Hierarchies of partitions are generally represented by dendrograms (direct representation). They can also be represented by saliency maps or minimum spanning trees. In this article, we precisely study the links between these three representations. In particular, we provide a new bijection between saliency maps and hierarchies based on quasi-flat zones as often used in image processing and we characterize saliency maps and minimum spanning trees as solutions to constrained minimization problems where the constraint is quasiflat zones preservation. In practice, these results make up a toolkit for designing new hierarchical methods where one can choose the most convenient representation. They also invite us to process non-image data with morphological hierarchies. More precisely, we show the practical interest of the proposed framework for: i) hierarchical watershed image segmentations, ii) combinations of different hierarchical segmentations, iii) hierarchicalizations of some non-hierarchical image segmentation methods based on regional dissimilarities, and iv) hierarchical analysis of geographical data.

Abstract-This article aims to understand the practical features of hierarchies of morphological segmentations, namely the quasi-flat zones hierarchy and watershed hierarchies, and to evaluate their potential in the context of natural image analysis. We propose a novel evaluation framework for hierarchies of partitions designed to capture various aspects of those representations: precision of their regions and contours, possibility to extract high quality horizontal cuts and optimal non-horizontal cuts for image segmentation, and ease of finding a set of regions representing a semantic object. This framework is used to assess and to optimize hierarchies with respect to the possible pre-and post-processing steps. We show that, used in conjunction with a state-of-the-art contour detector, watershed hierarchies are competitive with complex state-of-the-art methods for hierarchy construction. In particular, the proposed framework allows us to identify a watershed hierarchy based on a novel extinction value, the number of parent nodes, that outperforms the other hierarchies of morphological segmentations. This coupled with the fact that watershed hierarchies satisfy clear global optimality properties and can be efficiently computed on large data, make them valuable candidates for various computer vision tasks.Index Terms-mathematical morphology, hierarchy of partitions, watershed segmentation, image analysis.