Euler calculus

Euler calculus is a methodology from applied algebraic topology and integral geometry that integrates constructible functions and more recently definable functions[1] by integrating with respect to the Euler characteristic as a finitely-additive measure. In the presence of a metric, it can be extended to continuous integrands via the Gauss–Bonnet theorem.[2] It was introduced independently by Pierre Schapira[3][4][5] and Oleg Viro[6] in 1988, and is useful for enumeration problems in computational geometry and sensor networks.[7]

Euler integration for constructible functions

Euler calculus begins from the observation that the Euler characteristic with compact supports obeys one of the main properties of a measure: . As a result, for a suitably restricted class of functions, it is possible to define an integral with respect to this measure. One begins by selecting an o-minimal structure of definable sets in the topology, for instance, semialgebraic or subanalytic sets. The class of constructible functions consists of those functions such that is definable for all . A definition of the Euler integral follows:

Due to the properties of the o-minimal structure, the integral may also be computed by decomposing as a sum of indicator functions defined on a disjoint union of cells (giving a cell structure). If , then

See also

References

  1. ↑ Baryshnikov, Y.; Ghrist, R. Euler integration for definable functions, Proc. National Acad. Sci., 107(21), 9525–9530, 25 May 2010.
  2. ↑ McTague, Carl (1 Nov 2015). "A New Approach to Euler Calculus for Continuous Integrands". arXiv:1511.00257 [math.DG].
  3. ↑ Schapira, P. "Cycles Lagrangiens, fonctions constructibles et applications", Seminaire EDP, Publ. Ecole Polytechnique (1988/89)
  4. ↑ Schapira, P. Operations on constructible functions, J. Pure Appl. Algebra 72, 1991, 83–93.
  5. ↑ Schapira, Pierre. Tomography of constructible functions, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes Lecture Notes in Computer Science, 1995, Volume 948/1995, 427–435, doi:10.1007/3-540-60114-7_33
  6. ↑ Viro, O. Some integral calculus based on Euler characteristic, Lecture Notes in Math., vol. 1346, Springer-Verlag, 1988, 127–138.
  7. ↑ Baryshnikov, Y.; Ghrist, R. Target enumeration via Euler characteristic integrals, SIAM J. Appl. Math., 70(3), 825–844, 2009.
  • Van den Dries, Lou. Tame Topology and O-minimal Structures, Cambridge University Press, 1998. ISBN 978-0-521-59838-5
  • Arnold, V. I.; Goryunov, V. V.; Lyashko, O. V. Singularity Theory, Volume 1, Springer, 1998, p. 219. ISBN 978-3-540-63711-0
  • Ghrist, Robert. Euler Calculus video presentation, June 2009. published 30 July 2009.


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