Applications of Topological Graph Theory to 2-Manifold Learning

We show how, given a sufficiently large point cloud sampled from an embedded 2-manifold in \(\mathbb {R}^n\) , we may obtain a global representation of the embedded manifold as a cell complex with vertices given by a representative subset of the point cloud. We apply a known projection based approach (into approximations of tangent planes) from the studies of surface reconstruction and computer graphics, and we extend this with the application of techniques from topological graph theory for the purpose of topological identification of the sampled surface. We apply our methods to samples taken from orientable and nonorientable surfaces. Since our new approach is based on a cell complex rather than a triangulation, it does not derive from or apply Vornoi diagrams or any Delaunay structure.