Properties of simple sets in digital spaces. Contractions of simple sets preserving the homotopy type of a digital space

TL;DR

This paper introduces the concept of simple sets in digital spaces, showing how contracting these sets preserves topology and enables space compression, with applications in medical imaging and computer graphics.

Contribution

It defines simple sets based on contractible transformations and demonstrates their use in topology-preserving digital space thinning.

Findings

Contracting simple sets preserves homotopy type.

Digital spaces can be substantially compressed while maintaining topology.

Method has applications in medical imaging and pattern analysis.

Abstract

A point of a digital space is called simple if it can be deleted from the space without altering topology. This paper introduces the notion simple set of points of a digital space. The definition is based on contractible spaces and contractible transformations. A set of points in a digital space is called simple if it can be contracted to a point without changing topology of the space. It is shown that contracting a simple set of points does not change the homotopy type of a digital space, and the number of points in a digital space without simple points can be reduces by contracting simple sets. Using the process of contracting, we can substantially compress a digital space while preserving the topology. The paper proposes a method for thinning a digital space which shows that this approach can contribute to computer science such as medical imaging, computer graphics and pattern analysis.