Unified Smooth Vector Graphics: Modeling Gradient Meshes and Curve-based Approaches Jointly as Poisson Problem

TL;DR

This work addresses the long-standing separation between gradient-m mesh and curve-based smooth vector graphics by recasting both as Poisson problems solved over a patch-based domain. It introduces extended input primitives, an edge-graph for automatic intersection resolution, and a unified patch construction that assigns a boundary condition and a target Laplacian to each region, enabling per-patch Poisson rasterization. The approach delivers an open-source implementation, demonstrates qualitative results across diverse scenes, and shows compatibility with existing content creation workflows while enabling richer artistic control through unified boundary conditions and Laplacians. This unified framework paves the way for combined rasterization and vectorization tools that leverage the strengths of both gradient meshes and diffusion/Poisson curves in a single cohesive pipeline.

Abstract

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.

Figures (21)

• Figure 1: In this paper, we present an algorithm that unifies the image synthesis of smooth vector graphics containing, gradient meshes, diffusion curves, and Poisson curves in the same scene. Given a set of gradient meshes (a) and diffusion and Poisson curves (b), we first resolve the geometric intersections by forming an undirected edge graph (c), from which a unified patch representation is constructed that divides the domain into separate regions with well-defined Dirichlet and/or Neumann boundary conditions (d). Lastly, the image can be computed with an off-the-shelf Poisson solver (e).
• Figure 2: This figure shows the benefits of combining gradient meshes with diffusion curves. Gradient meshes are useful for controlling large multi-hue color gradients (the sky), while curve-based methods can add details (the Sun). From left to right, we show the input primitives (gradient meshes and diffusion curves) (a), the result of only rasterizing the gradient meshes (b), the result of only diffusing diffusion curves (c), and lastly our combined representation that contains both (d). Note that the sun diffuses into the sky, mimicking scattering.
• Figure 3: Overview of the three mesh and curve primitives that are used in our framework. Gradient meshes (a) are assembled from Ferguson patches, which contain smooth color gradients and are controlled by colors at the corners and tangent handles. Diffusion curves (b) define a color on the left and right side, typically by piecewise linear interpolation among color stops. Poisson curves (c) allow for the modeling of cusps and highlights by locally adjusting the Laplacian of the underlying color field.
• Figure 4: Schematic overview of our generalized smooth vector graphics pipeline. (a) Input to our method is a collection of (extended) gradient meshes, (extended) diffusion curves, and Poisson curves. (b) The input primitives are converted and inserted into an edge graph that resolves the intersection of input primitives. (c) From the edge graph a unified patch data structure is formed that defines patches as closed regions with boundary conditions according to the underlying input primitives. (d) The unified patch data structure specifies the ingredients for a Poisson problem that is solved to synthesize the final image.
• Figure 5: Input boundary curves consist of a spatial curve $\mathbf{x}(t)$ and boundary conditions on the left and right side, which may either be a Dirichlet boundary condition $\mathcal{B}^D$ (shown as colored line with control points) or a homogeneous Neumann boundary condition $\mathcal{B}^N$ (shown as a dashed line).