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

TL;DR

This paper introduces LCL decompositions for the plane that create variable density digital models preserving local topology, useful for applications like medical imaging, and provides an algorithm for converting image regions into topologically accurate digital spaces.

Contribution

It defines LCL decompositions for the plane, demonstrating their ability to produce topologically accurate digital models with variable density, and presents an algorithm for converting image regions into digital topological models.

Findings

Digital models are necessarily digital 2-manifolds.

LCL tilings can have arbitrary shapes and sizes.

The proposed algorithm effectively converts image regions into topological digital models.

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.