## Processing Manifold-valued Data on Finite Weighted Graphs

This project aims to investigate methods to perform image processing and data analysis on manifold-valued 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.

## Homogenization in Translation invariant periodic spaces

This project aims to investigate the discretization of elliptic partial differential equations with periodic boundary conditions on anisotropic spaces of translates. Of special interest are the influence of the anisotropic sampling lattice, the choice of the translation invariant space, convergence results and the investigation of periodic wavelets.

## 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.

## Manifold-valued Image Restoration Toolbox

In many application data is nonlinear, i.e. restricted in a certain range and equipped with a different distance measure. For example measuring angles in InSAR imaging or when working on the phase of complex valued wavelets. Other applications include denoising in several color spaces like RGB, HSV and CB. These data live on the circle (\mathbb S^1), the sphere (\mathbb S^2) and vector spaces of combined real valued and phase valued data ((\mathbb S^1)^m\times\mathbb R^n). Furthermore, in Diffusion Tensor Imaging (DTI) every data item of an image is given by an (r\times r) symmetric positive definite matrix, i.e. from the space Sym(r). Often, all these data are obstructed by noise due to measurement or data transfer.

## Multivariate Anisotropic Translation Invariant Spaces on the Torus

In my PhD thesis (PDF, 4.5 MB, in german) I investigated translation invariant spaces on anisotropic lattices, a corresponding fast Fourier transform as well as periodic wavelets and a fast wavelet transform. The corresponding wavelets are anisotropic and they are obtained by smapling smooth functions in the Fourier domain and hence have certain lkocalization properties in time. Especially a generalization of the de la Vallee Poussin mean to the anisotropic mutlivariate case is derived and their corresponding wavelets are derived.