Variable density preserving topology grids and the digital models for the plane

Abstract: We define LCL decompositions of the plane and investigate the advantages of using such decompositions in the context of digital topology. We show that discretization schemes based on such decompositions associate, to each LCL tiling of the plane, the digital model preserving the local topological structure of the object. We prove that for any LCL tiling of the plane, the digital model is necessarily a digital 2-manifold. We show that elements of an LCL tiling can be of an arbitrary shape and size. This feature generates a variable density grid with a required resolution in any region of interest, which is extremely important in medicine. Finally, we describe a simple algorithm, which allows transforming regions of interest produced by the image acquisition process into digital spaces with topological features of the regions.