Hausdorff distance

Definition

Let X and Y be two non-empty subsets of a metric space (M, d). We define their Hausdorff distance d_H(X, Y) by

d H ( X , Y ) = max { sup x ∈ X inf y ∈ Y d ( x , y ) , sup y ∈ Y inf x ∈ X d ( x , y ) } ,

where sup represents the supremum and inf the infimum.

Equivalently

d H ( X , Y ) = inf { ϵ ≥ 0 ;   X ⊆ Y ϵ   and   Y ⊆ X ϵ } ,

where

X ϵ := ⋃ x ∈ X { z ∈ M ;   d ( z , x ) ≤ ϵ } ,

that is, the set of all points within ϵ of the set X (sometimes called the ϵ -fattening of X or a generalized ball of radius ϵ around X ).

Remark

It is not true in general that if d H ( X , Y ) = ϵ , then

X ⊆ Y ϵ   and   Y ⊆ X ϵ .

For instance, consider the metric space of the real numbers R with the usual metric d induced by the absolute value,

d ( x , y ) := | y − x | , x , y ∈ R .

Take

X := ( 0 , 1 ] and Y := [ − 1 , 0 ) .

Then d H ( X , Y ) = 1   . However X ⊈ Y 1 because Y 1 = [ − 2 , 1 )   , but 1 ∈ X .

Properties

In general, d_H(X,Y) may be infinite. If both X and Y are bounded, then d_H(X,Y) is guaranteed to be finite.

d_H(X,Y) = 0 if and only if X and Y have the same closure.

For every point x of M and any non-empty sets Y, Z of M: d(x,Y) ≤ d(x,Z) + d_H(Y,Z), where d(x,Y) is the distance between the point x and the closest point in the set Y.

|diameter(Y)-diameter(X)| ≤ 2 d_H(X,Y).

If the intersection X∩Y has a non-empty interior, then there exists a constant r>0, such that every set X′ whose Hausdorff distance from X is less than r also intersects Y.

On the set of all non-empty subsets of M, d_H yields an extended pseudometric.

On the set F(M) of all non-empty compact subsets of M, d_H is a metric.

If M is complete, then so is F(M).

(Blaschke selection theorem) If M is compact, then so is F(M).

The topology of F(M) depends only on the topology of M, not on the metric d.

Motivation

The definition of the Hausdorff distance can be derived by a series of natural extensions of the distance function d(x, y) in the underlying metric space M, as follows:

Define a distance function between any point x of M and any non-empty set Y of M by:

For example, d(1, [3,6]) = 2 and d(7, [3,6]) = 1.

Define a distance function between any two non-empty sets X and Y of M by:

For example, d([1,7], [3,6]) = sup{d(1, [3,6]), d(7,[3,6])} = sup{d(1,3), d(7,6)} = 2.

If X and Y are compact then d(X,Y) will be finite; d(X,X)=0; and d inherits the triangle inequality property from the distance function in M. As it stands, d(X,Y) is not a metric because d(X,Y) is not always symmetric, and d(X,Y) = 0 does not imply that X = Y (It does imply that X ⊆ Y ). For example, d([1,3,6,7], [3,6]) = 2, but d([3,6], [1,3,6,7]) = 0. However, we can create a metric by defining the Hausdorff distance to be:

Applications

In computer vision, the Hausdorff distance can be used to find a given template in an arbitrary target image. The template and image are often pre-processed via an edge detector giving a binary image. Next, each 1 (activated) point in the binary image of the template is treated as a point in a set, the "shape" of the template. Similarly, an area of the binary target image is treated as a set of points. The algorithm then tries to minimize the Hausdorff distance between the template and some area of the target image. The area in the target image with the minimal Hausdorff distance to the template, can be considered the best candidate for locating the template in the target. In computer graphics the Hausdorff distance is used to measure the difference between two different representations of the same 3D object particularly when generating level of detail for efficient display of complex 3D models.

Related concepts

A measure for the dissimilarity of two shapes is given by Hausdorff distance up to isometry, denoted D_H. Namely, let X and Y be two compact figures in a metric space M (usually a Euclidean space); then D_H(X,Y) is the infimum of d_H(I(X),Y) along all isometries I of the metric space M to itself. This distance measures how far the shapes X and Y are from being isometric.

The Gromov–Hausdorff convergence is a related idea: we measure the distance of two metric spaces M and N by taking the infimum of d_H(I(M),J(N)) along all isometric embeddings I:M→L and J:N→L into some common metric space L.