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

(1)

and the Hausdorff distance is its symmetrization

(2)

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, .

When and are nonempty finite sets, the supremum and infimum become a maximum and minimum, respectively, giving

(3)

For example, for and in the real line with the Euclidean distance, , so .

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