Proceedings Eurographics/IEEE VGTC Symposium Point-Based Graphics (2005)
Stony Brook, NY, USA
June 20, 2005 to June 21, 2005
P.-T. Bremer , Univ. of Dlinois, Urbana, IL, USA
J.C. Hart , Univ. of Dlinois, Urbana, IL, USA
Recently, point set surfaces has been the focus of a large number of research efforts. Several different methods have been proposed to define surfaces from points and have been used in a variety of applications. However, so far little is know about the mathematical properties of the resulting surface. A central assumption for most algorithms is that the surface construction is well defined within a neighborhood of the samples. However, it is not clear that given an irregular sampling of a surface this is the case. The fundamental problem is that point based methods often use a weighted least squares fit of a plane to approximate a surface normal. If this minimization problem is ill-defined so is the surface construction. In this paper, we provide a proof that given reasonable sampling conditions the normal approximations are well defined within a neighborhood of the samples. Similar to methods in surface reconstruction, our sampling conditions are based on the local feature size and thus allow the sampling density to vary according to geometric complexity.
geometric complexity, sampling theorem, MLS surfaces, point set surface, weighted least squares fit, surface normal approximation, minimization problem, surface reconstruction
J. Hart and P. Bremer, "A sampling theorem for MLS surfaces," Point-Based Graphics 2005(PBG), Stony Brook, NY, USA, 2005, pp. 47-54.