One speaks of an inverse problem, if from certain observations the causual factors shall be computed. The range of applications is extremely large, since already, for instance, problems such as denoising or deblurring of images take the form of an inverse problem. Mathematically, it is typically modelled as an operator equation with the operator modelling the process leading to the observations. Depending on the functional analytic properties of the operator (and hence on the given application problem), an inverse problem is often ill-posed. This roughly means that a simple inversion is not possible or not stable.
Regularization of inverse problems is of key importance to derive stable recovery algorithms in this situation. Various approaches such as Tikhonov regularization are known and also very well understood. One rather novel idea is to regularize by imposing sparse approximation  constraints in the solution such as by minimizing over the l1 norm of the coefficients with respect to a suitably chosen orthonormal basis or, more generally, frame . For this, the area of applied harmonic analysis  in combination with dictionary learning techniques provides an abundance of different systems to apply here such as shearlets.
Some of our Research Topics
- Inverse Scatterig Problem, where an object shall be detected by the scattering of, for instance, acoustic waves.
- Inpainting, where missing parts of an image shall be recovered.
- Feature Extraction, where distinct features of data shall be extracted.
- Efficient data acquisition by subsampling.