2011/05/13 by Brandon Rowekamp, Rowekamp, Brandon
Computer Science · Mathematics · #Computational Geometry (cs.CG) #Computer Vision and Pattern Recognition (cs.CV) #Differential Geometry (math.DG) #FOS: Computer and information sciences #FOS: Mathematics #cs.CG #cs.CV #math.DG
paper · pdf · doi:10.48550/arxiv.1105.2831
arxiv created 2011/05/13 · arxiv updated 2011/05/17
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.