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.
