In machine learning and computer graphics, a fundamental task is the approximation of a probability density function through a well-dispersed collection of samples. Providing a formal metric for measuring the distance between probability measures on general spaces, Optimal Transport (OT) emerges as a pivotal theoretical framework within this context. However, the associated computational burden is prohibitive in most real-world scenarios. Leveraging the simple structure of OT in 1D, Sliced Optimal Transport (SOT) has appeared as an efficient alternative to generate samples in Euclidean spaces. This paper pushes the boundaries of SOT utilization in computational geometry problems by extending its application to sample densities residing on more diverse mathematical domains, including the spherical space Sd , the hyperbolic plane Hd , and the real projective plane Pd . Moreover, it ensures the quality of these samples by achieving a blue noise characteristic, regardless of the dimensionality involved. The robustness of our approach is highlighted through its application to various geometry processing tasks, such as the intrinsic blue noise sampling of meshes, as well as the sampling of directions and rotations. These applications collectively underscore the efficacy of our methodology.

翻译：在机器学习和计算机图形学中，一项基本任务是通过良好分布的样本集合来近似概率密度函数。最优传输（Optimal Transport, OT）为度量一般空间上概率测度之间的距离提供了形式化框架，在此背景下成为关键的理论基础。然而，其相关的计算负担在大多数实际场景中难以承受。利用一维最优传输的简单结构，切片最优传输（Sliced Optimal Transport, SOT）已成为欧几里得空间中生成样本的高效替代方案。本文通过将切片最优传输的应用扩展到更广泛数学域上的密度采样——包括球面空间S^d、双曲平面H^d及实射影平面P^d——从而突破了其在计算几何问题中的使用边界。此外，无论维度如何，该方法均能通过实现蓝噪声特性来确保样本质量。通过将本方法应用于多种几何处理任务（例如网格的内禀蓝噪声采样，以及方向与旋转的采样），其鲁棒性得到了突出体现。这些应用共同验证了我们方法的有效性。