Pixelations of planar semialgebraic sets and shape recognition

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.