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

Expressão da esterilidade feminina e da carpeloidia em mamoeiro sob diferentes ambientes de cultivo protegido.

O presente estudo teve por objetivo avaliar a ocorrência de esterilidade feminina e de carpeloidia em mamoeiros hermafroditas 'Baixinho-de-Santa Amália' cultivados sob manejo orgânico, em diferentes tipos de ambiente de proteção,e conduzido com ou sem bifurcação do tronco no transcorrer das quatro estações do ano. Foram construídos três tipos de estruturas de proteção contíguas: (i) estufa (cobertura de plástico); (ii) estufa sombreada (cobertura adicional de tela 'sombrite' - 30% sobre o plástico), e (iii) telado (cobertura exclusiva de tela 'sombrite' - 30%), ao lado de uma área de ambiente natural, a pleno sol. Nestes locais, foram cultivados, dentro das normas técnicas da agricultura orgânica, mamoeiros da cv. Baixinho-de-Santa-Amália. Em metade das plantas, abrangendo todos os ambientes de cultivo, a gema apical foi incisada, logo após a sexagem, visando à bifurcação do tronco. Para efeito de análise de variância, foram considerados quatro blocos por ambiente de cultivo, tendo cada bloco três repetições relativas ao modo de condução das plantas (com e sem bifurcação do tronco). Para análise estatística, procedeu-se à "análise conjunta de experimentos", no caso, os ambientes de cultivo. Nos mamoeiros com tronco bifurcado, houve diminuição do número de frutos carpeloides e aumento do número de flores fêmeas estéreis. No entanto, essa bifurcação não influenciou a frequência de frutos normais. Durante a primavera (setembro a dezembro), e notadamente na estufa, o maior número de frutos carpeloides por planta correlacionou-se a temperaturas mais elevadas, maior amplitude térmica e maior vigor vegetativo; já, a maior ocorrência de flores estaminadas correlacionou-se também a temperaturas elevadas, baixa luminosidade e menor vigor vegetativo. Por outro lado, essas mesmas condições ambientais e fenológicas favoráveis à carpeloidia aumentaram a quantidade de frutos normais, assim contribuindo positivamente para a produtividade do mamoeiro

Quantitative importance of staminodes for female reproductive success in Parnassia palustris under contrasting environmental conditions.

The five sterile stamens, or staminodes, in Parnassia palustris act both as false and as true nectaries. They attract pollinators with their conspicuous, but non-rewarding tips, and also produce nectar at the base. We removed staminodes experimentally and compared pollinator visitation rate and duration and seed set in flowers with and without staminodes in two different populations. We also examined the relative importance of the staminode size to other plant traits. Finally, we bagged, emasculated, and supplementary cross-pollinated flowers to determine the pollination strategy and whether reproduction was limited by pollen availability. Flowers in both populations were highly dependent on pollinator visitation for maximum seed set. In one population pollinators primarily cross-pollinated flowers, whereas in the other the pollinators facilitated self-pollination. The staminodes caused increased pollinator visitation rate and duration to flowers in both populations. The staminodes increased female reproductive success, but only when pollen availability constrained female reproduction. Simple linear regression indicated a strong selection on staminode size, multiple regression suggested that selection on staminode size was mainly caused by correlation with other traits that affected female fitness. [ABSTRACT FROM AUTHOR