Degree Reduction of Interval B-Spline Curves

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1 International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:14 No:03 1 Degree Reduction of Interval B-Spline Curves O. Ismail, Senior Member, IEEE Abstract An algorithmic approach to degree reduction of interval B-spline curve is presented in this paper. The curves that are useful in geometric modeling should have a relatively smooth shape and should be intuitively connected with the path of the sequence of control points. One family of curves satisfying this requirement is represented by the B-spline curves. spline curves) associated with the original interval B-spline curve are obtained. The control points of the four fixed Kharitonov's polynomials (four fixed B-spline curves) with lower degree are obtained by solving a set of linear equations. Finally, the interval control points of the required reduced interval B-spline curve are obtained from the fixed control points of the four reduced fixed Kharitonov's polynomials (four fixed reduced B-spline curves). An illustrative example is included in order to demonstrate the effectiveness of the proposed method. Index Terms Image processing, CAGD, degree reduction, interval B-spline curve. I. INTRODUCTION Computer-Aided Geometric Design (CAGD) is the basis for modern design in most branches of industry, from naval and aeronautic to textile industry and medical imaging. The technology using 3D graphics, virtual reality, animation techniques, etc., requires storing and processing complex images and complex geometric models of shapes (face, limbs, organs, etc.). It is necessary to understand better how to discretize geometric objects such as curves, surfaces, and volumes. Some of the design problems are handled by breaking the curves or surfaces into simpler pieces and then specifying how these pieces are joined together with some degree of smoothness. In Computer Aided Design and Geometric Modeling, there are considerable interests in approximating curves and surfaces with simpler forms of curves and surfaces. This problem arises whenever CAD data need to be shared across heterogeneous systems which use different proprietary data structures for model representations. For example, some systems restrict themselves to polynomial forms or limit the polynomial degree that they accommodate. Geometric modeling and computer graphics have been interesting and important subjects for many years from the point of view of scientists and engineers. One of the main and useful applications of these concepts is the treatment of curves and surfaces in terms of control points, a tool extensively used in CAGD. Parametric representations are widely used since they allow considerable flexibility for shaping and design. B- spline is among the most commonly used method for curve and surface design, and it has been widely used in practical The author is with Department of Computer Engineering, Faculty of Electrical and Electronic Engineering, University of Aleppo, Aleppo, CAD systems 1], 2]. In a system that uses B-spline 3], 4], 5], 6], 7], 8]. We should implement many practical algorithms, such as position and derivatives evaluation, knot insertion, knot deletion and degree elevation. A B-spline curve 4], 9], 10] is more widely and suitably used to represent a complex curve than higher degree Bezier curve because of its local control property and its ability to interpolate or approximate a curve with lower degree. The B-spline curve is a generalization of the Bezier curve and has more desired properties than the Bezier curves. The B-spline has the following important properties. First, it is a piecewise polynomial curve with a given degree. This property allows for designing a complex curve with lower degree polynomials, using multiple segments joined with certain continuity constraints. Second, the B-spline curve is contained in the convex hull of its control polyline. The polyline is defined by the B-spline control points; therefore, it can be used to represent the shape of the curve. Third, a change of the position of the control points only locally affects the curve. Thus the controllability is more flexible than the Bezier counterpart, and that is important for a curve design. Fourth, no straight line intersects the B-spline curve more times than it intersects the curve s control polyline. This results in a variation diminishing property. Last, affine transformation such as rotation, translation, and scaling can be applied to the B-spline control points quite easily instead of to the curve itself. This results in the affine invariance property. The B-spline curve overcomes the main disadvantages of the Bezier curve which are (1) the degree of the Bezier curve depends on the number of control points, (2) it offers only global control, and (3) individual segments are easy to connect with continuity, but is difficult to obtain. The B-spline curve is an approximating curve and is therefore defined by control points. However, in addition to the control points, the user has to specify the values of certain quantities called knots. They are real numbers that offer additional control over the shape of the curve. Degree reduction of parameter curves is one of the most common operations in Computer Aided Geometric Design (CAGD). It is used in data transfer and exchange between various CAGD systems, compression of shape data information, and so on. In some cases a designer has to link

2 International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:14 No:03 2 two or more B-spline curves of different degree to form a new curve or surface. In both situations the set of input curves must have a common degree. Such a problem can be solved either using degree elevation 11], 12] or reduction 13]. Both solutions should not affect the shape of curves. An interval B-spline curve is a B-spline curve whose control points are rectangles (the sides of which are parallel to coordinate axis) in a plane. Such a representation of parametric curves can account for error tolerances. Based upon the interval representation of parametric curves and surfaces, robust algorithms for many geometric operations such as curve/curve intersection were proposed 14]. The series of works by the authors of 14] indicate that using interval arithmetic will substantially increase the numerical stability in geometric computations and thus enhance the robustness of current CAD/CAM systems., - is minimum. The interval curves and have the same geometry and parameterization. spline curves), - associated with the original interval B- spline curve are: This paper is organized as follows. Section contains the basic results, whereas section presents a numerical example, and the final section offers conclusions. II. THE BASIC RESULTS For a given set of interval control points, - the uniform interval B-spline curve can be either the open or closed interval B-spline curves. In the case of closed interval B-spline curve. The closed uniform interval B-spline curve consists of connected interval segments of degree and defined by a linear combination of B-spline basis functions as: spline curves) can be written in matrix form as follows: ] ] with knot vector * +. ] ] The problem is to find a set of interval control points, - of the closed interval B-spline curve, that consists of connected interval segments of degree and is an integer greater than or equal to with a new knot vector * +. Each connected interval segment defined by a linear combination of B-spline basis functions as: ] spline curves), - associated with the reduced interval B- spline curve are: ] ] ] ] with new knot vector * +. such that: Similarly, the four fixed reduced Kharitonov's polynomials (four fixed reduced B-spline curves) can be written in matrix form as follows:

3 International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:14 No:03 3 and ] The normalized local support B-spline basis functions of degree are defined by the following deboor-cox recursive formula 16], 17]. { {, -, - } } with the convention that, where are the knots and. Now the problem can be converted into the following problem. For a given four fixed Kharitonov's polynomials (four fixed B-spline curves) associated with the original interval B-spline curve as in equation, find the corresponding four fixed reduced Kharitonov's polynomials (four fixed reduced B-spline curves) as in equation, such that: is minimum. In order to get the control point of B-spline, for we should get partial derivatives of from equation, that is: The control points of four fixed Kharitonov's polynomials (four fixed B-spline curves) with lower degree for and can be obtained by solving equation. Equation can be converted to the following form 18]: where,, - ] Therefore the control points for and of the reduced four fixed Kharitonov's polynomials (four fixed B-spline curves) can be obtained as: Finally, the interval control points of the required reduced interval B-spline curve are obtained as follows:, - ] and III. INTERVAL B-SPLINE DEGREE REDUCTION In order to find the reduced interval B-spline curve, some of the intermediate steps are needed to be computed. For interval B-spline degree reduction, the four fixed B-spline curves has to be used in finding the degree reduction of interval B-spline curve. Interval B-spline degree reduction can be created by the following steps: Algorithm for the Interval B-spline Degree Reduction 1. Find the four fixed B-spline curves of degree associated with the original interval B-spline curve. 2. Reduce degree of the four fixed B-spline curves as explained in section. 3. The reduced interval B-spline control points can be obtained from the four fixed reduced B-spline control points as follows:, - ] and IV. NUMERICAL EXAMPLE Consider the interval closed B-spline curve defined by four interval control points *, -+, this closed

4 y International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:14 No:03 4 uniform interval B-spline curve, consists of four connected interval cubic segments of order given as: with knot vector * +. The problem is to find the closed uniform interval B-spline curve, which consists of the four connected quadratic interval segments of order, with knot vector * +. As explained in section, the four fixed Kharitonov's polynomials (four fixed original B-spline curves) of order is written in matrix form as follows: Simulation results in Figure (1) shows the envelopes of the original interval B-spline curve and the reduced interval B-spline curve, respectively, where and indicate the initial control points of the original and reduced B-spline curves, respectively Fig.1:The original and reduced B-spline envelopes. o Original B-spline Curve. x Reduced B-spline Curve. 20 ] and the corresponding four fixed reduced Kharitonov's polynomials (four fixed reduced B-spline curves) of order is written in matrix form as follows: ] The control points for and ( ) of the reduced four fixed Kharitonov's polynomials (four fixed B-spline curves) are obtained as follows: and the interval control points of the required connected reduced interval B-spline segments are obtained: x V. CONCLUSIONS In this paper, an algorithmic approach to degree reduction of interval B-spline curve is presented. The B- spline stands as one of the most efficient curve representations, and possesses very attractive properties such as spatial uniqueness, boundedness and continuity, local shape controllability, and invariance to affine transformation. B-splines are parametric models that require a neighborhood relationship to be established between the data points prior to the construction of B-spline curve, as well as a topological (geometrical) meaningful assignment to the parameters. These properties indicate to the efficiency of evaluating the points on B-spline curves. Among several important properties of a non-rational curve, the degree reduction of a curve from into, (where is an integer greater than or equal to and ), is very useful for many applications to decrease the complexity of the polynomials. The proposed method of interval B-spline curve degree reduction is based on the whole continuous B- spline curve rather than the discrete re-sampled points used in almost all other methods that would introduce error and lead to imprecise results. The four fixed Kharitonov's polynomials (four fixed B-spline curves) associated with the original interval B-spline curve are obtained. The control points of the four fixed Kharitonov's polynomials (four fixed B-spline curves) with lower degree are obtained by solving a set of linear equations. Finally, the interval control points of the required reduced interval B-spline curve are obtained from the fixed control points of the four reduced fixed Kharitonov's polynomials (four fixed reduced B-spline curves).

5 International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:14 No:03 5 REFERENCES 1] L. Piegl and W. Tiller, The NURBS Book, Springer-Verlag, Edition, ] G. E. Farin, Curves and Surfaces for CAGD: A Practical Guide, Morgan Kaufmann; 5th edition, ] Y. Guand and T. Tjahjadi, Coarse-to-fine planar object identification using invariant curve features and B-spline modeling, Pattern Recognit. Vol. 33, pp , ] Z. Huang and F.S. Cohen, Affine-invariant B-spline moments for curve matching, IEEE Trans. Pattern Anal. Machine Intell. Vol. 5, No. 10, ] A. K. Klein, F. Lee and A. A. Amini, Quantitative coronary angiography with deformable spline models, IEEE Trans. Med. Imaging Vol. 16, No. 5, ] A. M. Baumberg and D. C. Hogg, Learning flexible models from image sequences, Eur. Conf. Comput. Vis. 94, pp , ] Y. Wang, E. K. Teoh and D. Shen, Lane detection using B-snake, in: IEEE International Conference on Information, Intelligence and Systems (ICIIS 99), Washington, DC, ] Y. Wang, E. K. Teoh and D. Shen, Structure-adaptive B-snake for segmenting complex objects, in: International Conference on Image Processing (ICIP 2001), Thessaloniki, Greece, pp , ] F. Mokhtarian and A. K. Mackworth, A theory of multiscale, curvature-based shape representation for planar curves, Pattern Analysis and Machine Intelligence, IEEE Transactions on, Vol. 14, No. 8, pp , ] T. B. Sebastian and B. B. Kimia, Curves vs. skeletons in object recognition, International Conference on Image Processing, Proceedings, Vol. 3, pp , ] O. Ismail, "Degree elevation of rational interval Bezier curves". 'Proc., the Third International Conference of E-Medical Systems, Fez, Morocco, ] O. Ismail, "Degree elevation of interval Bezier curves using Legendre- Bernstein basis transformations". International Journal of Video & Image Processing and Network Security (IJVIPNS), Vol. 10, No.6, pp. 6-9, ] O. Ismail, "Degree reduction of interval Bezier curves using Legendre- Bernstein basis transformations". The 1st Taibah University International Conference on Computing and Information Technology (ICCIT 2012), pp ] G. Shen and N. M. Patrikalakis, Numerical and geometrical properties of interval B-spline, International journal of shape modeling, Vol. 4, No. (1/2), pp , ] V. L. Kharitonov, "Asymptotic stability of an equilibrium position of a family of system of linear differential equations", Differential 'nye Urauneniya, vol. 14, pp , ] C. deboor, On calculating with B-splines, J. Approx. Theory, Vol. 6, pp. 50, ] M. G. Cox, The numerical evaluation of B-splines, J. Inst. Math. & Applic., Vol. 10, pp. 134, ] Y. Wang, Object segmentation and matching using B-spline model, Ph.D. Thesis, Nanyang Technological University, Singapore, O. Ismail (M 97 SM 04) received the B. E. degree in electrical and electronic engineering from the University of Aleppo, Syria in From 1987 to 1991, he was with the Faculty of Electrical and Electronic Engineering of that university. He has an M. Tech. (Master of Technology) and a Ph.D. both in modeling and simulation from the Indian Institute of Technology, Bombay, in 1993 and 1997, respectively. Dr. Ismail is a Senior Member of IEEE. Life Time Membership of International Journals of Engineering & Sciences (IJENS) and Researchers Promotion Group (RPG). His main fields of research include computer graphics, computer aided analysis and design (CAAD), computer simulation and modeling, digital image processing, pattern recognition, robust control, modeling and identification of systems with structured and unstructured uncertainties. He has published more than 62 refereed journals and conferences papers on these subjects. In 1997 he joined the Department of Computer Engineering at the Faculty of Electrical and Electronic Engineering in University of Aleppo, Syria. In 2004 he joined Department of Computer Science, Faculty of Computer Science and Engineering, Taibah University, K.S.A. as an associate professor for six years. He has been chosen for inclusion in the special 25th Silver Anniversary Editions of Who s Who in the World. Published in 2007 and Presently, he is with Department of Computer Engineering at the Faculty of Electrical and Electronic Engineering in University of Aleppo.

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