Total Variation and Mean Curvature PDEs on the Space of Positions and Orientations

Abstract

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 $\mathbb {M}=\mathbb {R}^{d}\rtimes S^{d\!-\!1}$ 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 $\mathbb {M}$. 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.