Planar Pixelations and Image Recognition

TL;DR

This paper introduces a Morse Theory-inspired algorithm that reconstructs geometric and topological features of shapes from pixelations, enabling accurate shape approximation as pixel size decreases.

Contribution

It presents a novel method to recover shape invariants from pixelated images, bridging pixelation and shape analysis with strong convergence guarantees.

Findings

Successfully recovers Betti numbers, area, perimeter, and curvature from pixelations

Provides a convergent approximation to the original shape as pixel size diminishes

Demonstrates robustness of the method in shape reconstruction

Abstract

Any subset of the plane can be approximated by a set of square pixels. This transition from a shape to its pixelation is rather brutal since it destroys geometric and topological information about the shape. Using a technique inspired by Morse Theory, we algorithmically produce a PL approximation of the original shape using only information from its pixelation. This approximation converges to the original shape in a very strong sense: as the size of the pixels goes to zero we can recover important geometric and topological invariants of the original shape such as Betti numbers, area, perimeter and curvature measures.