  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 neurons with spherical
decision surfaces and operates on point clouds. Such spherical neurons are
obtained by conformal embedding of Euclidean space and have recently been
revisited in the context of learning representations of point sets. Focusing on
3D geometry, we exploit the isometry property of spherical neurons and derive a
3D steerability constraint. After training spherical neurons to classify point
clouds in a canonical orientation, we use a tetrahedron basis to quadruplicate
the neurons and construct rotation-equivariant spherical filter banks. We then
apply the derived constraint to interpolate the filter bank outputs and, thus,
obtain a rotation-invariant network. Finally, we use a synthetic point set and
real-world 3D skeleton data to verify our theoretical findings. The code is
available at https://github.com/pavlo-melnyk/steerable-3d-neurons.
