[Paper Review] Poisson Vector Graphics (PVG) and Its Closed-Form Solver

Fei Hou, Qian Sun+6 more|arXiv (Cornell University)|Jan 16, 2017

Computer Graphics and Visualization Techniques12 references3 citations

TL;DR

This paper introduces Poisson Vector Graphics (PVG), an extension of diffusion curves that supports non-zero Laplacians via Poisson curves and regions, enabling photorealistic effects like highlights and translucency. It proposes a harmonic B-spline-based closed-form solver that enables resolution-independent rendering, random access, zooming, and anti-aliasing with less than 0.3% relative mean error.

ABSTRACT

This paper presents Poisson vector graphics, an extension of the popular first-order diffusion curves, for generating smooth-shaded images. Armed with two new types of primitives, namely Poisson curves and Poisson regions, PVG can easily produce photorealistic effects such as specular highlights, core shadows, translucency and halos. Within the PVG framework, users specify color as the Dirichlet boundary condition of diffusion curves and control tone by offsetting the Laplacian, where both controls are simply done by mouse click and slider dragging. The separation of color and tone not only follows the basic drawing principle that is widely adopted by professional artists, but also brings three unique features to PVG, i.e., local hue change, ease of extrema control, and permit of intersection among geometric primitives, making PVG an ideal authoring tool. To render PVG, we develop an efficient method to solve 2D Poisson's equations with piecewise constant Laplacians. In contrast to the conventional finite element method that computes numerical solutions only, our method expresses the solution using harmonic B-spline, whose basis functions can be constructed locally and the control coefficients are obtained by solving a small sparse linear system. Our closed-form solver is numerically stable and it supports random access evaluation, zooming-in of arbitrary resolution and anti-aliasing. Although the harmonic B-spline based solutions are approximate, computational results show that the relative mean error is less than 0.3%, which cannot be distinguished by naked eyes.

Motivation & Objective

• To extend diffusion curve graphics by enabling direct control over tone and shading through non-zero Laplacians.
• To address the limitations of existing diffusion curve models in controlling gradients, extrema, and complex effects like halos and translucency.
• To develop a numerically stable, closed-form solver for 2D Poisson equations with piecewise constant Laplacians that supports arbitrary zoom and anti-aliasing.
• To enable intuitive authoring by separating color (Dirichlet boundary condition) from tone (Laplacian control), improving usability over prior methods.

Proposed method

• Introduces two new primitives: Poisson curves (with specified Laplacian values) and Poisson regions (for localized tone control), enabling control over shading profiles.
• Represents the solution of the Poisson equation Δu = f using harmonic B-splines, where f is piecewise constant over subdomains.
• Constructs basis functions locally and solves a small sparse linear system to compute control coefficients, enabling closed-form evaluation.
• Supports random access evaluation and high-resolution zooming by evaluating the harmonic B-spline basis functions at any point.
• Uses quad-tree domain discretization to efficiently compute the harmonic B-spline coefficients and maintain numerical stability.
• Employs Green’s third identity and boundary element methods as a foundation, adapting them to handle non-zero Laplacians and area integrals.

Experimental results

Research questions

• RQ1 Can a vector graphics model extend diffusion curves by allowing non-zero Laplacians to enable richer shading effects such as specular highlights and translucency?
• RQ2 How can a closed-form solution for Poisson’s equation with piecewise constant Laplacians be constructed to support interactive authoring and high-resolution rendering?
• RQ3 Can the separation of color and tone in PVG improve user intuitiveness and control over local extrema and hue changes compared to existing diffusion curve methods?
• RQ4 How can a solver achieve both numerical stability and support for arbitrary zooming and anti-aliasing without pre-computation or iterative refinement?

Key findings

• The PVG framework successfully enables photorealistic rendering of effects such as specular highlights, core shadows, halos, and translucency using only sparse geometric primitives.
• The harmonic B-spline-based solver achieves a relative mean error of less than 0.3% compared to ground truth, which is imperceptible to the human eye.
• The solver supports random access evaluation, arbitrary zooming, and anti-aliasing without re-computing boundary conditions, outperforming iterative and pre-computed methods.
• The method avoids visual artifacts during zooming and panning, unlike Jacobi-iteration-based solvers that suffer from boundary inaccuracies.
• User studies confirm that PVG is more intuitive and easier to use than traditional diffusion curves and their biharmonic extensions.
• The approach is computationally efficient and stable, avoiding the ill-conditioning and numerical issues common in biharmonic solvers.