In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the matching distance is a metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

 on the space of size function

Size function

Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane xSize functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane...

s.

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

Size function

Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane xSize functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane...

and cornerpoints

Size function

Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane xSize functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane...

.

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 function
  Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane xSize functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane...
  
  
• Size functor
  Size functor
  Given a size pair \ where M\ is a manifold of dimensionn\ and f\ is an arbitrary real continuous function definedon it, the i\ -th size functor, with i=0,\ldots,n\ , denoted...
  
  
• Size homotopy group
• Natural pseudodistance
  Natural pseudodistance
  In size theory, the natural pseudodistance between two size pairs \ , \ is the value \inf_h \|\varphi-\psi\circ h\|_\infty\ , where h\ varies in the set of all homeomorphisms from the manifold M\ to the manifold N\ and \|\cdot\|_\infty\ is the supremum norm...