ABSTRACT
This chapter presents the tool that makes an idea practicable: mathematical structures called lattices, allowing us to compare weighted and nonweighted edges and vertices. It also presents several dilations and erosions, which always come in pairs. The chapter builds some morphological filters, called opening or closing. During the presentation of those various operators, the chapter gives an interpretation of classical graph operators in morphological terms. The chapter presents some connected operators based on pruning a tree representation of the image, illustrating their usage for image filtering and simplification. It deals with hierarchical segmentation. The chapter revisits the watershed, and shows that in the framework of edge-weighted graphs, the watershed has very strong links with the minimum spanning tree. The main principle of morphology, comparison, is rather different from the optimization paradigm. The chapter explores links between the morphological and the optimization approaches.
