Fully Steerable 3D Spherical Neurons

Emerging from low-level vision theory, steerable filters found their counterpart in deep learning. Earlier works used the steering theorems and presented convolutional networks equivariant to rigid transformations. In our work, we propose a steerable feed-forward learning-based approach that consists of spherical decision surfaces and operates on point clouds. Due to the inherent geometric 3D structure of our theory, we derive a 3D steerability constraint for its atomic parts, the hypersphere neurons. Exploiting the rotational equivariance, we show how the model parameters are fully steerable at inference time. The proposed spherical filter banks enable to make equivariant and, after online optimization, invariant class predictions for known synthetic point sets in unknown orientations.