Hausdorff metric

plural Hausdorff metrics

1. mathematical analysisIn the abstract metric space of all compact subsets of ℝⁿ, given a pair of compact sets A and B, the Hausdorff metric is h(A,B)= mbox maxρ(A,B),ρ(B,A) where ρ(A,B)= sup _(a∈A) inf _(b∈B),d(a,b), where d is the Euclidean metric in ℝⁿ.

   The Hausdorff metric topology was first discovered by D. Pompeiu and by F. Hausdorff in 1927. This topology has been used in a number of areas such as function space topologies, approximation theory, and optimisation (see, e.g., 22).

   2014, James F. Peters, Topology of Digital Images: Visual Pattern Discovery in Proximity Spaces, Springer Science & Business Media, →ISBN, page 195

   Synonyms: Hausdorff distance, Pompeiu-Hausdorff distance