RESEARCH
We perform research on matbematical methods in inverse problems and image processing, as well as mathematical modelling in biomedicine. The main focus of our group is on techniques using partial differential equations and variational methods. We are interested in all aspects of these techniques, including mathematical modeling, wellposedness analysis and efficient algorithms for sequential and parallel computer architectures. Below further information can be found on our work on the following topics:
Mathematical Imaging and Inverse Problems Mathematical Modelling Applications in Biomedicine 
Published Papers
Monographs & Edited Volumes Theses Preprints Talks & Posters 

PET Reconstruction with Prior Knowledge
SFB Molecular Cardiovascular Imaging, Subproject B2 (DFG, 20052009, 20092013)

INVERS
Segmentation and Cartoon Reconstruction in Optical Nanoscopy (BMBF, 20072010)

Sparsity and Bayesian Inversion
Sparsityconstrained inversion with tomographic applications (DFG, Inverse Problems Initiative, 20112013)

Regularization with Singular Energies
Regularisierung mit singulären Energien (DFG, 20082011)

Ion Channels
Multiscale Simulation of Ion Transport through Biological and Synthetic Channels (VWStiftung, 20092012)

4D Imaging
4D Imaging in Tomography and Optical Nanoscopy (Deutsche Telekom Stiftung, 20082010)

Semiconductor Design
Optimal Control of SelfConsistent Classical and Quantum Particle Systems (DFG, 20092012)
Motion Corrected Reconstruction via Registration
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Involved Group Members: Sebastian Suhr, Martin Burger.
Cooperation Partners: Jan Modersitzki (Lübeck)
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Processing of Color Images and Videos with Joint Edge Information
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Involved Group Members: EvaMaria Brinkmann, Martin Burger.
Cooperation Partners: Michael Möller, Tamara Seybold (arri)
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Computational Methods for l1type Regularizations
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Cooperation Partners: Stanley Osher
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Hyperelastic and MassPreserving Image Registration
Registration tasks are common in medical imaging.
Before comparing or combining the information of two images, their accurate alignment in one common frame has to be ensured.
Given two images  the template image and the reference image  image registration aims to establish pointtopoint correspondences between both images.
Due to the variety of possible transformations and the sensitivity against noise, registration results in a underdetermined and thus illposed inverse problem.
Involved Group Members: Sebastian Suhr, Martin Burger.
Cooperation Partners: Jan Modersitzki (Lübeck), Carsten Wolters (IBB UKM), Fabian Gigengack (EIMI Münster)
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Bayesian Inversion and Sparsity
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Cooperation Partners: Department of Mathematics, Helsinki University (Samuli Siltanen, Matti Lassas), Center for Industrial Mathematics, University of Bremen (Peter Maass), Department of Mathematics, Shanghai Jiao Tong University (Jianguo Huang)
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Segmentation
Segmentation tasks arise in various biomedical applications, e.g. the resolution of anatomical structures for further use in imaging or of cell structures for further analysis. We focus on segmentation tasks with particular issues such as thin structures (ventricular walls, brain sulci) or low contrast (cell microscopy, CT images), for which novel mathematical approaches incorporating apriori knowledge are developed. [...]
Cartoon Reconstruction and Total Variation
In many applications one is interested in computing structures contained in image (e.g. as an image decomposition in structure and texture, or as the only part to be reasonably approximated in an illposed inversion). Total variation minimization has emerged as a standard technique for this sake. Besides several applications our research in this area is focussing on the development and analysis of new (iterative) computational methods and on error estimates. Similar approaches are investigated for other regularizations based on singular (nondifferentiable and not strictly convex) energies. [...]
Motion Estimation and Optical Flow
A major task in computer vision is the extraction of object and motion information from a given image sequence. In this context, a frequently used concept is that of the optical flow: For consecutive frames, one determines a displacement field (resp. its time derivative), which sets points of equal brigthness into correspondence. The concept of the optical flow finds numerous applications, e.g. for compression of video image data, automatic retouching of movie sequences during the process of digitalization. Its most important applications, however, are connected with object recognition and motion estimation, e.g. motion registration in 4D medical imaging or obstacle detection in car traffic or robotic. [...]
Modeling and Simulation of Ion Channels
Ion channels are the main controllers of biological function by regulation flow in and out of cells. Their modelling and understanding is still quite open. Our aim is to derive improved approaches for the modelling and simulation by novel multiscale approaches coupling continuum PoissonNernstPlanck equations to microscopic simulation techniques. The NernstPlanck equations are discretized by hybrid mixed finite element methods, which allows a local microscale coupling in the spirit of the heterogeneous multiscale methods. [...]
Nonlocal Equations and Nonlinear Diffusion
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Nonlinear Degenerate CrossDiffusion Systems
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Swarming and Pedestrian Dynamics
The collective motion of human and animal groups is lively research topics with surprising similarities to models in molecular and cell biology. Our focus is the analysis and simulation of models including nonlinear diffusion and nonlocal aggregation terms. For this sake gradient flow formulations and optimal transport techniques are used. [...]
Optimal Control of Semiconductor Devicess
Classical biology predicts function from structure, which is very difficult in the important case of ion channels since structures are often unknown (and if only at low voltages). We consequently try to perform an inverse approach and to compute structure from function. This concerns in particular the permanent protein charge distributions, which we try to reconstruct from measured currentvoltage curves at different ion concentrations. In the same way we try to design synthetic channels, e.g. with maximal selectivity properties. Analogous inverse problems have been considered for semiconductor devices, where doping profiles are reconstructed from currentvoltage and capacitancevoltage measurements. [...]
Agentbased Models and Fokker–Planck Equations
A strong trend towards behavioural finance and agentbased market models can be observed in recent economic research. Similar approaches can be found in opinion dynamics. Such agentbased approaches are stochastic simulation approaches, which often lack a theoretical understanding and a the capability of theoretical predictions. We therefore follow a standard route in statistical physics and derive FokkerPlanck equations as scaling limits. The arising FokkerPlanck equations allow further insight and a more detailed analysis, e.g. concerning stationary solutions and trend to equilibria as well as the efficient computation of certain system properties (e.g. autocorrelations). We also investigate market equilibria in markets with heterogeneous agents based on related HamiltonJacobi equations. [...]
4D Positron Emission Tomography
PET in 4D (three spatial and one time dimension) is still a challenge with respect to modelling and computation. Our focus is on efficient implementations of EMtype algorithms and on incorporating apriori knowledge in order to cure problems caused by low signaltonoise ratios. [...]
Quantitative Molecular Imaging, Computation of Physiological Parameters
One of the major interests in modern molecular imaging (based e.g. on PET) is a quantitative analysis of physiological properties, which is not possible from higher resolution imaging techniques such as CT. For this sake it does not suffice to reconstruct images of tracer uptakes, but rather images of the physiological paramaters. The latter are connected to the tracer uptake via nonlinear differential equations. Our aim is to reconstruct these parameters directly from the PET data via the solution of nonlinear inverse problems. In this way also issues with the low SNR encountered for important tracers such as radioactive water can be overcome. [...]
Optical Nanoscopy
Light microscopy has been revolutionized over the last couple of years, changing the requirements of the accompanying data analytical methods. Observed high resolution images suffer from blurring and Poisson noise effects. Hence we focus on deconvolution problems with sparsity constraints. Particularly our research in this area includes the developement of cartoon reconstruction methods based on EMtype algorithms and TVregularization. New approaches to analyse optical nanoscopy data evolve by using anisotropic diffusion and segmentation techniques. [...]
Analysis of DNAdamage in Human Sperm by Raman Microspectroscopy
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Cooperation Partners: Center for Reproductive Medicine and Andrology (Stefan Schlatt, Con Mallidis, Victoria Sanchez), Institute for Applied Physics (Carsten Fallnich, Petra Gross), Horiba Yvon Gmbh
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Mathematical Modelling of Neuron Differentiation and Axon Polarization
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Cooperation Partners: Institute for Molecular Cell Biology (Andreas Püschel)
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Analysis of Cardiac Morphology by Pneumographic MicroCT
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Cooperation Partners: Department of Cardiac Surgery (Paul Lunkenheimer), ETH Zürich (Peter Niederer)
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Inverse Problems from ECG / BSPM data
Body surface potential mapping (BSPM) is an ECG technique that yields a high amount of data about the electric potential on the human torso. It is a classical inverse problem to compute the electrical potential on the epicardial surface, which is however difficult to solve due to its severe illposedness. In our research we focus on novel approaches for inverse problems related to the electrical activity of the heart, such as the early detection of ischemia or studies of arrythmic behaviour. In order to overcome the severe illposedness we try to incorporate additional prior information (e.g. based on bidomain models in the myocardium) or compute reduced information (e.g. vectorcardiography). [...]
Modeling and Inverse Problems in EEG / MEG
EEG and MEG recordings are used in many applications today, ranging from clinical testing to cognitive science. One aim in using EEG and MEG is to reconstruct brain activity by measuring the induced electromagnetic fields. This task involves challenging mathematical problems in different areas of mathematical modeling and imaging.
Cooperation Partners: Institute for Biomagnetism and Biosignalanalysis (PD Dr. Carsten Wolters)
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