Matching distance

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

Example: The matching distance between and is given by
Example: The matching distance between \ell_1=r+a+b and \ell_2=r'+a' is given by d_\text{match}(\ell_1, \ell_2)=\max\{\delta(r,r'),\delta(b,a'),\delta(a,\Delta)\}=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 \ell_1 and \ell_2, let C1 (resp. C2) be the multiset of all cornerpoints and cornerlines for \ell_1 (resp. \ell_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 \ell_1 and \ell_2 is given by d_\text{match}(\ell_1, \ell_2)=\min_\sigma\max_{p\in C_1}\delta (p,\sigma(p)) where σ varies among all the bijections between C1 and C2 and

\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\}.

Roughly speaking, the matching distance dmatch 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.


[edit] References

  1. ^ 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.

[edit] See also