A comparison of digitized rotations of neighborhood motion maps of closest neighbors on 2D regular grids

Abstract

Rigid motions on the plane are bijective isometries. Analogous motions also play an important role in image processing; however, usually, the digitized rigid motions are not injective and surjective. They are studied locally on each of the regular two-dimensional grids by neighborhood motion maps (see Pluta et al. for the square and the hexagonal grids and Avkan et al. for the triangular grid, respectively). In this paper, continuing the studies of the latter mentioned conference paper, we compare the bijectivity of the digitized rotations for the closest neighbors in all three regular grids. Rotations for every integer degree are studied for rotation centers at the corner, edge midpoint and at the center of the pixels. The experiment proves that, when the center of rotation is the center or the corner of a main pixel, then, the triangular and the square grids have a better behavior compared to the hexagonal grid, i.e., the number of bijective rotations is more for these two grids.