Geometric Optimisation Via Spectral Shifting

Abstract

We present a geometric optimisation framework that can recover fold-over free maps from non-injective initial states using popular flip-preventing distortion energies. Since flip-preventing energies are infinite for folded configurations, we propose a new regularisation scheme that shifts the singular values of the deformation gradient. This allow us to re-use many existing algorithms, especially locally injective methods for initially folded maps. Our regularisation is suitable for both singular value- and invariant-based formulations, and systematically contributes multiple stabilisers to the Hessian. In contrast to proxy-based techniques, we maintain second-order convergence. Compact expressions for the energy eigensystems can be obtained for our extended stretch invariants, enabling the use of fast projected Newton solvers. Although spectral shifting in general has no theoretical guarantees that the global minimum is an injection, extensive experiments show that our framework is fast and extremely robust in practice, and capable of generating high-quality maps from severely distorted, degenerate and folded initialisations.