Our main research activities
The classic regularisation theory is based on the assumption that the searched-for quantity is stationary during the data acquisition. However, in many applications from medicine to non-destructive testing, this assumption is not fulfilled, for example when investigating cross-scale structural changes of materials under load experiments or moving liquid fronts in porous media.
Applying the standard techniques from the literature provides a solution with low spatial or temporal resolution and introduces motion artefacts which impede a reliable diagnosis. Therefore, our research strives after developing a comprehensive regularization theory for dynamic inverse problems, new mathematical models with a specific treatment of the dynamics and new inversion schemes.
Developing new imaging modalities and new fields of applications necessitates novel approaches in mathematical modelling, data processing and image reconstruction. A special focus of our group lies in this respect on computerized tomography and microscopy.
The relentless progress in imaging technology and the exploitation of new fields of application require new mathematical models and numerical solution methods. The special focus of our group lies on classical modalities like computerized tomography or magnetic resonance imaging as well as on novel tomographic methods like magnetic particle imaging.
In order to extract features and properties of the studied medium, the reconstruction results are typically subject to further processing steps. Since an ever-increasing amount of data arises in modern applications, special algorithms are required to extract the desired information in an efficient and stable way. In this context, it is desirable to obtain this information directly from the data, i.e. in the form of a feature reconstruction.
Classical solution schemes for inverse problems are based on mathematical models which describe the link between data and searched-for quantity. For many classes of such model-based reconstruction methods, their convergence properties are well understood.
In novel, data-driven reconstruction methods, the sought-for quantity is determined by the evaluation of a neural network whose parameters have previously been trained on suitable test data. Thus, redundancies, structures and features in the test data can be identified and used for the solution step without explicit modeling.
A major difficulty of purely data-driven approaches is to establish a control over the output. Therefore, we investigate in the project iDeLIVER the combination of model-based and data-driven methods in order to enable a high acceleration of the reconstruction step without compromising the e.g. diagnostic validity.