This paper's main contribution is a new Point Set Surface (PSS) definition based on Moving Least Squares (MLS) fitting of algebraic spheres which can be expressed either by a projection procedure or as an implicit form. The paper defines a point set surface based on an MLS fitting of algebraic spheres, whose representation can be expressed as a projection operator or an implicit surface. The method has a more stable projection operator under low sampling and high curvature. It naturally handles planar point clouds. It provides an estimate of the surface's mean curvature at no additional cost and allows robust handling of sharp features and boundaries. Also presents a novel normal estimation technique. The paper proposes a MLS algorithm which outperforms previous approaches on dealing with undersampled regions. Instead of locally fitting a base plane and a polynomial, the authors use an algebraic spherical fitting to do projection, implicit function evaluation, and normal estimation. The authors also present methods to compute and propagate normal on a raw point set, as well as a tagging method to preserve features. The computation of the algebraic fitting is done by solving a least square problem, which is much simpler than solving the optimal geometrical fitting, and provides a good approximation of the signed distance for points close to the sphere. The paper proposes several extensions to Alexa et al.'s PSS paper: 1] The authors propose fitting an algebraic sphere (which can also characterize planes) to each point. Iteratively computing the best fit sphere in the vicinity of a point and then mapping the point on to the nearest point on the sphere provides an approximate projection approach, similar to the approach in the original work. 2] In the case that normals are unknown, the solution to the best sphere problem is obtained by solving a generalized eigenvalue problem, so that the parameters of the best fit sphere are only defined up to sign (and hence the implicit function cannot distinguish between interior and exterior). This does not affect the algorithm since the projection only requires knowing the zero-set (which is invariant to negation of parameter values). However, if the normals are given, then the solution is obtained by solving a least-squares problem, and the locally computed implicit function can be used to define a (consistent) global explicit. [Though the zero set of this global set is not equal to the fixed points of the projection operation, since that operation is only approximate and requires iteration.] 3] The authors propose a variation on Hoppe's normal estimation that uses the algebraic spheres to "transport" the normals to the neighbors, so that neighbors are not assumed to have parallel normals (making the method more robust near corners). 4] The authors propose an approach for marking and consistent propagation that can be used to reconstruct surfaces with sharp creases and corners. The key insight of this paper is that fitting an algebraic sphere, rather than a plane,
Guenebaud et al. REF presented APSS, algebraic point set surfaces.