DMesh++: An Efficient Differentiable Mesh for Complex Shapes (2412.16776v1)

Abstract: Recent probabilistic methods for 3D triangular meshes capture diverse shapes by differentiable mesh connectivity, but face high computational costs with increased shape details. We introduce a new differentiable mesh processing method in 2D and 3D that addresses this challenge and efficiently handles meshes with intricate structures. Additionally, we present an algorithm that adapts the mesh resolution to local geometry in 2D for efficient representation. We demonstrate the effectiveness of our approach on 2D point cloud and 3D multi-view reconstruction tasks. Visit our project page (https://sonsang.github.io/dmesh2-project) for source code and supplementary material.

• The paper introduces a minimum-ball algorithm that reduces computational complexity from O(N) to O(log N), enabling up to 32× faster 3D mesh processing.
• It presents an efficient reconstruction pipeline using the Reinforce-Ball method to adaptively generate high-fidelity 2D and 3D meshes.
• The work integrates advanced mesh subdivision and differentiable rendering with anti-aliasing for robust and scalable mesh optimization.

DMesh++: An Efficient Differentiable Mesh for Complex Shapes

The paper "DMesh++: An Efficient Differentiable Mesh for Complex Shapes" introduces a novel approach to accelerate and simplify the process of differentiable mesh handling. This solution aims to address the limitations posed by existing probabilistic methods for 3D triangular meshes, which often struggle with high computational costs when dealing with intricate shape details.

Background and Motivation: Computational efficiency and versatile shape representation are fundamental in 3D modeling. Meshes are often considered optimal for their efficiency and control over vertex position and connectivity. However, the discrete nature of mesh connectivity poses challenges in creating differentiable shape representations essential for machine learning applications. Prior approaches utilizing Transformer-based models for predicting mesh connectivity have encountered issues, such as robustness to outliers and computational inefficiency. The DMesh formulation proposed by Son et al. posed a probabilistic differentiable alternative by explicitly computing face combinations' probabilities to exist on the mesh. Nonetheless, DMesh faces computational roadblocks when scaling to complex geometries with detailed structures due to its reliance on Weighted Delaunay Triangulation (WDT).

Core Contributions of DMesh++: DMesh++ presents a significant enhancement of the original DMesh by addressing its computational inefficiencies. The main contributions of this paper can be categorized as follows:

1. Minimum-Ball Algorithm: This novel algorithm computes the probability of face existence through a condition based on the minimum bounding ball around face vertices, significantly reducing computational complexity. While the WDT approach necessitates a time complexity of $O(N)$, DMesh++ achieves a practical complexity of $O(\log N)$, enabling efficient handling of complex shapes with up to 32 times speed improvements demonstrated in practice for 3D scenarios.
2. Reconstruction and Adaptivity: The paper provides an efficient pipeline for reconstructing 2D and 3D meshes from point clouds or multi-view images. Additionally, the Reinforce-Ball algorithm is designed to eliminate redundant mesh faces in 2D, producing resolution-adaptive results without compromising on geometric fidelity.
3. Mesh Operations and Losses: Advanced mesh subdivision techniques and a differentiable rendering mechanism incorporating anti-aliasing are proposed. These operations ensure high-quality mesh outputs, well-suited for integration in differentiable optimization pipelines.

Potential Applications and Future Directions: DMesh++ holds promise for a range of applications beyond reconstruction. Its framework can be extended to larger scenes, leveraging real-world image datasets, thereby bridging the gap between high-quality neural representations, such as NeRF and practical mesh outputs, useful in simulations and other applications. Furthermore, its integration in training generative models, which comprehend mesh connectivity variably, opens avenues for innovations across diverse machine learning tasks. Future work may consider accelerating the Reinforce-Ball algorithm's convergence and expanding DMesh++ capabilities to more intricately structured datasets.

In conclusion, DMesh++ introduces an efficient and scalable approach for 3D mesh representations, retaining geometric accuracy while offering computational advantages. This paper lays a robust foundation for future exploration in both applied and theoretical avenues, iterating on the promise of differentiable, mesh-based modeling in sophisticated domains.