# Point Cloud Surfaces using Geometric Proximity Graphs

## Abstract

We present a new definition of an implicit surface over a noisy point cloud, based on the weighted least squares approach. It can be evaluated very fast, but artifacts are significantly reduced.

We propose to use a different kernel function that approximates geodesic distances on the surface by utilizing a geometric proximity graph. From a variety of possibilities, we have examined the Delaunay graph and the sphere-of-influence graph (SIG), for which we propose several extensions.

The proximity graph also allows us to estimate the local sampling density, which we utilize to automatically adapt the bandwidth of the kernel and to detect boundaries. Consequently, our method is able to handle point clouds of varying sampling density without manual tuning.

Our method can be integrated into other surface definitions, such as moving least squares, so that these benefits carry over.

Stichwörter: implicit surfaces, local polynomial regression, moving least squares, proximity graphs, surface approximation, weighted least squares

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## Bibtex

@ARTICLE{klein-2004-point, author = {Klein, J. and Zachmann, Gabriel}, title = {Point Cloud Surfaces using Geometric Proximity Graphs}, journal = {Computers and Graphics}, volume = {28}, number = {6}, year = {2004}, keywords = {implicit surfaces, local polynomial regression, moving least squares, proximity graphs, surface approximation, weighted least squares}, abstract = {We present a new definition of an implicit surface over a noisy point cloud, based on the weighted least squares approach. It can be evaluated very fast, but artifacts are significantly reduced. We propose to use a different kernel function that approximates geodesic distances on the surface by utilizing a geometric proximity graph. From a variety of possibilities, we have examined the Delaunay graph and the sphere-of-influence graph (SIG), for which we propose several extensions. The proximity graph also allows us to estimate the local sampling density, which we utilize to automatically adapt the bandwidth of the kernel and to detect boundaries. Consequently, our method is able to handle point clouds of varying sampling density without manual tuning. Our method can be integrated into other surface definitions, such as moving least squares, so that these benefits carry over.} }