Pixelations of planar semialgebraic sets and shape recognition

TL;DR

This paper introduces an algorithm that converts pixelated approximations of planar semialgebraic sets into piecewise linear sets, enabling the recovery of geometric and topological invariants through convergence of normal cycles.

Contribution

The paper presents a novel discretization method based on stratified Morse theory that ensures convergence of geometric invariants from pixel data to the original set.

Findings

Normal cycle of pixelated sets converges to that of the original set

Homotopy type and geometric invariants are recoverable from pixel data

Algorithm provides a bridge between pixelation and geometric analysis

Abstract

We describe an algorithm that associates to each positive real number $r$ and each finite collection $C_r$ of planar pixels of size $r$ a planar piecewise linear set $S_r$ with the following additional property: if $C_r$ is the collection of pixels of size $r$ that touch a given compact semialgebraic set $S$, then the normal cycle of $S_r$ converges to the normal cycle of $S$ in the sense of currents. 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.