Volume 3, Issue 1

We consider Hausdorff discretization from a metric space E to a discrete subspace D , which associates to a closed subset F of E any subset S of D minimizing the Hausdorff distance between F and S ; this minimum distance, called the Hausdorff radius of F and written r H ( F ), is bounded by the resolution of D . We call a closed set F separated if it can be partitioned into two non-empty closed subsets F 1 and F 2 whose mutual distances have a strictly positive lower bound. Assuming some minimal topological properties of E and D (satisfied in ℝ n and ℤ n ), we show that given a non-separated closed subset F of E , for any r > r H ( F ), every Hausdorff discretization of F is connected for the graph with edges linking pairs of points of D at distance at most 2 r . When F is connected, this holds for r = r H ( F ), and its greatest Hausdorff discretization belongs to the partial connection generated by the traces on D of the balls of radius r H ( F ). However, when the closed set F is separated, the Hausdorff discretizations are disconnected whenever the resolution of D is small enough. In the particular case where E = ℝ n and D = ℤ n with norm-based distances, we generalize our previous results for n = 2. For a norm invariant under changes of signs of coordinates, the greatest Hausdorff discretization of a connected closed set is axially connected. For the so-called coordinate-homogeneous norms, which include the L p norms, we give an adjacency graph for which all Hausdorff discretizations of a connected closed set are connected.

Image segmentation is the process of partitioning an image into a set of meaningful regions according to some criteria. Hierarchical segmentation has emerged as a major trend in this regard as it favors the emergence of important regions at different scales. On the other hand, many methods allow us to have prior information on the position of structures of interest in the images. In this paper, we present a versatile hierarchical segmentation method that takes into account any prior spatial information and outputs a hierarchical segmentation that emphasizes the contours or regions of interest while preserving the important structures in the image. Several applications are presented that illustrate the method versatility and efficiency.

Connected operators based on hierarchical image models have been increasingly considered for the design of efficient image segmentation and filtering tools in various application fields. Among hierarchical image models, component-trees represent the structure of grey-level images by considering their nested binary level-sets obtained from successive thresholds. Recently, a new notion of component-graph was introduced to extend the component-tree to any grey-level or multivalued images. The notion of shaping was also introduced as a way to improve the anti-extensive filtering by considering a two-layer component-tree for grey-level image processing. In this article, we study how component-graphs (that extend the component-tree from a spectral point of view) and shapings (that extend the component-tree from a conceptual point of view) can be associated for the effective processing of multivalued images. We provide structural and algorithmic developments. Although the contributions of this article are theoretical and methodological, we also provide two illustration examples that qualitatively emphasize the potential use and usefulness of the proposed paradigms for image analysis purposes.

The original version of the article was published in Mathematical Morphology - Theory and Applications 2 (2017) 55–75. Unfortunately, the original version contains a mistake: in the definition of Dif ( C 1 , C 2 ) in Section 3.6, max should be replaced by min. In this erratum we correct the formula defining Dif ( C 1 , C 2 ).