Annular filters for binary images

A binary annular filter removes isolated points in the foreground and the background of an image. Here, the adjective 'isolated' refers to an underlying adjacency relation between pixels, which may be different for foreground and background pixels. In this paper, annular filters are represented in terms of switch pairs. A switch pair consists of two operators which govern the removal of points from foreground and background, respectively. In the case of annular filters, switch pairs are completely determined by foreground and background adjacency. It is shown that a specific triangular condition in terms of both adjacencies is required to establish idempotence of the resulting annular filter

Climbing: A Unified Approach for Global Constraints on Hierarchical Segmentation

International audienceThe paper deals with global constraints for hierarchical segmentations. The proposed framework associates, with an input image, a hierarchy of segmentations and an energy, and the subsequent optimization problem. It is the first paper that compiles the different global constraints and unifies them as Climbing energies. The transition from global optimization to local optimization is attained by the h-increasingness property, which allows to compare parent and child partition energies in hierarchies. The laws of composition of such energies are established and examples are given over the Berkeley Dataset for colour and texture segmentation

Correspondence between Topological and Discrete Connectivities in Hausdorff Discretization

We consider \emph{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 \emph{Hausdorff radius} of $F$ and written $r_H(F)$, is bounded by the resolution of $D$. We call a closed set $F$ \emph{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 $2r$. 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 \emph{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

Shopping centre siting and modal choice in Belgium: a destination based analysis

Although modal split is only one of the elements considered in decision-making on new shopping malls, it remarkably often arises in arguments of both proponents and opponents. Today, this is also the case in the debate on the planned development of three major shopping malls in Belgium. Inspired by such debates, the present study focuses on the impact of the location of shopping centres on the travel mode choice of the customers. Our hypothesis is that destination-based variables such as embeddedness in the urban fabric, accessibility and mall size influence the travel mode choice of the visitors. Based on modal split data and location characteristics of seventeen existing shopping centres in Belgium, we develop a model for a more sustainable siting policy. The results show a major influence of the location of the shopping centre in relation to the urban form, and of the size of the mall. Shopping centres that are part of a dense urban fabric, measured through population density, are less car dependent. Smaller sites will attract more cyclists and pedestrians. Interestingly, our results deviate significantly from the figures that have been put forward in public debates on the shopping mall issue in Belgium

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