  Total variation regularization and total variation flows (TVF) have been
widely applied for image enhancement and denoising. To include a generic
preservation of crossing curvilinear structures in TVF we lift images to the
homogeneous space $M = \mathbb{R}^d \rtimes S^{d-1}$ of positions and
orientations as a Lie group quotient in SE(d). For d = 2 this is called 'total
roto-translation variation' by Chambolle & Pock. We extend this to d = 3, by a
PDE-approach with a limiting procedure for which we prove convergence. We also
include a Mean Curvature Flow (MCF) in our PDE model on M. This was first
proposed for d = 2 by Citti et al. and we extend this to d = 3. Furthermore,
for d = 2 we take advantage of locally optimal differential frames in
invertible orientation scores (OS). We apply our TVF and MCF in the
denoising/enhancement of crossing fiber bundles in DW-MRI. In comparison to
data-driven diffusions, we see a better preservation of bundle boundaries and
angular sharpness in fiber orientation densities at crossings. We support this
by error comparisons on a noisy DW-MRI phantom. We also apply our TVF and MCF
in enhancement of crossing elongated structures in 2D images via OS, and
compare the results to nonlinear diffusions (CED-OS) via OS.
