Hausdorff Distance

The Hausdorff distance is a distance between nonempty subsets of a metric space. Let be a metric space, and define the distance from a point to a nonempty subset by . The directed Hausdorff distance from to is

and the Hausdorff distance is its symmetrization

On the collection of nonempty compact subsets of , is a metric called the Hausdorff metric. For more general subsets it can be infinite, and distinct sets with the same closure can have Hausdorff distance zero. For nonempty closed intervals and in the real line, .

The Hausdorff distance records the largest nearest-neighbor discrepancy between the two sets. The related chamfer distance instead averages nearest-neighbor discrepancies.