Efficient Approximation of Curve-Shaped Objects in ${\mathbb Z}^2$ Based on the Maximum Difference Between Discrete Curvature Values

Abstract

In this paper, we propose a novel algorithm for the cubic approximation of digital curves and curve-shaped objects in $\mathbb {Z}^{2}$. At first, the discrete curvature value is computed for each point of the given digital curve, C, using the improved k-curvature estimation technique. Based on the estimated k-curvature value, the points are selected from C to obtain the resultant set of reduced points, $C^{'}$. We use a set of cubic B-splines for the approximation of the given digital curve C. For the selection of control points, our algorithm works on a new parameter, threshold, defined as the maximum difference between discrete curvature values, based on which the control points are selected from the given digital curve, C, such that the maximum discrete curvature difference from the last selected point and the next point to be selected do not exceed the threshold. Further adjustments are made in the selection of control points based on the principle that high curvature areas of a digital curve represent more information whereas low curvature areas represent less information. Experimental results and comparisons with the existing algorithm on various digital objects demonstrate our approach’s effectiveness. It has been observed that our algorithm generates better output for approximating real-world curves in which there are large number of control points, and the rate of curvature change is fast. Our algorithm also takes less computational time since it selects the control points of a digital curve in a single iteration.