Thinning grayscale well-composed images

Usual approaches for constructing topological maps on discrete structures are based on cellular complexes topology. This paper aims to construct a coherent topological map defined on a square grid from a watershed transformation. The main idea behind the proposed approach is to impose some constraints on the original image in order to obtain good properties of the resulting watershed. We propose a definition of well-composed grayscale images based on the well-composed set theory and the cross-section topology. Properties of two different thinning algorithms are then studied and we show how to obtain a thin crest network. We derive an efficient algorithm that permits the construction of a meaningful topological map, resulting in a topological segmentation, i.e. a segmentation that describes in a coherent framework faces and contours. Finally, we demonstrate the usefulness of this algorithm for multilevel image segmentation.