Topological Properties of Hausdorff Discretizations
Mohamed Tajine, Christian Ronse · Kluwer Academic Publishers eBooks · 2005
We study a new framework for discretization of closed sets based on Hausdorff metric as described in [15, 16, 23, 24]. Let F be a non-empty closed subset of ℝn, S $$ \subseteq $$ ℤn is a Hausdorff discretization of F if it minimizes the Hausdorff distance to F. We study the properties of Hausdorff discretization for homogeneous metrics. For such metrics the popular covering discretizations are Hausdorff discretizations. We also study some topological properties of Hausdorff discretizations. Actually, a Hausdorff discretization of a connected closed set is 8-connected, its maximal Hausdorff discretization is 4-connected, and a Hausdorff discretization “preserves” the homotopy for a class of closed sets and a class of homogeneous metrics. Under some general condition, a Hausdorff discretization is “homeomorphic” to the original set.