JuliaManifolds since 2019

Manifolds.jl implements a library of manifolds and provides combinators to construct new manifolds from these. Examples are the product manifold of two manifolds, the power manifold, the tangent bundle, and Lie groups. Using a trait-based system, any manifold may be augmented with additional geometric structure, including various metrics, without sacrificing efficiency.

Manopt.jl Optimization on Riemannian manifolds since 2016

This toolbox provides an easy access to image processing tasks for such data, and optimization on manifolds in Julia. In general the toolbox combines ideas from manopt and pymanopt and can be used with all manifolds from Manifolds.jl

Processing Manifold-valued Data on Finite Weighted Graphs
since 2016

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
  • D. Merkert

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
MVIRT 2014—2018
  • J. Persch

The MVIRT Matlab toolbox provides an easy access to image processing tasks for data, where the values of each measurement are points on a manifold. This project is superseeded by Manopt.jl.

Multivariate Anisotropic Translation Invariant Spaces on the Torus PhD project
MPAWL 2009—2013

In my PhD thesis we 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 Vallée Poussin mean to the anisotropic multivariate case is derived and their corresponding wavelets are derived.