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Preprints

Bergmann, R, Herzog, R, Tenbrinck, D and Vidal Núñez, J (2019). FenchelDuality for Convex Optimization and a Primal Dual Algorithm
on Riemannian Manifolds
This paper introduces a new duality theory that generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. This notion of conjugation even yields a more general Fenchel conjugate for the case where the manifold is a vector space. We investigate its properties, e.g., the FenchelYoung inequality and the characterization of the convex subdifferential using the analogue of the FenchelMoreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primaldual optimization algorithm, and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions. Numerical results illustrate the performance of the algorithm, which competes with the recently derived DouglasRachford algorithm on manifolds of nonpositive curvature. Furthermore we show that our novel algorithm numerically converges on manifolds of positive curvature.@online{BergmannHerzogTenbrinckVidalNunez2019, author = {Bergmann, Ronny and Herzog, Roland and Tenbrinck, Daniel and Vidal Núñez, José}, title = {FenchelDuality for Convex Optimization and a Primal Dual Algorithm on Riemannian Manifolds}, year = {2019}, eprint = {1908.02022}, eprinttype = {arXiv} }
Journal Papers
2020

Bergmann, R, Herrmann, M, Herzog, R, Schmidt, S and Vidal Núñez, J (2020). Discrete Total Variation of the Normal Vector Field as Shape Prior with Applications in Geometric Inverse Problems. Inverse Problems
accepted for publication.
An analogue of the total variation prior for the normal vector field along the boundary of piecewise flat shapes in 3D is introduced. A major class of examples are triangulated surfaces as they occur for instance in finite element computations. The analysis of the functional is based on a differential geometric setting in which the unit normal vector is viewed as an element of the twodimensional sphere manifold. It is found to agree with the discrete total mean curvature known in discrete differential geometry. A split Bregman iteration is proposed for the solution of discretized shape optimization problems, in which the total variation of the normal appears as a regularizer. Unlike most other priors, such as surface area, the new functional allows for piecewise flat shapes. As two applications, a mesh denoising and a geometric inverse problem of inclusion detection type involving a partial differential equation are considered. Numerical experiments confirm that polyhedral shapes can be identified quite accurately.@article{BergmannHerrmannHerzogSchmidtVidalNunez20202, author = {Bergmann, Ronny and Herrmann, Marc and Herzog, Roland and Schmidt, Stephan and Vidal Núñez, José}, title = {Discrete Total Variation of the Normal Vector Field as Shape Prior with Applications in Geometric Inverse Problems}, year = {2020}, journal = {Inverse Problems}, note = {accepted for publication.}, doi = {10.1088/13616420/ab6d5c}, eprint = {1908.07916}, eprinttype = {arXiv} }

Bergmann, R, Herrmann, M, Herzog, R, Schmidt, S and Vidal Núñez, J (2020). Total Variation of the Normal Vector Field as Shape Prior. Inverse Problems
accepted for publication.
An analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. The analysis of the total variation of the normal vector field is based on a differential geometric setting in which the unit normal vector is viewed as an element of the twodimensional sphere manifold. It is shown that spheres are stationary points when the total variation of the normal is minimized under an area constraint. Shape calculus is used to characterize the relevant derivatives. Since the total variation functional is nondifferentiable whenever the boundary contains flat regions, an extension of the split Bregman method to manifold valued functions is proposed.@article{BergmannHerrmannHerzogSchmidtVidalNunez2019, author = {Bergmann, Ronny and Herrmann, Marc and Herzog, Roland and Schmidt, Stephan and Vidal Núñez, José}, title = {Total Variation of the Normal Vector Field as Shape Prior}, year = {2020}, journal = {Inverse Problems}, note = {accepted for publication.}, doi = {10.1088/13616420/ab6d5b}, eprint = {1902.07240}, eprinttype = {arXiv} }

Bergmann, R and Merkert, D (2020). FFTbased homogenization on periodic anisotropic translation invariant spaces. Applied and Computational Harmonic Analysis. 48 266–92
In this paper we derive a discretisation of the equation of quasistatic elasticity in homogenization in form of a variational formulation and the socalled LippmannSchwinger equation, in anisotropic spaces of translates of periodic functions. We unify and extend the truncated Fourier series approach, the constant finite element ansatz and the anisotropic lattice derivation. The resulting formulation of the LippmannSchwinger equation in anisotropic translation invariant spaces unifies and analyses for the first time both the Fourier methods and finite element approaches in a common mathematical framework. We further define and characterize the resulting periodised Green operator. This operator coincides in case of a Dirichlet kernel corresponding to a diagonal matrix with the operator derived for the Galerkin projection stemming from the truncated Fourier series approach and to the anisotropic lattice derivation for all other Dirichlet kernels. Additionally, we proof the boundedness of the periodised Green operator. The operator further constitutes a projection if and only if the space of translates is generated by a Dirichlet kernel. Numerical examples for both the de la Vallée Poussin means and Box splines illustrate the flexibility of this framework.@article{BergmannMerkert2020, author = {Bergmann, Ronny and Merkert, Dennis}, title = { {FFT}based homogenization on periodic anisotropic translation invariant spaces}, year = {2020}, eprint = {1701.04685}, eprinttype = {arXiv}, journal = {Applied and Computational Harmonic Analysis}, volume = {48}, number = {1}, pages = {266292}, doi = {10.1016/j.acha.2018.05.003} }
2019

Bergmann, R and Herzog, R (2019). Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds. SIAM Journal on Optimization. 29 2423–44
KarushKuhnTucker (KKT) conditions for equality and inequality constrained optimization problems on smooth manifolds are formulated. Under the Guignard constraint qualification, local minimizers are shown to admit Lagrange multipliers. The linear independence, MangasarianFromovitz, and Abadie constraint qualifications are also formulated, and the chain “LICQ implies MFCQ implies ACQ implies GCQ” is proved. Moreover, classical connections between these constraint qualifications and the set of Lagrange multipliers are established, which parallel the results in Euclidean space. The constrained Riemannian center of mass on the sphere serves as an illustrating numerical example.@article{BH18, author = {Bergmann, Ronny and Herzog, Roland}, title = {Intrinsic formulation of {KKT} conditions and constraint qualifications on smooth manifolds}, journal = {SIAM Journal on Optimization}, year = {2019}, eprint = {1804.06214}, eprinttype = {arXiv}, doi = {10.1137/18M1181602}, volume = {29}, number = {4}, pages = {24232444} }
2018

Bergmann, R and Gousenbourger, PY (2018). A variational model for data fitting on manifolds by minimizing the acceleration of a Bézier curve. Frontiers in Applied Mathematics and Statistics
We derive a variational model to fit a composite Bézier curve to a set of data points on a Riemannian manifold. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closedform, numerically stable and efficient algorithm to compute the gradient of a Bézier curve on manifolds with respect to its control points, expressed as a concatenation of socalled adjoint Jacobi fields. Several examples illustrate the capabilites and validity of this approach both for interpolation and approximation. The examples also illustrate that the approach outperforms previous works tackling this problem.@article{BG18, author = {Bergmann, Ronny and Gousenbourger, PierreYves}, title = {A variational model for data fitting on manifolds by minimizing the acceleration of a Bézier curve}, year = {2018}, journal = {Frontiers in Applied Mathematics and Statistics}, doi = {10.3389/fams.2018.00059}, eprint = {1807.10090}, eprinttype = {arXiv} }

Bergmann, R, Fitschen, J H, Persch, J and Steidl, G (2018). Priors with coupled first and second order differences for manifoldvalued image processing. Journal of Mathematical Imaging and Vision. 60 1459–81
We generalize discrete variational models involving the infimal convolution (IC) of first and second order differences and the total generalized variation (TGV) to manifoldvalued images. We propose both extrinsic and intrinsic approaches. The extrinsic models are based on embedding the manifold into an Euclidean space of higher dimension with manifold constraints. An alternating direction methods of multipliers can be employed for finding the minimizers. However, the components within the extrinsic IC or TGV decompositions live in the embedding space which makes their interpretation difficult. Therefore we investigate two intrinsic approaches: for Lie groups, we employ the group action within the models; for more general manifolds our IC model is based on recently developed absolute second order differences on manifolds, while our TGV approach uses an approximation of the parallel transport by the pole ladder. For computing the minimizers of the intrinsic models we apply gradient descent algorithms. Numerical examples demonstrate that our approaches work well for certain manifolds.@article{BFPS18, author = {Bergmann, Ronny and Fitschen, Jan Henrik and Persch, Johannes and Steidl, Gabriele}, title = {Priors with coupled first and second order differences for manifoldvalued image processing}, journal = {Journal of Mathematical Imaging and Vision}, doi = {10.1007/s108510180840y}, year = {2018}, volume = {60}, issue = {9}, pages = {14591481}, eprint = {1709.01343}, eprinttype = {arXiv} }

Bergmann, R and Merkert, D (2018). A framework for FFTbased homogenization on anisotropic lattices. Computers & Mathematics with Applications. 76 125–40
In order to take structural anisotropies of a given composite and different shapes of its unit cell into account, we generalize the Basic Scheme in homogenization by Moulinec and Suquet to arbitrary sampling lattices and tilings of the ddimensional Euclidean space. We employ a Fourier transform for these lattices by introducing the corresponding set of sample points, the so called pattern, and its frequency set, the generating set, both representing the anisotropy of both the shape of the unit cell and the chosen preferences in certain sampling directions. In several cases, this Fourier transform is of lower dimension than the space itself. For the so called rank1lattices it even reduces to a onedimensional Fourier transform having the same leading coefficient as the fastest Fourier transform implementation available. We illustrate the generalized Basic Scheme on an anisotropic laminate and a generalized ellipsoidal Hashin structure. For both we give an analytical solution to the elasticity problem, in two and three dimensions, respectively. We then illustrate the possibilities of choosing a pattern. Compared to classical grids this introduces both a reduction of computation time and a reduced error of the numerical method. It also allows for anisotropic subsampling, i.e. choosing a sub lattice of a pixel or voxel grid based on anisotropy information of the material at hand.@article{BM18, author = {Bergmann, Ronny and Merkert, Dennis}, title = {A framework for FFTbased homogenization on anisotropic lattices}, year = {2018}, eprint = {1605.05712}, eprinttype = {arXiv}, volume = {76}, issue = {1}, pages = {125140}, journal = {Computers \& Mathematics with Applications.}, doi = {10.1016/j.camwa.2018.04.008} }

Bergmann, R and Tenbrinck, D (2018). A graph framework for manifoldvalued data. SIAM Journal on Imaging Sciences. 11 325–60
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 \(p\)Laplacian 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{BT18, author = {Bergmann, Ronny and Tenbrinck, Daniel}, title = {A graph framework for manifoldvalued data}, eprint = {1702.05293}, eprinttype = {arXiv}, year = {2018}, volume = {11}, issue = {1}, pages = {325360}, journal = {SIAM Journal on Imaging Sciences}, doi = {10.1137/17M1118567} }
2017

Bergmann, R, Fitschen, J H, Persch, J and Steidl, G (2017). Iterative multiplicative filters for data labeling. International Journal of Computer Vision. 123 435–53
Based on an idea of Åström et. al. [2017, JMIV] we propose a new iterative multiplicative filtering algorithm for label assignment matrices which can be used for the supervised partitioning of data. Starting with a rownormalized matrix containing the averaged distances between prior features and observed ones, the method assigns in a very efficient way labels to the data. We interpret the algorithm as a gradient ascent method with respect to a certain function on the product manifold of positive numbers followed by a reprojection onto a subset of the probability simplex consisting of vectors whose components are bounded away from zero by a small constant. While such boundedness away from zero is necessary to avoid an arithmetic underflow, our convergence results imply that they are also necessary for theoretical reasons. Numerical examples show that the proposed simple and fast algorithm leads to very good results. In particular we apply the method for the partitioning of manifoldvalued images.@article{BFP17, author = {Bergmann, Ronny and Fitschen, Jan Henrik and Persch, Johannes and Steidl, Gabriele}, title = {Iterative multiplicative filters for data labeling}, year = {2017}, volume = {123}, issue = {3}, pages = {435453}, journal = {International Journal of Computer Vision}, eprint = {1604.08714}, eprinttype = {arXiv}, doi = {10.1007/s1126301709959} }
2016

Bergmann, R and Weinmann, A (2016). A second order TVtype approach for inpainting and denoising higher dimensional combined cyclic and vector space data. Journal of Mathematical Imaging and Vision. 55 401–27
In this paper we consider denoising and inpainting problems for higher dimensional combined cyclic and linear space valued data. These kind 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 (TV) type problems for denoising, inpainting as well as the combination of both. We provide a convergence analysis and we apply the algorithms to concrete problems.@article{BW16, author = {Bergmann, Ronny and Weinmann, Andreas}, title = {A second order {TV}type approach for inpainting and denoising higher dimensional combined cyclic and vector space data}, journal = {Journal of Mathematical Imaging and Vision}, year = {2016}, volume = {55}, number = {3}, pages = {401427}, doi = {10.1007/s1085101506273}, eprint = {1501.02684}, eprinttype = {arXiv} }

Bačák, M, Bergmann, R, Steidl, G and Weinmann, A (2016). A second order nonsmooth variational model for restoring manifoldvalued images. SIAM Journal on Scientific Computing. 38 A567–A597
We introduce a new nonsmooth variational model for the restoration of manifoldvalued data which includes second order differences in the regularization term. While such models were successfully applied for realvalued 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 machinery of Jacobi fields. For the \(n\)sphere and the manifold of symmetric positive definite matrices, we demonstrate the performance of our algorithm in practice. We prove the convergence of the proposed exact and inexact variant of the cyclic proximal point algorithm in Hadamard spaces. These results which are of interest on its own include, e.g., the manifold of symmetric positive definite matrices.@article{BBSW16, author = {Ba{\v{c}}{\'a}k, Miroslav and Bergmann, Ronny and Steidl, Gabriele and Weinmann, Andreas}, title = {A second order nonsmooth variational model for restoring manifoldvalued images}, journal = {SIAM Journal on Scientific Computing}, year = {2016}, volume = {38}, number = {1}, pages = {A567A597}, doi = {10.1137/15M101988X}, eprint = {1506.02409}, eprinttype = {arXiv} }

Bergmann, R, Persch, J and Steidl, G (2016). A parallel Douglas–Rachford algorithm for minimizing ROFlike functionals on images with values in symmetric Hadamard manifolds. SIAM Journal on Imaging Sciences. 9 901–37
We are interested in restoring images having values in a symmetric Hadamard manifold by minimizing a functional with a quadratic data term and a total variation like regularizing term. To solve the convex minimization problem, we extend the DouglasRachford algorithm and its parallel version to symmetric Hadamard manifolds. The core of the DouglasRachford algorithm are reflections of the functions involved in the functional to be minimized. In the Euclidean setting the reflections of convex lower semicontinuous functions are nonexpansive. As a consequence, convergence results for KrasnoselskiMann iterations imply the convergence of the DouglasRachford algorithm. Unfortunately, this general results does not carry over to Hadamard manifolds, where proper convex lower semicontinuous functions can have expansive reflections. However, splitting our restoration functional in an appropriate way, we have only to deal with special functions namely, several distancelike functions and an indicator functions of a special convex sets. We prove that the reflections of certain distancelike functions on Hadamard manifolds are nonexpansive which is an interesting result on its own. Furthermore, the reflection of the involved indicator function is nonexpansive on Hadamard manifolds with constant curvature so that the DouglasRachford algorithm converges here.
Several numerical examples demonstrate the advantageous performance of the suggested algorithm compared to other existing methods as the cyclic proximal point algorithm or halfquadratic minimization. Numerical convergence is also observed in our experiments on the Hadamard manifold of symmetric positive definite matrices with the affine invariant metric which does not have a constant curvature.@article{BPS16, author = {Bergmann, Ronny and Persch, Johannes and Steidl, Gabriele}, title = {A parallel {D}ouglas–{R}achford algorithm for minimizing {ROF}like functionals on images with values in symmetric {H}adamard manifolds}, journal = {SIAM Journal on Imaging Sciences}, year = {2016}, volume = {9}, number = {3}, pages = {901937}, doi = {10.1137/15M1052858}, eprint = {1512.02814}, eprinttype = {arXiv} } 
Bergmann, R, Chan, R H, Hielscher, R, Persch, J and Steidl, G (2016). Restoration of manifoldvalued images by halfquadratic minimization. Inverse Problems and Imaging. 10 281–304
The paper addresses the generalization of the halfquadratic minimization method for the restoration of images having values in a complete, connected Riemannian manifold. We recall the halfquadratic minimization method using the notation of the \(c\)transform and adapt the algorithm to our special variational setting. We prove the convergence of the method for Hadamard spaces. Extensive numerical examples for images with values on spheres, in the rotation group \(\operatorname{SO}(3)\), and in the manifold of positive definite matrices demonstrate the excellent performance of the algorithm. In particular, the method with \(\operatorname{SO}(3)\)valued data shows promising results for the restoration of images obtained from Electron Backscattered Diffraction which are of interest in material science.@article{BCHPS16, author = {Bergmann, Ronny and Chan, Raymond H. and Hielscher, Ralf and Persch, Johannes and Steidl, Gabriele}, title = {Restoration of manifoldvalued images by halfquadratic minimization}, journal = {Inverse Problems and Imaging}, year = {2016}, volume = {10}, number = {2}, pages = {281304}, doi = {10.3934/ipi.2016001}, eprint = {1505.07029}, eprinttype = {arXiv} }
2015

Bergmann, R and Prestin, J (2015). Multivariate periodic wavelets of de la Vallée Poussin type. Journal of Fourier Analysis and Applications. 21 342–69
In this paper we present a general approach to multivariate periodic wavelets generated by scaling functions of de la Vallée Poussin type. These scaling functions and their corresponding wavelets are determined by their Fourier coefficients, which are sample values of a function, that can be chosen arbitrarily smooth, even with different smoothness in each direction. This construction generalizes the onedimensional de la Vallée Poussin means to the multivariate case and enables the construction of wavelet systems, where the set of dilation matrices for the twoscale relation of two spaces of the multiresolution analysis may contain shear and rotation matrices. It further enables the functions contained in each of the function spaces from the corresponding series of scaling spaces to have a certain direction or set of directions as their focus, which is illustrated by detecting jumps of certain directional derivatives of higher order.@article{BP15, author = {Bergmann, Ronny and Prestin, Jürgen}, title = {Multivariate periodic wavelets of de la Vallée Poussin type}, journal = {Journal of Fourier Analysis and Applications}, year = {2015}, volume = {21}, number = {2}, pages = {342–369}, doi = {10.1007/s000410149372z}, eprint = {1402.3710}, eprinttype = {arXiv} }
2014

Bergmann, R, Laus, F, Steidl, G and Weinmann, A (2014). Second order differences of cyclic data and applications in variational denoising. SIAM Journal on Imaging Sciences. 7 2916–53
In many image and signal processing applications, such as interferometric synthetic aperture radar (SAR), electroencephalogram (EEG) data analysis, groundbased astronomy, and color image restoration, in HSV or LCh spaces the data has its range on the onedimensional sphere \(\mathbb S^1\). Although the minimization of total variation (TV) regularized functionals is among the most popular methods for edgepreserving image restoration, such methods were only very recently applied to cyclic structures. However, as for Euclidean data, TV regularized variational methods suffer from the socalled 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 \(\mathbb S^1\)valued data in a sound way which is independent of the chosen representation system on the circle. Our absolute cyclic first order difference is just the geodesic distance between points. Similar to the geodesic distances, the absolute cyclic second order differences have only values in \([0,\pi]\). We update the cyclic variational TV approach by our new cyclic second order differences. To minimize the corresponding functional we apply a cyclic proximal point method which was recently successfully proposed for Hadamard manifolds. Choosing appropriate cycles this algorithm can be implemented in an efficient way. The main steps require the evaluation of proximal mappings of our cyclic differences for which we provide analytical expressions. Under certain conditions we prove the convergence of our algorithm. Various numerical examples with artificial as well as realworld data demonstrate the advantageous performance of our algorithm.@article{BLSW14, author = {Bergmann, Ronny and Laus, Friederike and Steidl, Gabriele and Weinmann, Andreas}, title = {Second order differences of cyclic data and applications in variational denoising}, journal = {SIAM Journal on Imaging Sciences}, year = {2014}, volume = {7}, number = {4}, pages = {2916–2953}, doi = {10.1137/140969993}, eprint = {1405.5349}, eprinttype = {arXiv} }
2013

Bergmann, R (2013). The fast Fourier transform and fast wavelet transform for patterns on the torus. Applied and Computational Harmonic Analysis. 35 39–51
We introduce a fast Fourier transform on regular ddimensional 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 onedimensional 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.@article{Ber13, author = {Bergmann, Ronny}, title = {The fast Fourier transform and fast wavelet transform for patterns on the torus}, journal = {Applied and Computational Harmonic Analysis}, year = {2013}, volume = {35}, number = {1}, pages = {39–51}, doi = {10.1016/j.acha.2012.07.007}, eprint = {1107.5415v2}, eprinttype = {arXiv} }
Book Chapters

Bergmann, R, Laus, F, Persch, J and Steidl, G (2019). Recent advances in denoising of manifoldvalued images. Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2. Elsevier. 20 553–78
@incollection{BergmannLausPerschSteidl2019, author = {Bergmann, Ronny and Laus, Friederike and Persch, Johannes and Steidl, Gabriele}, title = {Recent advances in denoising of manifoldvalued images}, editor = {Kimmel, Ron and Tai, XueCheng}, booktitle = {Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2}, year = {2019}, chapter = {14}, pages = {553578}, series = {Handbook of Numerical Analysis}, volume = {20}, publisher = {Elsevier}, doi = {10.1016/bs.hna.2019.07.005} }
Peerreviewed Conference Proceedings
2019

Bergmann, R, Herzog, R, LoayzaRomera, E and Welker, K (2019). Shape optimization: what to do first, optimize or discretize? 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM). WileyBlackwell
In general, it is not easy to decide whether to use a discretize–then–optimize or optimize–then–discretize approach to solve PDEconstrained optimization problems. Shape optimization problems are particularly challenging due to the infinite dimensional, nonlinear structure of shape spaces. One option to overcome this drawback is to choose finite dimensional shape spaces, i.e., to discretize first. Nevertheless, this limits the amount of available tools coming from function spaces. In this note, we aim to show advantages and disadvantages of both approaches based on a simple shape optimization problem.@inproceedings{BergmannHerzogLoayzaWelker2019, title = {Shape optimization: what to do first, optimize or discretize?}, author = {Bergmann, Ronny and Herzog, Roland and LoayzaRomera, Estefanía and Welker, Kathrin}, doi = {10.1002/pamm.201900067}, crossref = {pamm2019}, year = {2019} }

(2019). 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM). WileyBlackwell
@proceedings{pamm2019, year = {2019}, editor = , title = {90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM)}, publisher = {WileyBlackwell} }

Herzog, R, Herrmann, M, Bergmann, R, Schmidt, S and VidalNúñez, J (2019). Geometry processing problems using the total variation of the normal vector field. 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM). WileyBlackwell
This paper presents a novel approach to the formulation and solution of discrete geometry processing problems including mesh denoising. The main quantity of interest is the piecewise constant unit normal vector field, which we consider on piecewise flat, triangulated surfaces. In a similar fashion as one does for images defined on ’flat’ domains, our goal is to remove noise while preserving shape features such as sharp edges. To this end, we model a denoising problem via a quadratic vertex tracking term and a regularizer based on the total variation of the normal vector field. Since the latter has values on the unit sphere, its total variation reduces to a finite sum of the geodesic distances of neighboring normals. We solve the model numerically by applying splitBregmann iterations and provide results for the ’fandisk’ benchmark.@inproceedings{HerzogHerrmannBergmannSchmidtVidalNunez2019, title = {Geometry processing problems using the total variation of the normal vector field}, author = {Herzog, Roland and Herrmann, Marc and Bergmann, Ronny and Schmidt, Stephan and VidalN{\'u}{\~n}ez, Jos{\'e}}, doi = {10.1002/pamm.201900189}, crossref = {pamm2019}, year = {2019} }
2017

Bergmann, R (2017). MVIRT, A toolbox for manifoldvalued image restoration. IEEE International Conference on Image Processing, IEEE ICIP 2017, Beijing, China, September 17–20, 2017
In many real life application measured data takes its values on Riemannian manifolds. For the special case of the Euclidean space this setting includes the classical grayscale and color images. Like these classical images, manifoldvalued data might suffer from measurement errors in form of noise or missing data. In this paper we present the manifoldvalued image restoration toolbox (MVIRT) that provides implementations of classical image processing tasks. Based on recent developments in variational methods for manifoldvalued image processing methods, like total variation regularization, the toolbox provides easy access to work with these algorithms. The toolbox is implemented in Matlab, open source, and easily extendible, e.g. with own manifolds, noise models or further algorithms. This paper introduces the main mathematical methods as well as numerical examples.@inproceedings{Ber17, author = {Bergmann, Ronny}, title = { {MVIRT}, A toolbox for manifoldvalued image restoration}, year = {2017}, booktitle = {IEEE International Conference on Image Processing, IEEE ICIP 2017, Beijing, China, September 1720, 2017}, doi = {10.1109/ICIP.2017.8296271} }

Bergmann, R and Tenbrinck, D (2017). Nonlocal inpainting of manifoldvalued data on finite weighted graphs. Geometric Science of Information – 3rd Conference on Geometric Science of Information. Springer International Publishing, Cham. 604–12
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} }

Bergmann, R and Merkert, D (2017). Approximation of periodic PDE solutions with anisotropic translation invariant spaces. 2017 International Conference on Sampling Theory and Applications (SampTA). 396–9
We approximate the quasistatic equation of linear elasticity in translation invariant spaces on the torus. This unifies different FFTbased discretisation methods into a common framework and extends them to anisotropic lattices. We analyse the connection between the discrete solution spaces and demonstrate the numerical benefits. Finite element methods arise as a special case of periodised Box spline translates.@inproceedings{BM17a, author = {Bergmann, Ronny and Merkert, Dennis}, title = {Approximation of periodic PDE solutions with anisotropic translation invariant spaces}, year = {2017}, pages = {396399}, booktitle = {2017 International Conference on Sampling Theory and Applications (SampTA)}, doi = {10.1109/SAMPTA.2017.8024347} }

Bergmann, R, Fitschen, J H, Persch, J and Steidl, G (2017). Infimal Convolution Coupling of First and Second Order Differences on ManifoldValued Images. Scale Space and Variational Methods in Computer Vision: 6th International Conference, SSVM 2017, Kolding, Denmark, June 48, 2017, Proceedings. Springer International Publishing, Cham. 447–59
Recently infimal convolution type functions were used in regularization terms of variational models for restoring and decomposing images. This is the first attempt to generalize the infimal convolution of first and second order differences to manifoldvalued images. We propose both an extrinsic and an intrinsic approach. Our focus is on the second one since the summands arising in the infimal convolution lie on the manifold themselves and not in the higher dimensional embedding space. We demonstrate by numerical examples that the approach works well on the circle, the 2sphere, the rotation group, and the manifold of positive definite matrices with the affine invariant metric.@inproceedings{BFPS17b, author = {Bergmann, Ronny and Fitschen, Jan Henrik and Persch, Johannes and Steidl, Gabriele}, editor = {Lauze, Fran{\c{c}}ois and Dong, Yiqiu and Dahl, Anders Bjorholm}, title = {Infimal Convolution Coupling of First and Second Order Differences on ManifoldValued Images}, booktitle = {Scale Space and Variational Methods in Computer Vision: 6th International Conference, SSVM 2017, Kolding, Denmark, June 48, 2017, Proceedings}, year = {2017}, publisher = {Springer International Publishing}, address = {Cham}, pages = {447459}, isbn = {9783319587714}, doi = {10.1007/9783319587714_36} }
2015

Oezguen, N, Schubert, K J, Bergmann, R, Bennewitz, R and Strauss, D J (2015). Relating tribological stimuli to somatosensory electroencephalographic responses. Engineering in Medicine and Biology Society (EMBC), 37th Annual International Conference of the IEEE. 8115–8
The present study deals with the extraction of neural correlates evoked by tactile stimulation of the human fingertip. A reciprocal sliding procedure was performed using a homebuilt tribometer while simultaneously electroencephalographic (EEG) data from the somatosensory cortex was recorded. The tactile stimuli were delivered by a sliding block with equidistant, perpendicular ridges. The experiments were designed and performed in a fully passive way to prevent attentional locked influences from the subjects. In order to improve the signaltonoise ratio (SNR) of event related singletrials (ERPs), nonlocal means in addition to 2Danisotropic denoising schemes based on tight Gabor frames were applied. This novel approach allowed for an easier extraction of ERP alternations. A negative correlation between the latency of the P100 component of the resulting brain responses and the intensity of the underlying lateral forces was found. These findings lead to the conclusion that an increasing stimulus intensity results in a decreasing latency of the brain responses.@inproceedings{OSBBS15, author = {Oezguen, Novaf and Schubert, Kristof J. and Bergmann, Ronny and Bennewitz, Roland and Strauss, Daniel J.}, booktitle = {Engineering in Medicine and Biology Society (EMBC), 37th Annual International Conference of the IEEE}, title = {Relating tribological stimuli to somatosensory electroencephalographic responses}, year = {2015}, pages = {81158118}, doi = {10.1109/EMBC.2015.7320277}, month = aug }

Bergmann, R and Weinmann, A (2015). Inpainting of cyclic data using first and second order differences. Energy Minimization Methods in Computer Vision and Pattern Recognition, 10th International Conference on, EMMCVPR 2015, Hong Kong. Springer. 155–68
Cyclic data arise in various image and signal processing applications such as interferometric synthetic aperture radar, electroencephalogram data analysis, and color image restoration in HSV or LCh spaces. In this paper we introduce a variational inpainting model for cyclic data which utilizes our definition of absolute cyclic second order differences. Based on analytical expressions for the proximal mappings of these differences we propose a cyclic proximal point algorithm (CPPA) for minimizing the corresponding functional. We choose appropriate cycles to implement this algorithm in an efficient way. We further introduce a simple strategy to initialize the unknown inpainting region. Numerical results both for synthetic and realworld data demonstrate the performance of our algorithm.@inproceedings{BW15, author = {Bergmann, Ronny and Weinmann, Andreas}, title = {Inpainting of cyclic data using first and second order differences}, year = {2015}, pages = {155–168}, editor = {Tai, X.C. and Bae, E. and Chan, T. F. and Leung, S. Y. and Lysaker, M.}, publisher = {Springer}, booktitle = {Energy Minimization Methods in Computer Vision and Pattern Recognition, 10th International Conference on, EMMCVPR 2015, Hong Kong}, doi = {10.1007/9783319146126_12}, eprint = {1410.1998}, eprinttype = {arXiv} }
2014

Bergmann, R and Prestin, J (2014). Multivariate anisotropic interpolation on the torus. Approximation Theory XIV: San Antonio 2013. 27–44
We investigate the error of periodic interpolation, when sampling a function on an arbitrary pattern on the torus. We generalize the periodic StrangFix conditions to an anisotropic setting and provide an upper bound for the error of interpolation. These conditions and the investigation of the error especially take different levels of smoothness along certain directions into account.@inproceedings{BP14, author = {Bergmann, Ronny and Prestin, Jürgen}, title = {Multivariate anisotropic interpolation on the torus}, year = {2014}, pages = {27–44}, editor = {Fasshauer, G. and Schumaker, L.}, booktitle = {Approximation Theory XIV: San Antonio 2013}, doi = {10.1007/9783319064048_3}, eprint = {1309.3432}, eprinttype = {arXiv} }
Theses

Bergmann, R (2018). Variational Methods for ManifoldValued Image Processing. Cumulative Habilitation thesis. Technische Universität Kaiserslautern
@thesis{Bergmann2018Habil, author = {Bergmann, Ronny}, school = {Technische Universit{\"a}t Kaiserslautern}, title = , year = {2018}, type = {Cumulative Habilitation thesis} }

Bergmann, R (2013). Translationsinvariante Räume Multivariater Anisotroper Funktionen Auf Dem Torus. Dissertation. Universität zu Lübeck
in german. Similarily: Shaker Verlag, ISBN 9783844022667, 2013.
Die translationsinvarianten Räume sind seit Ende der 1980er Jahre ein wichtiges Werkzeug in der Zerlegung und Analyse von Daten und Funktionen. Sie stellen eine Funktion in verschiedenen Detailstufen dar, wobei sich auf jeder Stufe lokale Eigenschaften der Funktion in den Koeffizienten der Translate wiederfinden. Dazu ist es notwendig, dass die Funktionen einer solchen Zerlegung gut lokalisiert sind. Dies überträgt sich dann auf die von den Translaten der Wavelets aufgespannten orthogonalen Komplemente innerhalb der gestaffelten Räume. Neben diesen theoretischen Eigenschaften haben vor allem die schnellen Algorithmen der Wavelet Transformation zur großen Verbreitung und Anwendung der Wavelets geführt. In den 1990er Jahren wurden periodische Wavelets entwickelt und in den letzten Jahren mehrdimensionale Wavelets, wie die Shearlets oder Curvelets, die insbesondere spezielle Richtungen in der Zerlegung einer Funktion bevorzugen.
Diese Arbeit widmet sich den mehrdimensionalen periodischen translationsinvarianten Räumen, die insbesondere allgemeiner sind als diejenigen, die lediglich durch Tensorproduktbildung aus dem Eindimensionalen hervorgehen, und somit eine Anisotropie in ihren Fourier Koeffizienten besitzen und gewisse Richtungseigenschaften haben. Diese Richtungspräferenz spiegelt sich auch in den Mustern wider, mit denen die Translate der Funktionen die hier betrachteten Räume bilden.
Zunächst wird für die periodischen translationsinvarianten Räume auf dem Torus der Interpolationsfehler betrachtet. Dazu werden die anisotropen periodischen StrangFixBedingungen eingeführt und mit ihnen Fehlerabschätzungen angegeben, welche die Räume bezüglich der Approximationsgüte für Funktionen mit bestimmten Glattheitseigenschaften charakterisieren.
Für die anisotrope periodische WaveletTransformation werden Algorithmen vorgestellt, die in ihrer Komplexität den schnellen Algorithmen der eindimensionalen Wavelets entsprechen und dabei insbesondere dimensionsunabhängig sind. Wichtigste Werkzeuge sind dazu die schnelle FourierTransformation und die den Wavelets zu Grunde liegenden ZweiSkalenGleichungen der MultiskalenAnalyse. Außerdem wird durch diese Beschreibung eine Richtungsklassifikation der betrachteten Wavelets auf den Mustern möglich.
Ausgehend von den de la Vallée PoussinMitteln wird dann eine Verallgemeinerung vorgestellt, bei der lokalisierte anisotrope periodische Wavelets konstruiert werden, deren FourierKoeffizienten als Abtastung einer beliebig glatten Funktion gegeben sind, deren Träger endlich ist. Dies verallgemeinert sowohl die eindimensionalen de la Vallée PoussinWavelets als auch die multivariaten DirichletWavelets. Für spezielle glatte Funktionen gelingt es, in der Konstruktion anstelle der rekursiven Definition eine explizite Darstellung der FourierKoeffizienten anzugeben und somit für diese lokalisierten Wavelets eine gesamte MultiskalenAnalyse zu konstruieren.@thesis{B13Diss, author = {Bergmann, Ronny}, school = {Universität zu Lübeck}, title = , year = {2013}, type = {Dissertation}, language = {german}, note = {in german. Similarily: Shaker Verlag, ISBN 9783844022667, 2013.} } 
Bergmann, R (2009). Interaktive Und Automatisierte Hypergraphenvisualisierung Mittels NURBSKurven. Diploma thesis. Universität zu Lübeck
In german.
Für wissenschaftliche Arbeiten in der Graphentheorie, vor allem bei den allgemeineren Hypergraphen, sind Darstellungen zur Erklärung und Er läuterung von großer Bedeutung. Diese Arbeit schafft daher die Grundlagen einer Darstellung von Hypergraphen, die in dem Programm „Gravel“, einem Editor für ebendiese, umgesetzt wurden.
Für Bilder von Hypergraphen werden in dieser Arbeit theoretische Grundlagen entwickelt, die auf Basis der NURBSKurven eine Darstellung der Hyperkanten ermöglicht. Dazu werden die Eigenschaften von periodischen NURBSKurven betrachtet und Algorithmen zur interaktiven Modifikati on vorgestellt. Der Begriff des Hyperkantenumrisses und seiner Gültigkeit wird eingeführt, um formelle Anforderungen an die Darstellung zu schaffen. Auf Grundlage der genannten Begriffe wird das Programm „Gravel“ zum Zeichnen von Graphen und Hypergraphen entwickelt und implementiert.@thesis{B09Dipl, author = {Bergmann, Ronny}, school = {Universität zu Lübeck}, title = {Interaktive und automatisierte Hypergraphenvisualisierung mittels NURBSKurven}, year = {2009}, type = {Diploma thesis}, language = {german}, note = {In german.} }
Miscellaneous

Bergmann, R, Laus, F, Persch, J and Steidl, G (2017). Manifoldvalued Image Processing. Siam News. 50 1&3
@article{BLPS17, author = {Bergmann, Ronny and Laus, Friederike and Persch, Johannes and Steidl, Gabriele}, title = {Manifoldvalued Image Processing}, journal = {Siam News}, volume = {50}, number = {8}, month = oct, year = {2017}, pages = {1\&3}, keywords = {misc} }