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.
There is a great amount of interest in extending the notion of equivariance in neural networks from 
just translations to other groups. In particular, in the past year a sequence of papers have appeared 
starting with (Cohen, Geiger et al.) on "spherical CNNs" that achieve equivariance to rotations for images 
painted on the surface of the unit sphere.
The present paper extends these ideas to volumetric data in the unit ball (rather than just the sphere) 
by the use of Zernike polynomials. Since Zernike polynomials can be expressed as the product of spherical 
harmonics with a radial function, this is essentially the same as adding a radial component to a spherical 
CNN. 
The main result of the paper appears to be Theorem 1, which shows that what the authors define as 
volumetric convolution is equivariant to rotations. This is split across Sections 4.2 and 4.3.. However, 
apart from the radial component, this result is bascally the same as the SO3-equivariance of spherical CNNs, 
as discussed in three very recent spherical CNN papers: (Cohen, Geiger et al.) (Esteves Allen-Blanchette et al) 
and (Kondor, Lin and Trivedi). However, the somewhat more abstract, representation theoretic approach 
of some of these works allows a more compact derivation than the one in the present paper.
The authors also fail to cite recent work on SE(2) and SE(3) equivariant neural networks. SE(3) comprises 
all rotations and translations of R^3, so the latter, in particular, encapuslates SO(3) equivariance as 
a special case. In particular, part of the construction in (Weiler, Hamprecht and Storath, CVPR 2018) 
is to add  Gaussian radial functions to SO(2) equivariant filters, which is just the 2D analog of 
what is happening in the present paper. Then in (Weiler, Geiger, et al., 2018) the same is done in 3D, 
except of course they go further by also adding translation equivariance. Admittedly, these papers are 
very new, so the authors might not have known about them.
I also find some of the mathematical details a little puzzling:
1. As explained in the spherical CNN papers, taking the cross correlation of two functions on the sphere 
(by extension, in the unit ball) naturally results in a function that lives on the rotation group SO(3), i.e. 
the cross-correlation (or convolution) is parametrized by three Euler angles. I don't understand why the 
authors restrict themselves to considering just two angles, forcing their filters to be polar, as 
derived in Section 5. This seems like an artificial restriction that will limit the power of their approach. 
2. The paper mentions that Zernike polynomials are "orthogonal and complete in B^3". I think what they mean 
is that they are an ortogonal and complete basis for an appropriate space of functions on B^3, and that 
space of functions is L_2(B^3). However, this is still not enough. For (3) to hold, one also needs the 
basis to be orthonormal. Please be more precise.
3. In the same vein, at one point in the proof, the authors mention that "rotations are unitary operators 
in a Hilbert space". This is not true of Hilbert spaces in general. It requires the above orthonormality etc.. 
4. Exactly as a consequence of orthonorlamity, Equation 5 is essentially just a generalized Fourier transform 
on B^3, hence, in principle, it can be inverted analytically. I understand that the fact that the input 
image is rasterized complicates things and in a practical implementation it might be expedient to invert (5) 
by using the pseudo-inverse. However, this is a one-time operations and is really just a hack. It seems strange 
that the authors use a special iterative method just to invert a close to unitary matrix.
The mathematical shortcomings of the paper could be compensated by amazingly good experimental results. 
The actual results are good, but still not best-in-class, possibly because at the end of the day the network 
is still only rotationally equivariant and does not take into account translations. 
The spherical CNN and SE(n) equivariance papers generally apply the group equivariant operations consistently 
across multiple layers. In contrast, the present paper only applies it in the first layer, and then uses a 
combination of tricks like multiple viewpoints and bilinear pooling to boost performance. Unfortunately, the 
benefits of this additional conceptual complexity are not entirely borne out in the experimental results.