Sampling is a core component of many applications including: imaging, rendering, geometry processing and visualization. Prior research has primarily focused on blue noise sampling with a single class of samples. Limited research has been done on multi-class sampling. Multi-class blue noise sampling is still a challenging problem when the density distribution of every class is non-uniform and different from each other. In this paper, we present a Wasserstein blue noise sampling algorithm for multi-class sampling by throwing samples as the Wasserstein barycenter of multiple density distributions. We employ a more general representation of the optimal transport problem to break up the partition necessary in other optimal sampling. Moreover, an adaptive blue noise distribution property for every individual class is guaranteed, as well as their combined class. The sampling efficiency is also improved by applying the Wasserstein distance with entropic constraints. Our method can be applied to multi-class sampling on the point set surface. We also demonstrate applications in object distribution and color stippling.
Implicitizing rational surfaces is a fundamental computational task in Algorithmic Algebraic Geometry. Although the resultant of three moving planes corresponding to a ¼-basis for a rational surface is guaranteed to contain the implicit equation of the surface as a factor, ¼-bases for rational surfaces are difficult to compute. Moreover, ¼-bases often have high degrees, so these resultants generally contain many extraneous factors. Here we develop fast algorithms to implicitize rational tensor product surfaces by computing the resultant of three moving planes corresponding to three syzygies with low degrees. These syzygies are easy to compute, and the resultants of the corresponding moving planes generally contain fewer extraneous factors than the resultants of the moving planes corresponding to ¼-bases. We predict and compute all the possible extraneous factors that may appear in these resultants. Examples are provided to clarify and illuminate the theory.