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

Diego Patiño, Knut Peterson, Kostas Daniilidis, David K. Han

TL;DR

This paper introduces Electrostatics-Inspired Surface Reconstruction (EISR), a PDE-driven approach that replaces the nonlinear Eikonal equation with Poisson's equation to recover 3D surfaces. By expressing the solution as the electrostatic potential of a positive charge distribution and leveraging Green's functions, the method obtains a closed-form implicit field that is a linear superposition of Gaussian charges. Learning optimizes the charge parameters under boundary and interiority constraints, enabling high-frequency detail with a compact prior set and without requiring surface normals. The approach yields competitive results against strong baselines on standard datasets, demonstrates insightful behavior through Fourier analysis of the implicit field, and highlights practical trade-offs between the number of priors and reconstruction fidelity.

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.

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

TL;DR

This paper introduces Electrostatics-Inspired Surface Reconstruction (EISR), a PDE-driven approach that replaces the nonlinear Eikonal equation with Poisson's equation to recover 3D surfaces. By expressing the solution as the electrostatic potential of a positive charge distribution and leveraging Green's functions, the method obtains a closed-form implicit field that is a linear superposition of Gaussian charges. Learning optimizes the charge parameters under boundary and interiority constraints, enabling high-frequency detail with a compact prior set and without requiring surface normals. The approach yields competitive results against strong baselines on standard datasets, demonstrates insightful behavior through Fourier analysis of the implicit field, and highlights practical trade-offs between the number of priors and reconstruction fidelity.

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.
Paper Structure (19 sections, 1 theorem, 25 equations, 7 figures, 3 tables)

This paper contains 19 sections, 1 theorem, 25 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Proposed Method
  4. Electrostatic Potential-based SDF
  5. Solving Poisson Equation with Green's Functions
  6. Surface Reconstruction as a Linear Combination of Gaussian Charges
  7. Learning the Charge Distribution
  8. Experiments
  9. Experimental Setup
  10. Surface Reconstruction Accuracy
  11. Comparative Results
  12. Sensitivity to the Charge Initialization
  13. Limitations
  14. Conclusions
  15. The Minimum Principle
  16. ...and 4 more sections

Key Result

Proposition 1.1

Let $\Omega$ be a closed region with boundary $\partial \Omega$, and let $f({\bm{x}})$ be a solution to the PDE The minimum principle states that, if $g({\bm{x}}) < 0\; \forall {\bm{x}} \in \Omega$, then $f({\bm{x}})$ takes its minimum value somewhere on $\partial \Omega$ and not in the interior of the region.

Figures (7)

  • Figure 1: Overview of our proposed method. We take inspiration from electrostatic theory and represent the implicit scalar field to reconstruct a target shape as a solution of Poisson's equation. We solve the PDE analytically using Green's functions and parametrize the charge density function to fit the desired shape.
  • Figure 2: Surface reconstruction vs. number of charges. Our method achieves accurate results with only $1k$ charges when reconstructing a shape's overall geometry. Increasing the charges significantly improves the reconstruction's high-frequency details.
  • Figure 3: Qualitative results on the Stanford 3D Scanning repository and the Williams et al. Williams2018DeepGP dataset employed in IGR Gropp2020ImplicitGR. Notice how our method effectively captures fine-resolution details from the target shapes.
  • Figure 4: Reconstruction accuracy at different initializations of the charge density spread. We initialize the charges with a normal distribution centered at zero and vary their standard deviation.
  • Figure 5: Cross-section of scalar fields computed with our method. Note how the zero-level set aligns consistently with the target surface.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Proposition 1.1: Minimum principle
  • proof