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, well-posedness 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

  Variational Regularization Methods for Inverse Problems

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Cooperation Partners: Elena Resmerita
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  Sparsity, Compressed Sensing, and Ill-Posedness

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Cooperation Partners: Stanley Osher
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  Computational Methods for l1-type Regularizations

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Cooperation Partners: Stanley Osher
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  Image Registration

Registration tasks are common in medial 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 point-to-point correspondences between both images. Due to the variety of possible transformations and the sensitivity against noise, registration results in a under-determined and thus ill-posed inverse problem.
Cooperation Partners: Department of Computer Science (Xiaoyi Jiang, Fabian Gigengack), Institute for Biomagnetism and Biosignal Analysis (Carsten Wolters), MIC University of Lübeck (Bernd Fischer, Jan Modersitzki)
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  Bayesian Inversion and Sparsity

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Cooperation Partners: Department of Mathematics, Helsinki University (Samuli Siltanen, Matti Lassas)
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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 a-priori 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 ill-posed 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 Poisson-Nernst-Planck equations to microscopic simulation techniques. The Nernst-Planck 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 Cross-Diffusion 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 current-voltage 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 current-voltage and capacitance-voltage measurements.  [...]

  Agent-based Models and Fokker–Planck Equations

A strong trend towards behavioural finance and agent-based market models can be observed in recent economic research. Similar approaches can be found in opinion dynamics. Such agent-based 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 Fokker-Planck equations as scaling limits. The arising Fokker-Planck 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 Hamilton-Jacobi 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 EM-type algorithms and on incorporating a-priori knowledge in order to cure problems caused by low signal-to-noise 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 EM-type algorithms and TV-regularization. New approaches to analyse optical nanoscopy data evolve by using anisotropic diffusion and segmentation techniques.  [...]

  Analysis of DNA-damage 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 Micro-CT

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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 ill-posedness. 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 ill-posedness we try to incorporate additional prior information (e.g. based on bidomain models in the myocardium) or compute reduced information (e.g. vectorcardiography). [...]

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