Stochastic Computation of Barycentric Coordinates

This paper presents a practical and general approach for computing barycentric coordinates through stochastic sampling. Our key insight is a reformulation of the kernel integral defining barycentric coordinates into a weighted least-squares minimization that enables Monte Carlo integration without sacrificing linear precision. Our method can thus compute barycentric coordinates directly at the points of interest, both inside and outside the cage, using just proximity queries to the cage such as closest points and ray intersections. As a result, we can evaluate barycentric coordinates for a large variety of cage representations (from quadrangulated surface meshes to parametric curves) seamlessly, bypassing any volumetric discretization or custom solves. To address the archetypal noise induced by sample-based estimates, we also introduce a denoising scheme tailored to barycentric coordinates. We demonstrate the efficiency and flexibility of our formulation by implementing a stochastic generation of harmonic coordinates, mean-value coordinates, and positive mean-value coordinates.