On the equivalence between hierarchical segmentations and ultrametric watersheds

TL;DR

This paper establishes a theoretical link between hierarchical segmentations and ultrametric watersheds in edge-weighted graphs, providing a practical framework for efficient computation of hierarchies in image segmentation.

Contribution

It introduces ultrametric watersheds and proves their bijection with hierarchical segmentations, enabling practical algorithms for segmentation hierarchies.

Findings

Established a bijection between ultrametric watersheds and hierarchical segmentations.

Provided an efficient watershed-based algorithm for computing segmentation hierarchies.

Demonstrated the framework's application in constrained connectivity segmentation.

Abstract

We study hierarchical segmentation in the framework of edge-weighted graphs. We define ultrametric watersheds as topological watersheds null on the minima. We prove that there exists a bijection between the set of ultrametric watersheds and the set of hierarchical segmentations. We end this paper by showing how to use the proposed framework in practice in the example of constrained connectivity; in particular it allows to compute such a hierarchy following a classical watershed-based morphological scheme, which provides an efficient algorithm to compute the whole hierarchy.