DiffusionNet: Discretization Agnostic Learning on Surfaces (2012.00888v3)

Abstract: We introduce a new general-purpose approach to deep learning on 3D surfaces, based on the insight that a simple diffusion layer is highly effective for spatial communication. The resulting networks are automatically robust to changes in resolution and sampling of a surface -- a basic property which is crucial for practical applications. Our networks can be discretized on various geometric representations such as triangle meshes or point clouds, and can even be trained on one representation then applied to another. We optimize the spatial support of diffusion as a continuous network parameter ranging from purely local to totally global, removing the burden of manually choosing neighborhood sizes. The only other ingredients in the method are a multi-layer perceptron applied independently at each point, and spatial gradient features to support directional filters. The resulting networks are simple, robust, and efficient. Here, we focus primarily on triangle mesh surfaces, and demonstrate state-of-the-art results for a variety of tasks including surface classification, segmentation, and non-rigid correspondence.

DiffusionNet: Discretization Agnostic Learning on Surfaces

The paper "DiffusionNet: Discretization Agnostic Learning on Surfaces" introduces a novel approach to geometric deep learning on 3D surfaces, addressing core challenges in surface-based methods, especially concerning robustness to changes in discretization and sampling. The method capitalizes on a diffusion layer for inter-point communication, demonstrating significant advantages in terms of generalization and scalability, complemented by spatial gradient features for directional filtering.

Methodology

DiffusionNet reinterprets spatial convolution, traditionally complex in non-Euclidean domains, through diffusion processes, aligning with discrete differential geometry principles. It encompasses three core components:

1. Pointwise Functions via MLPs: At the foundation are multilayer perceptrons applied independently at each point, allowing flexible scalar transformations crucial for surface feature learning.
2. Learned Diffusion Layers: The model employs diffusion as a conduit for information flow, represented discretely using weak Laplacian matrices. These layers facilitate spatial interactions while maintaining robustness across varied surface discretizations. Notably, diffusion time is a learnable parameter, allowing optimization of spatial range during training, thereby circumventing manual neighborhood definitions.
3. Spatial Gradient Features: Gradient features address the lack of directional sensitivity inherent in pure diffusion models. By capturing geometric anisotropy, these features contribute to directional context, enhancing the expressive power of the network.

Contributions and Implications

DiffusionNet presents several notable contributions to surface learning:

• Discretization Robustness: Its architecture inherently supports invariance to surface discretization changes, demonstrated by consistent performance across mesh resolutions and representations. This trait surmounts a profound limitation of existing methods, paving the way for practical applications not constrained by sampling schemes or mesh quality.
• State-of-the-Art Results: The network achieves competitive or superior accuracy across benchmarks in classification, segmentation, and non-rigid correspondence tasks, marking its efficacy compared to traditional convolution-based surface networks.
• Computational Efficiency: The reliance on fundamental geometry operations—void of complex canonical mappings—facilitates scalable implementations accommodating real-world data sizes while retaining accuracy.
• Cross-representation Generalization: Beyond robustness to single representation variants, DiffusionNet learns transferable features across different types of geometric data like triangle meshes and point clouds, bolstering applicability across mixed-source datasets.

Future Perspectives

The approach opens avenues for universal representation learning within geometric domains, enhancing interoperability of learned models. In practice, its deployment could result in more seamless adaptations to dynamically changing terrain data or heterogeneous CAD inputs without re-training. Speculatively, embedding additional layers exploiting volumetric or higher-order spectral information could further expand its application to indirect representations like depth maps or implicit surfaces.

In summary, "DiffusionNet" introduces a paradigm shift in surface deep learning methodologies through its discretization-agnostic architecture, laying foundations for universally adaptable geometric models. Its robustness and efficiency stand to reshape practical engagements in graphics and computational geometry, potentially influencing broader AI domains requiring nuanced spatial interpretations.