Electrostatics-Inspired Surface Reconstruction (EISR): Recovering 3D Shapes as a Superposition of Poisson's PDE Solutions

TL;DR

This paper introduces a novel surface reconstruction method that models 3D shapes as superpositions of solutions to Poisson's equation, inspired by electrostatics, leading to improved detail recovery with fewer priors.

Contribution

It proposes encoding surface reconstruction as a solution to Poisson's equation and uses Green's functions for a closed-form solution, connecting physics and PDEs for shape modeling.

Findings

Enhanced approximation of high-frequency details.

Effective with limited shape priors.

Closed-form parametric expression for solutions.

Abstract

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.