A Khalimsky-Like Topology on the Triangular Grid

Abstract

It is well known that there are topological paradoxes in digital geometry and in digital image processing. The most studied such paradoxes are on the square grid, causing the fact that the digital version of the Jordan curve theorem needs some special care. In a nutshell, the paradox can be interpreted by lines, e.g., two different color diagonals of a chessboard that go through each other without sharing a pixel. The triangular grid also has a similar paradox, here diamond chains of different directions may cross each other without having an intersection trixel (triangle pixel). In this paper, a new topology is offered on the triangular grid, which gives a solution to the topological problems in the triangular grid analogous to the Khalimsky's solution on the square grid.