Learning shape correspondence with anisotropic convolutional neural networks (1605.06437v1)

Abstract: Establishing correspondence between shapes is a fundamental problem in geometry processing, arising in a wide variety of applications. The problem is especially difficult in the setting of non-isometric deformations, as well as in the presence of topological noise and missing parts, mainly due to the limited capability to model such deformations axiomatically. Several recent works showed that invariance to complex shape transformations can be learned from examples. In this paper, we introduce an intrinsic convolutional neural network architecture based on anisotropic diffusion kernels, which we term Anisotropic Convolutional Neural Network (ACNN). In our construction, we generalize convolutions to non-Euclidean domains by constructing a set of oriented anisotropic diffusion kernels, creating in this way a local intrinsic polar representation of the data (`patch'), which is then correlated with a filter. Several cascades of such filters, linear, and non-linear operators are stacked to form a deep neural network whose parameters are learned by minimizing a task-specific cost. We use ACNNs to effectively learn intrinsic dense correspondences between deformable shapes in very challenging settings, achieving state-of-the-art results on some of the most difficult recent correspondence benchmarks.

• The paper introduces an intrinsic CNN architecture that generalizes convolutions to manifolds using anisotropic heat kernels for dense shape correspondence.
• It employs anisotropic diffusion kernels to create local, deformation-invariant representations that efficiently handle non-rigid shapes.
• Experimental results on FAUST and SHREC'16 demonstrate that ACNN significantly reduces geodesic errors compared to state-of-the-art methods.

Learning Shape Correspondence with Anisotropic Convolutional Neural Networks

This paper addresses the challenging problem of establishing dense intrinsic correspondence between non-rigid shapes using machine learning techniques. The proposed solution involves the development of Anisotropic Convolutional Neural Networks (ACNNs), which leverage anisotropic diffusion kernels to expand the capabilities of conventional CNNs from Euclidean to non-Euclidean domains.

Key Contributions

The authors introduce an intrinsic CNN architecture that generalizes convolutions to apply on manifolds. By employing anisotropic heat kernels, the approach provides local intrinsic representations, thereby facilitating efficient and effective learning of dense correspondences in deformable shapes. This is particularly valuable in applications like texture mapping and animation, where dealing with non-rigid transformations, topological noise, and missing parts is crucial.

Theoretical and Implementation Insights

• Anisotropic Diffusion Kernels: The use of anisotropic heat kernels allows for location-dependent local weighting functions, leading to spatially defined filters that maintain invariance under deformations.
• Network Architecture: The ACNN architecture includes intrinsic convolutional layers, fully connected layers, and softmax output layers. This setup supports deep learning architectures that can adapt to the geometric complexity of 3D shapes, addressing feature transformations and deformation challenges.
• Comparison with Existing Methods: The ACNN outperforms methods such as Geodesic CNN (GCNN), Random Forests (RF), and Blended Intrinsic Maps (BIM) on various benchmarks, highlighting its capability to produce fewer errors in correspondence assignments.

Experimental Results

The paper demonstrates the effectiveness of ACNN on the FAUST and SHREC'16 datasets, where it achieves superior performance in state-of-the-art benchmarks. Notably, it achieves nearly perfect correspondences, with negligible artifacts on complex datasets characterized by non-isometric deformations and high levels of difficulty.

• Performance Metrics: ACNN shows a significant enhancement in geodesic error metrics compared to baseline methods, particularly excelling in scenarios involving partial shapes with missing parts.

Implications and Future Directions

The introduction of ACNN is a step forward in the field of geometry processing, demonstrating potential for broader application in computer graphics, vision, and beyond. Here are some impending advancements and theoretical implications:

• Generalization to Various Geometric Problems: The intrinsic nature of this CNN architecture has potential applications in other tasks, such as shape retrieval, recognition, or any scenario requiring robust feature correspondence under complex transformations.
• Handling Diverse Data Formats: The ability of ACNN to work with different 3D data representations, including point clouds and meshes, opens avenues for integrating with modern 3D sensing technologies and furthering advancements in augmented reality and virtual modeling.
• Potential Improvements: Future work could focus on optimizing computational efficiency and extending the method to handle more extreme deformations.

In summary, the paper provides a comprehensive framework for learning shape correspondence using anisotropic CNNs, emphasizing its contribution to handling non-Euclidean domains through a principled approach to deep learning on manifolds. The simplicity combined with effectiveness positions ACNN as a powerful tool in the landscape of computational geometry and related fields.