Planar Pixelations and Shape Reconstruction

Any subset of the plane can be approximated by a set of square pixels. Unfortunately, while this pixelation looks similar to the original set, it does not resemble the original set closely from a mathematical perspective. 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.