- See also
- References
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 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 
and 
, let 
(resp. 
) be the multiset of
all cornerpoints and cornerlines for (resp. ) counted with their
multiplicities, augmented by adding a countable infinity of points of the
diagonal 
.
The matching distance between and is given by
where 
varies among all the bijections between and and
Roughly speaking, the matching distance 
between two size functions is the minimum, over all the matchings
between the cornerpoints of the two size functions, of the maximum
of the 
-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 
. Moreover, the definition of 
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
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.
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