Giesen, Joachim ;
Madhusudan, Manjunath
Minimizing Absolute Gaussian Curvature Locally
Abstract
One of the remaining challenges when reconstructing a surface from a
finite sample is recovering nonsmooth 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.
BibTeX  Entry
@InProceedings{giesen_et_al:DSP:2009:2031,
author = {Joachim Giesen and Manjunath Madhusudan},
title = {Minimizing Absolute Gaussian Curvature Locally},
booktitle = {Computational Geometry},
year = {2009},
editor = {Pankaj Kumar Agarwal and Helmut Alt and Monique Teillaud},
number = {09111},
series = {Dagstuhl Seminar Proceedings},
ISSN = {18624405},
publisher = {Schloss Dagstuhl  LeibnizZentrum fuer Informatik, Germany},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2009/2031},
annote = {Keywords: Absolute Gaussian curvature, surface reconstruction, mesh smoothing}
}
2009
Keywords: 

Absolute Gaussian curvature, surface reconstruction, mesh smoothing 
Seminar: 

09111  Computational Geometry

Issue date: 

2009 
Date of publication: 

2009 