New characterizations of minimum spanning trees and of saliency maps based on quasi-flat zones

We study three representations of hierarchies of partitions: dendrograms (direct representations), saliency maps, and minimum spanning trees. We provide a new bijection between saliency maps and hierarchies based on quasi-flat zones as used in image processing and characterize saliency maps and minimum spanning trees as solutions to constrained minimization problems where the constraint is quasi-flat zones preservation. In practice, these results form a toolkit for new hierarchical methods where one can choose the most convenient representation. They also invite us to process non-image data with morphological hierarchies

On morphological hierarchical representations for image processing and spatial data clustering

Hierarchical data representations in the context of classi cation and data clustering were put forward during the fties. Recently, hierarchical image representations have gained renewed interest for segmentation purposes. In this paper, we briefly survey fundamental results on hierarchical clustering and then detail recent paradigms developed for the hierarchical representation of images in the framework of mathematical morphology: constrained connectivity and ultrametric watersheds. Constrained connectivity can be viewed as a way to constrain an initial hierarchy in such a way that a set of desired constraints are satis ed. The framework of ultrametric watersheds provides a generic scheme for computing any hierarchical connected clustering, in particular when such a hierarchy is constrained. The suitability of this framework for solving practical problems is illustrated with applications in remote sensing

A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research

Constructive links between some morphological hierarchies on edge-weighted graphs

International audienceIn edge-weighted graphs, we provide a unified presentation of a family of popular morphological hierarchies such as component trees, quasi flat zones, binary partition trees, and hierarchical watersheds. For any hierarchy of this family, we show if (and how) it can be obtained from any other element of the family. In this sense, the main contribution of this paper is the study of all constructive links between these hierarchies

On the equivalence between hierarchical segmentations and ultrametric watersheds

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.Comment: 19 pages, double-colum

Automatic Selection of Stochastic Watershed Hierarchies

International audienceThe segmentation, seen as the association of a partition with an image, is a difficult task. It can be decomposed in two steps: at first, a family of contours associated with a series of nested partitions (or hierarchy) is created and organized, then pertinent contours are extracted. A coarser partition is obtained by merging adjacent regions of a finer partition. The strength of a contour is then measured by the level of the hierarchy for which its two adjacent regions merge. We present an automatic segmentation strategy using a wide range of stochastic watershed hierarchies. For a given set of homogeneous images, our approach selects automatically the best hierarchy and cut level to perform image simplification given an evaluation score. Experimental results illustrate the advantages of our approach on several real-life images datasets

Adjunctions on the lattice of hierarchies

24 pagesHierarchical segmentation produces not a fixed partition but a series of nested partitions, also called hierarchy. The structure of a hierarchy is univocally expressed by an ultrametric 1/2-distance. The lattice structure of hierarchies is equivalent with the lattice structure of their ultrametric 1/2-distances. The hierarchies form a complete sup- and inf- generated lattice on which an adjunction can be defined

Adjunctions on the lattice of dendrograms and hierarchies

56 pagesMorphological 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