  Unpaired image-to-image translation has been applied successfully to natural
images but has received very little attention for manifold-valued data such as
in diffusion tensor imaging (DTI). The non-Euclidean nature of DTI prevents
current generative adversarial networks (GANs) from generating plausible images
and has mainly limited their application to diffusion MRI scalar maps, such as
fractional anisotropy (FA) or mean diffusivity (MD). Even if these scalar maps
are clinically useful, they mostly ignore fiber orientations and therefore have
limited applications for analyzing brain fibers. Here, we propose a
manifold-aware CycleGAN that learns the generation of high-resolution DTI from
unpaired T1w images. We formulate the objective as a Wasserstein distance
minimization problem of data distributions on a Riemannian manifold of
symmetric positive definite 3x3 matrices SPD(3), using adversarial and
cycle-consistency losses. To ensure that the generated diffusion tensors lie on
the SPD(3) manifold, we exploit the theoretical properties of the exponential
and logarithm maps of the Log-Euclidean metric. We demonstrate that, unlike
standard GANs, our method is able to generate realistic high-resolution DTI
that can be used to compute diffusion-based metrics and potentially run fiber
tractography algorithms. To evaluate our model's performance, we compute the
cosine similarity between the generated tensors principal orientation and their
ground-truth orientation, the mean squared error (MSE) of their derived FA
values and the Log-Euclidean distance between the tensors. We demonstrate that
our method produces 2.5 times better FA MSE than a standard CycleGAN and up to
30% better cosine similarity than a manifold-aware Wasserstein GAN while
synthesizing sharp high-resolution DTI.
