Dilation and Erosion on the Triangular Tessellation: An Independent Approach

In this paper, a new idea for morphological operations, i.e., dilation and erosion on the regular triangular tessellation is presented. The triangles have two orientations; they are addressed by zero-sum and one-sum triplets and called even and odd pixels, respectively. The triangular grid is not a lattice, that is, there are grid vectors that do not translate the grid to itself. Different sets of vectors translate the even and odd pixels into the grid: for even pixels vectors with sum 0 and 1 can be used, while for odd pixels vectors with sum 0 and -1 are appropriate. Based on this fact, we introduce a technique in which one can work "independently" with the even and the odd pixels in morphological operations. Examples and various properties of the "independent" dilation and erosion are analyzed.