Discrete & Computational Geometry

, Volume 43, Issue 4, pp 893–913

# On 2-dimensional Simple Sets in n-dimensional Cubic Grids

Article

## Abstract

Preserving topological properties of objects during reduction procedures is an important issue in the field of discrete image analysis. Such procedures are generally based on the notion of simple point, the exclusive use of which may result in the appearance of “topological artifacts.” This limitation leads to consider a more general category of objects, the simple sets, which also enable topology-preserving image reduction. A study of two-dimensional simple sets in two-dimensional spaces has been proposed recently. This article is devoted to the study of two-dimensional simple sets in spaces of higher dimension (i.e., n-dimensional spaces, n≥3). In particular, several properties of minimal simple sets (i.e., which do not strictly include any other simple sets) are proposed, leading to a characterisation theorem. It is also proved that the removal of a two-dimensional simple set from an object can be performed by only considering the minimal ones, thus authorising the development of efficient thinning algorithms.

## Keywords

Digital topology Thinning Topology preservation Simple sets Cubical complexes n-dimensional spaces

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