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

# Projects

## 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.

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## 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.

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## 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.

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## Manifold-valued Image Restoration Toolbox

In many application measured data appears nonlinear, i.e. restricted in a certain range or equipped with a different distance measure.

This toolbox provides an easy access to image processing tasks for such data, where the values of each measurement are points on a manifold. You can get started by downloading the source code from Github or by cloning the git-repository using

1
git clone git@github.com:kellertuer/MVIRT.git


Examples are InSAR imaging or when working with phase data, i.e. on the circle $\mathbb S^1$, or Diffusion Tensor Imaging (DTI), where every data items are an $n\times n$ symmetric positive definite matrices, $\SPD{n}$..

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## 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 sampling smooth functions in the Fourier domain and hence have certain localization properties in time. Especially a generalization of the de la Vallee Poussin mean to the anisotropic multivariate case is derived and their corresponding wavelets are derived.

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