Efficient piecewise higher-order parametrization of discrete surfaces with local and global injectivity

Abstract The parametrization of triangle meshes, in particular by means of computing a map onto the plane, is a key operation in computer graphics. Typically, a piecewise linear setting is assumed, i.e., the map is linear per triangle. We present a method for the efficient computation and optimization of piecewise nonlinear parametrizations, with higher-order polynomial maps per triangle. We describe how recent advances in piecewise linear parametrization, in particular efficient second-order optimization based on majorization, as well as practically important constraints, such as local injectivity, global injectivity, and seamlessness, can be generalized to this higher-order regime. Not surprisingly, parametrizations of higher quality, i.e., lower distortion, can be obtained that way, as we demonstrate on a variety of examples.