Tessellations are an important tool to model the microstructure of cellular and polycrystalline materials. Classical tessellation models include the Voronoi diagram and Laguerre tessellation whose cells are polyhedra. Due to the convexity of their cells, those models may be too restrictive to describe data that includes possibly anisotropic grains with curved boundaries. Several generalizations exist. The cells of the generalized balanced power diagram are induced by elliptic distances leading to more diverse structures. So far, methods for computing the generalized balanced power diagram are restricted to discretized versions in the form of label images. In this work, we derive an analytic representation of the vertices and edges of the generalized balanced power diagram in 2d. Based on that, we propose a novel algorithm to compute the whole diagram.

翻译：分割是模拟细胞和多晶材料微观结构的重要工具。经典分割模型包括沃罗诺伊图和拉盖尔分割，其胞体均为多面体。由于这些模型中的胞体具有凸性，可能过于严格而难以描述包含具有弯曲边界的各向异性晶粒的数据。为此已有若干推广模型。广义平衡幂图由椭圆距离诱导产生，其胞体结构更加多样。目前，计算广义平衡幂图的方法仅限于标签图像形式的离散化版本。本文推导了二维广义平衡幂图顶点与边的解析表示，并据此提出了一种计算完整广义平衡幂图的新算法。