki j1 . This result is used in the next iteration for evaluating ki j2 . The same must be done forki j3 , ki j4 , ki j5 and ki j6 .ki j1 = τ f′ (xi j (piτ))ki j2 = τ f′(xi j (piτ)+14ki j1)ki j3 = τ f′(xi j (piτ)+18ki j1 +18ki j2)ki j4 = τ f′(xi j (piτ)− 12ki j2 + ki j3)(23)ki j5 = τ f′(xi j (piτ)+316ki j1 +916ki j4)ki j6 = ∆t f(xi j (piτ)− 37ki j1 +27ki j2 +127ki j3 −127ki j4 +87ki j5)The final integration is a weighted sum of the five calculated derivatives:xi j ((n+1)τ) =190(7ki j1 +32ki j3 +12ki j4 +32ki j5 +7ki j6)(24)5 Simulated ResultsThis Hole-filler template has been simulated using Pentium IV Machine with 3.0 Ghz. speed usingdifferent Numerical integration algorithms. The Settling time T and integration time Ts is obtainedwith various step sizes is to be displayed below in the Table-1. The settling time Ts describes the timefrom start of integration until the last cell leaves the interval [-1.0, 1.0] which is based on certain limit(e.g.,∣∣dxdt < 0.01∣∣). The simulation shows the desired output for every cell. We use +1 and -1 to indicatethe black and white pixels, respectively.372 V. Murugesh, Krishnan Batri(a) (b)Figure 2: Image Before and After Hole FillingStep SizeEuler Algorithm RK-Gill Algorithm RK-Butcher AlgorithmSettlingTime(T)IntegrationTime(Ts)SettlingTime(T)IntegrationTime(Ts)SettlingTime(T)IntegrationTime(Ts)0.5 6.5 2.5 6.8 2.4 5.8 2.40.6 15.5 12.7 16.4 13.7 11.4 12.50.7 32.5 28.3 32.0 27.4 30.0 27.20.8 35.0 30.7 34.6 30.0 32.4 29.60.9 36.8 32.6 36.6 32.0 34.2 31.61.0 37.9 33.6 37.6 33.0 36.0 32.81.5 44.8 36.8 45.7 36.9 41.1 36.02.0 47.4 43.2 48.2 43.6 46.2 42.82.5 50.6 45.6 52.6 44.5 48.3 44.73.0 53.5 49.3 54.8 50.2 52.3 49.2Table 1: Simulated Results of Hole-Filler Template DesignExampleThe templates A, B and I are given as follows:A = 0 1.0 01.0 3.0 1.00 1.0 0.0 , B =0 0 00 4 00 0 0 , I =−1.0Using the simulation program developed in C++, the input image is shown in Figure-2(a) and theoutput image in Figure-2(b). The obtained result is represented in Table-1. From the table-1, we findthat RK-Butcher algorithm yields less settling time and integration time compared to Euler and RK-Gillalgorithms.6 ConclusionIt is shown that the cellular neural network based hole-filler template could be designed from itsdynamic behavior using different numerical algorithms, and also the template for other cellular neuralnetwork can similarly be designed. The hole is filled and the outside image remains the same. Thetemplates of the cellular neural network are not unique and this is important in its implementation.An Efficient Numerical Integration Algorithm for Cellular Neural Network BasedHole-Filler Template Design 373Bibliography[1] L. O. Chua, L. Yang, Cellular Neural Networks: Theory, IEEE Transactions on Circuits and Sys-tems, Vol. 35, pp. 1257 - 1272, 1988.[2] L. O. Chua, L. Yang, Cellular Neural Networks: Applications, IEEE Transactions on Circuits andSystems, Vol. 35, pp. 1273 - 1290, 1988.[3] J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and Gen-eral Linear Methods, Chichester: John Wiley, 1987.[4] J. C. Butcher, Numerical Methods for Ordinary Differential Equations, Chichester: John Wiley,2003.[5] M. Bader, A comparative study of new truncation error estimates and intrinsic accuracies of somehigher order Runge-Kutta algorithms, Computers & Chemistry, Vol. 11 pp. 121-124, 1987.[6] M. Bader, A new technique for the early detection of stiffness in coupled differential equationsand application to standard Runge-Kutta algorithms, Theoretical Chemistry Accounts, Vol. 99, pp.215-219, 1988.[7] G. Devarajan, V. Murugesh, K. Murugesan, Numerical Solution of Second-order Robot Arm ControlProblem using Runge-Kutta Butcher Algorithm, International Journal of Computer Mathematics,Vol. 83 pp. 345-356, 2006.[8] S. C. Oliveira, Evaluation of effectiveness factor of immobilized enzymes using Runge-Kutta-Gillmethod: how to solve mathematical undetermination at particle center point?, Bio Process Engi-neering, Vol. 20, pp. 85-187, 1999.[9] C. L. Yin, J. L. Wan, H. Lin, W. K. Chen, The cloning template design of a cellular neural network,Journal of the Franklin Institute, Vol. 336, pp. 903-909, 1999.[10] V. Murugesh, K. Murugesan, Comparison of Numerical Integration Algorithms in Raster CNNSimulation, Lecture Notes in Computer Science, Vol. 3285 pp. 115-122, 2004.[11] V. Murugesh, K. Murugesan, Simulation of Cellular Neural Networks using the RK-Butcher algo-rithm, International Journal of Management and Systems, Vol. 21, pp. 65-78, 2005.[12] V. Murugesh, K. Murugesan, Simulation of Time-Multiplexing Cellular Neural Networks with Nu-merical Integration Algorithms, Lecture Notes in Computer Science, Vol. 3991, pp. 115-122, 2005.V. MurugeshDepartment of Information and Communication EngineeringHannam University133 Ojung-dong Daeduk-gu, Daejeon 306-791, Republic of KoreaE-mail: murugesh72@gmail.comK. BatriDepartment of Computer Science and EngineeringMuthyammal Engineering CollegeRasipuram 637 408, IndiaE-mail: krishnan.batri@gmail.comReceived: July 25, 2007374 V. Murugesh, Krishnan BatriDr. V. Murugesh obtained his Bachelor of Science in Com-puter Science and Master of Computer Applications degree fromBharathiar University, Coimbatore, India during 1992 and 1995respectively. Completed his PhD in Computer Science fromBharathidasan University, Tiruchirappalli, India during 2006. Hehas held various positions at National Institute of Technology,Tiruchirappalli, India and Sona College of Technology, Salem,India. Currently, he is working as Assistant Professor in theDepartment of Information and Communication Engineering atHannam University, Daejeon, Republic of Korea. His fields ofinterest are in Neural Network based Image Processing and Sci-entific computing. He has published more than 30 technical pa-pers in International, National journals and conferences.Krishnan Batri received the M.E. from Madurai Kamarj Uni-versity in 2003. He is a Research Scholar with the Depart-ment of Computer Science and Engineering in the National Insti-tute of Technology Tiruchirapalli, Tamil Nadu, India. Currentlyhe is working as a Assistant Professor with the department ofComputer Science and Engineering at Muthayammal Engineer-ing College, Rasipuram, Tamilnadu, India. His research inter-ests include Information Retrieval, Data fusion and Genetic algo-rithms.

An Efficient Numerical Integration Algorithm for Cellular Neural Network Based Hole-Filler Template Design

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An Efficient Numerical Integration Algorithm for Cellular Neural Network Based Hole-Filler Template Design

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