Thinning on Quadratic, Triangular, and Hexagonal Cell Complexes

This paper deals with a thinning algorithm proposed in 2001 by Kovalevsky, for 2D binary images modelled by cell complexes, or, equivalently, by Alexandroff T0 spaces. We apply the general proposal of Kovalevsky to cell complexes corresponding to the three possible normal tilings of congruent convex polygons in the plane: the quadratic, the triangular, and the hexagonal tilings. For this case, we give a theoretical foundation of Kovalevsky’s thinning algorithm: We prove that for any cell, local simplicity is sufficient to satisfy simplicity, and that both are equivalent for certain cells. Moreover, we show that the parallel realization of the algorithm preserves topology, in the sense that the numbers of connected components both of the object and of the background, remain the same. The paper presents examples of skeletons obtained from the implementation of the algorithm for each of the three cell complexes under consideration.