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

TL;DR

This paper extends total variation and mean curvature PDE models to higher dimensions on the space of positions and orientations, improving crossing structure preservation in image and DW-MRI data denoising.

Contribution

It introduces a PDE framework for total variation and mean curvature flows on $\,\mathbb{R}^d \rtimes S^{d-1}$, extending previous 2D models to 3D with convergence proofs.

Findings

Better preservation of crossing fiber bundles in DW-MRI compared to data-driven methods.

Enhanced crossing elongated structures in 2D images with improved boundary and angular sharpness.

Demonstrated effectiveness through error comparisons on noisy DW-MRI phantom.

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