Piecewise linear mapping optimization based on the complex view

We present an efficient modified Newton iteration for the optimization of nonlinear energies on triangle meshes. Noting that the linear mapping between any pair of triangles is a special case of harmonic mapping, we build upon the results of Chen and Weber [CW17]. Based on the complex view of the linear mapping, we show that the Hessian of the isometric energies has a simple and compact analytic expression. This allows us to analytically project the per-element Hessians to positive semidefinite matrices for efficient Newton iteration. We show that our method outperforms state-of-the-art methods on 2D deformation and parameterization. Further, we inspect the spectra of the per triangle energy Hessians and show that given an initial mapping, simple global scaling can shift the energy towards a more convex state. This allows Newton iteration to converge faster than starting from the given initial state. Additionally, our formulations support adding an energy smoothness term to the optimization with little additional effort, which improves the mapping results such that concentrated distortions are reduced.