We propose a variational technique to optimize for generalized barycentric coordinates that offers additional control compared to existing models. Prior work represents barycentric coordinates using meshes or closed-form formulae, in practice limiting the choice of objective function. In contrast, we directly parameterize the continuous function that maps any coordinate in a polytope's interior to its barycentric coordinates using a neural field. This formulation is enabled by our theoretical characterization of barycentric coordinates, which allows us to construct neural fields that parameterize the entire function class of valid coordinates. We demonstrate the flexibility of our model using a variety of objective functions, including multiple smoothness and deformation-aware energies; as a side contribution, we also present mathematically-justified means of measuring and minimizing objectives like total variation on discontinuous neural fields. We offer a practical acceleration strategy, present a thorough validation of our algorithm, and demonstrate several applications.

翻译：我们提出了一种变分技术，用于优化广义重心坐标，与现有模型相比，该方法提供了额外的控制能力。先前的工作通常使用网格或闭式公式表示重心坐标，这在实际中限制了目标函数的选择。相比之下，我们直接通过神经场参数化连续函数，该函数将多胞体内部的任意坐标映射为其重心坐标。这一公式基于我们对重心坐标的理论刻画，使我们能够构建参数化整个有效坐标函数类的神经场。我们通过多种目标函数（包括多种光滑性和变形感知能量）展示了模型的灵活性；作为附带的贡献，我们还提出了在非连续神经场上测量和最小化全变差等目标的数学合理方法。我们提供了一种实用的加速策略，对我们的算法进行了全面的验证，并展示了多个应用场景。