Minimizing Absolute Gaussian Curvature Locally

Abstract

One of the remaining challenges when reconstructing a surface from a
  finite sample is recovering non-smooth surface features like sharp
  edges. There is practical evidence showing that a two step approach
  could be an aid to this problem, namely, first computing a
  polyhedral reconstruction isotopic to the sampled surface, and
  secondly minimizing the absolute Gaussian curvature of this
  reconstruction globally. The first step ensures topological
  correctness and the second step improves the geometric accuracy of
  the reconstruction in the presence of sharp features without
  changing its topology. Unfortunately it is computationally hard to
  minimize the absolute Gaussian curvature globally. Hence we study a
  local variant of absolute Gaussian curvature minimization problem
  which is still meaningful in the context of surface
  fairing. Absolute Gaussian curvature like Gaussian curvature is
  concentrated at the vertices of a polyhedral surface embedded into
  $mathbb{R}^3$. Local optimization tries to move a single vertex in
  space such that the absolute Gaussian curvature at this vertex is
  minimized. We show that in general it is algebraically hard to find
  the optimal position of a vertex. By algebraically hard we mean that
  in general an optimal solution is not constructible, i.e., there
  exist no finite sequence of expressions starting with rational numbers, 
where each expression is either the sum, difference,
  product, quotient or $k$'th root of preceding expressions and the
  last expressions give the coordinates of an optimal solution. Hence
  the only option left is to approximate the optimal position.  We
  provide an approximation scheme for the minimum possible value of
  the absolute Gaussian curvature at a vertex.