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.

翻译：理论上，扩散曲线承诺为无限分辨率矢量图形提供复杂的颜色渐变。然而，现有实现存在缩放性能差、离散化伪影或对丰富边界条件支持不足等问题。先前边界元法在扩散曲线中的应用依赖于多边形近似，这要么牺牲了贝塞尔曲线的高阶光滑性，要么在极精细的多边形近似下导致必须求解庞大且计算成本高昂的方程组。本文采用边界积分方程方法，精确高效地求解底层偏微分方程。针对所需分辨率与视窗，我们随后对该解进行插值，并利用边界元法进行渲染。我们将这种混合方法与基于非均匀四叉树的快速多极算法相结合以实现高效计算。此外，我们引入自适应策略，真正实现可无限缩放的扩散曲线。