This project aims to investigate methods to perform image processing and data analysis on manifoldvalued data defined on a graph. This reaches from defining meaningful models and differential operators for data on Riemannian manifolds, to deriving numerical optimization schemes to solve the related problems, to the investigation of theoretical properties as convergence rates and upper bounds, to finally applying derived algorithms to real world applications, e.g., in medical imaging.
References

Bergmann, R., & Tenbrinck, D. (2017). A graph framework for manifoldvalued data. SIAM Journal on Imaging Sciences.
To appear.
Graphbased methods have been proposed as a unified framework for discrete calculus of local and nonlocal image processing methods in the recent years. In order to translate variational models and partial differential equations to a graph, certain operators have been investigated and successfully applied to realworld applications involving graph models. So far the graph framework has been limited to real and vectorvalued functions on Euclidean domains. In this paper we generalize this model to the case of manifoldvalued data. We introduce the basic calculus needed to formulate variational models and partial differential equations for manifoldvalued functions and discuss the proposed graph framework for two particular families of operators, namely, the isotropic and anisotropic graph pLaplacian operators, p≥1. Based on the choice of p we are in particular able to solve optimization problems on manifoldvalued functions involving total variation (p=1) and Tikhonov (p=2) regularization. Finally, we present numerical results from processing both synthetic as well as realworld manifoldvalued data, e.g., from diffusion tensor imaging (DTI) and light detection and ranging (LiDAR) data.@article{BT17, author = {Bergmann, Ronny and Tenbrinck, Daniel}, title = {A graph framework for manifoldvalued data}, year = {2017}, eprint = {1702.05293}, eprinttype = {arXiv}, journal = {SIAM Journal on Imaging Sciences}, note = {To appear.} }

Bergmann, R., & Tenbrinck, D. (2017). Nonlocal inpainting of manifoldvalued data on finite weighted graphs. In F. Nielsen & F. Barbaresco (Eds.), Geometric Science of Information – 3rd Conference on Geometric Science of Information (pp. 604–612). Cham: Springer International Publishing.
Recently, there has been a strong ambition to translate models and algorithms from traditional image processing to nonEuclidean domains, e.g., to manifoldvalued data. While the task of denoising has been extensively studied in the last years, there was rarely an attempt to perform image inpainting on manifoldvalued data. In this paper we present a nonlocal inpainting method for manifoldvalued data given on a finite weighted graph. We introduce a new graph infinityLaplace operator based on the idea of discrete minimizing Lipschitz extensions, which we use to formulate the inpainting problem as PDE on the graph. Furthermore, we derive an explicit numerical solving scheme, which we evaluate on two classes of synthetic manifoldvalued images.@inproceedings{BT17a, author = {Bergmann, Ronny and Tenbrinck, Daniel}, title = {Nonlocal inpainting of manifoldvalued data on finite weighted graphs}, year = {2017}, booktitle = {Geometric Science of Information  3rd Conference on Geometric Science of Information}, editor = {Nielsen, Frank and Barbaresco, Fr{\'e}d{\'e}ric}, eprint = {1704.06424}, eprinttype = {arXiv}, doi = {10.1007/9783319684451}, address = {Cham}, publisher = {Springer International Publishing}, pages = {604612}, isbn = {9783319684451} }