Geometry of unorganized 3D point clouds
The reliable extraction of quantitative geometric information from a discrete cloud of points sampled from a surface is an important task in computer vision and computer graphics.
The dominant paradigm is to fit a smooth parameterized or implicit surface, or a polygonal mesh to the point cloud, and then to apply the standard differential geometric formulae on the smooth surface or one of the numerous methods for estimating curvature and curvature lines on a polygonal surface. These approaches generally perform badly and far from real-time on sparse, noisy, non-organized data and on scenes involving multiple objects, occlusions, and partial views.
We present a new method for defining neighborhoods, and assigning principal curvature frames, and mean and Gauss curvatures to the points of an unorganized oriented point-cloud. The neighborhoods are estimated by measuring implicitly the surface distance between points. The 3D shape recovery is based on conformal geometry, works directly on the cloud, and does not rely on the generation of polygonal or smooth models.