Ultrametric Watersheds

IEEE Trans. Pattern Anal. Mach. Intell.

et al. 2011

272

In this work, we extend a common framework for graph-based image segmentation that includes the graph cuts, random walker, and shortest path optimization algorithms. Viewing an image as a weighted graph, these algorithms can be expressed by means of a common energy function with differing choices of a parameter q acting as an exponent on the differences between neighboring nodes. Introducing a new parameter p that fixes a power for the edge weights allows us to also include the optimal spanning forest algorithm for watershed in this same framework. We then propose a new family of segmentation algorithms that fixes p to produce an optimal spanning forest but varies the power q beyond the usual watershed algorithm, which we term the power watershed. In particular, when q=2, the power watershed leads to a multilabel, scale and contrast invariant, unique global optimum obtained in practice in quasi-linear time. Placing the watershed algorithm in this energy minimization framework also opens new possibilities for using unary terms in traditional watershed segmentation and using watershed to optimize more general models of use in applications beyond image segmentation.

Section: Resultsmentioning

confidence: 99%

Power Watershed: A Unifying Graph-Based Optimization Framework

Couprie

Grady

IEEE Trans. Pattern Anal. Mach. Intell.

et al. 2011

272

“…Thus M(F ) is a segmentation of G. -As F is a topological watershed, we have by Pr. 4 that for any v = {x, y} ∈ E, F (x, y) = F (v). In particular, if there exist X and…”

Section: Topological Watersheds On Edge-weighted Graphsmentioning

confidence: 99%

“…Apart when otherwise mentionned, and to the best of the author's knowledge, all the properties and theorems formally stated in this paper are new. This paper is an extended version of [4].…”

Section: Introductionmentioning

confidence: 99%

On the Equivalence Between Hierarchical Segmentations and Ultrametric Watersheds

2011

J Math Imaging Vis

Self Cite

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 on 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.

“…3. Similarly, there exists a bijection between the set of hierarchies of connected partitions and the set of ultrametric watersheds [81,82]. In [84], it is proposed a generic algorithm for computing hierarchies and their associated ultrametric watershed.…”

Section: Ultrametric Watersheds: From Hierarchical Segmentations To Smentioning

confidence: 99%

“…None of these three tools allows for an easy visualisation of a given hierarchy as an image. We now introduce ultrametric watershed [81,82] as a tool that helps visualising a hierarchy: we stack the contours of the regions of the hierarchy; thus, the more a contour of a region is present in the hierarchy, the more visible it is. Ultrametric watershed is the formalisation and the caracterisation of a notion introduced under the name of saliency map [60].…”

Section: Ultrametric Watersheds: From Hierarchical Segmentations To S...mentioning

confidence: 99%

On Morphological Hierarchical Representations for Image Processing and Spatial Data Clustering

Soille¹,

Applications of Discrete Geometry and Mathematical Morphology

2012

Self Cite

Abstract. Hierarchical data representations in the context of classification and data clustering were put forward during the fifties. 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 satisfied. 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.