We introduce PoissonNet, an architecture for shape reconstruction that
addresses the challenge of recovering 3D shapes from points. Traditional deep
neural networks face challenges with common 3D shape discretization techniques
due to their computational complexity at higher resolutions. To overcome this,
we leverage Fourier Neural Operators (FNOs) to solve the Poisson equation and
reconstruct a mesh from oriented point cloud measurements. PoissonNet exhibits
two main advantages. First, it enables efficient training on low-resolution
data while achieving comparable performance at high-resolution evaluation,
thanks to the resolution-agnostic nature of FNOs. This feature allows for
one-shot super-resolution. Second, our method surpasses existing approaches in
reconstruction quality while being differentiable. Overall, our proposed method
not only improves upon the limitations of classical deep neural networks in
shape reconstruction but also achieves superior results in terms of
reconstruction quality, running time, and resolution flexibility. Furthermore,
we demonstrate that the Poisson surface reconstruction problem is well-posed in
the limit case by showing a universal approximation theorem for the solution
operator of the Poisson equation with distributional data utilizing the Fourier
Neuronal Operator, which provides a theoretical foundation for our numerical
results. The code to reproduce the experiments is available on:
https://github.com/arsenal9971/PoissonNet.

PoissonNet 是一种用于从点云中恢复三维形状的架构，通过利用 Fourier 神经算子 (FNOs) 来解决 Poisson 方程并从定向点云测量中重建网格，其具有效率高、重建质量较好、运行时间短以及分辨率灵活等优势，并通过数值结果的理论定理证明 Poisson 表面重建问题在极限情况下是明确的。