System and method for establishing correspondence, matching and repairing three dimensional surfaces of arbitrary genus and arbitrary topology in two dimensions using global parameterization

Abstract

A system, method and computer program product for parameterizing a 3D surface of genus g and arbitrary topology to the 2D plane. The method includes receiving an input 3D mesh of genus g; computing n discrete harmonic one-forms on the surface of the mesh to obtain n numbers for every edge in the graph represented by an input mesh; selecting two compatible linearly independent discrete harmonic one-forms for parameterization; assigning an arbitrary vertex in the mesh as an origin, and then summing each of the two harmonic one-forms along the edges of the graph using a traversal strategy to form 2D coordinates for every other vertex of the mesh to form the 2D parameterization of the 3D surface; detecting all singularities in the 2D parameterization; and, removing faces connected to the singularity and parameterizing the vertices of the removed faces using a 2D parameterization method, and connecting these vertices using a 2D polygonalization algorithm. Using the 3D mapping methodology employed, a correspondence function is computed that can be used to match two 3D object surfaces and repair meshes of arbitrary genus.