Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations
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Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations Bart M. N. Smets1 · Jim Portegies1 · Etienne St-Onge2 · Remco Duits1 Received: 23 October 2019 / Accepted: 31 August 2020 © The Author(s) 2020
Abstract Two key ideas have greatly improved techniques for image enhancement and denoising: the lifting of image data to multiorientation distributions and the application of nonlinear PDEs such as total variation flow (TVF) and mean curvature flow (MCF). These two ideas were recently combined by Chambolle and Pock (for TVF) and Citti et al. (for MCF) for twodimensional images. In this work, we extend their approach to enhance and denoise images of arbitrary dimension, creating a unified geometric and algorithmic PDE framework, relying on (sub-)Riemannian geometry. In particular, we follow a different numerical approach, for which we prove convergence in the case of TVF by an application of Brezis–Komura gradient flow theory. Our framework also allows for additional data adaptation through the use of locally adaptive frames and coherence enhancement techniques. We apply TVF and MCF to the enhancement and denoising of elongated structures in 2D images via orientation scores and compare the results to Perona–Malik diffusion and BM3D. We also demonstrate our techniques in 3D in the denoising and enhancement of crossing fiber bundles in DW-MRI. In comparison with data-driven diffusions, we see a better preservation of bundle boundaries and angular sharpness in fiber orientation densities at crossings. Keywords Total variation · Mean curvature · Sub-Riemannian geometry · Roto-translations · Denoising · Fiber enhancement
1 Introduction In the last decade, many PDE-based image-analysis techniques for tracking and enhancement of curvilinear structures in images took advantage of lifting image data, typically defined on Rd , to a multi-orientation distribution (e.g., an orientation score) defined on the homogeneous space Md of d-dimensional positions and orientations, see Fig. 1 and [5,8,11,14,20,53]. After lifting the image to a multiorientation distribution, the distribution is taken as an initial condition of a PDE flow. After solving a limited number of iterations of the PDE model, one obtains a regularized version of the original distribution, and by integration over all orientations, one obtains a regularized version of the original image.
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Bart M. N. Smets [email protected], [email protected]
1
CASA, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
2
Sherbrooke Connectivity Imaging Lab (SCIL), Sherbrooke, Canada
The key advantage of lifting the images from Rd to the homogeneous space Md is that the PDE flow can act differently on substructures with different orientations [5,11,24]. For instance, if the image contains two crossing lines, the PDE can regularize the two lines independently, rather than regularizing the whole crossing. Similarly, if the image contains a corner, the corner is preserved in the regularized image. This idea
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