Volume 14, issue 6 (2014)

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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
Abstract

We describe an algorithm that associates to each positive real number $\epsilon$ and each finite collection ${C}_{\epsilon }$ of planar pixels of size $\epsilon$ a planar piecewise linear set ${S}_{\epsilon }$ with the following property: If ${C}_{\epsilon }$ is the collection of pixels of size $\epsilon$ that touch a given compact semialgebraic set $S$, then the normal cycle of ${S}_{\epsilon }$ 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.

Keywords
semialgebraic sets, pixelations, normal cycle, total curvature, Morse theory
Mathematical Subject Classification 2010
Primary: 53A04
Secondary: 53C65, 58A35