Algebraic & Geometric Topology

Pixelations of planar semialgebraic sets and shape recognition

Liviu Nicolaescu and Brandon Rowekamp

Full-text: Open access

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

Subjects
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]

Keywords
semialgebraic sets pixelations normal cycle total curvature Morse theory

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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