Topology preservation on the triangular grid

There are exactly three regular planar grids, which are formed by tiling the 2-dimensional Euclidean space with regular triangles, squares, and hexagons. The topology of the square grid is well-understood, but it cannot be said of the remaining two regular sampling schemes. This work deals with the topological properties of digital binary pictures sampled on the triangular grid. Some characterizations of simple pixels and sufficient conditions for topology preserving operators are reported. These results provide the theoretical background to various topological algorithms including thinning, shrinking, generating discrete Voronoi diagrams, and contour smoothing on the triangular grid.