CubeNet: Equivariance to 3D Rotation and Translation (1804.04458v1)

Abstract: 3D Convolutional Neural Networks are sensitive to transformations applied to their input. This is a problem because a voxelized version of a 3D object, and its rotated clone, will look unrelated to each other after passing through to the last layer of a network. Instead, an idealized model would preserve a meaningful representation of the voxelized object, while explaining the pose-difference between the two inputs. An equivariant representation vector has two components: the invariant identity part, and a discernable encoding of the transformation. Models that can't explain pose-differences risk "diluting" the representation, in pursuit of optimizing a classification or regression loss function. We introduce a Group Convolutional Neural Network with linear equivariance to translations and right angle rotations in three dimensions. We call this network CubeNet, reflecting its cube-like symmetry. By construction, this network helps preserve a 3D shape's global and local signature, as it is transformed through successive layers. We apply this network to a variety of 3D inference problems, achieving state-of-the-art on the ModelNet10 classification challenge, and comparable performance on the ISBI 2012 Connectome Segmentation Benchmark. To the best of our knowledge, this is the first 3D rotation equivariant CNN for voxel representations.

• The paper presents a novel group CNN architecture achieving linear equivariance to translations and right-angle rotations in 3D voxel data.
• It utilizes cube symmetries and group convolutions to preserve both global and local shape signatures while reducing the need for extensive data augmentation.
• Empirical results on ModelNet10 and ISBI datasets demonstrate state-of-the-art performance and improved generalization with less training data.

CubeNet: Equivariance to 3D Rotation and Translation

In the paper "CubeNet: Equivariance to 3D Rotation and Translation," Worrall and Brostow delve into the issue of sensitivity in 3D convolutional neural networks (CNNs) to transformations such as translations and rotations. The authors propose CubeNet, a novel group CNN architecture that achieves linear equivariance to megalithic 3D transformations—including translations and right-angle rotations—by exploiting the symmetry of a cube. This architecture preserves the global and local signatures of 3D shapes while accounting for changes in pose, thereby overcoming significant challenges that standard CNNs face when dealing with voxelized 3D data.

The authors situate CubeNet within the broader trend of integrating geometric transformations into neural network architectures to enhance interpretability and performance on tasks impacted by such transformations. Previous approaches have typically involved augmenting input data with various transformations or elaborating on rotation-equivariant CNNs primarily in 2D contexts. However, CubeNet is distinguished by its implementation in a 3D context, applied specifically to voxel representations, marking it as pioneering in this domain.

CubeNet's methodological approach hinges on group convolution, which extends traditional convolution to include not only shifts but also rotations as group transformations. Specifically, CubeNet employs linear equivariance rather than invariance, maintaining transformation information throughout the network. This is realized through the use of the cube group, depicted by symmetries of a cube, which consists of 24 right-angle rotations. However, for computational tractability, the authors often focus on Klein's four-group and the tetrahedral group, which offer a reduced set of rotations.

The realization of CubeNet involves implementing a group-convolutional operation that becomes permuted when an input experiences transformation. This permutation is guided by a Cayley table that elucidates the relationships among transformations within a group. Consequently, CubeNet manages to balance the computational load with the architectural complexity required to model and predict 3D transformations effectively.

The paper highlights CubeNet's performance through empirical evaluation on benchmarks such as ModelNet10 and the ISBI 2012 Connectome Segmentation, achieving state-of-the-art results in the former. The ModelNet10 experiments demonstrate CubeNet's enhanced accuracy compared to traditional volumetric models, underscoring its reduced reliance on comprehensive data augmentation and rotation averaging in both training and testing phases. For the ISBI challenge, CubeNet shows competitive performance without extensive model search or post-processing, reinforcing its practicality for 3D visual tasks.

A key takeaway is CubeNet's potential for improved generalization with less training data—a notable advantage in real-world applications where data augmentation is often extensive or computationally expensive. The authors infer that by embedding equivariance directly into the network architecture, CubeNet makes a substantive leap toward more efficient and interpretable models for 3D data. It opens avenues for future exploration of continuous rotations, other transformations such as scaling, and applications in different geometric contexts.

CubeNet represents a significant development for tasks requiring 3D transformation equivariance, challenging existing paradigms in deep learning with its elegant, mathematically grounded approach to modeling 3D voxel data. As the field evolves, further exploration and application of such architectures could meaningfully impact computational efficiency and efficacy in broader machine learning and artificial intelligence communities.