A Novel Half-way Shifting Bezier Curve Model with Improved Error Bound Ferdous A. Sohel, Gour C. Karmakar, and Laurence S. Dooley {Ferdous.Sohel, Gour.Karmakar, Laurence.Dooley}@infotech.monash.edu.au Gippsland School of Computing and Information Technology Monash University, Churchill, Victoria 3842, Australia. Abstract— Bezier curves are fundamental to a wide array of applications like computer aided design, object shape description, calligraphic character description and surface mapping. However, as Bezier curves consider only the global information of its control points, it leads to erroneous representations in these applications. This error can be minimized if the local information of the Bezier control points is incorporated with the global information. This paper presents one such technique namely a novel half-way shifting Bezier curve (HSBC) model with improved error bound which will incorporate the local information of control points along with the global information of the control points by making a half way shifting of the Bezier curve points towards the control polygon. It has been proven that HSBC model is bounded by the Bezier curve and its control polygon and thus has an improved error bound with out increase in the computational order time requirement. I. INTRODUCTION Bezier curves were independently developed by Casteljau [1] and Bézier [2] in the early sixties. Since then these have been applied in many computer-aided design (CAD) applications. While the origin of the Bezier curves can be traced back to the design of car body shapes in the motor companies, their usage is no longer confined to this field only. Indeed, their robustness in curve and surface representation the Bezier curves’ usage now pervades many recent areas of image processing technology including shape description of characters [3-4] and objects [5], shape error concealment for MPEG-4 objects [6] and surface mapping [7-8]. Bezier curves are defined by a set of control points. The number and the orientation of the control points govern the shape of the curve. Bezier curves however, only consider global information [9] of their control points and calculate the curve points in a linear recursive approach starting with the edges of the control polygon. As a result, there is often a large gap between the curve and its control polygon especially in the central region of the curve. It ultimately restricts the maximum curve length for a given number of control points. A composite Bezier strategy comprising multiple curve segments can be used to address this shortcoming. However when describing a shape using a Bezier curve, most of the control points are required to be defined outside the original shape which will not necessarily be inside the coordinate system and also in the case of composite Bezier the number of control points is more. Thus it increases the computational overhead in many applications. Degree elevation [10] has been exploited to form a curve with an increased number of control points though all these points, except the two end points have to be recalculated so incurring a significant computational overhead. Subdivision and refinement techniques have also been used to minimize the gap between the Bezier curve and its control polygon by increasing the number of curve segments. When the control points of a curve are known, two sets of new control points that are closer to the curve can be calculated using subdivision algorithms such as midpoint subdivision [11] or arbitrary subdivision [12]. All these algorithms however increase the number of curve segments and thus the number of control points. Moreover, to ensure that curve segments are conjoint, the number of subdivisions has also always to be constrained. All the aforementioned algorithms minimize the gap between the Bezier curve and its control polygon by increasing the number of control points. In communication applications this means a higher coding and transmission cost to represent a particular shape. To overcome this problem, this paper presents a novel half-way shifting Bezier curve (HSBC) model with improved error bound, which incorporates localized information within the classical Bezier curve framework by shifting the curve point at the midpoint between the curve point and the control polygon, with no increase in computational complexity. It is particularly noteworthy that this new model can be seamlessly integrated into Bezier refinement algorithms such as degree elevation and subdivision and it will be shown that HSBC retains many of the core properties of the classical Bezier curve. The performance of the model as a generic shape descriptor for a number of arbitrary shapes, has been extensively analyzed using both quantitative and qualitative metrics, with results clearly confirming its superiority compared with the original Bezier curve [1-2]. The remainder of the paper is structured as follows: Section II provides a short overview of the classical Bezier curve identifying problems due to the global control, while Section III discusses the theoretical basis of the new HSBC model along with proofs that many of the key properties of the traditional Bezier curve are retained and also the improvements in error bound. Section IV presents some experimental results confirming the superior performance of HSBC model relative to the original Bezier curve, with some conclusions given in Section V. II. OVERVIEW OF THE CLASSICAL BEZIER CURVE The Bezier curve is a recursive linear weighted subdivision of the edges of the generated polygon starting with a set of points to form the initial polygon and ending when the final point is generated for a particular weight t . The set of 1 + N starting points is referred to as the control points which govern the characteristics of the Bezier curve of degree N . The polygon connecting the control points is called the control polygon. The Casteljau form of the Bezier curve for an ordered set of control points { } 1 2 1 , , , + = N v v v V K is defined as:- ⎪ ⎩ ⎪ ⎨ ⎧ ≤ ≤ − = + = + − = = − + − 1 0 ; , , 1 ; 1 , , 2 ); ( ) ( ) 1 ( ; 1 ; ) ( 1 1 1 u r N i N r t tv t v t r if v V, of member i t v r i r i i th r i L L (1) where t is the weight of subdivision which determines the number of points on the Bezier curve. The final generation ) ( 1 1 t v N + is the Bezier curve.