Planar Pixelations and Image Recognition
  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. 
Brandon Rowekamp
math.DG cs.CG cs.CV