A Higher Order Manifold-Valued Convolutional Neural Network with Applications to Diffusion MRI Processing

In this paper, we present a novel generalization of the Volterra Series, which can be viewed as a higher-order convolution, to manifold-valued functions. A special case of the manifold-valued Volterra Series (MVVS) gives us a natural extension of the ordinary convolution to manifold-valued functions that we call, the manifold-valued convolution (MVC). We prove that these generalizations preserve the equivariance properties of the Euclidean Volterra Series and the traditional convolution operator. We present novel deep network architectures using the MVVS and the MVC operations which are then validated via two experiments. These include, (i) movement disorder classification from diffusion magnetic resonance images (dMRI), and (ii) fiber orientation distribution function (fODF) reconstruction from compressed sensed dMRIs. In both the experiments, MVVS and MVC networks outperform the state-of-the-art.