An Optimal Transport-Based Restoration Method for Q-Ball Imaging

We propose a variational approach for edge-preserving total variation (TV)-based regularization of Q-ball data from high angular resolution diffusion imaging (HARDI). While total variation is among the most popular regularizers for variational problems, its application to orientation distribution functions (ODF), as they naturally arise in Q-ball imaging, is not straightforward. We propose to use an extension that specifically takes into account the metric on the underlying orientation space. The key idea is to write the difference quotients in the TV seminorm in terms of the Wasserstein statistical distance from optimal transport. We combine this regularizer with a matching Wasserstein data fidelity term. Using the Kantorovich-Rubinstein duality, the variational model can be formulated as a convex optimization problem that can be solved using a primal-dual algorithm. We demonstrate the effectiveness of the proposed framework on real and synthetic Q-ball data.