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Pixelations of planar semialgebraic sets and shape recognition

Liviu I Nicolaescu and Brandon Rowekamp

Algebraic & Geometric Topology 14 (2014) 3345–3394

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.

semialgebraic sets, pixelations, normal cycle, total curvature, Morse theory
Mathematical Subject Classification 2010
Primary: 53A04
Secondary: 53C65, 58A35
Received: 5 August 2013
Revised: 22 April 2014
Accepted: 24 April 2014
Published: 15 January 2015
Liviu I Nicolaescu
Department of Mathematics
University of Notre Dame
255 Hurley
Notre Dame, IN 46556-4618
Brandon Rowekamp
Department of Mathematics & Statistics
Minnesota State University, Mankato
273 Wissink Hall
Mankato, MN 56001