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Total variation regularization and total variation flows (TVF) have beenwidely applied for image enhancement and denoising. To include a genericpreservation of crossing curvilinear structures in TVF we lift images to thehomogeneous space $M = \mathbb{R}^d \rtimes S^{d-1}$ of positions andorientations as a Lie group quotient in SE(d). For d = 2 this is called 'totalroto-translation variation' by Chambolle & Pock. We extend this to d = 3, by aPDE-approach with a limiting procedure for which we prove convergence. We alsoinclude a Mean Curvature Flow (MCF) in our PDE model on M. This was firstproposed 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 ininvertible orientation scores (OS). We apply our TVF and MCF in thedenoising/enhancement of crossing fiber bundles in DW-MRI. In comparison todata-driven diffusions, we see a better preservation of bundle boundaries andangular sharpness in fiber orientation densities at crossings. We support thisby error comparisons on a noisy DW-MRI phantom. We also apply our TVF and MCFin enhancement of crossing elongated structures in 2D images via OS, andcompare the results to nonlinear diffusions (CED-OS) via OS.

Total Variation and Mean Curvature PDEs on $\mathbb{R}^d \rtimes S^{d-1}$