用静电场原理重构3D形状，细节更清晰，少量先验即可出效果。
Electrostatics-Inspired Surface Reconstruction (EISR): Recovering 3D Shapes as a Superposition of Poisson's PDE Solutions
- 将3D表面重建视为泊松方程解的叠加，利用格林函数获得闭式表达。
- 在少量形状先验下仍能精确还原高频细节，优于传统方法。
- 适合需要高保真几何重建的场景，如数字孪生、逆向工程。
隐式形状表示（如SDF）通过标量场的等值面来恢复3D形状表面。现有方法利用SDF是光楔型偏微分方程（PDE）解的性质，通过机器学习近似求解。本文提出一种新方法，将表面重建建模为代理PDE——泊松方程的解。我们探索了泊松方程与物理的联系，例如正电荷密度产生的静电势。通过格林函数推导出该方程解的闭式参数表达，并利用代理方程的线性特性，将目标形状的隐式场表示为多个解的叠加。实验表明，即使在少量形状先验条件下，本方法也能更好逼近高频细节。
原文摘要 · Abstract (English)
Implicit shape representation, such as SDFs, is a popular approach to recover the surface of a 3D shape as the level sets of a scalar field. Several methods approximate SDFs using machine learning strategies that exploit the knowledge that SDFs are solutions of the Eikonal partial differential equation (PDEs). In this work, we present a novel approach to surface reconstruction by encoding it as a solution to a proxy PDE, namely Poisson's equation. Then, we explore the connection between Poisson's equation and physics, e.g., the electrostatic potential due to a positive charge density. We employ Green's functions to obtain a closed-form parametric expression for the PDE's solution, and leverage the linearity of our proxy PDE to find the target shape's implicit field as a superposition of solutions. Our method shows improved results in approximating high-frequency details, even with a small number of shape priors.
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