Research on smooth vector graphics is separated into two independent research threads: one on interpolation-based gradient meshes and the other on diffusion-based curve formulations. With this paper, we propose a mathematical formulation that unifies gradient meshes and curve-based approaches as solution to a Poisson problem. To combine these two well-known representations, we first generate a non-overlapping intermediate patch representation that specifies for each patch a target Laplacian and boundary conditions. Unifying the treatment of boundary conditions adds further artistic degrees of freedoms to the existing formulations, such as Neumann conditions on diffusion curves. To synthesize a raster image for a given output resolution, we then rasterize boundary conditions and Laplacians for the respective patches and compute the final image as solution to a Poisson problem. We evaluate the method on various test scenes containing gradient meshes and curve-based primitives. Since our mathematical formulation works with established smooth vector graphics primitives on the front-end, it is compatible with existing content creation pipelines and with established editing tools. Rather than continuing two separate research paths, we hope that a unification of the formulations will lead to new rasterization and vectorization tools in the future that utilize the strengths of both approaches.

翻译：关于平滑矢量图形的研究分为两个独立的研究方向：一是基于插值的梯度网格，二是基于扩散的曲线表述。本文提出了一种数学表述，将梯度网格和基于曲线的方法统一为泊松问题的解。为了融合这两种经典表示方法，我们首先生成一种非重叠的中间面片表示，为每个面片指定目标拉普拉斯算子和边界条件。统一处理边界条件为现有表述增添了更多艺术创作自由度，例如在扩散曲线上应用诺伊曼边界条件。为在给定输出分辨率下合成光栅图像，我们分别对面片的边界条件和拉普拉斯算子进行光栅化处理，并通过求解泊松问题计算最终图像。我们在包含梯度网格和基于曲线图元的多种测试场景中评估了该方法。由于我们的数学表述在前端采用成熟的平滑矢量图形基元，因此与现有内容创作流程及编辑工具兼容。我们期望通过统一这两种表述，能够催生未来充分利用两者优势的新型光栅化与矢量化工具，而非延续两条独立的研究路径。