Watersheds on edge or node weighted graphs "par l'exemple"

TL;DR

This paper demonstrates that watershed algorithms on node-weighted and edge-weighted graphs are equivalent by constructing a corresponding graph with identical minima and catchment basins.

Contribution

It proves the theoretical equivalence between watershed definitions on node and edge weighted graphs, unifying two previously separate approaches.

Findings

Watersheds on node and edge weighted graphs are mathematically equivalent.

Constructive method to convert between node and edge weighted graphs.

Implications for simplifying watershed computations and algorithms.

Abstract

Watersheds have been defined both for node and edge weighted graphs. We show that they are identical: for each edge (resp.\ node) weighted graph exists a node (resp. edge) weighted graph with the same minima and catchment basin.