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

2026-02-12

AI Summary

This work addresses the challenge of high-fidelity 3D surface reconstruction from sparse observations, particularly in recovering high-frequency geometric details. The authors propose modeling implicit shape representations as solutions to the Poisson equation, introducing this partial differential equation as a surrogate model for surface reconstruction for the first time. Leveraging a physical analogy to electrostatic potentials, they derive a closed-form parametric representation via Green’s functions. The method constructs surfaces through linear superposition of fundamental solutions, enabling high-quality reconstructions with only minimal shape priors. Experimental results demonstrate that the approach significantly outperforms existing techniques in preserving fine-scale geometric details while maintaining both accuracy and computational efficiency.

Technology Category

• Computer Vision: 3D Computer Vision
• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• surface reconstruction
• 3D shape recovery
• high-frequency details
• implicit representation
• Poisson's equation

Innovation

• Poisson's equation
• surface reconstruction
• implicit representation
• Green's function
• electrostatic analogy