Digital continuity of rotations in the 2D regular grids

Abstract

A digitized rigid motion is called digitally continuous if two neighbor pixels still stay neighbors after the motion. This concept plays important role when people or computers (artificial intelligence, machine vision) need to recognize the object shown in the image. In this paper, digital rotations of a pixel with its closest neighbors are of our interest. We compare the neighborhood motion map results among the three regular grids, when the center of rotation is the midpoint of a main pixel, a grid point (corner of a pixel) or an edge midpoint. The first measure about the quality of digital rotations is based on bijectivity, e.g., measuring how many of the cases produce bijective and how many produce not bijective neighborhood motion maps (Avkan et. al, 2022). Now, a second measure is investigated, the quality of bijective digital rotations is measured by the digital continuity of the resulted image: we measure how many of the cases are bijective and also digitally continuous. We show that rotations on the triangular grid prove to be digitally continuous at many more real angles and also as a special case, many more integer angles compared to the square grid or to the hexagonal grid with respect to the three different rotation centers.