In medical applications, the diffusivity of water in tissues that exhibit fibrous microstructures, such as muscle fibres or axons in cerebral white matter, contains valuable information about the fiber architecture in the living organism. Diffusion-weighted (DW) magnetic resonance imaging (MRI) is well-established as a way of measuring the main diffusion directions. A widely used reconstruction scheme for DW-MRI data is Q-ball imaging where the quantity of interest is the marginal probability of diffusion in a given direction, the orientation distribution function (ODF).
In joint work with J. Lellmann (Vogt, Lellmann: An Optimal Transport-Based Restoration Method for Q-Ball Imaging, accepted for SSVM 2017), we propose a variational approach for edge-preserving total variation (TV)-based regularization of Q-ball data. While total variation is among the most popular regularizers for variational problems, its application to ODFs is not straightforward. We propose to write the difference quotients in the TV seminorm in terms of the Wasserstein statistical distance from optimal transport and 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.
Current work is focussing on how to apply this to the reconstruction of Q-ball data from so called high angular resolution diffusion imaging (HARDI). Furthermore, the mathematically precise formulation of the function spaces for the continuous model as well as the existence of minimizers of the variational formulation are interesting theoretical questions for future work.