Algebraic & Geometric Topology

Pixelations of planar semialgebraic sets and shape recognition

Liviu Nicolaescu and Brandon Rowekamp

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We describe an algorithm that associates to each positive real number ε and each finite collection Cε of planar pixels of size ε a planar piecewise linear set Sε with the following property: If Cε is the collection of pixels of size ε that touch a given compact semialgebraic set S, then the normal cycle of Sε converges in the sense of currents to the normal cycle of S. In particular, in the limit we can recover the homotopy type of S and its geometric invariants such as area, perimeter and curvature measures. At its core, this algorithm is a discretization of stratified Morse theory.

Article information

Algebr. Geom. Topol., Volume 14, Number 6 (2014), 3345-3394.

Received: 5 August 2013
Revised: 22 April 2014
Accepted: 24 April 2014
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A04: Curves in Euclidean space
Secondary: 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx] 58A35: Stratified sets [See also 32S60]

semialgebraic sets pixelations normal cycle total curvature Morse theory


Nicolaescu, Liviu; Rowekamp, Brandon. Pixelations of planar semialgebraic sets and shape recognition. Algebr. Geom. Topol. 14 (2014), no. 6, 3345--3394. doi:10.2140/agt.2014.14.3345.

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