Matching distance

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

Example: The matching distance between ${\displaystyle \ell _{1}=r+a+b}$ and ${\displaystyle \ell _{2}=r'+a'}$ is given by ${\displaystyle d_{\text{match}}(\ell _{1},\ell _{2})=\max\{\delta (r,r'),\delta (b,a'),\delta (a,\Delta )\}=4}$

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

The matching distance between ${\displaystyle \ell _{1}}$ and ${\displaystyle \ell _{2}}$ is given by ${\displaystyle d_{\text{match}}(\ell _{1},\ell _{2})=\min _{\sigma }\max _{p\in C_{1}}\delta (p,\sigma (p))}$ where ${\displaystyle \sigma }$ varies among all the bijections between ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$ and

${\displaystyle \delta \left((x,y),(x',y')\right)=\min \left\{\max\{|x-x'|,|y-y'|\},\max \left\{{\frac {y-x}{2}},{\frac {y'-x'}{2}}\right\}\right\}.}$

Roughly speaking, the matching distance ${\displaystyle d_{\text{match}}}$ between two size functions is the minimum, over all the matchings between the cornerpoints of the two size functions, of the maximum of the ${\displaystyle L_{\infty }}$-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 ${\displaystyle \Delta }$. Moreover, the definition of ${\displaystyle \delta }$ implies that matching two points of the diagonal has no cost.

See also

• Size theory
• Size function
• Size functor
• Size homotopy group
• Natural pseudodistance

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.
2. 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.