Matching Distance

In mathematics, the matching distanceMichele 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.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. is a metric on the space of size functions.

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 $C_1$ (resp. $C_2$) 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 $\$.

The ''matching distance'' between $\ell_1$ and $\ell_2$ is given by
$d_\text(\ell_1, \ell_2)=\min_\sigma\max_\delta (p,\sigma(p))$
where $\sigma$ varies among all the bijections between $C_1$ and $C_2$ and

: $\delta\left((x,y),(x',y')\right)=\min\left\.$

Roughly speaking, the matching distance $d_\text$
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 $\Delta$. 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

References

{{DEFAULTSORT:Matching Distance
Topology