  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.
