Matching distance explained

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

The core of the definition of matching distance is the observation that theinformation 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 C _{1} (resp. C _{2} ) be the multiset ofall cornerpoints and cornerlines for \ell _{1} (resp. \ell _{2} ) counted with theirmultiplicities, augmented by adding a countable infinity of points of thediagonal \{(x,y)\in \R ^{2:} x = y\} .

The matching distance between

\ell _{1} and \ell _{2} is given byd _{match(\ell} _{1,} \ell _{2)=min} _{\sigmamax} ^{p\in C _{1}} \delta (p,\sigma(p)) where \sigma varies among all the bijections between C _{1} and C _{2} and\delta\left((x,y),(x',y')\right) =min \left\{ max \{|x - x'|,|y - y'|\}, max \left\{ (y - x)/(, 2) (y' - x')/(2) \right\}\right\}.

Roughly speaking, the matching distance

d _{match} between two size functions is the minimum, over all the matchingsbetween the cornerpoints of the two size functions, of the maximumof the L _{infty} -distances between two matched cornerpoints. Sincetwo size functions can have a different number of cornerpoints,these can be also matched to points of the diagonal \Delta . Moreover, the definition of \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

Notes and 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.