Euler calculus is a methodology from applied
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
and
integral geometry In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformation ...
that integrates
constructible function In complexity theory, a time-constructible function is a function ''f'' from natural numbers to natural numbers with the property that ''f''(''n'') can be constructed from ''n'' by a Turing machine in the time of order ''f''(''n''). The purpose of ...
s and more recently
definable functions by integrating with respect to the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
as a finitely-additive
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
. In the presence of a metric, it can be extended to continuous integrands via the
Gauss–Bonnet theorem
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology.
In the simplest application, the case of a ...
. It was introduced independently by
Pierre Schapira and
Oleg Viro
Oleg Yanovich Viro (russian: Олег Янович Виро) (b. 13 May 1948, Leningrad, USSR) is a Russian mathematician in the fields of topology and algebraic geometry, most notably real algebraic geometry, tropical geometry and knot theory.
...
in 1988, and is useful for enumeration problems in
computational geometry
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...
and
sensor network
Wireless sensor networks (WSNs) refer to networks of spatially dispersed and dedicated sensors that monitor and record the physical conditions of the environment and forward the collected data to a central location. WSNs can measure environmental c ...
s.
[Baryshnikov, Y.; Ghrist, R]
Target enumeration via Euler characteristic integrals
SIAM ''J. Appl. Math.'', 70(3), 825–844, 2009.
See also
*
Topological data analysis
In applied mathematics, topological based data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challengin ...
References
*Van den Dries, Lou
''Tame Topology and O-minimal Structures'' Cambridge University Press, 1998.
* Arnold, V. I.;
Goryunov, V. V.; Lyashko, O. V
''Singularity Theory'', Volume 1 Springer, 1998, p. 219.
External links
* Ghrist, Robert
Euler Calculusvideo presentation, June 2009. published 30 July 2009.
Algebraic topology
Computational topology
Integral geometry
Measure theory
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