Nonlinear mass-preserving registration for magnetic resonance imaging (MRI) and positron emission tomography (PET)

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Image registration is an important task in medical imaging. The problem is to automatically establish point-to-point correspondences between objects sensed at different times, with different devices, or from different perspectives. Image registration is essential for applications such as motion correction or data fusion. As image registration is an ill-posed inverse problem, proper modeling and additional constraints on the wanted transformation field become inevitable. Mass-preservation (MP) is one of those constraints that proved to be crucial for the correction of image distortions due to field inhomogeneities in Magnetic Resonance Imaging (MRI) and the respiratory and cardiac motion correction in thoracic Positron Emission Tomography (PET). A wide utilization of MP techniques in those medical applications requires, however, the solution of challenging mathematical questions to be addressed in this project. Existing MP techniques will hence be optimized with respect to numerical efficiency and implementation. Problem specific data fidelities will yield an improved robustness against noise and even simplifications of the algorithms. A key component yielding existence of solutions in MP setting is a regularization energy based on nonlinear elasticity. This outstanding regularizer and its numerical discretization will hence be further investigated in this project. A deep mathematical understanding of the developed models will be gained by using regularization theory for inverse problems.