Total Generalized Variation in Diffusion Tensor Imaging

We study the extension of total variation (TV), total deformation (TD), and (second-order) total generalised variation (TGV) to symmetric tensor fields. We show that for a suitable choice of finite-dimensional norm, these variational semi-norms are rotation-invariant in a sense natural and well-suited for application to diffusion tensor imaging (DTI). Combined with a positive definiteness constraint, we employ these novel semi-norms as regularisers in ROF-type denoising of medical in-vivo brain images. For the numerical realisation, we employ the Chambolle-Pock algorithm, for which we develop a novel duality-based stopping criterion which guarantees error bounds with respect to the functional values. Our findings indicate that TD and TGV, both of which employ the symmetrised differential, provide improved results compared to other evaluated approaches.