Theoretical Foundation of the Stretch Energy Minimization for Area-Preserving Simplicial Mappings

The stretch energy is a fully nonlinear energy functional that has been applied to the numerical computation of area-preserving mappings. However, this approach lacks theoretical support and the analysis is complicated due to the full nonlinearity of the functional. In this paper, we establish a theoretical foundation of stretch energy minimization (SEM) for the computation of area-preserving mappings: the sufficient and necessary conditions for the energy minimizers are mappings being area-preserving. In addition, we derive a neat gradient formula of the functional and develop the associated line search gradient descent method of SEM with theoretically guaranteed convergence. Also, a simple post-processing technique is developed to guarantee the bijectivity of produced map-pings. Furthermore, we generalized the theoretical results to the stretch energy for arbitrary area measures and the balanced energy so that the mass-preserving and distortion-balancing mappings can also be computed by minimizing the generalized stretch energy and the balanced energy, respectively. Numerical experiments and comparisons to another state-of-the-art algorithm are demonstrated to validate the effectiveness, accuracy, and robustness of SEM for computing area-preserving mappings.