A compression of digital images derived from a Khalimsky topological structure

Abstract

The paper introduces the notion of $$A$$ -retract to compress digital images derived from a Khalimsky topological structure. Compared with a Khalimsky retract, an $$A$$ -retract is very flexible and efficient for the process of compressing (or thinning) (see Fig. 10). More precisely, to study digital images (or spaces) derived from a Khalimsky topological structure, a Khalimsky continuous map and a Khalimsky homeomorphism have been often used in digital topology. However, these maps are so rigid that they have some limitations of both a rotation and a parallel translation from the viewpoint of geometry (see Examples 3.4, 4.1 and 5.1 as motivating examples). To overcome these limitations, the paper generalizes a Khalimsky continuous map, a Khalimsky adjacency map and a homeomorphism between Khalimsky topological spaces so that we can establish new maps such as an $$A$$ -map and an $$A$$ -isomorphism. Using these notions, we can substantially study and classify digital images induced by a Khalimsky topological structure (see the table in Fig. 5 explaining some merits of an $$A$$ -map compared with both a Khalimsky continuous map and a Khalimsky adjacency map). As an application, for digital images derived from a Khalimsky topological structure $$X$$ , the paper proposes a method of thinning or compressing by using an $$A$$ -retract of $$X$$ , which shows that this approach can contribute to computer science such as image analysis, image processing, computer graphics, mathematical morphology and so forth.