Subdivision Matrices and Iterated Function Systems for Parametric Interval Bezier Curves

Transcription

1 International Journal of Video&Image Processing Network Security IJVIPNS-IJENS Vol:7 No:0 Subdivision Matrices Iterated Function Systems for Parametric Interval Bezier Curves O. Ismail, Senior Member, IEEE Abstract Fractals are famous both for their strange appearance for their odd geometric properties. The problem of modeling is very simple when one has mathematical description of the fractal he wants to model. A set of transformations that generates a fractal by iteration is called an iterated function systems (IFS). An iterated function system maps the corresponding fractal onto itself as a collection of smaller self-similar copies. Fractals are often defined as fixed points of iterated function systems because when applied to the fractal the transformations that generate a fractal do not alter the fractal. In this paper we are going to show that the parametric interval Bezier curves are indeed fractals if the four associated with the original interval Bezier curves are fractals. The four fixed Kharitonov's polynomials (four fixed Bezier curves) associated with the original parametric interval Bezier curve are obtained. Iterated function systems (IFS) for the four are constructed so that starting with any compact set, not ust the four control polygons, iterating the transformations, the resulting sets converge in the limit to the given four fixed Kharitonov's polynomials (four fixed Bezier curves) curves. The transformation matrices in the iterated function systems are constructed to mimic the subdivision procedure, so there is indeed a deep connection between subdivision algorithms fractal procedures. The fixed control points that subdivide the four curves into two four fixed Kharitonov's polynomials (four fixed Bezier curves) curves are affine combinations of the original four control points. Thus we can represent the four fixed Kharitonov's polynomials (four fixed Bezier curves) subdivision by two square matrices whose entries are the coefficients in these affine combinations. A numerical example is included in order to demonstrate the effectiveness of the proposed method. Index Terms Subdivision matrices, iterated function systems, interval Bezier curve, CAGD. I. INTRODUCTION Parametric representations are widely used since they allow considerable flexibility for shaping design. In computer aided design geometric modeling, there are considerable interests in approximating curves surfaces with simpler forms of curves surfaces. Parametric Bezier curves are interactive. It is possible to control the shape of the Bezier curve by moving the control points by smoothly connecting individual segments. The Bezier curve possesses a characteristic intuitiveness in expressing the desired shape through property of convexity of control polygon. Interestingly during the same time, de Castelau too adopted Bernstein Basis for his work which was The author is with Department of Computer Engineering, Faculty of Electrical Electronic Engineering, University of Aleppo, Aleppo, ( oismail@ieee.org). focused on the property of non-negativity partition of unity of the basis function associated with the control points. Subdivision of the Bezier curve is required to break the curve into number of small segments for various applications like curve fitting, segmentation, interpolation, so. Besides some key Bernstein basis properties that constraints the behavior of Bezier curve like symmetry, recursion, nonnegativity, unity of partition, unimodality, relation to monomial basis, lower upper bounds, variation diminishing property, derivatives integrals there are some algorithms based properties like degree elevation, degree reduction composition that have consistently been evaluated analyzed over a period of decades in order to broaden its applications. There are several kinds of polynomial curves in CAGD, e.g., Bezier [], [], [3], [4] Said-Ball [5], Wang-Ball [6], [7], [8], B-spline curves [9] DP curves [0], []. These curves have some common different properties. All of them are defined in terms of the sum of product of their blending functions control points. They are ust different in their own basis polynomials. In order to compare these curves, we need to consider these properties. The common properties of these curves are control points, weights, their number of degrees. Control points are the points that affect to the shape of the curve. Weights can be treated as the indicators to control how much each control point influences to the curve. Number of degree determines the maximum degree of polynomials. In different curves, these properties are not computed by the same method. To compare different kinds of curves we need to convert these curves into an intermediate form. A parametric interval Bezier curve [-8] is a Bezier curve whose control points are rectangles (the sides of which are parallel to coordinate axis) in a plane. The control points that subdivide a Bezier curve into two Bezier curves are affine combinations of the original Bezier control points. Thus we can represent Bezier subdivision by two square matrices whose entries are the coefficients in these affine combinations. Fractals are always associated with obects that have strange pathological properties. Taking into account that fact the one that linear segments can also be constructed using suitable (IFS). Fractals are identified with attractors of some contractive mappings because such identification is the best one from the computer graphics point of view. Indeed, the mappings give us some coefficients which are commonly IJVIPNS-IJENS April 07 IJENS

2 International Journal of Video&Image Processing Network Security IJVIPNS-IJENS Vol:7 No:0 used in image coding. Fractals are attractors-fixed points of iterated function systems (IFS) [9], [0], []. The fractal methods became very important tools in many disciplines found very wide practical applications e.g., image compression, generating shore lines, mountains, clouds, pattern recognition, image processing, computer graphics or in medicine economy. In computer graphics, there is a need for succinct description of a complex obect s geometry an efficient way to vary the geometry of an obect to make movies. [] associated with the original parametric interval Bezier curve are: P n (u) = p 0 B n 0 (u) + p B n (u) + p + B n (u) + p + 3 B n 3 (u) + p 4 B n 4 (u) + p 5 B n 5 (u) + α 0,n B n 0 (u) + α,n B n (u) + α,n B n (u) + + α n,n B n n (u) P n (u) = p 0 B n 0 (u) + p + B n (u) + p + B n (u) + p 3 B n 3 (u) + p 4 B n 4 (u) + p + 5 B n 5 (u) + α 0,n B n 0 (u) + α,n B n (u) + α,n B n (u) + + α n,n B n n (u) In this paper it has been shown that if the four fixed Kharitonov's polynomials (four fixed Bezier curves) associated with the original parametric interval Bezier curve are fractals; then the given parametric interval Bezier curve is fractal. This paper is organized as follows. Section II contains the basic results, whereas section III shows a numerical example, the final section offers conclusions. II. THE BASIC RESULTS Parametric interval Bezier curves are segments of parametric interval polynomial curves. Each piece of a parametric interval polynomial curve is ust like any other piece of a parametric interval polynomial curve, so each segment of a parametric interval Bezier curve is itself a parametric interval Bezier curve. Thus, parametric interval Bezier curves are self-similar. Therefore, even though the parametric interval Bezier curves are infinitely differentiable, the interval Bezier curves are also fractal interval curves. Let {[p i, p + n i ]} be a given set of interval control points which defines the interval Bezier curve: n P I n (u) = [p i, p + i ] B n i (u) n = ([x i, x + i ], [y i, y + i ]) B n i (u) 0 u () of degree n where {B n n k (u)} k=0 are Bernstein polynomials formed by: P 3 n (u) = p + 0 B n 0 (u) + p + B n (u) + p B n (u) + p 3 B n 3 (u) + p + 4 B n 4 (u) + p + 5 B n 5 (u) + α 3 0,n B n 0 (u) + α 3,n B n (u) + α 3,n B n (u) + + α 3 n,n B n n (u) P 4 n (u) = p + 0 B n 0 (u) + p B n (u) + p B n (u) + p + 3 B n 3 (u) + p + 4 B n 4 (u) + p 5 B n 5 (u) + α 4 0,n B n 0 (u) + α 4,n B n (u) + α 4,n B n (u) + + α 4 n,n B n n (u) (3) The four fixed Kharitonov's polynomials (four fixed Bezier curves) can be written as follows: n P n (u) = α i,n B i n (u) for all u [0,] ( =,,3,4) (4) The de Castelau evaluation algorithm for the four fixed Kharitonov's polynomials (four fixed Bezier curves) is also a subdivision procedure, if we run the de Castelau evaluation algorithm at the midpoint of the parameter interval [a, b], then the fixed control points of the four fixed Kharitonov's polynomials (four fixed Bezier curves) that subdivide the four at u = (a + b) emerge along the left right lateral edges of the de Castelau triangle as shown below in Figure. Moreover, the labels along each edge in this triangle are, independent of the choice of the parameter interval. Our initial goal is to develop explicit expressions for the fixed control points q 0, q,, q n r 0, r,, r n of the four fixed Kharitonov's polynomials (four fixed Bezier curves) that subdivide the four fixed Kharitonov's polynomials (four fixed Bezier curves) of degree n in terms of the original control points α 0, α,, α n of the four fixed Kharitonov's polynomials (four fixed Bezier curves). Such formulas are easy to find. B k (u) = ( k ) ( u)( k) u k, (k = 0,,, ) () ( k ) =! k! ( k)! q q q 3 r0 r r is a binomial coefficient. q0 0 3 r 3 The sufficient condition for the given parametric interval Bezier curves to be fractals is that the four fixed Kharitonov's polynomials (four fixed Bezier curves) associated with the parametric interval Bezier curve should be fractals. Now the interval Bezier curve P n I (u) for u [0,] is fractal, if the four P n (u) for u [0,] =,,3,4 are fractals. The four α n k Fig. : The de Castelau subdivision algorithm for the four fixed Kharitonov's polynomials (four fixed Bezier curves) for =,,3,4. In general, since all the paths from α k to q l from to r n l are identical: IJVIPNS-IJENS April 07 IJENS

3 International Journal of Video&Image Processing Network Security IJVIPNS-IJENS Vol:7 No:0 3 { q l = l l p(l, m)α m r n l m=0 l = l p(l, m)α n m m=0 } ( =,,3,4) where p(l, m) is the number of paths from α m to q l or equivalently from α n m to r n l. But the number of paths from the m th position at the base to the apex of a triangle is the same as the number of paths from the apex to the m th position at the base of a triangle. The number of paths from the apex to any node in a triangle is given by the values in the nodes of Pascal s triangle that is, by binomial coefficients. Therefore, p(l, m) = ( l ), so m { q l = l l ( l m ) α m r n l m=0 l = l ( l m ) α n m m=0 ( =,,3,4), l = 0,,, n, l = 0,,, n } (5) (6) Using Equation (6), we can represent the fixed points q 0, q,, q n r 0, r,, r n that subdivide the four fixed Kharitonov's polynomials (four fixed Bezier curves) using matrix multiplication. Let H(n) be the (n + ) (n + ) matrix defined by: ( =,,3,4) (8) We can split H(n) into two square (n + ) (n + ) matrices L(n) R(n), where L(n) consists of the first (n + ) rows R(n) the last (n + ) rows of H(n). Thus, ( l L(n) = = m ) l n [ n n n] (l, m = 0,,, n) ( =,,3,4) n n n n R(n) = 0 n n l ( n = n m ) n l [ 0 0 ] (l, m = 0,,, n) ( =,,3,4) (9) (0) where, the matrices L(n) R(n), represent left right subdivision for the four fixed Kharitonov's polynomials (four fixed Bezier curves). H(n) = n n n n 0 n n [ 0 0 ] ( =,,3,4) Then, using equation (6), we can write: α 0 H(n) α = [ α n ] [ q 0 q q n = r 0 r r n ] (7) Therefore, we can write: IJVIPNS-IJENS April 07 IJENS 0 0 α 0 α 0 q 0 0 α α q L(n) = = [ α n n ] [ n n n] [ α n ] [ q n ] ( =,,3,4) () n α 0 n n n α 0 r 0 α α R(n) = 0 n r n = [ α n ] [ α [ 0 0 ] n ] [ r n ] ( =,,3,4) () Thus L(n) R(n) can be used to subdivide the four. We notice that L(n) R(n) share a common row: the last

4 International Journal of Video&Image Processing Network Security IJVIPNS-IJENS Vol:7 No:0 4 row of L(n) is the same as the first row of R(n) because q n = r n. Starting with the original four fixed Kharitonov's polynomials (four fixed Bezier curves) control points applying these matrices repeatedly generates a sequence of fixed control polygons that converge to the original four fixed Kharitonov's polynomials (four fixed Bezier curves) [3], [4]. [α 0 Suppose, however, that the matrices α = α α n ] T are invertible. Let L p (n) = [α ] L(n) α { R p (n) = [α ] R(n) α } (3) ( =,,3,4) Then, α L p (n) = α ([α ] L(n) α ) = L(n) α { α R p (n) = α ([α ] R(n) α ) = R(n) α } ( =,,3,4) (4) Moreover, iterating the transformations L p (n) R p (n) multiplying now on the right instead of on the left generates the same sequence of four fixed Kharitonov's polynomials (four fixed Bezier curves) control polygons as iterating the matrices L(n) R(n) on the control polygons α for ( =,,3,4). But, this is the key point, it is easy to show that the matrices L p (n) R p (n), represent contractive maps. Therefore {L p (n), R p (n)} for ( =,,3,4) are iterated function systems, so we can start with any compact set the iteration will still converge to the four fixed Kharitonov's polynomials (four fixed Bezier curves) with fixed control points α 0, α,, α n for ( =,,3,4). For the four fixed Kharitonov's polynomials (four fixed Bezier curves) of degree n, the subdivision matrices L(n) R(n), are (n + ) (n + ) matrices. To form the matrices L p (n) R p (n) for ( =,,3,4) the coordinates of the four fixed Kharitonov's polynomials (four fixed Bezier curves) control points α = [α 0 α α n ] T must constitute invertible (n + ) (n + ) matrices. If the fixed points α 0, α,, α n lie in an n dimensional affine space, then we can use homogeneous coordinates to form (n + ) (n + ) matrices. α = [ α 0 α α n ] ( =,,3,4) (5) The matrices α for ( =,,3,4) are then invertible precisely when the fixed points α 0, α,, α n are affinely independent. Typically, however, the degree n of the four is larger than the dimension d of the ambient space of the fixed control points. For example, for planar cubic curves, T n = 3 but d =. Nevertheless, in general, we can still form the matrices L p (n) R p (n), by lifting the fixed control points α = [α 0 dimensions IJVIPNS-IJENS April 07 IJENS α α n ] T to higher For planar four fixed Kharitonov's polynomials (four fixed Bezier curves), we can proceed in the following fashion. Let (x k, y k ) for ( =,,3,4) be the rectangular coordinates of α k, for k = 0,,, n. Then we can lift α to higher dimensions by introducing homogeneous coordinates lifting each four fixed Kharitonov's polynomials (four fixed Bezier curves) control point α k to dimension k, 3 k n, by annexing zeroes ones to the coordinates of α k for ( =,,3,4) k = 0,,, n. For example, for cubics we introduce one additional coordinate set: α = [ α 0 α α α = x 0 x x ] [ ( =,,3,4) x 3 y 0 y y y ] (6) for quartics we introduce two additional coordinates set: α x 0 y α = [ α α α 3 α = x x x 3 ] [ ( =,,3,4) x 4 y y y 3 y ] (7) so on. We notice that in each case, the matrix α is invertible if: α 0 det [ α α ] 0 ( =,,3,4) (8) Thus, α is invertible if the points α 0, α, α, are affinely independent; that is, if the points α 0, α, α, are not collinear. Of course, we could choose any other three non collinear fixed control points this lifting technique would work equally well. To generate the arbitrary degree n four fixed Kharitonov's polynomials (four fixed Bezier curves) using the iterated function systems {L p (n), R p (n)} for ( =,,3,4), we can now proceed in the following way: (i) lift the four fixed Kharitonov's polynomials (four fixed Bezier curves) control points α from the ambient affine space of dimension d to the ambient affine space of dimension n ; (ii) generate the corresponding higher dimensional four fixed Kharitonov's polynomials (four fixed Bezier curves) using the iterated function systems {L p (n), R p (n)} for ( =,,3,4);

5 International Journal of Video&Image Processing Network Security IJVIPNS-IJENS Vol:7 No:0 5 (iii) proect the resulting n dimensional four fixed Kharitonov's polynomials (four fixed Bezier curves) orthogonally back down to the original dimension d. III. NUMERICAL EXAMPLE Consider the case of quadratic parametric interval Bezier curve P I (u) for 0 u. P I (u) = [p i, p + i ] B i (u) = ([x i, x + i ], [y i, y + i ]) B i (u) 0 u Here, n =, so the subdivision matrices L() R() are 3 3 matrices, which at least are the right size matrices for generating fractals in the plane. As explained in section II the matrices L() R() are obtained as follows: 0 0 L() = , R() = 0 [ 4 4 4] [ 0 0 ] Unfortunately, it is not clear what happens to an arbitrary point when we multiply by these matrices on the right. But suppose that instead of the matrices L() R(), we let: α = [ α 0 α α ] = [ x 0 x ( =,,3,4) x y 0 y y ] be the matrices of the four fixed Kharitonov's polynomials (four fixed Bezier curves) control points we consider the matrices: L p () = [α ] L () α { R p () = [α ] R () α } ( =,,3,4) Applying these matrices to the four fixed Kharitonov's polynomials (four fixed Bezier curves) control points α on the right yields: α L p () = α ([α ] L() α ) = L() α { α R p () = α ([α ] L() α ) = R() α } ( =,,3,4) Furthermore, if we iterate this procedure by continuing to apply the matrices L p () R p () on the right, then we get exactly the same points that are generated by applying the matrices L() R() to the four fixed Kharitonov's polynomials (four fixed Bezier curves) control polygons α on the left. Thus the fixed points generated by applying the matrices L p () R p () on the right to the four fixed Kharitonov's polynomials (four fixed Bezier curves) control points α converge to the four fixed Kharitonov's polynomials (four fixed Bezier curves) for the control points α IJVIPNS-IJENS April 07 IJENS. The matrices L p () R p () represent affine transformations, since the third column in both matrices is [0 0 ] T. because [ ] T is the last column of α, it follows that: 0 [α ] [ ] = [ 0] ( =,,3,4) L() [ ] = [ ] R() [ ] = [ ] { } Therefore, the last column of both L p () R p () is [0 0 ] T because: 0 [α ] L() [ ] = [α ] [ ] = [ 0] 0 [α ] R() [ ] = [α ] [ ] = [ 0] { ( =,,3,4) } The matrices {L p (), R p ()} for ( =,,3,4) form an iterated function systems. The eigenvalues of L() R() are.00, 0.50, 0.5, so the matrices L p () R p (), which are similar to L() R() therefore have the same eigenvalues. (The eigenvalue corresponds to translationthat is, to the column [0 0 ] T, so it is only necessary to consider the other two eigenvalues; that is, the eigenvalues of the upper submatrices of L p () R p ()). The limit curve in this case the quadratic four fixed Kharitonov's polynomials (four fixed Bezier curves) are fractals! Using the iterated function systems, we can start with any compact set still converge to the same fixed points. Thus we are no longer constrained to start the iteration with the four fixed Kharitonov's polynomials (four fixed Bezier curves) control polygons. Since the quadratic four fixed Kharitonov's polynomials (four fixed Bezier curves) are fractals; the given quadratic parametric interval Bezier curve is fractal. IV. CONCLUSIONS In this paper it has been seen that the parametric interval Bezier curves are fractals. Fractals are attractors-fixed points of iterated function systems. A parametric interval Bezier curves are interval polynomials-linear combinations of Bernstein basis functions. It has been shown that if the four

6 International Journal of Video&Image Processing Network Security IJVIPNS-IJENS Vol:7 No:0 6 associated with the original parametric interval Bezier curve are fractals; then the given parametric interval Bezier curve is fractal. The four fixed Kharitonov's polynomials (four fixed Bezier curves) associated with the original parametric interval Bezier curve are obtained. The de Castelau subdivision algorithm is used to show that the four fixed Kharitonov's polynomials (four fixed Bezier curves) associated with the original parametric interval Bezier curve are also attractors. Subdivision has the look feel of a fractal procedure. In the stard fractal algorithm, we start with a compact set, iterate a collection of contractive transformations, converge in the limit to a fractal shape. In the four fixed Kharitonov's polynomials (four fixed Bezier curves) subdivision, we start with the four fixed Kharitonov's polynomials (four fixed Bezier curves) control polygons, iterate the de Castelau subdivision algorithm, converge in the limit to the four fixed Kharitonov's polynomials (four fixed Bezier curves) curves. The four fixed Kharitonov's polynomials (four fixed Bezier curves) subdivision can be represented by two square matrices whose entries are the coefficients in these affine combinations. Iterated function systems are constructed to generate polynomial piecewise polynomial curves built from these subdivision matrices. The proposed approach is based on proecting from higher dimensions, the control points are lift to higher dimensions then apply orthogonal proections. To construct the four fixed Kharitonov's polynomial (four fixed Bezier curves) curves using the iterated function systems {L p (n), R p (n)}, we start with any compact subset of n dimensional affine space iterate these transformations. The result is proected orthogonally into two dimensions. Since the coordinates behave independently under recursive subdivision since we do not change the x, y coordinates when we lift the control points to higher dimensions, these orthogonal proections converge to the four fixed Kharitonov's polynomials (four fixed Bezier curves). Thus, somewhat surprisingly, the four fixed Kharitonov's polynomials (four fixed Bezier curves). This means that the given parametric interval Bezier curve is fractal. REFERENCES [] P. Bezier, "Definition Numerique Des Courbes et Surfa ces I", Automatisme, Vol., pp , 966. [] P. Bezier, "Definition Numerique Des Courbes et Surfa ces II", Automatisme, Vol., pp. 7, 967. [3] P. Bezier, Numerical control, Mathematics Applications, New York: Wiley, 97. [4] P. Bezier, The Mathematical Basis of the UNISURF CAD System, Butterworth, London, 986. [5] H. B. Said, "A generalized Ball curve its recursive algorithm", ACM. Transaction on Graphics, Vol. 8, No. 4, pp , 989. [6] A. A. Ball, CONSURF Part : Introduction to conic lofting tile, Computer Aided Design, Vol. 6, No. 4, pp , 974. [7] A. A. Ball, CONSURF Part : Description of the algorithms, Computer Aided Design, Vol. 7, No. 4, pp. 37 4, 975. [8] A. A. Ball, CONSURF Part 3: How the program is used, Computer Aided Design, Vol. 9, No., pp. 9, 977. [9] G. J. Wang, "Ball curve of high degree its geometric properties", Applied. Mathematics: A Journal of Chinese Universities Vol., pp. 6-40, 987. [0] J. Delgado J. M. Pena, A linear complexity algorithm for the Bernstein basis, Proceedings of the 003 International Conference on Geometric Modeling Graphics (GMAG 03), pp. 6 67, 003. [] J. Delgado J. M. Pena, "A shape preserving representation with an evaluation algorithm of linear complexity, Computer Aided Geometric Design, pp. -0, 003. [] O. Ismail," L Degree Reduction of Interval Bezier Curves", International Journal of Research Reviews in Computer Science (IJRRCS), Vol., No., pp. 4-46, 00. [3] O. Ismail, "Degree Elevation of Interval Bezier Curves Using Legendre- Bernstein Basis Transformations", International Journal of Video & Image Processing Network Security (IJVIPNS), Vol. 0, No. 6, pp. 6-9, 00. [4] O. Ismail, "Degree Elevation of Interval Bezier Curves", International Journal of Video & Image Processing Network Security (IJVIPNS), Vol. 3, No., pp. 8-, 03. [5] O. Ismail, "Reparametrization Subdivision of Interval Bezier Curves", International Journal of Video & Image Processing Network Security (IJVIPNS), Vol. 4, No. 4, pp. -6, 04. [6] O. Ismail, "A smooth Connection of Interval Bezier Curve Segments", International Journal of Video & Image Processing Network Security (IJVIPNS), Vol. 5, No. 4, pp. -5, 05. [7] O. Ismail, "Regularity of Interval Bezier Curves", International Journal of Video & Image Processing Network Security (IJVIPNS), Vol. 5, No. 6, pp. -4, 05. [8] O. Ismail, "Degree Reduction of Parametric Interval Bezier Curves", Proc., International Conference on Sciences of Electronic, Technologies of Information Telecommunications (SETIT 06), Hammamet, Tunisia, 06. [9] M. Barnsley, Fractals everywhere, (Second Edition), Academic Press, Boston, Mass., 993. [0] W. Kotarski A. Lisowska, On Bezier-fractal modeling of D shapes, International Journal of Pure Applied Mathematics, Vol. 4, No., pp. 9-30, 005. [] C. T. John, All Bezier curves are attractors of iterated function systems, New York Journal of Mathematics, Vol. 3, pp. 07-5, 007. [] V. L. Kharitonov, "Asymptotic stability of an equilibrium position of a family of system of linear differential equations", Differential 'nye Urauneniya, vol. 4, pp , 978. [3] R. Goldman, Pyramid algorithms: A dynamic programming approach to curves surfaces for geometric modeling, Morgan Kaufmann, San Francisco, 00. [4] J. Lane R. Riesenfeld, A theoretical development for the computer generation display of piecewise polynomial surfaces, IEEE Trans. on Pattern Anal. Mach. Intell., Vol., pp , 980. O. Ismail (M 97 SM 04) received the B. E. degree in electrical electronic engineering from the University of Aleppo, Syria in 986. From 987 to 99, he was with the Faculty of Electrical Electronic Engineering of that university. He has an M. Tech. (Master of Technology) a Ph.D. both in modeling simulation from the Indian Institute of Technology, Bombay, in , respectively. Dr. Ismail is a Senior Member of IEEE. Life Time Membership of International Journals of Engineering & Sciences (IJENS) Researchers Promotion Group (RPG). He is an Academic Member of the Athens Institute for Education Research, belonging to the Computer Research Unit the Electrical Engineering Research Unit. His main fields of research include computer graphics, computer aided analysis design (CAAD), computer simulation modeling, digital image processing, pattern recognition, robust control, modeling identification of systems with structured unstructured uncertainties. He has published more than 74 refereed international ournals conferences papers on these subects. In 997 he oined the Department of Computer Engineering at the Faculty of Electrical Electronic Engineering in University of Aleppo, Syria. In 004 he oined Department of Computer Science, Faculty of Computer Science Engineering, Taibah University, K.S.A. as an associate professor for six years. He has been chosen for inclusion in the special 5th Silver Anniversary Editions of Who s Who in the World. Published in Presently, he is working as a professor in Department of Computer Engineering at the Faculty of Electrical Electronic Engineering, University of Aleppo IJVIPNS-IJENS April 07 IJENS