  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.
