On topology preservation of mixed operators in triangular, square, and hexagonal grids

A crucial issue in digital topology is to ensure topology preservation of binary image operators. Several sufficient conditions have already been proposed (i.e., for operators that never change a white pixel to black one and a black pixel to a white one, respectively), however not much attention has been paid on mixed operators, which can modify both black and white pixels. In this paper, we intend to fill in this gap by presenting some sufficient criteria on the latter family of operators considering all the possible regular partitionings of the plane (i.e., triangular, square, and hexagonal grids).