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

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

Point-wise mutual information-based video segmentation with high temporal consistency

In this paper, we tackle the problem of temporally consistent boundary detection and hierarchical segmentation in videos. While finding the best high-level reasoning of region assignments in videos is the focus of much recent research, temporal consistency in boundary detection has so far only rarely been tackled. We argue that temporally consistent boundaries are a key component to temporally consistent region assignment. The proposed method is based on the point-wise mutual information (PMI) of spatio-temporal voxels. Temporal consistency is established by an evaluation of PMI-based point affinities in the spectral domain over space and time. Thus, the proposed method is independent of any optical flow computation or previously learned motion models. The proposed low-level video segmentation method outperforms the learning-based state of the art in terms of standard region metrics

Watersheds, waterfalls, on edge or node weighted graphs

We present an algebraic approach to the watershed adapted to edge or node weighted graphs. Starting with the flooding adjunction, we introduce the flooding graphs, for which node and edge weights may be deduced one from the other. Each node weighted or edge weighted graph may be transformed in a flooding graph, showing that there is no superiority in using one or the other, both being equivalent. We then introduce pruning operators extract subgraphs of increasing steepness. For an increasing steepness, the number of never ascending paths becomes smaller and smaller. This reduces the watershed zone, where catchment basins overlap. A last pruning operator called scissor associates to each node outside the regional minima one and only one edge. The catchment basins of this new graph do not overlap and form a watershed partition. Again, with an increasing steepness, the number of distinct watershed partitions contained in a graph becomes smaller and smaller. Ultimately, for natural image, an infinite steepness leads to a unique solution, as it is not likely that two absolutely identical non ascending paths of infinite steepness connect a node with two distinct minima. It happens that non ascending paths of a given steepness are the geodesics of lexicographic distance functions of a given depth. This permits to extract the watershed partitions as skeletons by zone of influence of the minima for such lexicographic distances. The waterfall hierarchy is obtained by a sequence of operations. The first constructs the minimum spanning forest which spans an initial watershed partition. The contraction of the trees into one node produces a reduced graph which may be submitted to the same treatment. The process is iterated until only one region remains. The union of the edges of all forests produced constitutes a minimum spanning tree of the initial graph

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

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

Hierarchical Image Representation Simplification Driven by Region Complexity

International audienceThis article presents a technique that arranges the elements of hierarchical representations of images according to a coarseness attribute. The choice of the attribute can be made according to prior knowledge about the content of the images and the intended application. The transformation is similar to filtering a hierarchy with a non-increasing attribute, and comprises the results of multiple simple filterings with an increasing attribute. The transformed hierarchy can be used for search space reduction prior to the image analysis process because it allows for direct access to the hierarchy elements at the same scale or a narrow range of scales