Ronny Bergmann

A Second Order Nonsmooth Variational Model for Restoring Manifold-Valued Images

We introduce a new nonsmooth variational model for the restoration of manifold-valued data which includes second order differences in the regularization term. While such models were successfully applied for real-valued 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 machin...

A Second-Order TV-Type Approach for Inpainting and Denoising Higher Dimensional Combined Cyclic and Vector Space Data

In this paper, we consider denoising and inpainting problems for higher dimensional combined cyclic and linear space-valued data. These kinds 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-type problems for denoising, inpainting as well as the combination of both. We provide a convergence analysis and apply the algorithms to concrete problems.

The fast Fourier Transform and fast Wavelet Transform for Patterns on the Torus

Abstract We introduce a fast Fourier transform on regular d-dimensional lattices. We investigate properties of congruence class representants, i.e. their ordering, to classify directions and derive a Cooley–Tukey algorithm. Despite the fast Fourier techniques itself, there is also the advantage of this transform to be parallelized efficiently, yielding faster versions than the one-dimensional Fourier transform. These properties of the lattice can further be used to perform a fast multivariate wavelet decomposition, where the wavelets are given as trigonometric polynomials. Furthermore the preferred directions of the decomposition itself can be characterized.

Second Order Differences of Cyclic Data and Applications in Variational Denoising

In many image and signal processing applications, such as interferometric synthetic aperture radar (SAR), electroencephalogram (EEG) data analysis, ground-based astronomy, and color image restoration, in HSV or LCh spaces the data has its range on the one-dimensional sphere

Restoration of manifold-valued images by half-quadratic minimization

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Inpainting of Cyclic Data Using First and Second Order Differences

. Although the minimization of total variation (TV) regularized functionals is among the most popular methods for edge-preserving image restoration , such methods were only very recently applied to cyclic structures. However, as for Euclidean data, TV regularized variational methods suffer from the so-called 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

A Parallel Douglas–Rachford Algorithm for Minimizing ROF-like Functionals on Images with Values in Symmetric Hadamard Manifolds

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Multivariate Periodic Wavelets of de la Vallée Poussin Type

-valued data in a sound way which is independent of the chosen representation system on the circle. Our abso...

Multivariate Anisotropic Interpolation on the Torus

The paper addresses the generalization of the half-quadratic minimization method for the restoration of images having values in a complete, connected Riemannian manifold. We recall the half-quadratic minimization method using the notation of the

A Graph Framework for Manifold-Valued Data

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