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| In [[mathematics]], the '''Hausdorff distance''', or '''Hausdorff metric''', also called '''[[Dimitrie Pompeiu|Pompeiu]]–Hausdorff distance''',<ref>[[R. Tyrrell Rockafellar]], [[Roger J-B Wets]], Variational Analysis, Springer-Verlag, 2005, ISBN 3-540-62772-3, ISBN 978-3-540-62772-2, pg.117.</ref> measures how far two [[subset]]s of a [[metric space]] are from each other. It turns the set of [[non-empty set|non-empty]] [[compact space|compact]] subsets of a metric space into a metric space in its own right. It is named after [[Felix Hausdorff]].
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| 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 you can be forced to travel by an adversary who chooses a point in one of the two sets, from where you 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.
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| It seems that this distance was first introduced by Hausdorff in his book ''[[Grundzüge der Mengenlehre]]'', first published in 1914.
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| == Definition ==
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| [[Image:Hausdorff distance sample.svg|thumb|250px|right|Components of the calculation of the Hausdorff distance between the green line ''X'' and the blue line ''Y''.]]
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| Let ''X'' and ''Y'' be two non-empty subsets of a metric space (''M'', ''d''). We define their Hausdorff distance {{nowrap|''d''<sub> H</sub>(''X'', ''Y'')}} by
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| : <math> d_{\mathrm H}(X,Y) = \max\{\,\sup_{x \in X} \inf_{y \in Y} d(x,y),\, \sup_{y \in Y} \inf_{x \in X} d(x,y)\,\}\mbox{,} \! </math>
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| where ''sup'' represents the [[supremum]] and ''inf'' the [[infimum]].
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| Equivalently
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| :<math>d_H(X,Y) = \inf\{\epsilon \geq 0\,;\ X \subseteq Y_\epsilon \ \mbox{and}\ Y \subseteq X_\epsilon\}</math>,<ref>Munkres, James; ''Topology'' (2nd edition). Prentice Hall, 1999. Pages 280--281.</ref> | |
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| where
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| :<math> X_\epsilon := \bigcup_{x \in X} \{z \in M\,;\ d(z,x) \leq \epsilon\}</math> ,
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| that is, the set of all points within <math>\epsilon</math> of the set <math>X</math> (sometimes called the <math>\epsilon</math>-fattening of <math>X</math> or a generalized ball of radius <math>\epsilon</math> around <math>X</math>).
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| === Remark === | |
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| It is not true in general that if <math> d_H(X,Y) = \epsilon </math>, then
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| :<math> X\subseteq Y_\epsilon \ \mbox{and} \ Y\subseteq X_\epsilon</math> .
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| For instance, consider the metric space of the real numbers <math>\mathbb{R}</math> with the usual metric <math>d</math> induced by the absolute value,
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| :<math>d(x,y) := |y - x|, \quad x,y \in \mathbb{R}</math> .
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| Take
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| :<math>X := \{1/n\,;\ n \in \mathbb{N}\} \quad \mbox{and} \quad Y := \{-1/n\,;\ n \in \mathbb{N}\}</math> .
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| Then <math>d_H(X,Y) = 1\ </math>. However <math>X \nsubseteq Y_1</math> because <math>Y_1 \subset [-2,1)\ </math>, but <math>1 \in X</math>.
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| ==Properties==
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| *In general, ''d''<sub>H</sub>(''X'',''Y'') may be infinite. If both ''X'' and ''Y'' are [[bounded set|bounded]], then ''d''<sub>H</sub>(''X'',''Y'') is guaranteed to be finite.
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| *We have ''d''<sub>H</sub>(''X'',''Y'') = 0 if and only if ''X'' and ''Y'' have the same [[closure (topology)|closure]].
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| *On the set of all non-empty subsets of ''M'', ''d''<sub>H</sub> yields an extended [[pseudometric space|pseudometric]].
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| *On the set ''F''(''M'') of all non-empty compact subsets of ''M'', ''d''<sub>H</sub> is a metric.
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| **If ''M'' is [[complete metric space|complete]], then so is ''F''(''M'').<ref>[http://www-math.mit.edu/phase2/UJM/vol1/HAUSF.PDF] Completeness and Total Boundedness of the Hausdorff Metric</ref>
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| **''([[Blaschke selection theorem]])'' If ''M'' is compact, then so is ''F''(''M'').
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| **The [[topological space|topology]] of ''F''(''M'') depends only on the topology of ''M'', not on the metric ''d''.
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| ==Motivation==
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| 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:<ref>{{cite book
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| | last = Barnsley
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| | first = Michael
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| | authorlink = Michael Barnsley
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| | title = Fractals Everywhere
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| | publisher = Morgan Kaufmann
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| | year = 1993
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| | pages = Ch. II.6
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| | isbn = 0-12-079069-6}}
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| </ref>
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| *Define a distance function between any point ''x'' of ''M'' and any non-empty set ''Y'' of ''M'' by:
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| ::<math>d(x,Y)=\inf \{ d(x,y) | y \in Y \}\ </math>.
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| :For example, ''d''(1, [3,6]) = 2 and ''d''(7, [3,6]) = 1.
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| *Define a distance function between any two non-empty sets ''X'' and ''Y'' of ''M'' by:
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| ::<math>d(X,Y)=\sup \{ d(x,Y) | x \in X \}\ </math>.
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| :For example, ''d''([1,7], [3,6]) = ''d''(1, [3,6]) = 2.
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| *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 {{nowrap|1=''d''(''X'',''Y'') = 0}} does not imply that {{nowrap|1=''X'' = ''Y''}} (It does imply that <math> X \subseteq Y</math>). For example, {{nowrap|1=''d''([1,3,6,7], [3,6]) = 2}}, but {{nowrap|1=''d''([3,6], [1,3,6,7]) = 0}}. However, we can create a metric by defining the '''Hausdorff distance''' to be:
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| ::<math>d_{\mathrm H}(X,Y) = \max\{d(X,Y),d(Y,X) \} \, .</math>
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| == Applications ==
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| 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 detection|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.<ref>[http://www.cs.cornell.edu/vision/hausdorff/hausmatch.html Hausdorff-Based Matching<!-- Bot generated title -->]</ref>
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| In [[Computer Graphics]] the Hausdorff distance is used to measure the difference between two different representations of the same 3D object<ref>P. Cignoni, C. Rocchini, R. Scopigno, "Metro: Measuring Error on Simplified Surfaces", Computer Graphics Forum, Volume 17, Number 2, June 1998, pp. 167-174</ref> particularly when generating [[level of detail]] for efficient display of complex 3D models.
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| ==Related concepts==
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| A measure for the dissimilarity of two [[shapes]] is given by ''Hausdorff distance up to isometry'', denoted ''D''<sub>H</sub>. Namely, let ''X'' and ''Y'' be two compact figures in a metric space ''M'' (usually a [[Euclidean space]]); then ''D''<sub>H</sub>(''X'',''Y'') is the infimum of ''d''<sub>H</sub>(''I''(''X''),''Y'') along all [[isometry|isometries]] ''I'' of the metric space ''M'' to itself. This distance measures how far the shapes ''X'' and ''Y'' are from being isometric.
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| 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''<sub>H</sub>(''I''(''M''),''J''(''N'')) along all isometric embeddings ''I'':''M''→''L'' and ''J'':''N''→''L'' into some common metric space ''L''. | |
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| == See also ==
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| * [[Wijsman convergence]]
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| * [[Kuratowski convergence]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * http://planetmath.org/encyclopedia/HausdorffMetric.html
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| * [http://www-math.mit.edu/phase2/UJM/vol1/HAUSF.PDF Completeness and Total Boundedness of the Hausdorff Metric] (pdf)
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| * http://cgm.cs.mcgill.ca/~godfried/teaching/cg-projects/98/normand/main.html
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| * [http://meshlabstuff.blogspot.com/2010/01/measuring-difference-between-two-meshes.html Using MeshLab to measure difference between two surfaces] A short tutorial on how to compute and visualize the Hausdorff distance between two triangulated 3D surfaces using the open source tool [[MeshLab]].
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| * MATLAB code for Hausdorff distance: [http://www.mathworks.com/matlabcentral/fileexchange/27905-hausdorff-distance]
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| {{DEFAULTSORT:Hausdorff Distance}}
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| [[Category:Metric geometry]]
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