Mathias Feinler

Learned Methods in Imaging

In the field of imaging, measurement data is acquired in order to reconstruct the desired quantity - an image of the interior of the object being measured. This requires to solve an inverse problem. Usually, the measurement is accelerated such that only a limited number of data points can be acquired. Therefore, there is no “inverse” of the forward operator. The optimal solution which can be derived by this measured data must be formulated in probabilities. Thus, the best possible reconstruction is the one that is the most probable, considering all measured data. This solution is referred to as the MAP (maximum a posteriori) solution. However, knowledge of the distribution is necessary for the calculation. For the example of brain-MRIs, this means that all biologically and morphologically reasonable characteristics of a (human) brain must be known. Since this quantity is not known, neural networks are used to estimate this distribution or properties thereof, using a finite training data set.

Motion Estimation for Dynamic Inverse Problems in Imaging

Perturbations can occur during data acquisition. The most prominent ones are movements by the patient. Usually an MRI fails if the patient moves during the measurement process. Mathematically, this problem can be modeled by a non-linear forward operator. The challenge here is to extract these deformation fields from the data. These can be used to correct measurement data and compensate for motion artefacts. This in turn requires knowledge of the joint distribution of image and deformation. Neural networks can be used to solve such a non-linear inverse problem. These are implanted in iterative solvers in order to approximate individual projection steps. This synergistically combines classical data-consistent reconstruction approaches with deep learning-based methods.