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

Remco Duits, Etienne St-Onge, Jim Portegies, Bart Smets

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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 tothe homogeneous space $M = \mathbb{R}^d \rtimes S^{d-1}$ of positionsand orientations as a Lie group quotient in SE(d). For d = 2 this iscalled 'total roto-translation variation' by Chambolle & Pock. Weextend this to d = 3, by a PDE-approach with a limiting procedure forwhich 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 oflocally optimal differential frames in invertible orientation scores(OS). We apply our TVF and MCF in the denoising/enhancement of crossingfiber bundles in DW-MRI. In comparison to data-driven diffusions, we seea better preservation of bundle boundaries and angular sharpness infiber orientation densities at crossings. We support this by errorcomparisons on a noisy DW-MRI phantom. We also apply our TVF and MCF inenhancement of crossing elongated structures in 2D images via OS, andcompare the results to nonlinear diffusions (CED-OS) via OS.
Originele taal-2Engels
Artikelnummer1902.08145
Aantal pagina's17
TijdschriftarXiv
Volume2019
StatusGepubliceerd - 1 feb 2019

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