Globally Injective Flattening via a Reduced Harmonic Subspace

Abstract

We present a highly efficient-and-robust method for free-boundary flattening of disk-like triangle meshes in a globally injective manner. We show that by restricting the solution to a low-dimensional subspace of harmonic maps, we can dramatically accelerate the process while obtaining a low-distortion result. The algorithm consists of two main steps. A linear subspace construction, and a nonlinear nonconvex optimization for finding a low-distortion globally injective map within that subspace. The complexity of the first step dominates the algorithm's runtime and is merely that of solving a linear system. We combine recent results for computing locally-and-globally injective maps with that of harmonic maps into a conceptually simple algorithm that guarantees global injectivity. We demonstrate the great efficiency of our method over a dataset of 100 large scale models with more than 2M triangles each. Our algorithm is 10 times faster on average compared to the state-of-the-art Efficient Bijective Parameterizations (EBP) method [Su et al. 2020], on these high-resolution meshes, and more than 20 times faster on challenging examples (Figures 1,11). The speedup over [Jiang et al. 2017; Smith and Schaefer 2015] is even more dramatic.