A new vision for the measure of circularity using a certain representation space
Michel Goeb, Jean-Marie Becker · Journal of Physics Conference Series · 2008
Detecting and characterizing families of plane circles is a fundamental concern in image processing. Many methods have been developed for these purposes, often on ad hoc bases, for example in the context of metrology. This paper is based on the consideration of circles as topological points in a certain 3D representation space (RS). Instead of the natural RS with (a,b,R) coordinates, (a, b) for the center and R for the radius of the circle, another RS, denoted Σ, is studied, with the third coordinate changed from R to c with c = a2 + b2 − R2. Z is connected with Voronoi closest et farthest point diagrams and their lifted version. Morever, Σ possesses a canonical measure, introduced by Stoka. This paper is focused on families of circles S(P,Q) constrained to have two fixed sets of internal (Pi) and external (Qj) points, representable as polyhedra in Σ. The Stoka measure of these polyhedra is shown to be an adequate measure of circularity expressed by a simple formula. The power of this approach is shown on two applications : circularity assessment as used in metrology, and Fillmore formula. This approach has several ramifications, for example in Hyperbolic Geometry.