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CUBIC APPROXIMATION OF CURVE-SHAPED OBJECTS IN Z2 : A GENERALIZED APPROACH BASED ON DISCRETE CURVATURE SHYAMOSREE PAL AND PARTHA BHOWMICK INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR, INDIA

Abstract. Approximation of arbitrary curves and curve-shaped objects on the digital plane, Z2 , is a captivating problem with potential usages in many computer-aided applications, such as image processing and image analysis, pattern recognition, computer vision, etc. The simplest approximation is linear in nature, and for improving the quality of approximation, higher order curves are used. Opposed to the continuous nature of a curve in the real space, a digital curve is inherently discrete in nature, which requires special techniques for realization of an efficient approximation in the digital space. Thus, given a digital curve C, defined as a sequence of integer points, our task is to select an appropriate subsequence C0 of (possibly non-contiguous) points from C, such that the set of curves interpolating the points in C0 gives a reconstructed output that “almost resembles” the original curve. To obtain the desired approximation, we have used a set of cubic B-splines, the reasons being as follows: • Polynomials of cubic degree are widely used in applications related computer graphics and image processing for their simple but effective nature. The quadratic polynomial does not offer much flexibility, and for polynomials with higher degrees, there is always a trade-off between the polynomial’s complexity and the computational burden. • Cubic B-splines possess C 0 , C 1 , and C 2 continuities, which ensure the smoothness and the optimal exactness of the fitted curve against the given set of control points (A curve that is only C 0 continuous may have a “kink” in the “knot point” between two consecutive segments. Hence a curve is made to be C d continuous so that all d derivatives of the curve are continuous.). • Since a set of B-splines is piecewise continuous, the order of the curve that interpolates a set of m(≥ 4) control points is not dependent on m. Each B-spline segment requires four control points in order, whose relative positions decide the curve pattern. Uniform spacing of control points, as selected from C, results in varying deviation of the fitted B-splines from the original curve C when its curvature varies appreciably. More precisely, at a high-curvature region, the deviation of a B-spline from the original curve segment becomes unexpectedly large if the spacing between the control points is high. On the other hand, if the spacing between the control points is low, then in an area of low curvature there arises redundancy of control points. So we have to strike a balance in the selection of control points that constitute C0 , so that the output (spline-fitted curve) does not deviate at areas of high curvature and C0 does not include extraneous control points at areas of low curvature. To achieve this, we select control points in accordance with the curvature in such a way that control points appear densely in areas of high curvature and sparsely in areas of low curvature. Evidently, the notion of discrete curvature plays a crucial role in deciding the density of control points along the digital curve. To suit the requirements in the digital paradigm, the definition of real curvature (y 00 /(1 + y 02 )3/2 , as defined for a curve y = f (x) ∈ R2 ) is modified in our work to estimate the discrete curvature at each point on an arbitrary digital curve whose underlying equation is not known. An obvious strategy is to replace the first- and second-order derivatives by the corresponding numerical differences, which, however, produces inadmissible errors in curvature estimation. For, the slope angle between two consecutive points, (x, y) and (x0 , y 0 ), of a digital curve can differ only by a multiple of 450 , as max(|x − x0 |, |y − y 0 |) = 1. To circumvent this, we use a generalized approach based on paired differences over a contiguous subsequence of 2k(k > 1) of points in C centered at the point p where the curvature is estimated. Certain geometric properties of digital straightness have been used in tandem to estimate the degree of bending of C in and around the point p. In effect, a smoothed version of discrete curvature is estimated, where k can be viewed as a smoothing parameter. Exhaustive testing and experimental results demonstrate the strength and elegance of our method. 1