Discretization in Hausdorff Space
Journal of Mathematical Imaging and Vision
Topological properties of Hausdorff discretization, and comparison to other discretization schemes
Theoretical Computer Science
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
Hausdorff Discretizations of Algebraic Sets and Diophantine Sets
DGCI '00 Proceedings of the 9th International Conference on Discrete Geometry for Computer Imagery
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Our theory of Hausdorff discretization has been given in the following framework [10]. Assume an arbitrary metric space (E, d) (E can be a Euclidean space) and a nonvoid proper subspace D of E (the discrete space) such that: (1) D is boundedly finite, that is every bounded subset of D is finite, and (2) the distance fromp oints of E to D is bounded; we call this bound the covering radius, it is a measure of the resolution of D. For every nonvoid compact subset K of E, any nonvoid finite subset S of D such that the Hausdorff distance between S and K is minimal is called a Hausdorff discretizing set (or Hausdorff discretization) of K; among such sets there is always a greatest one (w.r.t. inclusion), which we call the maximal Hausdorff discretization of K. The distance between a compact set and its Hausdorff discretizing sets is bounded by the covering radius, so that these discretizations converge to the original compact set (for the Hausdorff metric) when the resolution of D tends to zero. Here we generalize this theory in two ways. First, we relax condition (1) on D: we assume simply that D is boundedly compact, that is every closed bounded subset of D is compact. Second, the set K to be discretized needs not be compact, but boundedly compact, or more generally closed (cfr. [15] in the particular case where E = Rn and D = Zn).