  The physical and clinical constraints surrounding diffusion-weighted imaging
(DWI) often limit the spatial resolution of the produced images to voxels up to
8 times larger than those of T1w images. Thus, the detailed information
contained in T1w imagescould help in the synthesis of diffusion images in
higher resolution. However, the non-Euclidean nature of diffusion imaging
hinders current deep generative models from synthesizing physically plausible
images. In this work, we propose the first Riemannian network architecture for
the direct generation of diffusion tensors (DT) and diffusion orientation
distribution functions (dODFs) from high-resolution T1w images. Our integration
of the Log-Euclidean Metric into a learning objective guarantees, unlike
standard Euclidean networks, the mathematically-valid synthesis of diffusion.
Furthermore, our approach improves the fractional anisotropy mean squared error
(FA MSE) between the synthesized diffusion and the ground-truth by more than
23% and the cosine similarity between principal directions by almost 5% when
compared to our baselines. We validate our generated diffusion by comparing the
resulting tractograms to our expected real data. We observe similar fiber
bundles with streamlines having less than 3% difference in length, less than 1%
difference in volume, and a visually close shape. While our method is able to
generate high-resolution diffusion images from structural inputs in less than
15 seconds, we acknowledge and discuss the limits of diffusion inference solely
relying on T1w images. Our results nonetheless suggest a relationship between
the high-level geometry of the brain and the overall white matter architecture.
