A Variational Taxonomy for Surface Reconstruction from Oriented Points

The problem of reconstructing a watertight surface from a finite set of oriented points has received much attention over the last decades. In this paper, we propose a general higher order framework for surface reconstruction. It is based on the idea that position and normal defined by each oriented point can be used to construct an implicit local description of the unknown surface. On the one hand, this allows us to systematically explain and relate several popular methods, for example implicit moving least squares, smooth signed distance surface reconstruction as well as (screened) Poisson surface reconstruction. On the other hand, it allows to derive and discuss a number of new approaches for reconstructing either the signed distance or the indicator function of the sought object. All of these approaches are able to achieve competitive results but one of them turns out to be especially promising. To improve reconstructions in difficult real world scenarios where point clouds have been estimated from colour images, we introduce a hull constraint that encourages the surface to stay within a given region. Our framework is implemented on the GPU using a recent cyclic scheme called Fast Jacobi, which combines low implementational effort with high efficiency.