Total Generalized Variation in Diffusion Tensor Imaging

We study the extension of total variation (TV), total deformation (TD), and (second-order) total generalized variation ($\TGV^2$) to symmetric tensor fields. We show that for a suitable choice of finite-dimensional norm, these variational seminorms 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 seminorms as regularizers in Rudin--Osher--Fatemi (ROF) type denoising of medical in vivo brain images. For the numerical realization, 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 $\TGV^2$, both of which employ the symmetrized differential, provide improved results compared to other evaluated approaches.

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