Algebraic & Geometric Topology

Pixelations of planar semialgebraic sets and shape recognition

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513716043

Digital Object Identifier
doi:10.2140/agt.2014.14.3345

Mathematical Reviews number (MathSciNet)
MR3302965

Zentralblatt MATH identifier
1311.53004

Citation

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. https://projecteuclid.org/euclid.agt/1513716043

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