Polyhedral Metrics in Surface Reconstruction: Tight Triangulations

摘要

The paper deals with surface modelling based on a given three-dimensional dataset (regular or scattered). The first problem which the authors need to confront, is to make a suitable triangulation. It is well known that a triangulation strongly influences the 'quality' of a resulting surface. The most typical questions concerning a triangulation are the following ones: Is it possible to determine an 'appropriate' triangulation; How does the shape of a surface from which the authors draw their data set, influence the choice of triangulation; The triangulation itself can be considered as the simplist C(O) approximation, or in other words, a polyhedral approximation. The authors discuss advantages and disadvantages of polyhedral approximations, obtained from possible triangulations, from a geometric point of view. The authors use the notion of manifolds with polyhedral metrics (or simply polyhedra), i.e., metric spaces, where every point has a neighborhood isometric to some neighbourhood of a two-dimensional cone vertex. Making use of the 'polyhedral' analogues of well-known concepts of classical differential geometry, such as the Gaussian curvature, mean curvature, etc., the authors propose a pure geometric approach to choose the 'best' (in a certain sense) triangulation of a given three-dimensional data set.