Convolution is an efficient technique to obtain abstract feature representations using hierarchical layers in deep networks. Although performing convolution in Euclidean geometries is fairly straightforward, its extension to other topological spaces---such as a sphere S^2 or a unit ball B^3---entails unique challenges. In this work, we propose a novel `"volumetric convolution" operation that can effectively convolve arbitrary functions in B^3. We develop a theoretical framework for "volumetric convolution" based on Zernike polynomials and efficiently implement it as a differentiable and an easily pluggable layer for deep networks. Furthermore, our formulation leads to derivation of a  novel formula to measure the symmetry of a function in B^3 around an arbitrary axis, that is useful in 3D shape analysis tasks. We demonstrate the efficacy of proposed volumetric convolution operation on a possible use-case i.e., 3D object recognition task.
A novel convolution operator for automatic representation learning inside unit ball This work is related to the recent spherical CNN and SE(n) equivariant network papers and extends previous ideas to volumetric data in the unit ball. Proposes using volumetric convolutions on convolutions networks in order to learn unit ball and discusses methodology and results of process.