An elementary algorithm for digital arc segmentation

Abstract

International audienceThis paper concerns the digital circle recognition problem, especially in the form of the circular separation problem. General fundamentals, based on classical tools, as well as algorithmic details are given (the latter by providing pseudo-code for major steps of the algorithm). After recalling the geometrical meaning of the separating circle problem, we present an incremental algorithm to segment a discrete curve into digital arcs

Analytical Description of Digital Circles

“…In the space that is dual to the parameters space, a plane separating two sets of n 3D points is searched using tools coming from computational geometry [22]. In [18,9,14,27,23] a 2D point belonging to the intersection of n 2 half-planes is searched in the original plane using either brute-force algorithms [9,14,27] or tools coming from computational geometry [18,23].…”

Circularity Measuring in Linear Time

“…These 2 methods used a sophisticated tool in linear programming [11]. So, in the contexte of circular object recognition, the proposed method is better than the others: Kim [7] (O(n 3 )), Kim [8] (O(n 2 )), Coeurjolly [2] (O(n 4/3 log n)), Fisk [5] (O(n 2 )), . .…”

“…Later, he [8] reduced this complexity to O(n 2 ). Coeurjolly [2] transformed the problem of circle recognition into the search of a 2D point that belongs to the intersection of n 2 half-planes. It can be solved in O(n 4/3 log n) time.…”

Generalized Perpendicular Bisector and exhaustive discrete circle recognition

“…Other papers project into the (C x , C y , r)-plane. The problem consists then in searching for a 2D point belonging to the intersection of n 2 half-planes (let us cite [10,11] among the different papers with this approach). When considering the dual of the parameter spaces, the problem corresponds to a separation problem of two sets in 3D by a plane [7,12].…”