- The paper introduces a differentiable mesh representation that leverages weighted Delaunay triangulation for dynamic optimization of both geometry and topology.
- It employs a probability-based selection of mesh faces from tetrahedral decompositions to facilitate gradient-descent optimization.
- The approach achieves improved fidelity at sharp features and enhanced computational efficiency using a CUDA-optimized implementation for real-time applications.
DMesh: Differentiable Representation for General 3D Triangular Meshes
Overview of DMesh Representation
DMesh introduces a novel approach to represent general 3D triangular meshes differentiably. The representation is based on the Weighted Delaunay Triangulation (WDT) and allows for dynamic optimization of both the geometry and topology of the meshes. DMesh represents a substantial enhancement in handling a variety of mesh types, including both open and closed surfaces, optimizing the mesh structure using gradient-based techniques.
Methodology
The core innovation in DMesh lies in the differentiation of mesh representation through the use of tetrahedra divided by Weighted Delaunay Triangulation (WDT). Mesh topology changes dynamically by selecting a subset of triangular faces ("real part") from these tetrahedra while others ("imaginary part") support the structure without contributing to the actual mesh. This approach uses probability assignments for face existence, enabling differentiable manipulations.
Differentiable Formulation
- Weighted Delaunay Triangulation: DMesh utilizes WDT to partition the domain and develop a superset of potential mesh faces.
- Face Existence Probability: A probabilistic layer determines which faces from the WDT belong to the mesh. This is managed through a differentiable evaluation of face existence probabilities, facilitating the gradient-descent optimization.
Comparative Evaluation
Comparisons to traditional methods like typical mesh CNNs and implicit surface approaches reveal that DMesh has advantages in handling varying topology without predetermined constraints. Importantly, the representation directly outputs mesh structures, maintaining the fidelity at sharp features better than approaches reliant on high-density polygon meshes.
Computational Efficiency
The implementation includes an optimization in CUDA, enhancing computational speed, essential for managing the complex calculations involved in real-time applications. Techniques to assess the probability of face existence and rendering calculations contribute significantly to the method's efficiency.
Practical Applications and Theoretical Implications
Practical applications of DMesh span from enhanced 3D object reconstructions in computer vision to dynamic animations in CGI and gaming, where complex topologies are common. Theoretically, the approach pushes forward our understanding of integrating probabilistic models with geometric deep learning frameworks and opens new avenues for research into differentiable geometric representations.
Future Directions
Future enhancements could focus on:
- Scaling and Optimization: Further reducing computational overhead, possibly through more efficient algorithms for WDT within GPUs.
- Handling Non-manifold Geometry: Addressing the challenge of non-manifold edges that may occur due to the flexible nature of the topology definition in DMesh.
- Extended Applications: Exploring applications beyond static mesh generation, such as dynamic mesh deformation in real-time simulations.
DMesh sets a new standard for mesh representation, balancing between flexibility in topology and efficiency in computation, suitable for both academic research and practical applications in industry.