Image Processing and Recognition

HAUSDORFF DISTANCE

The Hausdroff distance is defined on the subsets of a space. Suppose that the space has a distance function d(x,y). This is a symmetric definite positive function defined on the pairs of points of the space:

d(x,y) > 0 if x and y are different
d(x,x) = 0 for any x
d(x,y) = d(y,x) for every x and y

Exemple of Hausdorff
distance The directed Hausdorff distance from a set A to a set B is (see figure)

h(A,B) = max_a min_b d(a,b)

The Hausdorf distance between two sets is

H(A,B) = max { h(A,B), h(B,A) }

For image processing applications is has proven usefulto apply also a slightly different distance, the modified Hausdorff distance,

h'(A,B) = (1/|A|) ∑_a min_b d(a,b)

This distance takes the average of the single point distances, and decreases the impacts of outliers.

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