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

1. Problem Statement

Implicit Surface Reconstruction (ISR) is a dominant approach for recovering 3D geometry from inputs like point clouds or images. Most state-of-the-art methods represent shapes as the zero-level set of a Signed Distance Function (SDF).

• Current Limitations:
  • Supervision Requirements: Standard ISR methods (e.g., DeepSDF, IGR) require dense supervision (point-distance pairs) to learn the SDF, which is often unavailable in real-world scenarios.
  • PDE Constraints: Many methods enforce the Eikonal equation ($\|\nabla \phi(x)\| = 1$) via regularization losses to ensure the field behaves like an SDF. However, the Eikonal equation is non-linear and difficult to solve analytically. This forces a trade-off between reconstruction loss and PDE regularization, leading to slow convergence and potential instability.
  • High-Frequency Details: Existing methods often struggle to capture fine geometric details without massive numbers of parameters or complex architectures.

2. Methodology: Electrostatics-Inspired Surface Reconstruction (EISR)

The authors propose replacing the non-linear Eikonal equation with Poisson's equation, drawing inspiration from electrostatic theory.

A. Theoretical Foundation

Instead of modeling the shape as a solution to the Eikonal equation, EISR models the implicit scalar field $\phi(x)$ as the electrostatic potential generated by a positive charge density $\rho(x)$ distributed within the object.

• Governing Equation: Poisson's Equation:
  $$\Delta \phi(x) = -\frac{\rho(x)}{\epsilon_0}$$
  where $\Delta$ is the Laplacian, $\rho(x) > 0$ is the charge density, and $\epsilon_0$ is the permittivity (set to 1).
• Boundary Conditions: The surface boundary $\partial \Omega$ is defined as the iso-surface where $\phi(x) = \tau$ (a constant positive value).
• Validity: The authors prove that if $\rho(x) > 0$, the Minimum Principle ensures that $\phi(x)$ achieves its minimum on the boundary $\partial \Omega$ and is strictly greater inside the domain. This satisfies the requirement for the field to be positive inside the object and negative outside (relative to the iso-surface), making it suitable for Marching Cubes extraction.

B. Analytical Solution via Green's Functions

A key advantage of Poisson's equation is its linearity, allowing for an analytical solution using Green's functions.

• The solution $\phi(x)$ is expressed as the convolution of the Green's function $G(x, s)$ and the charge density $\rho(s)$:
  $$\phi(x) = -\frac{1}{\epsilon_0} \int_{\mathbb{R}^3} G(x - s) \rho(s) \, ds$$
• For the 3D Laplacian, the Green's function is $G(x, s) = \frac{1}{4\pi \|x - s\|}$.

C. Parametric Representation (Shape Priors)

To make the solution computationally tractable and differentiable, the charge density $\rho(x)$ is parameterized as a sum of isotropic Gaussian distributions (shape priors):
$$\rho_\theta(x) = \sum_{i=1}^K Q_i \cdot \mathcal{N}(x; s_i, \sigma_i)$$
where $s_i$ is the location, $Q_i$ is the magnitude, and $\sigma_i$ is the spread of the $i$-th Gaussian charge.

Due to the linearity of the PDE, the total potential is the superposition of the potentials generated by individual Gaussian charges. The closed-form solution for a single Gaussian charge is derived as:
$$\phi_i(x) = \frac{Q_i}{4\pi \|x - s_i\|} \text{erf}\left( \frac{\|x - s_i\|}{\sqrt{2}\sigma_i} \right)$$
The final implicit field is the sum of these $K$ terms.

D. Optimization Strategy

The method optimizes the parameters $\theta = \{s_i, Q_i, \sigma_i\}$ for $K$ charges to fit the target shape using two loss functions:

1. Boundary Condition Loss ($L_{bc}$): Minimizes the difference between the predicted potential and the target threshold $\tau$ on sampled surface points.
   $$L_{bc} = \frac{1}{|X|} \sum_{x \in X} \|\phi_\Omega(x) - \tau\|^2$$
2. Charge Restriction Loss ($L_{cr}$): Ensures charges remain inside the object to maintain the validity of the Minimum Principle. It penalizes the distance of charges from the interior of the shape.
   $$L_{cr} = \frac{1}{K} \sum_{i=1}^K d_{\min}(s_i, Y)^2$$
   where $Y$ is a set of points inside the object.

The total loss is $L = L_{bc} + \lambda L_{cr}$. Notably, no PDE regularization loss is required because the PDE is solved analytically.

3. Key Contributions

1. Proxy PDE: Introduces Poisson's equation as a linear, analytically solvable alternative to the non-linear Eikonal equation for implicit shape representation.
2. Closed-Form Superposition: Leverages Green's functions to derive a closed-form expression for the implicit field as a superposition of Gaussian charge potentials, eliminating the need for iterative PDE solvers or complex neural network architectures for the field itself.
3. High-Frequency Efficiency: Demonstrates that a relatively small number of Gaussian parameters (shape priors) can capture high-frequency geometric details effectively, outperforming methods that require dense supervision or massive parameter counts.

4. Experimental Results

The method was evaluated on the Stanford 3D Scanning Repository and the Williams et al. dataset, comparing against Implicit Geometric Regularization (IGR) and classical Poisson Reconstruction.

• Accuracy: EISR achieved competitive Chamfer distances ($d_C$) and Hausdorff distances ($d_H$), often surpassing IGR and classical Poisson reconstruction (even when the latter used ground-truth normals).
• Detail Recovery: The method successfully reconstructed high-frequency details (e.g., folds on an Armadillo, facial features on a Buddha) using as few as 5,000 Gaussian charges.
• Parameter Efficiency: The representation is compact. Increasing the number of charges progressively refines the surface, with 50k charges yielding near-perfect IoU and F1 scores.
• Initialization Sensitivity: Experiments showed that initializing charges with smaller spreads ($\sigma$) favors high-frequency detail capture, while larger spreads capture low-frequency global shapes.

5. Significance and Limitations

Significance:

• Physics-Informed Learning: Successfully integrates physical laws (electrostatics) directly into the reconstruction pipeline, reducing reliance on data-driven regularization.
• Simplicity & Interpretability: The method avoids complex neural network architectures for the implicit field, offering a mathematically transparent and interpretable representation.
• Reduced Supervision: By solving the PDE analytically, it removes the need for the difficult balance between reconstruction loss and Eikonal regularization loss.

Limitations:

• Full Supervision: Currently requires full surface supervision (point clouds from the surface) for optimization, limiting immediate application to single-view RGB images without extension.
• Flat/Concave Surfaces: Reconstructing perfectly flat or highly concave surfaces theoretically requires infinite charges; however, the method provides good approximations with finite charges.
• Computational Cost: The current implementation requires computing pairwise distances between query points and all charge centers, which can be a bottleneck for very large numbers of charges (though future work suggests using spatial data structures to optimize this).

In conclusion, EISR offers a novel, mathematically grounded framework for 3D reconstruction that bridges the gap between physical modeling and machine learning, achieving high-fidelity results with a compact, interpretable parameterization.