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

Local Turn-Boundedness, a curvature control for continuous curves with application to digitization

This article focuses on the classical problem of the control of information loss during the digitization step. The properties proposed in the literature rely on smoothness hypotheses that are not satisfied by the curves including angular points. The notion of turn introduced by Milnor in the article On the Total Curvature of Knots generalizes the notion of integral curvature to continuous curves. Thanks to the turn, we are able to define the local turn-boundedness. This promising property of curves does not require smoothness hypotheses and shares several properties with the par(r)-regularity, in particular well-composed digitizations. Besides, the local turn-boundedness enables to constrain spatially the continuous curve as a function of its digitization

Some representations of real numbers using integer sequences

The paper describes three models of the real field based on subsets of the integer sequences. The three models are compared to the Harthong-Reeb line. Two of the new models, contrary to the Harthong-Reeb line, provide accurate integer "views" on real numbers at a sequence of growing scales B^n (B ≥ 2)

Some representations of real numbers using integer sequences

The paper describes three models of the real field based on subsets of the integer sequences. The three models are compared to the Harthong-Reeb line. Two of the new models, contrary to the Harthong-Reeb line, provide accurate integer "views" on real numbers at a sequence of growing scales B^n (B ≥ 2)

Some representations of real numbers using integer sequences

The paper describes three models of the real field based on subsets of the integer sequences. The three models are compared to the Harthong-Reeb line. Two of the new models, contrary to the Harthong-Reeb line, provide accurate integer "views" on real numbers at a sequence of growing scales B^n (B ≥ 2)

Some representations of real numbers using integer sequences

The paper describes three models of the real field based on subsets of the integer sequences. The three models are compared to the Harthong-Reeb line. Two of the new models, contrary to the Harthong-Reeb line, provide accurate integer "views" on real numbers at a sequence of growing scales B^n (B ≥ 2)