In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance,
measures how far two
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s of a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
are from each other. It turns the set of
non-empty compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
subsets of a metric space into a metric space in its own right. It is named after
Felix Hausdorff
Felix Hausdorff ( , ; November 8, 1868 – January 26, 1942) was a German mathematician, pseudonym Paul Mongré (''à mogré' (Fr.) = "according to my taste"), who is considered to be one of the founders of modern topology and who contributed sig ...
and
Dimitrie Pompeiu.
Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set. The Hausdorff distance is the longest distance someone can be forced to travel by an adversary who chooses a point in one of the two sets, from where they then must travel to the other set. In other words, it is the greatest of all the distances from a point in one set to the closest point in the other set.
This distance was first introduced by Hausdorff in his book ''
Grundzüge der Mengenlehre'', first published in 1914, although a very close relative appeared in the doctoral thesis of
Maurice Fréchet
Maurice may refer to:
*Maurice (name), a given name and surname, including a list of people with the name
Places
* or Mauritius, an island country in the Indian Ocean
* Maurice, Iowa, a city
* Maurice, Louisiana, a village
* Maurice River, a t ...
in 1906, in his study of the space of all continuous curves from
.
Definition
Let
be a
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
. For each pair of non-empty subsets
and
, the Hausdorff distance between
and
is defined as
where
represents the
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
operator,
the
infimum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...
operator, and where
quantifies the distance from a point
to the subset
.
An equivalent definition is as follows. For each set
let
which is the set of all points within
of the set
(sometimes called the
-fattening of
or a generalized
ball
A ball is a round object (usually spherical, but sometimes ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for s ...
of radius
around
).
Then, the Hausdorff distance between
and
is defined as
Equivalently,
where
is the smallest distance from the point
to the set
.
Remark
It is not true for arbitrary subsets
that
implies
:
For instance, consider the metric space of the real numbers
with the usual metric
induced by the absolute value,
:
Take
:
Then
. However
because
, but
.
But it is true that
and
; in particular it is true if
are closed.
Properties
* In general,
may be infinite. If both ''X'' and ''Y'' are bounded set, bounded, then
is guaranteed to be finite.
*
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 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'').
** If ''M'' is compact, then so is ''F''(''M'').
** The
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
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
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
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 does not imply that (it does imply that
). For example, , but . However, we can create a metric by defining the Hausdorff distance to be
, 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
. 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
the Hausdorff distance is used to measure the difference between two different representations of the same 3D object
is around 2,704.8 km.
A measure for the dissimilarity of two shapes is given by ''Hausdorff distance up to isometry'', denoted ''D''
. Namely, let ''X'' and ''Y'' be two compact figures in a metric space ''M'' (usually a
''I'' of the metric space ''M'' to itself. This distance measures how far the shapes ''X'' and ''Y'' are from being isometric.
The
is a related idea: measuring the distance of two metric spaces ''M'' and ''N'' by taking the infimum of
into some common metric space ''L''.
A short tutorial on how to compute and visualize the Hausdorff distance between two triangulated 3D surfaces using the open source tool