Higher-dimensional power diagrams for semi-discrete optimal transport

Abstract: Efficient algorithms for solving optimal transport problems are important for measuring and optimizing distances between functions. In the $L^2$ semi-discrete context, this problem consists of finding a map from a continuous density function to a discrete set of points so as to minimize the transport cost, using the squared Euclidean distance as the cost function. This has important applications in image stippling, clustering, resource allocation and in generating blue noise point distributions for rendering. Recent algorithms have been developed for solving the semi-discrete problem in $2d$ and $3d$, however, algorithms in higher dimensions have yet to be demonstrated, which rely on the efficient calculation of the power diagram (Laguerre diagram) in higher dimensions. Here, we introduce an algorithm for computing power diagrams, which extends to any topological dimension. We first evaluate the performance of the algorithm in $2d-6d$. We then restrict our attention to four-dimensional settings, demonstrating that our power diagrams can be used to solve optimal quantization and semi-discrete optimal transport problems, whereby a prescribed mass of each power cell is achieved by computing an optimized power diagram.