Optimal Time Computation of the Tangent of a Discrete Curve: Application to the Curvature

Abstract

With the definition of discrete lines introduced by Réveillès [REV91], there has been a wide range of research in discrete geometry and more precisely on the study of discrete lines. By the use of the linear time segment recognition algorithm of Debled and Réveillès [DR94], Vialard [VIA96a] has proposed a O(l) algorithm for computing the tangent in one point of a discrete curve where l is the average length of the tangent. By applying her algorithm to n points of a discrete curve, the complexity becomes O(n.l). This paper proposes a new approach for computing the tangent. It is based on a precise study of the tangent evolution along a discrete curve. The resulting algorithm has a O(n) complexity and is thus optimal. Some applications in curvature computation and a tombstones contours study are also presented.

Lyndon Christoffel digitally convex

“…The implementation of our algorithm to check digital convexity was compared to the method of Debled-Rennesson et al [9], implemented in linear time with the optimization of [10,11]. The results (see Fig.…”

“…Digital convexity is then achieved if and only if the induced sequence of slopes is monotonous. From this observation, a linear -and thus optimal -time algorithm is obtained using optimal time methods for computing the tangential cover of a digital contour, which relies on a moving digital straight line recognition algorithm [10,11]. It is worthy to note that digital convex sets and hulls have specific properties that are not shared by their Euclidean counterpart.…”

Curvature Estimation for Discrete Curves Based on Auto-adaptive Masks of Convolution

“…There exists many methods to estimate the length [4], or the curvature of discrete curves [11,7,14,3] such as the discretized graph of a function. A method is to recognize maximal straight segments [6] but it is sensitive to noise.…”

“…An important problem is to estimate derivatives of digital functions, for example the tangent space at a point or the curvature of the shape. There exists many approaches based on line segmentation [11,7,14] or on filtering [19,13,8]. New approaches to derivative estimation [12,8] are based on convolution product with a diffusion kernel and give striking estimations of derivatives on noisy curves.…”

Discrete Contour Extraction from Reference Curvature Function

“…The selection of the more probable shape is defined from geometric constraints extracted from the tangential cover [13]. The notion of tangential cover is illustrated on the figure 1(a) which shows all the maximal segments of a discrete shape.…”