Numerical investigation of dictionary-based regularization for inverse problems and approximation problems on spheres and balls - with applications to seismic tomography and high-dimensional geophysical modelling
Final Report Abstract
Non-invasive methods for the investigation of the functionality of the human brain are used in the neurosciences. For this purpose, neuronal currents are reconstructed based on their measurable effects outside the head in order to achieve insights to the human brain and its activity. Measurable data are changes in the electric potential on the scalp (electroencephalography, EEG) and of the magnetic flux density outside of the head (magnetoencephalography, MEG). The reconstruction of the neuronal current from MEG and EEG data is a severely ill-posed inverse problems, since particular parts of the current are invisible for both measurement techniques and even small changes in the measurements can have a huge impact on the reconstructed neuronal current. In order to achieve reliable reconstructions, we further investigated the theory of both problems and we developed numerical methods for their stable resolution. For this purpose, the human brain is modelled via a ball and the surrounding head tissues are modelled by spherical shells in the very beginning. The effects of the neuronal current on the measurements can be described via integral equations. We analyzed the problem deeply and characterized the parts of the neuronal current which are silent for the sensors. Therewith, we were able to describe in detail that the reconstructable parts of the neuronal current from both measurement methods are complementary to each other. Hence, within a simultaneous joint inversion of MEG and EEG data, no more information can be achieved as with the combination of the independent reconstructions. After having analyzed the problem deeply, we constructed numerical methods for the reconstruction of the current from the given measurements. We mainly used the regularized functional matching pursuit (RFMP), which reconstructs the current iteratively. Based on the iterated approximation, a trial function is chosen from a set of functions (dictionary) and added to the approximation in such a way that the error between the simulated and measured data is minimized effciently. Thereby, we take into account that the approximated current is not chosen randomly in order to compensate for small noise on the data. The RFMP has the advantage that a suitable dictionary containing global and localized trial functions can reconstruct coarse structures in the first steps and the algorithm adds more localized details of the current in the later steps. Due to the possibility of assembling the dictionary freely, advantages of further reconstruction methods can be combined. We compared the results of the RFMP reconstruction and its enhancements with further reconstruction methods in the context of synthetically generated (noisy) data. We conclude that the RFMP yields the best reconstruction but requires the most time for computation. Eventually, we applied the RFMP to a set of real data collected by our research partner in Cambridge (UK). The reconstructed neuronal current satisfies all physiological expectations and suits the data.
Publications
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On the null space of a class of Fredholm integral equations of the first kind. In: J. Inverse Ill-Posed Probl. 24 (2016), 687-710
V. Michel und S. Orzlowski
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On the convergence theorem for the regularized functional matching pursuit (RFMP) algorithm. In: Int. J. Geomath. 8 (2017), 183-190
V. Michel und S. Orzlowski
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On the non-uniqueness of gravitational and magnetic eld data inversion (survey article). In: Handbook of Mathematical Geodesy. Hrsg. von W. Freeden und M. Z. Nashed. Basel: Birkhäuser, 2018, 883-919
S. Leweke, V. Michel und R. Telschow
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The Inverse Magneto-electroencephalography Problem for the Spherical Multipleshell Model-Theoretical Investigations and Numerical Aspects. Diss. Universität Siegen, Department Mathematik, AG Geomathematik, 2018
S. Leweke
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Vectorial Slepian functions on the ball. In: Numer. Func. Anal. Opt. 39 (2018), 1120-1152
S. Leweke, V. Michel und N. Schneider
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Electro-magnetoencephalography for a spherical multipleshell model: novel integral operators with singular-value decompositions. In: Inverse Probl. 36 (2020), 035003
S. Leweke, V. Michel und A. S. Fokas
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Vector-valued spline method for the spherical multiple-shell electro-magnetoencephalography problem. In: Inverse Probl. 38 (2022), 085001
S. Leweke, O. Hauk und V. Michel
