Novel morphological algorithms for dominating sets on graphs with applications to image analysis

Abstract

In this paper, we extend the morphological operators defined for graphs by Cousty et al. to use structuring elements. We then apply these extended operators to develop algorithms for Minimum Dominating Set (MDS) and Minimum Independent Dominating Set (MIDS) on incomplete grid graphs which correspond to binary images with 4-connected neighbourhoods. We show that our algorithm performs as well as the best known heuristic for Minimum Independent Dominating Set. We apply the extended morphological graph operators and algorithms to various image analysis tasks such as distance transforms, skeletons and clustering. In particular, we propose a novel MIDS Skeleton that may potentially reduce the time for reconstructing the original objects. A hierarchical clustering algorithm (also using MIDS) is proposed. This algorithm is analogous to the conventional algorithms that use a distance threshold for clustering. We illustrate the proposed algorithms on several example images and conclude that they are useful in image analysis. © 2012 Springer-Verlag.