Nonlinear Multigrid for Injective Embedding

Computing injective, flip-free embeddings of simplicial meshes is a fundamental challenge in geometry processing. While the Total Lifted Content (TLC) energy provides a smooth functional with theoretically guaranteed injective global minima, its non-convex nature presents significant optimization hurdles. In this work, we propose a Nonlinear Multigrid (MG/Opt) framework specifically tailored for efficient TLC energy minimization. Our approach leverages a topology-preserving mesh hierarchy to rapidly eliminate low-frequency errors on coarse scales while maintaining rigorous gradient consistency between hierarchical levels through linear correction terms. To address the persistent challenge of local inversions, we introduce a targeted smoothing strategy and an adaptive parameter scheduling mechanism that balances global convergence speed with local injectivity precision. Experimental evaluations across extensive 2D and 3D benchmark datasets demonstrate the robustness of our approach. Our method achieves a 100% convergence success rate across all tested 2D cases and all but one 3D dataset (the VOLMAP benchmark). Furthermore, our MG/Opt framework delivers substantial performance gains, achieving up to 63 × speedup on high-resolution subdivided datasets compared to traditional solvers. These results confirm that our multigrid strategy provides a scalable and reliable solution for large-scale injective mapping problems. Code and data for this paper are at https://github.com/distmeshM/MGOpt_TLC