  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 ($\mathbb{S}^2$) or a unit ball
($\mathbb{B}^3$)---entails unique challenges. In this work, we propose a novel
`\emph{volumetric convolution}' operation that can effectively convolve
arbitrary functions in $\mathbb{B}^3$. We develop a theoretical framework for
\emph{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 $\mathbb{B}^3$ 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.

Machine Learning;Computer Vision and Pattern Recognition;Machine Learning