Hausdorff Discretization and Its Comparison to Other Discretization Schemes

Abstract

We study the problem of discretization in a Hausdorff space followed in [WTR 98]. We recall the definitions and properties of the Hausdorff discretization of a compact set. We also study the relationship between the covering discretizations and the Hausdorff discretization. For a cellular metric every covering discretization minimizes the Hausdorff distance, and conversely, if the supercover discretization minimizes the Hausdorff distance then the metric is cellular. The supercover discretization is the Hausdorff discretization iff the metric is proportional to d ∞. We compare also the Hausdorff discretization and the Bresenham discretization [Bres 65]. Actually, the Bresenham discretization of a segment of IR 2 is not always a good discretization relatively to a Hausdorff metric.

Two Discrete-Euclidean Operations Based on the Scaling Transform

“…We can notice that the object topology is modified: a hole can appear and then disappear. In [13], authors provide a theorem linking topology modifications and grid size. This gives only a general bound because the recon- Fig.…”

Hausdorff Discretization for Cellular Distances and Its Relation to Coverand Supercover Discretizations

“…This covering assumption excludes schemes such as the discretization sampling used in [18] (where W ( p) = {p} for each p ∈ D), as well as the grid-intersect discretization (where each window consists of a cross made of two half-open segments of unit length centered about the point). In particular, in [27,28] we extended the study initiated in [29] of the grid-intersect discretization of line segments (namely, Bresenham's line), showing that it has too few points to give a short Hausdorff distance between the Euclidean segment and its discretization. Note that instead of taking the whole discretization by dilation, one can also take W -discretizing sets.…”

“…In the particular case where E = R n and D = Z n , we have some further results. In [27,28] we considered the Hausdorff discretization of line segments and its comparison to the Bresenham discretization; we study the topological properties of Hausdorff discretization in [25,27]; finally, the decidability of discretizing algebraic sets, and its realization as diophantine discrete sets, is studied in [26]. Further works will be needed on the topological and geometrical properties of Hausdorff discretization and on the discretization of operators.…”

“…Similar to [19], this paper is a generalization of [29], and a preliminary exposition (without proofs) or some of our results, in the restricted case of the Euclidean space R n and its discretization in Z n , can be found in [28,30]. The paper is organized as follows.…”

“…We do in some way the converse, start with global conditions, see which types of discretization satisfy them, and finally compare them to already existing discretization schemes. In particular, we have started to analyze the topological properties of our new discretization approach [25,27].…”