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 that integrates
constructible functions 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's ...
as a finitely-additive
measure. In the presence of a metric, it can be extended to continuous integrands via the
Gauss–Bonnet theorem. It was introduced independently by
Pierre Schapira and
Oleg Viro in 1988, and is useful for enumeration problems in
computational geometry 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 ...
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 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 challenging. TDA ...
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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