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 nonsmooth, 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 primaldual optimization methods are the stateoftheart 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 nonsmooth optimization which were recently extensively exploited in realvalued image processing with manifoldvalued settings. This leaves high potential for future research.
In our project we want to generalize convex models for the restoration of realvalued images to cyclic and manifoldvalued 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 manifoldvalued 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

Bergmann, R., Chan, R. H., Hielscher, R., Persch, J., & Steidl, G. (2016). Restoration of manifoldvalued images by halfquadratic minimization. Inverse Problems and Imaging, 10(2), 281–304.
The paper addresses the generalization of the halfquadratic minimization method for the restoration of images having values in a complete, connected Riemannian manifold. We recall the halfquadratic minimization method using the notation of the ctransform and adapt the algorithm to our special variational setting. We prove the convergence of the method for Hadamard spaces. Extensive numerical examples for images with values on spheres, in the rotation group \operatornameSO(3), and in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm. In particular, the method with \operatornameSO(3)valued data shows promising results for the restoration of images obtained from Electron Backscattered Diffraction which are of interest in material science.@article{BCHPS16, author = {Bergmann, R. and Chan, R. H. and Hielscher, R. and Persch, J. and Steidl, G.}, title = {Restoration of manifoldvalued images by halfquadratic minimization}, journal = {Inverse Problems and Imaging}, year = {2016}, volume = {10}, number = {2}, pages = {281304}, doi = {10.3934/ipi.2016001}, eprint = {1505.07029}, eprinttype = {arXiv} }

Bergmann, R., & Weinmann, A. (2016). A second order TVtype approach for inpainting and denoising higher dimensional combined cyclic and vector space data. Journal of Mathematical Imaging and Vision, 55(3), 401–427.
In this paper we consider denoising and inpainting problems for higher dimensional combined cyclic and linear space valued data. These kind of data appear when dealing with nonlinear color spaces such as HSV, and they can be obtained by changing the space domain of, e.g., an optical flow field to polar coordinates. For such nonlinear data spaces, we develop algorithms for the solution of the corresponding second order total variation (TV) type problems for denoising, inpainting as well as the combination of both. We provide a convergence analysis and we apply the algorithms to concrete problems.@article{BW16, author = {Bergmann, R. and Weinmann, A.}, title = {A second order {TV}type approach for inpainting and denoising higher dimensional combined cyclic and vector space data}, journal = {Journal of Mathematical Imaging and Vision}, year = {2016}, volume = {55}, number = {3}, pages = {401427}, doi = {10.1007/s1085101506273}, eprint = {1501.02684}, eprinttype = {arXiv} }

Bačák, M., Bergmann, R., Steidl, G., & Weinmann, A. (2016). A second order nonsmooth variational model for restoring manifoldvalued images. SIAM Journal on Scientific Computing, 38(1), A567–A597.
We introduce a new nonsmooth variational model for the restoration of manifoldvalued data which includes second order differences in the regularization term. While such models were successfully applied for realvalued images, we introduce the second order difference and the corresponding variational models for manifold data, which up to now only existed for cyclic data. The approach requires a combination of techniques from numerical analysis, convex optimization and differential geometry. First, we establish a suitable definition of absolute second order differences for signals and images with values in a manifold. Employing this definition, we introduce a variational denoising model based on first and second order differences in the manifold setup. In order to minimize the corresponding functional, we develop an algorithm using an inexact cyclic proximal point algorithm. We propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machinery of Jacobi fields. For the nsphere and the manifold of symmetric positive definite matrices, we demonstrate the performance of our algorithm in practice. We prove the convergence of the proposed exact and inexact variant of the cyclic proximal point algorithm in Hadamard spaces. These results which are of interest on its own include, e.g., the manifold of symmetric positive definite matrices.@article{BBSW16, author = {Ba{\v c}{\'a}k, M. and Bergmann, R. and Steidl, G. and Weinmann, A.}, title = {A second order nonsmooth variational model for restoring manifoldvalued images}, journal = {SIAM Journal on Scientific Computing}, year = {2016}, volume = {38}, number = {1}, pages = {A567A597}, doi = {10.1137/15M101988X}, eprint = {1506.02409}, eprinttype = {arXiv} }

Bergmann, R., Persch, J., & Steidl, G. (2016). A parallel Douglas–Rachford algorithm for minimizing ROFlike functionals on images with values in symmetric Hadamard manifolds. SIAM Journal on Imaging Sciences, 9(3), 901–937.
We are interested in restoring images having values in a symmetric Hadamard manifold by minimizing a functional with a quadratic data term and a total variation like regularizing term. To solve the convex minimization problem, we extend the DouglasRachford algorithm and its parallel version to symmetric Hadamard manifolds. The core of the DouglasRachford algorithm are reflections of the functions involved in the functional to be minimized. In the Euclidean setting the reflections of convex lower semicontinuous functions are nonexpansive. As a consequence, convergence results for KrasnoselskiMann iterations imply the convergence of the DouglasRachford algorithm. Unfortunately, this general results does not carry over to Hadamard manifolds, where proper convex lower semicontinuous functions can have expansive reflections. However, splitting our restoration functional in an appropriate way, we have only to deal with special functions namely, several distancelike functions and an indicator functions of a special convex sets. We prove that the reflections of certain distancelike functions on Hadamard manifolds are nonexpansive which is an interesting result on its own. Furthermore, the reflection of the involved indicator function is nonexpansive on Hadamard manifolds with constant curvature so that the DouglasRachford algorithm converges here. Several numerical examples demonstrate the advantageous performance of the suggested algorithm compared to other existing methods as the cyclic proximal point algorithm or halfquadratic minimization. Numerical convergence is also observed in our experiments on the Hadamard manifold of symmetric positive definite matrices with the affine invariant metric which does not have a constant curvature.@article{BPS16, author = {Bergmann, R. and Persch, J. and Steidl, G.}, title = {A parallel {D}ouglas–{R}achford algorithm for minimizing {ROF}like functionals on images with values in symmetric {H}adamard manifolds}, journal = {SIAM Journal on Imaging Sciences}, year = {2016}, volume = {9}, number = {3}, pages = {901937}, doi = {10.1137/15M1052858}, eprint = {1512.02814}, eprinttype = {arXiv} }

Bergmann, R., Laus, F., Steidl, G., & Weinmann, A. (2014). Second order differences of cyclic data and applications in variational denoising. SIAM Journal on Imaging Sciences, 7(4), 2916–2953.
In many image and signal processing applications, such as interferometric synthetic aperture radar (SAR), electroencephalogram (EEG) data analysis, groundbased astronomy, and color image restoration, in HSV or LCh spaces the data has its range on the onedimensional sphere \mathbb S^1. Although the minimization of total variation (TV) regularized functionals is among the most popular methods for edgepreserving image restoration , such methods were only very recently applied to cyclic structures. However, as for Euclidean data, TV regularized variational methods suffer from the socalled staircasing effect. This effect can be avoided by involving higher order derivatives into the functional. This is the first paper which uses higher order differences of cyclic data in regularization terms of energy functionals for image restoration. We introduce absolute higher order differences for \mathbb S^1valued data in a sound way which is independent of the chosen representation system on the circle. Our absolute cyclic first order difference is just the geodesic distance between points. Similar to the geodesic distances, the absolute cyclic second order differences have only values in [0,\pi]. We update the cyclic variational TV approach by our new cyclic second order differences. To minimize the corresponding functional we apply a cyclic proximal point method which was recently successfully proposed for Hadamard manifolds. Choosing appropriate cycles this algorithm can be implemented in an efficient way. The main steps require the evaluation of proximal mappings of our cyclic differences for which we provide analytical expressions. Under certain conditions we prove the convergence of our algorithm. Various numerical examples with artificial as well as realworld data demonstrate the advantageous performance of our algorithm.@article{BLSW14, author = {Bergmann, R. and Laus, F. and Steidl, G. and Weinmann, A.}, title = {Second order differences of cyclic data and applications in variational denoising}, journal = {SIAM Journal on Imaging Sciences}, year = {2014}, volume = {7}, number = {4}, pages = {2916–2953}, doi = {10.1137/140969993}, eprint = {1405.5349}, eprinttype = {arXiv} }

Bergmann, R., & Weinmann, A. (2015). Inpainting of cyclic data using first and second order differences. In X.C. Tai, E. Bae, T. F. Chan, S. Y. Leung, & M. Lysaker (Eds.), Energy Minimization Methods in Computer Vision and Pattern Recognition, 10th International Conference on, EMMCVPR 2015, Hong Kong (pp. 155–168). Springer.
Cyclic data arise in various image and signal processing applications such as interferometric synthetic aperture radar, electroencephalogram data analysis, and color image restoration in HSV or LCh spaces. In this paper we introduce a variational inpainting model for cyclic data which utilizes our definition of absolute cyclic second order differences. Based on analytical expressions for the proximal mappings of these differences we propose a cyclic proximal point algorithm (CPPA) for minimizing the corresponding functional. We choose appropriate cycles to implement this algorithm in an efficient way. We further introduce a simple strategy to initialize the unknown inpainting region. Numerical results both for synthetic and realworld data demonstrate the performance of our algorithm.@inproceedings{BW15, author = {Bergmann, R. and Weinmann, A.}, title = {Inpainting of cyclic data using first and second order differences}, year = {2015}, pages = {155–168}, editor = {Tai, X.C. and Bae, E. and Chan, T. F. and Leung, S. Y. and Lysaker, M.}, publisher = {Springer}, booktitle = {Energy Minimization Methods in Computer Vision and Pattern Recognition, 10th International Conference on, EMMCVPR 2015, Hong Kong}, doi = {10.1007/9783319146126_12}, eprint = {1410.1998}, eprinttype = {arXiv} }

Oezguen, N., Schubert, K. J., Bergmann, R., Bennewitz, R., & Strauss, D. J. (2015). Relating tribological stimuli to somatosensory electroencephalographic responses. In Engineering in Medicine and Biology Society (EMBC), 37th Annual International Conference of the IEEE (pp. 8115–8118).
The present study deals with the extraction of neural correlates evoked by tactile stimulation of the human fingertip. A reciprocal sliding procedure was performed using a homebuilt tribometer while simultaneously electroencephalographic (EEG) data from the somatosensory cortex was recorded. The tactile stimuli were delivered by a sliding block with equidistant, perpendicular ridges. The experiments were designed and performed in a fully passive way to prevent attentional locked influences from the subjects. In order to improve the signaltonoise ratio (SNR) of event related singletrials (ERPs), nonlocal means in addition to 2Danisotropic denoising schemes based on tight Gabor frames were applied. This novel approach allowed for an easier extraction of ERP alternations. A negative correlation between the latency of the P100 component of the resulting brain responses and the intensity of the underlying lateral forces was found. These findings lead to the conclusion that an increasing stimulus intensity results in a decreasing latency of the brain responses.@inproceedings{OSBBS15, author = {Oezguen, Novaf and Schubert, Kristof J. and Bergmann, Ronny and Bennewitz, Roland and Strauss, Daniel J.}, booktitle = {Engineering in Medicine and Biology Society (EMBC), 37th Annual International Conference of the IEEE}, title = {Relating tribological stimuli to somatosensory electroencephalographic responses}, year = {2015}, pages = {81158118}, doi = {10.1109/EMBC.2015.7320277}, month = aug }