DualConvMesh-Net: Joint Geodesic and Euclidean Convolutions on 3D Meshes (2004.01002v1)

Abstract: We propose DualConvMesh-Nets (DCM-Net) a family of deep hierarchical convolutional networks over 3D geometric data that combines two types of convolutions. The first type, geodesic convolutions, defines the kernel weights over mesh surfaces or graphs. That is, the convolutional kernel weights are mapped to the local surface of a given mesh. The second type, Euclidean convolutions, is independent of any underlying mesh structure. The convolutional kernel is applied on a neighborhood obtained from a local affinity representation based on the Euclidean distance between 3D points. Intuitively, geodesic convolutions can easily separate objects that are spatially close but have disconnected surfaces, while Euclidean convolutions can represent interactions between nearby objects better, as they are oblivious to object surfaces. To realize a multi-resolution architecture, we borrow well-established mesh simplification methods from the geometry processing domain and adapt them to define mesh-preserving pooling and unpooling operations. We experimentally show that combining both types of convolutions in our architecture leads to significant performance gains for 3D semantic segmentation, and we report competitive results on three scene segmentation benchmarks. Our models and code are publicly available.

• The paper demonstrates that integrating geodesic and Euclidean convolutions enhances 3D semantic segmentation performance.
• It employs mesh-preserving pooling with Vertex Clustering and Quadric Error Metrics to maintain geometric detail across resolutions.
• Experimental results on ScanNet v2, S3DIS, and Matterport3D show state-of-the-art mIoU improvements, validating the hybrid approach.

An Analysis of DualConvMesh-Nets: Joint Geodesic and Euclidean Convolutions on 3D Meshes

In the paper titled "DualConvMesh-Nets: Joint Geodesic and Euclidean Convolutions on 3D Meshes," the authors explore a novel architecture for processing 3D geometric data efficiently, focusing particularly on 3D semantic segmentation tasks. They introduce DualConvMesh-Nets (DCM-Nets), which leverage both geodesic and Euclidean convolutions to enhance the semantic segmentation of 3D meshes. This innovative hybrid approach aims to address the limitations of using solely geodesic or Euclidean convolutions by capturing the complementary benefits of both.

Dual Convolution Mechanism

The paper outlines the integration of two distinct types of convolutions:

1. Geodesic Convolutions: These are specifically designed for mesh surfaces or graphs, utilizing kernel weights that follow the local geometry of the mesh. They are adept at capturing features pertinent to object surfaces that may be disconnected in Euclidean space but are logically connected over the mesh surface.
2. Euclidean Convolutions: Independent of any specific mesh structure, these convolutions rely on local affinity representations based on Euclidean distances between points in 3D space. They are effective in capturing interactions between spatially close objects, even if these objects are geodesically separated.

By combining these convolutions, the authors argue that DCM-Nets can leverage spatial closeness to propagate relevant contextual information while simultaneously utilizing the detailed geometric information provided by the mesh structure.

Architecture and Methodology

The authors propose a deep hierarchical architecture that is capable of processing multiple resolutions of 3D mesh data. Key elements of the proposed methodology include:

• Mesh Simplification for Multi-Resolution Processing: The paper adapts two mesh simplification techniques, Vertex Clustering (VC) and Quadric Error Metrics (QEM), to perform mesh-preserving pooling and unpooling operations. This is critical for maintaining meaningful mesh structures across different levels of abstraction.
• Pooling Trace Maps: This novel approach tracks vertex connectivity, enabling effective pooling and unpooling operations across different mesh levels.
• Random Edge Sampling (RES): Introduced for efficiently sampling graph neighborhoods, RES aims to mitigate the computational overhead in densely populated regions by probabilistically subsampling edges based on neighborhood size.

Experimental Results

The experimental section of the paper provides evidence for the efficacy of DCM-Nets through competitive results on several 3D semantic segmentation benchmarks, namely ScanNet v2, S3DIS, and Matterport3D. The results demonstrate state-of-the-art performance for graph convolutional methods, with notable improvements in mIoU scores, establishing DCM-Nets as a robust technique for 3D scene understanding.

Implications and Future Directions

The authors discuss implications for both theoretical advancement and practical application. By effectively combining geodesic and Euclidean convolutions, DCM-Nets offer a scalable solution that can be extended to various tasks within 3D geometric analysis.

Future work suggested includes adapting this approach for instance segmentation and exploring point convolution methods further. These expansions could leverage DCM-Nets' ability to maintain geometric consistency while integrating spatial proximity information.

Conclusion

The introduction of DualConvMesh-Nets represents a significant advance in 3D mesh processing, combining the strengths of geodesic and Euclidean convolutions to improve semantic segmentation outcomes. The proposed architecture and techniques provide a comprehensive framework capable of addressing complexities inherent in 3D geometric data, signaling potential applications across broader areas in computational geometry and computer vision.