On the Angles of Change of the Neighborhood Motion Maps on the Triangular Grid

Abstract

Rotations of digital images are important geometric transformations which are considered in various digital grids. These digitized rotations have different properties than the analogous Euclidean rotations. On the triangular grid, we consider the digitized rotations of a pixel (referred as the main pixel) together with its closest neighbor pixels. Given a main pixel, of a certain distance d from the origin, we calculate the angles of the rotations where the neighborhood motion map of the main pixel changes. The neighborhood motion map changes when the respective locations of the closest neighbors change. We also differentiate the cases when the neighborhood motion map is injective and not injective. The former case is connected to the bijective digital rotations, while the latter case is connected to the cases when some image information is lost (in the neighborhood of the main pixel).