CHS: Small: Novel Method for Vectorization of Arbitrary Natural Images and Its Applications

矢量图形提供了一种紧凑和无损的图像表示，具有诸如几何可编辑性、分辨率独立性、显著节省存储和网络带宽、以急剧变化的分辨率显示图像以及易于动画等优点。 本研究的目的是显着推进传统的边界图像矢量化的偏微分方程（PDE）的基础上，其内在的联系与绿色的功能和谐波B样条（作为PDE的基本解决方案），尚未探索矢量图形，图像建模，图像数据拟合和分析。 如果成功的话，项目成果将包括一个新的矢量图像建模方法及其在图像矢量化和创作以及实体纹理和动画方面的应用。 在这个项目的理论基础的核心是偏微分方程和他们的无网格封闭形式求解器的基础上的基本解决方案。 新的表示预计将优于传统的扩散曲线和梯度网格为基础的表示。 新的建模方案将能够在理论上表示任意图像与任意不连续性。 因此，本研究将推动矢量图形学在理论和实践上的发展。 除了传统的视觉计算领域外，由于本研究完全建立在偏微分方程及其基本解决方案的基础上，预计其他学科，如应用数学，物理科学，机械工程和地球/空间科学将直接受益于项目成果。本项目将探索一种新颖的图像矢量化建模方案：泊松矢量图形（PVG），通过一组稀疏的几何图元和颜色约束来计算复杂的颜色梯度。 具体的研究工作包括：（1）建立了PVG的非零Laplacian理论基础，建立了以绿色函数为基本解的偏微分方程及其无网格闭合解的方法，（2）利用绿色函数与调和B样条函数之间的内在联系，推导了Poisson方程的闭合解;（3）提出了一种基于数值优化的PVG矢量化方法，使其能够高精度地表示任意自然图像：（4）设计了一个PVG的创作工具，该工具采用了新的Poisson曲线和Poisson区域隐喻，使用户能够比传统的一阶或二阶扩散曲线更灵活地设计矢量图像;（5）证明了新的PVG可应用于动画。 与当前最先进的框架进行全面的定性和定量比较，以展示新框架的优越性。 最后，本计画的整合方法结合了扩散曲线与梯度网格的优点，将向量图形的应用范围大幅扩展至以数值量测为主流的视觉资讯建模、分析与处理。

Vector graphics offers a compact and lossless image representation with advantages such as geometric editability, resolution independence, significant saving in storage and in network bandwidth, image display at drastically varying resolutions, and ease of animation. This research aims to significantly advance the traditional boundary of image vectorization based on partial differential equations (PDEs) and their intrinsic connection with Green's functions and harmonic B-splines (serving as fundamental solutions for PDEs), which has not yet been explored for vector graphics, image modeling, image data fitting, and analysis. If successful, project outcomes will include a novel vector image modeling methodology and its application for image vectorization and authoring as well as solid texture and animation. At the core of this project's theoretical foundation are PDEs and their meshless closed-form solvers based on fundamental solutions. The novel representation is expected to outperform the conventional diffusion curve based and gradient mesh based representations. The new modeling scheme will be capable in theory of expressing arbitrary images with arbitrary discontinuities. Consequently, this research will advance the state of the art in both the theory and practice of vector graphics. Beyond the conventional frontier of visual computing, since this research is solely built upon PDEs and their fundamental solutions, it is anticipated that other disciplines such as applied mathematics, the physical sciences, mechanical engineering, and the earth/space sciences will directly benefit from project outcomes.This project will explore a novel image vectorization modeling scheme: Poisson Vector graphics (PVG), which computes complex color gradients via a sparse set of geometric primitives and color constraints. Detailed research activities include: (1) articulation of a sound theoretic foundation for PVG with non-zero Laplacians, the methodology to be founded upon PDEs and their meshless closed-form solvers by taking advantage of Green's functions serving as their fundamental solutions; (2) derivation of a closed-form solution for Poisson equations based on the intrinsic connection between Green's function and harmonic B-splines; (3) development of a method for vectorization of arbitrary natural images by PVG based on numerical optimization, so that they can be represented with high precision; (4) design of an authoring tool for PVG with new Poisson curve and Poisson region metaphors, so that users will be able to design vector images with much more flexibility than conventional first or second order diffusion curves; and (5) demonstration that the novel PVG is applicable to animation. Comprehensive qualitative and quantitative comparison with the current state-of-the-art will be carried out to showcase the new framework's superiority. Ultimately, this project's integrated approach combines the merits of diffusion curve and gradient mesh, which is capable of drastically expanding the applied scope of vector graphics to visual information modeling, analysis, and processing, where numerical measurements are prevalent.