Adjunctions on the lattice of dendrograms and hierarchies
Résumé
Morphological image processing uses two types of trees. The min-tree represents the relations between the regional minima and the various lakes during flooding. As the level of flooding increases in the various lakes, the flooded domain becomes larger. A second type of tree is used in segmentation and is mainly associated to the watershed transform. The watershed of a topographic surface constitutes a partition of its support. If the relief is flooded, then for increasing levels of floodings, catchment basins merge. The relation of the catchment basins during flooding also obeys a tree structure. We start by an axiomatic definition of each type of tree, min and max tree being governed by a single axiom ; for nested catchment basins, a second axiom is required. There is a one to one correspondance between the trees and an ultrametric half distance, as soon one introduces a total order compatible with the inclusion. Hierarchies obey a complete lattice structure, on which several adjunctions are defined, leading to the construction of morphological filters. Hierarchies are particular useful for interactive image segmentation, as they constitute a compact representation of all contours of the image, structured in a way that interesting contours are easily extracted. The last part extends the classical connections and partial connections to the multiscale case and introduces taxonomies.
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