# Topology on Digital Label Images

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## Abstract

In digital imaging, after several decades devoted to the study of topological properties of binary images, there is an increasing need of new methods enabling to take into (topological) consideration *n*-ary images (also called label images). Indeed, while binary images enable to handle one object of interest, label images authorise to simultaneously deal with a plurality of objects, which is a frequent requirement in several application fields. In this context, one of the main purposes is to propose topology-preserving transformation procedures for such label images, thus extending the ones (e.g., growing, reduction, skeletonisation) existing for binary images. In this article, we propose, for a wide range of digital images, a new approach that permits to locally modify a label image, while preserving not only the topology of each label set, but also the topology of any arrangement of the labels understood as the topology of any union of label sets. This approach enables in particular to unify and extend some previous attempts devoted to the same purpose.

### Keywords

Digital imaging Topology Label images Homotopy Simple points### References

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