An Adaptive Fast-Multipole-Accelerated Hybrid Boundary Integral Equation Method for Accurate Diffusion Curves

In theory, diffusion curves promise complex color gradations for infinite-resolution vector graphics. In practice, existing realizations suffer from poor scaling, discretization artifacts, or insufficient support for rich boundary conditions. Previous applications of the boundary element method to diffusion curves have relied on polygonal approximations, which either forfeit the high-order smoothness of B\'ezier curves, or, when the polygonal approximation is extremely detailed, result in large and costly systems of equations that must be solved. In this paper, we utilize the boundary integral equation method to accurately and efficiently solve the underlying partial differential equation. Given a desired resolution and viewport, we then interpolate this solution and use the boundary element method to render it. We couple this hybrid approach with the fast multipole method on a non-uniform quadtree for efficient computation. Furthermore, we introduce an adaptive strategy to enable truly scalable infinite-resolution diffusion curves.