Electrostatics-Inspired Surface Reconstruction (EISR): Recovering 3D Shapes as a Superposition of Poisson's PDE Solutions
This work addresses surface reconstruction in computer vision and graphics, offering a method that enhances detail recovery, though it appears incremental by adapting PDE-based strategies.
The paper tackles the problem of 3D surface reconstruction by proposing a novel approach that encodes shapes as solutions to Poisson's equation, leveraging its linearity to represent implicit fields as superpositions of solutions, resulting in improved approximation of high-frequency details with fewer shape priors.
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.