Reparametrization of Interval Curves on Rectangular Domain

Transcription

1 International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:15 No:05 1 Reparametrization of Interval Curves on Rectangular Domain O. Ismail, Senior Member, IEEE Abstract The parameter varies normally in the range, -. It is, however, easy to reparametrize the curve such that its parameter varies in an arbitrary range, -, where and are real and. If an interval curve, - is defined by interval control points, - in the range, - and if we select an interval, -, how can we calculate the interval control points, - such that the reparametrized interval curve, - based on them will go from, - to, - [i.e.,, -, - and, -, - ] and will be identical to, - in that interval. In this paper this concept has been discussed to form a new curve over rectangular domain such that its parameter varies in an arbitrary range, - instead of standard parameter, -. We also want that curve gets generated within the given error tolerance limit. The four fixed Kharitonov's polynomials (four fixed curves) associated with the original interval curve are obtained. A new parameterization is applied to the four fixed Kharitonov's polynomials (four fixed curves). Finally, the required interval control points are obtained from the fixed control points of the four fixed Kharitonov's polynomials. Using matrix representation, it has been shown how to determine the control polygon that covers an arbitrary interval, ] of the given interval curve. A numerical example is included in order to demonstrate the effectiveness of the proposed method. Index Term Reparametrization, interval curve, rectangular domain, image processing, CAGD. I. INTRODUCTION Geometric representations are ways to describe the arrangement of 3D vertices to create 3D shapes. There are three basic primitives or fundamental ways to create a 3D shape: points, curves, and polygons. Points are the simplest primitive; a 3D shape can be composed of a large amount of points to cover the whole 3D surface, where each point can be represented by a 3D vertex. A more practical and compact way of representing 3D shapes is using curves, which can be approximated by lines or points. A curve can be represented as a collection of points lying on a 2D or 3D plane. If the points are properly spaced, they can be interconnected using short straight line segments, yielding an adequate visual representation of the curve. Curves could be represented as a list of points; the higher the number of points used to represent a curve, the more accurate and smooth will be its visualization. There are several kinds of polynomial curves in CAGD, e.g., Bezier [1], [2], [3], [4] Said-Ball [5], Wang-Ball [6], [7], [8], B-spline curves [9] and DP curves [10], [11]. The author is with Department of Computer Engineering, Faculty of Electrical and Electronic Engineering, University of Aleppo, Aleppo, ( oismail@ieee.org). These curves have some common and different properties. All of them are defined in terms of the sum of product of their blending functions and control points. They are just 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, and 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. In parametric form, the coordinates of each point in a curve are represented as functions of a single parameter that varies across a predefined range of values. Since a point on a parametric curve is specified by a single value of the parameter, the parametric form is axis-independent. The curve endpoints and length are fixed by the parameter range. In many cases it is convenient to normalize the parameter range for the curve segment of interest to. The shape of a parametric curve can be controlled by using the control points of the curve, and the sequence of line segments defined by the control points is called the control polygon. A curve can be formed by smoothing its control polygon. In other words, by specifying a control polygon, a designer can easily design curves which he imagines and their shapes can also be easily modified. Compared with the algebraic representation, the parametric representation is more advantageous. This is why parametric curves are mostly used in CAD systems. Parametric representation for curves is important in computer-aided geometric design, medical imaging, computer vision, computer graphics, shape matching, and face/object recognition. They are far better alternatives to free form representation, which are plagued with unboundedness and stability problems. Parametric representations are widely used since they allow considerable flexibility for shaping and design. A curve that actually passes through each control point is called an interpolating curve; a curve that passes near to the control points but not necessarily through them is called an approximating curve. An interval is a simply a range of numbers defined by an upper and lower bound. For example, interval, - denotes the set * +. Intervals allow for calculations to be performed when the precise value of a variable is uncertain, but a bound on the possible values can be obtained. The rules

2 International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:15 No:05 2 governing such calculations define an interval arithmetic. Interval equations are easy to evaluate, but the resulting axisaligned rectangular regions are generally loose bounds. Much tighter bounds can be found, but there is a tradeoff between evaluation cost and bounding efficiency. To provide bound that very tight, but computationally much simpler than working directly with the curve or surface, we introduce the concept of a geometric interval. A geometric interval is a twoor three-dimensional region that is relatively easy to evaluate, but provides a tighter bound than axis-aligned rectangles. Standard interval arithmetic techniques form the framework for the geometric interval methods [12]. A scalar interval is a closed set of real values. Intervals are written, -, where, - * + Intervals can be thought of as numbers for which the exact value is uncertain. Viewed in this way, we can define the standard arithmetic operations for intervals. These operations form an interval arithmetic. If is one of the binary arithmetic operations in * + and we apply to the intervals, - and, -, the result is [13],[14]:, -, - *, -, -+ An interval curve is a 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. Any parametric curve can be reparametrized, then the reparametrization has the effect of changing the range over which the curve segment is defined. Parametrization aids computation in the sense that it provides a built-in parameter space for direct evaluation of quantities like tangents, normal, surface/ plane intersects and projections. Same curve can be represented by multiple parametrizations. Hence, in free form design reparametrization can be used to reconcile parametrization of different curve segments (or surface patches) that have been defined independently. Reparametrization of a curve means to change how a curve is parametrized, i.e., to change which parameter value is assigned to each point on the curve. Reparametrization can be performed by a parameter substitution. In curve design, the problem often is how to balance the desire for constructing a particular shape for a curve and obtaining a proper parametrization. Most often, we may construct the initial curve to interpolate/approximate the given data points with an initial parametrization using one of the various known techniques. However, the curves are refined to achieve the desired shape by various modifications of weights and/or control points. But, our parametrization is lost! That is the small changes in curve shape might lead to a bad or improper parametrization, which if used to construct surfaces results in badly parametrized surfaces. Hence, it is necessary to reparametrize the curve/surface to correct such situations where the shape is right and the parametrization is wrong. The problem of parametric interval curve reparametrization consists of changing the current parameter of a curve with another parameter using a reparametrization function [15], [16], [17], [18], [19]. It should be noted that the shape of the curve remains unchanged during this process; only the way the curve is described is altered. If it is important that the degree of the curve should be kept unchanged, we may choose a linear reparametrization function. This paper is organized as follows. Section contains the basic results, whereas section shows a numerical example and the final section offers conclusions. II. THE BASIC RESULTS An interval polynomial is a polynomial whose coefficients are interval. We shall denote such polynomials in the form to distinguish them from ordinary (single-valued) polynomials. In general we express an interval polynomial of degree in the form:, -, -, -, - for are interval control points (rectangular intervals) and are the blending functions. If is odd the blending functions recursively defined by: If { is even, the blending functions are defined by: } Vector-valued interval in the most general terms is defined as any compact set of points dimensions as tensor products of scalar intervals:, -, - *, -, -+ Such vector-valued intervals are clearly just rectangular regions in plane [20]. For each, -, the value of the interval curve is an interval vector that has the following significance: For any fixed curve whose control points satisfy, - for, we have. Likewise, the entire interval curve defines a region in the plane that contains all curves whose control points satisfy, - for.

3 International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:15 No:05 3 The given interval curve, - with standard parameter range, - will reparametrize as, - in the new parameter range, -. This means, -., - ( )/, where it has to satisfy the condition, -, - and, -, -. The four fixed Kharitonov's polynomials (four fixed curves), - associated with the original interval curve are: Typically, CAGD curve constructions require use of parametric variable, defined for a curve domain from to to represent curves. However, it can be transformed to another domain of parametric variable, by using reparameterization matrix,. The curve which has domain of parametric variable from to, is called uniform curve. On the other hand, the curve with domain of parametric variable, can be called non-uniform curve. For the conversions between non-uniform and CAGD curves, it is necessary to use the reparameterization matrix in order to obtain the same domains of parametric variables. Therefore, the four fixed Kharitonov's polynomials (four fixed curves) can be rewritten in the following way: The four fixed Kharitonov's polynomials (four fixed curves) can be written as follows:, -, -, - Now, the problem can be converted into: the four fixed curve, - for associated with the original interval curve, - with standard parameter range, - will reparametrize as, - in the new parameter range, -. Therefore, we can write: { } The reparameterization matrix, can be defined as follows:, - ( ), - ( ) or ( ), - In CAGD curves, the reparameterization matrix can be used to transform the control points of the four fixed Kharitonov's polynomials (four fixed curves) denoted by { } and, with parametric variable, into the control points of the corresponding four fixed Kharitonov's polynomials (four fixed curves) denoted by { }, with parametric variable, as follows:

4 y International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:15 No: is a monomial coefficient matrix given in equation (12) as follows: Finally, the interval control points *, -+ with parametric variable,, can be obtained as follows:, - [ ] On the other hand, the reparameterization matrix can be used to transform the control points of the four fixed Kharitonov's polynomials (four fixed curves), denoted by { } and, with parametric variable, into the control points of the corresponding four fixed Kharitonov's polynomials (four fixed curves), denoted by { }, with parametric variable, [ as follows: ], - [ ]., - / i.e., (, -)., - / and (, -)., -/ and will be identical in shape to, - in that interval. As explained in section, the four fixed Kharitonov's polynomials (four fixed curves) are found, and the reparameterization matrix and the monomial coefficient matrix are obtained as: and the interval control points with parametric variable,, are obtained as follows: Simulation results in Figure (1), shows the reparametrization of the interval curve on rectangular domain in the range, -. and the interval control points {[ ]} with parametric variable,, can be found as: [ ] [ ( ) ( ) ] Fig.1:Reparametrization of interval curve on rectangular domain. III. NUMERICAL EXAMPLE Example: Consider the interval curve, where, defined by four interval control points: Interval Curve Envelopes for 0 < = u < = 1. o Interval Curve Envelopes for 1 < = u < = x The problem is to find an interval curve, -, where, i.e., (, -, -) that s defined by four interval control points *, -+ such that the curve, - based on them will go from., -/ to IV. CONCLUSIONS We can break a curve down into smaller segments by truncating or subdividing it. There are many reasons for doing this. For example, we may truncate to isolate and extract that part of a curve surviving a model modification process, or subdivide it to compute points for displaying it. To truncate, subdivide, or change the direction of parameterization of a

5 International Journal of Video&Image Processing and Network Security IJVIPNS-IJENS Vol:15 No:05 5 curve ordinarily requires a mathematical operation called reparameterization. Ideally, this operation produces a change in the parametric interval so that neither the shape nor the position of the curve is changed. This effect is often referred to as shape invariance under parameterization and reparameterization. For example, reversing the direction of parameterization is the simplest form of reparameterization. This transformation does not change the shape or position of the curve, nor does it change any of the curve's analytic properties. In this paper a new representation forms of parametric interval curves is presented. This concept has been discussed to form a new curve, - over rectangular domain such that its parameter varies in an arbitrary range, - where and are real, instead of that its parameter varies in the range, -. The problem is to find an interval curve, -, where, that s defined by interval control points *, -+ such that the curve, - based on them will go from., -/ to., - / i.e., and (, -)., - / and will be identical in shape to, - in that interval. The four fixed Kharitonov's polynomials (four fixed curves) associated with the original interval curve are obtained. A new parameterization is applied to the four fixed Kharitonov's polynomials (four fixed curves). Finally, the required interval control points are obtained from the fixed control points of the four reparametrized Kharitonov's polynomials. Using matrix representation, it has been shown how to determine the control polygon that covers an arbitrary interval, - of the given interval curve. Also, we saw that any change in shape of the curve often causes the initial parametrization to be lost and hence reparametrization which preserves the shape of the curve is required, which can be done in terms of certain weight relations that are preserved by reparametrization. REFERENCES [1] P. Bezier, "Definition Numerique Des Courbes et I", Automatisme, Vol. 11, pp , [2] P. Bezier, "Definition Numerique Des Courbes et II", Automatisme, Vol. 12, pp , [3] P. Bezier, Numerical control, Mathematics and Applications, New York: Wiley, [4] P. Bezier, The Mathematical Basis of the UNISURF CAD System, Butterworth, London, [5] H. B. Said, "A generalized Ball curve and its recursive algorithm", ACM. Transaction on Graphics, Vol. 8, No. 4, pp , [6] A. A. Ball, CONSURF Part 1: Introduction to conic lofting tile, Computer Aided Design, Vol. 6, No. 4, pp , [7] A. A. Ball, CONSURF Part 2: Description of the algorithms, Computer Aided Design, Vol. 7, No. 4, pp , [8] A. A. Ball, CONSURF Part 3: How the program is used, Computer Aided Design, Vol. 9, No. 1, pp. 9 12, [9] G. J. Wang, "Ball curve of high degree and its geometric properties", Applied. Mathematics: A Journal of Chinese Universities Vol. 2, pp , [10] J. Delgado and J. M. Pena, A linear complexity algorithm for the Bernstein basis, Proceedings of the 2003 International Conference on Geometric Modeling and Graphics (GMAG 03), pp , [11] J. Delgado and J. M. Pena, "A shape preserving representation with an evaluation algorithm of linear complexity, Computer Aided Geometric Design, pp. 1-10, [12] R. E. Moore, Interval analysis. Prentice-Hall, Englewood Cliffs, N.J., [13] T. J. Hickey, Q. Ju,, and M. H. van Emden, Interval arithmetic: From principles to implementation, Journal of the ACM Vol. 48, No. 5 pp , [14] D. Ratz, Inclusion isotone extended interval arithmetic, Tech. Rep. D-76128, Institut f ur Angewandte Mathematik, Universit at Karlsruhe, May [15] P. Hartley, and C. Judd, "Parametrization of Bezier-type B-spline curves," Computer Aided Design, Vol. 10, No. 2, pp , [16] P. Hartley, and C. Judd, "Parametrization and shape of B-spline curves", Computer Aided Design, Vol. 12, No. 5, pp , [17] A. A. Ball, "Reparametrization and its application in computeraided geometric design." International Journal for Numerical Methods in Engineering, Vol. 20, pp , [18] J. Hands, "Reparametrization of rational surfaces, In The Mathematics of Surfaces II, ed. R. Martin. Oxford University, [19] L. Alt, "Rational linear reparametrization of NURBS and the blossoming principle, Computer Aided Geometric Design, Vol. 10, No. 5, pp. 465, [20] T. W. Sederberg and R. T. Farouki, Approximation by interval Bezier curves, IEEE Comput. Graph. Appl., No, 2, Vol. 15, pp , [21] V. L. Kharitonov, "Asymptotic stability of an equilibrium position of a family of system of linear differential equations", Differential 'nye Urauneniya, vol. 14, pp , 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 70 international 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.