$\DeclareMathOperator{\arccosh}{arccosh} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator{\Exp}{Exp} \newcommand{\geo}[2]{\gamma_{\overset{\frown}{#1,#2}}} \newcommand{\geoS}{\gamma} \newcommand{\geoD}[2]{\gamma_} \newcommand{\geoL}[2]{\gamma(#2; #1)} \newcommand{\gradM}{\nabla_{\M}} \newcommand{\gradMComp}[1]{\nabla_{\M,#1}} \newcommand{\Grid}{\mathcal G} \DeclareMathOperator{\Log}{Log} \newcommand{\M}{\mathcal M} \newcommand{\N}{\mathcal N} \newcommand{\mat}[1]{\mathbf{#1}} \DeclareMathOperator{\prox}{prox} \newcommand{\PT}[3]{\mathrm{PT}_{#1\to#2}#3} \newcommand{\R}{\mathbb R} \newcommand{\SPD}[1]{\mathcal{P}(#1)} \DeclareMathOperator{\Tr}{Tr} \newcommand{\tT}{\mathrm{T}} \newcommand{\vect}[1]{\mathbf{#1}}$

# Non-Smooth Variational Models with Second Order and Local Anisotropy Priors for Restoring Cyclic and Manifold-Valued Images

Variational methods in imaging are nowadays developing towards a quite universal and flexible tool, allowing for highly successful approaches on various imaging tasks. Many useful techniques rely on non-smooth, convex functionals. Combinations of first and second order derivatives in regularization functionals or the incorporation of anisotropies steered by the local structures of the image have led to very powerful image restoration techniques. Splitting algorithms together with primal-dual optimization methods are the state-of-the-art techniques for minimizing these functionals. Their strength consists in the splitting of the original problem into a sequence of proximal mappings which can be computed efficiently.

In various applications in image processing and computer vision the functions of interest take values on the circle or in manifolds. Although manifolds play an important role in these fields for a long time, there are only few papers which combine results on non-smooth optimization which were recently extensively exploited in real-valued image processing with manifold-valued settings. This leaves high potential for future research.

In our project we want to generalize convex models for the restoration of real-valued images to cyclic and manifold-valued images. We want to focus on symmetric spaces having applications in image processing. For Hadamard spaces the models are still convex which is in general, e.g., for spheres, not the case. A specific feature of our models is that their regularization terms will incorporate first and second order differences or directional anisotropies. The challenges of our project include the appropriate construction of restoration models for manifold-valued signals and images, the analysis of the models, and the development of efficient minimization algorithms, including convergence results. There is a rich potential for applications of the methods which will be developed within our project. Among others we will use our models for the analysis of Electroencephalographical data and of Electron Backscattered Diffraction data. A publicly available software package is planed as well.

# References

1. Bergmann, R and Weinmann, A (2016). A second order TV-type approach for inpainting and denoising higher dimensional combined cyclic and vector space data. Journal of Mathematical Imaging and Vision. 55 401–27
2. Bačák, M, Bergmann, R, Steidl, G and Weinmann, A (2016). A second order non-smooth variational model for restoring manifold-valued images. SIAM Journal on Scientific Computing. 38 A567–A597
3. Bergmann, R, Persch, J and Steidl, G (2016). A parallel Douglas–Rachford algorithm for minimizing ROF-like functionals on images with values in symmetric Hadamard manifolds. SIAM Journal on Imaging Sciences. 9 901–37
4. Bergmann, R, Chan, R H, Hielscher, R, Persch, J and Steidl, G (2016). Restoration of manifold-valued images by half-quadratic minimization. Inverse Problems and Imaging. 10 281–304
5. Bergmann, R, Laus, F, Steidl, G and Weinmann, A (2014). Second order differences of cyclic data and applications in variational denoising. SIAM Journal on Imaging Sciences. 7 2916–53

1. Bergmann, R and Weinmann, A (2015). Inpainting of cyclic data using first and second order differences. Energy Minimization Methods in Computer Vision and Pattern Recognition, 10th International Conference on, EMMCVPR 2015, Hong Kong. Springer. 155–68
2. Oezguen, N, Schubert, K J, Bergmann, R, Bennewitz, R and Strauss, D J (2015). Relating tribological stimuli to somatosensory electroencephalographic responses. Engineering in Medicine and Biology Society (EMBC), 37th Annual International Conference of the IEEE. 8115–8