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.

翻译：隐式形状表示（如符号距离函数）是一种流行的三维形状表面重建方法，它将表面恢复为标量场的等值面。已有多种方法利用符号距离函数是Eikonal偏微分方程解这一知识，采用机器学习策略对其进行近似。本文提出了一种新颖的表面重建方法，将其编码为代理偏微分方程——即泊松方程——的解。随后，我们探讨了泊松方程与物理学（例如由正电荷密度产生的静电势）之间的联系。我们利用格林函数获得该偏微分方程解的闭式参数表达式，并借助代理偏微分方程的线性特性，通过解的叠加来得到目标形状的隐式场。实验表明，即使在形状先验数量较少的情况下，我们的方法在近似高频细节方面也取得了更好的效果。