Voronoi tessellation, also known as Voronoi diagram, is an important computational geometry technique that has applications in various scientific disciplines. It involves dividing a given space into regions based on the proximity to a set of points. Autodifferentiation is a powerful tool for solving optimization tasks. Autodifferentiation assumes constructing a computational graph that allows to compute gradients using backpropagation algorithm. However, often the Voronoi tessellation remains the only non-differentiable part of a pipeline, prohibiting end-to-end differentiation. We present the method for autodifferentiation of the 2D Voronoi tessellation. The method allows one to construct the Voronoi tessellation and pass gradients, making the construction end-to-end differentiable. We provide the implementation details and present several important applications. To the best of our knowledge this is the first autodifferentiable realization of the Voronoi tessellation providing full set of Voronoi geometrical parameters in a differentiable way.

翻译：Voronoi 剖分，亦称 Voronoi 图，是一种重要的计算几何技术，在众多科学领域均有应用。其核心在于根据一组给定点的邻近关系将空间划分为若干区域。自动微分是解决优化任务的有力工具，它通过构建计算图，使得能够利用反向传播算法计算梯度。然而，在许多流程中，Voronoi 剖分往往是唯一不可微分的环节，这阻碍了端到端的微分计算。本文提出了一种针对二维 Voronoi 剖分的自动微分方法。该方法允许构建 Voronoi 剖分并传递梯度，从而使整个构建过程实现端到端可微。我们提供了实现细节，并展示了若干重要应用。据我们所知，这是首个以可微方式提供完整 Voronoi 几何参数集的自动可微 Voronoi 剖分实现。