Hierarchical Segmentations with Graphs: Quasi-flat Zones, Minimum Spanning Trees, and Saliency Maps

Abstract

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.

Evaluation of Hierarchical Watersheds

“…They are deeply linked to singlelinkage clustering and to the notion of a minimum spanning tree [23]. A quasi-flat zone of the weighted graph (G, W ) at level λ is a maximal set of vertices such that there exists a path of maximal weight λ between any two of its vertices.…”

“…This definition has been formalized in the context of minimum spanning forests that already enabled to define watershed cuts as an optimal solution to a combinatorial problem related to minimum spanning tree [21]. It has also been shown that hierarchies of watersheds are linked to the quasiflat zones hierarchies [22], to the single-linkage clustering problem [23], and to connective segmentation [14], [24].…”

Partition and Inclusion Hierarchies of Images: A Comprehensive Survey

“…It is accompanied by a paper which presents efficient quasi-linear algorithms based on the Kruskal Minimum Spanning Tree algorithm for computing these hierarchies [53], and establishes the relations between the α-tree of an image with other Mathematical Morphology hierarchies and segmentation from markers. Recently, these links were studied further in [54] and put additionally in relation with saliency maps, images which can be interpreted as characteristic maps of hierarchies used for visualization, accompanied by example applications both from the field of image analysis and outside of it.…”

“…As such, further discussion exceeds the scope of this paper and the interested reader is referred to [74,75] for the theory of braids of partitions, and to [76] for their application to multimodal image segmentation. We also direct the reader interested in combining different hierarchies of the same image into a new hierarchy (rather than a braid or some other structure) to [54] where such a method was proposed for the hierarchies of partitions. …”

“…Such a hierarchy was first explored by [93] where different segmentation methods based on the principal hierarchy were also proposed, as well as a possibility of different valuations of the graph edges. The concept was reintroduced to the community and formalized under the name of BPT by Altitude Ordering by Najman et al [53] and Cousty et al [54] where the relations to other hierarchies were established.…”