Fully Steerable 3D Spherical Neurons

Emerging from low-level vision theory, steerable filters found their counterpart in prior work on steerable convolutional neural 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. Focusing on 3D geometry, we derive a 3D steerability constraint for hypersphere neurons, which are obtained by conformal embedding of Euclidean space and have recently been revisited in the context of learning representations of point sets. Exploiting the rotational equivariance, we show how our model parameters are fully steerable at inference time. We use a synthetic point set and real-world 3D skeleton data to show how the proposed spherical filter banks enable making equivariant and, after online optimization, invariant class predictions for known point sets in unknown orientations.