Poisson Manifold Reconstruction - Beyond Co-dimension One

Screened Poisson Surface Reconstruction creates 2D surfaces from sets of oriented points in 3D (and can be extended to co-dimension one surfaces in arbitrary dimensions). In this work we generalize the technique to manifolds of co-dimension larger than one. The reconstruction problem consists of finding a vector-valued function whose zero set approximates the input points. We argue that the right extension of screened Poisson Surface Reconstruction is based on exterior products: the orientation of the point samples is encoded as the exterior product of the local normal frame. The goal is to find a set of scalar functions such that the exterior product of their gradients matches the exterior products prescribed by the input points. We show that this setup reduces to the standard formulation for co-dimension 1, and leads to more challenging multi-quadratic optimization problems in higher co-dimension. We explicitly treat the case of co-dimension 2, i.e., curves in 3D and 2D surfaces in 4D. We show that the resulting bi-quadratic problem can be relaxed to a set of quadratic problems in two variables and that the solution can be made effective and efficient by leveraging a hierarchical approach.