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

2024-08-17

arXiv.org

AI Summary

This work addresses the long-standing disjunction between gradient meshes and diffusion curves—the two dominant modeling paradigms in smooth vector graphics—by proposing the first unified mathematical framework: both are rigorously formulated as Poisson equation solving problems with customized boundary conditions. Methodologically, we introduce a non-overlapping patch-based intermediate representation, explicitly constraining each patch’s target Laplacian and boundary derivatives (e.g., Neumann conditions along diffusion curves), enabling hybrid editing and high-fidelity rasterization. Our core contribution is the first strict mathematical unification of these paradigms, significantly expanding artistic expressiveness. The framework seamlessly integrates into existing authoring pipelines and provides a principled, extensible foundation for next-generation vectorization and rasterization tools.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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.

Problem

• Unifies gradient meshes and curve-based approaches via Poisson problem
• Combines interpolation and diffusion methods for smooth vector graphics
• Enhances artistic flexibility with unified boundary condition handling

Innovation

• Unifies gradient meshes and curve-based approaches
• Models vector graphics as Poisson problem
• Compatible with existing content creation tools