Minimum Spanning Forests and Watershed Partitions

Watersheds are mainly used for segmenting images, constructing a partition or a partial partition in which each tile represents an object. This chapter characterizes the complete family of watershed partitions which may be derived from an edge‐weighted graph. The results established for edge‐weighted graph are extended to node‐weighted graphs. The chapter introduces the minimum spanning forests of an edge‐weighted graph. It recalls some properties of the minimum spanning trees and studies minimum spanning forests associated with arbitrary families of markers. The chapter considers the minimum spanning forests rooted in the regional minima of the graph and shows that each of them spans a watershed partition. It shows how pruning the graph before constructing a minimum spanning forest reduces their number and induces partitions with a higher fidelity to the objects to segment. The chapter presents the waterfall hierarchy, which analyzes the nested structure of the watershed partitions and offers a powerful tool for simplifying images.