Shape As Points: A Differentiable Poisson Solver (2106.03452v2)

Abstract: In recent years, neural implicit representations gained popularity in 3D reconstruction due to their expressiveness and flexibility. However, the implicit nature of neural implicit representations results in slow inference time and requires careful initialization. In this paper, we revisit the classic yet ubiquitous point cloud representation and introduce a differentiable point-to-mesh layer using a differentiable formulation of Poisson Surface Reconstruction (PSR) that allows for a GPU-accelerated fast solution of the indicator function given an oriented point cloud. The differentiable PSR layer allows us to efficiently and differentiably bridge the explicit 3D point representation with the 3D mesh via the implicit indicator field, enabling end-to-end optimization of surface reconstruction metrics such as Chamfer distance. This duality between points and meshes hence allows us to represent shapes as oriented point clouds, which are explicit, lightweight and expressive. Compared to neural implicit representations, our Shape-As-Points (SAP) model is more interpretable, lightweight, and accelerates inference time by one order of magnitude. Compared to other explicit representations such as points, patches, and meshes, SAP produces topology-agnostic, watertight manifold surfaces. We demonstrate the effectiveness of SAP on the task of surface reconstruction from unoriented point clouds and learning-based reconstruction.

• The paper presents a differentiable Poisson solver that transforms oriented point clouds into high-quality, watertight 3D meshes with Shape-As-Points.
• It leverages GPU-optimized spectral methods over uniform grids to achieve orders-of-magnitude speed improvements and robust noise handling.
• Experimental evaluations show SAP outperforming traditional neural methods in both efficiency and reconstruction accuracy in diverse 3D modeling tasks.

Shape As Points: A Differentiable Poisson Solver

The paper, "Shape As Points: A Differentiable Poisson Solver," addresses the limitations of neural implicit representations in the domain of 3D shape reconstruction and proposes an innovative method named Shape-As-Points (SAP). This work revisits the traditional point cloud representation and introduces a GPU-accelerated, differentiable Poisson Surface Reconstruction (DPSR) layer. This layer bridges explicit oriented point clouds with implicit 3D meshes, presenting an interpretable and lightweight alternative to neural implicit representations.

Technical Contributions and Methodology

The primary contribution of the paper is the introduction of a differentiable Poisson solver that efficiently solves for an indicator function from oriented point clouds. This is leveraged to create a novel shape representation—Shape-As-Points—which facilitates effective surface reconstructions. The core of SAP relies on a differentiable formulation of Poisson Surface Reconstruction, allowing gradients to be back-propagated from surface reconstruction losses such as Chamfer distance, thus enabling efficient optimization of 3D shape reconstructions.

SAP's differentiable Poisson solver employs spectral methods over uniform grids, in contrast to traditional finite-element methods that demand complex data structures like octrees. This choice allows for significant leveraging of GPU optimizations. The solver computes high-quality, watertight meshes in a fraction of the time required by neural implicit representations, offering orders of magnitude improvement in computational efficiency.

Comparative and Experimental Analysis

The paper provides an extensive evaluation on multiple datasets, demonstrating that SAP outperforms traditional representations and modern neural methods in both efficiency and accuracy. Experiments include optimization-based 3D reconstruction from unoriented point clouds, as well as learning-based reconstruction on the ShapeNet dataset. SAP excels, producing meshes that are free from self-intersections and topology-agnostic, unlike the fixed topologies of mesh-based models.

Quantitatively, SAP is demonstrated to reduce inference times by a significant margin compared to neural networks, making it feasible for real-time applications. Moreover, its ability to handle high noise levels and outliers robustly showcases its practical applicability in real-world scenarios, including those with noisy sensor data or incomplete scans.

Implications and Future Directions

Practically, the development of SAP holds implications for industries relying on 3D modeling and computer vision, potentially transforming workflows in architectural modeling, robotics, and autonomous navigation by providing rapid and robust shape reconstructions. Theoretically, the introduction of a differentiable Poisson solver opens avenues for further research into differentiable PDE solvers within the field of computer graphics and vision.

Speculatively, future work might explore integrating SAP into larger-scale scene reconstructions or adapting it through sliding-window techniques or hierarchical grids to overcome the memory limitations inherent in 3D volumetric data processing. The integration of this methodology with differentiable rendering pipelines could also enhance applications in neural rendering, augmenting its utility across diverse computational fields.

Overall, the paper makes a compelling case for the 'Shape-As-Points' paradigm, promoting a paradigm shift from implicit neural representations to an efficient, interpretable alternative capable of high-fidelity surface reconstructions.