In this paper, we propose a phase-field model to partition a curved surface into path-connected segments with minimal boundary length. Phase-fields offer a powerful tool to represent diffuse interfaces with controlled width and to optimize them in a variational framework. We demonstrate how the multiplicative combination of phase-field functions can be used to effectively compute a hierarchical partition of unity. This induces an associated hierarchy of atlases, whose charts naturally overlap and thus are well-suited for applications such as texture mapping. Furthermore, we obtain distortion minimizing segmentations via a PDE-constraint optimization approach where the phase-field model allows direct use of Lagrangian calculus. Following Sharp and Crane (2018), the Yamabe equation, which allows computing the distortion induced by segment flattening, is considered as the constraint. This way, we obtain end-to-end diffuse formulations of variational problems in surface segmentation that are straightforward to treat computationally. Various examples illustrate the flexibility and robustness of this approach.

在本文中，我们提出了一种相场模型，将曲面划分为具有最小边界长度的路径连接段。相场提供了一种强大的工具来表示具有受控宽度的漫反射界面并在变分框架中对其进行优化。我们演示了如何使用相场函数的乘法组合来有效地计算统一的分层分区。这导致了相关的地图集层次结构，其图表自然重叠，因此非常适合纹理映射等应用程序。此外，我们通过 PDE 约束优化方法获得失真最小化分割，其中相场模型允许直接使用拉格朗日微积分。继 Sharp 和 Crane (2018) 之后，Yamabe 方程可以计算由段展平引起的失真，被视为约束。通过这种方式，我们获得了表面分割中变分问题的端到端漫反射公式，这些公式可以直接通过计算进行处理。各种示例说明了这种方法的灵活性和稳健性。