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In mathematics, the matching distance^[1]^[2] is a metric on the space of size functions.

Example: The matching distance between $ℓ_{1} = r + a + b$ and $ℓ_{2} = r^{'} + a^{'}$ is given by $d_{match} (ℓ_{1}, ℓ_{2}) = \max {δ (r, r^{'}), δ (b, a^{'}), δ (a, Δ)} = 4$

The core of the definition of matching distance is the observation that the information contained in a size function can be combinatorially stored in a formal series of lines and points of the plane, called respectively cornerlines and cornerpoints.

Given two size functions $ℓ_{1}$ and $ℓ_{2}$ , let $C_{1}$ (resp. $C_{2}$ ) be the multiset of all cornerpoints and cornerlines for $ℓ_{1}$ (resp. $ℓ_{2}$ ) counted with their multiplicities, augmented by adding a countable infinity of points of the diagonal ${(x, y) \in ℝ^{2} : x = y}$ .

The matching distance between $ℓ_{1}$ and $ℓ_{2}$ is given by $d_{match} (ℓ_{1}, ℓ_{2}) = \min_{σ} \max_{p \in C_{1}} δ (p, σ (p))$ where $σ$ varies among all the bijections between $C_{1}$ and $C_{2}$ and

δ ((x, y), (x^{'}, y^{'})) = \min {\max {| x - x^{'} |, | y - y^{'} |}, \max {\frac{y - x}{2}, \frac{y^{'} - x^{'}}{2}}} .

Roughly speaking, the matching distance $d_{match}$ between two size functions is the minimum, over all the matchings between the cornerpoints of the two size functions, of the maximum of the $L_{\infty}$ -distances between two matched cornerpoints. Since two size functions can have a different number of cornerpoints, these can be also matched to points of the diagonal $Δ$ . Moreover, the definition of $δ$ implies that matching two points of the diagonal has no cost.

References

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+In [[mathematics]],  the '''matching distance'''<ref name="dAFrLa06">Michele d'Amico, Patrizio Frosini, Claudia Landi, ''Using matching distance in Size Theory: a survey'', International Journal of Imaging Systems and Technology, 16(5):154–161, 2006.</ref><ref name="dAFrLa10">Michele d'Amico, Patrizio Frosini, Claudia Landi, ''Natural pseudo-distance and optimal matching between reduced size functions'', Acta Applicandae Mathematicae, 109(2):527-554, 2010.</ref> is a [[metric (mathematics)|metric]] on the space of [[size function]]s.
+[[Image:DistMatchWiki1.png|thumb|500px|Example: The matching distance between <math>\ell_1=r+a+b</math> and <math>\ell_2=r'+a'</math> is given by <math>d_\text{match}(\ell_1, \ell_2)=\max\{\delta(r,r'),\delta(b,a'),\delta(a,\Delta)\}=4</math>]]
+The core of the definition of matching distance is the observation that the
+information contained in a size function can be combinatorially stored in a formal series of lines and points of the plane, called respectively ''[[size function|cornerlines]]''  and ''[[size function|cornerpoints]]''.
+Given two size functions <math>\ell_1</math> and <math>\ell_2</math>,  let  <math>C_1</math> (resp. <math>C_2</math>) be the multiset of
+all cornerpoints and cornerlines for <math>\ell_1</math> (resp. <math>\ell_2</math>) counted with their
+multiplicities, augmented by adding a countable infinity of points  of the
+diagonal <math>\{(x,y)\in \R^2: x=y\}</math>.
+The ''matching distance'' between <math>\ell_1</math> and <math>\ell_2</math> is given by
+<math>d_\text{match}(\ell_1, \ell_2)=\min_\sigma\max_{p\in C_1}\delta (p,\sigma(p))</math>
+where <math>\sigma</math> varies among all the bijections between <math>C_1</math> and <math>C_2</math> and
+: <math>\delta\left((x,y),(x',y')\right)=\min\left\{\max \{|x-x'|,|y-y'|\}, \max\left\{\frac{y-x}{2},\frac{y'-x'}{2}\right\}\right\}.</math>
+Roughly speaking, the matching distance <math>d_\text{match}</math>
+between two size functions is the minimum, over all the matchings
+between the cornerpoints of the two size functions, of the maximum
+of the <math>L_\infty</math>-distances between two matched cornerpoints. Since
+two size functions can have a different number of cornerpoints,
+these can be also matched to points of the diagonal <math>\Delta</math>. Moreover, the definition of <math>\delta</math> implies that matching two points of the diagonal has no cost.
+==References==
+{{Reflist}}
+==See also==
+* [[Size theory]]
+* [[Size function]]
+* [[Size functor]]
+* [[Size homotopy group]]
+* [[Natural pseudodistance]]
+{{DEFAULTSORT:Matching Distance}}
+[[Category:Topology]]

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Latest revision as of 13:47, 28 August 2013

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