Thorsten M. Buzug

Wolfgang Erb, Andreas Weinmann, Mandy Ahlborg et al.

Magnetic particle imaging (MPI) is a promising new in-vivo medical imaging modality in which distributions of super-paramagnetic nanoparticles are tracked based on their response in an applied magnetic field. In this paper we provide a mathematical analysis of the modeled MPI operator in the univariate situation. We provide a Hilbert space setup, in which the MPI operator is decomposed into simple building blocks and in which these building blocks are analyzed with respect to their mathematical properties. In turn, we obtain an analysis of the MPI forward operator and, in particular, of its ill-posedness properties. We further get that the singular values of the MPI core operator decrease exponentially. We complement our analytic results by some numerical studies which, in particular, suggest a rapid decay of the singular values of the MPI operator.

Laura Hellwege, Johann Christopher Engster, Moritz Schaar et al.

Assume you encounter an inverse problem that shall be solved for a large number of data, but no ground-truth data is available. To emulate this encounter, in this study, we assume it is unknown how to solve the imaging problem of Computed Tomography (CT). An unsupervised deep learning approach is introduced, that leverages the inherent similarities between deep neural network training, deep image prior (DIP) and unrolled optimization schemes. We demonstrate the feasibility of reconstructing images from measurement data by pure network inference, without relying on ground-truth images in the training process or additional gradient steps for unseen samples. Our method is evaluated on the two-dimensional 2DeteCT dataset, showcasing superior performance in terms of mean squared error (MSE) and structural similarity index (SSIM) compared to traditional filtered backprojection (FBP) and maximum likelihood (ML) reconstruction techniques as well as similar performance compared to a supervised DL reconstruction. Additionally, our approach significantly reduces reconstruction time, making it a promising alternative for real-time medical imaging applications. Future work will focus on extending this methodology for adaptability of the projection geometry and other use-cases in medical imaging.

Wolfgang Erb, Christian Kaethner, Mandy Ahlborg et al.

Motivated by an application in Magnetic Particle Imaging, we study bivariate Lagrange interpolation at the node points of Lissajous curves. The resulting theory is a generalization of the polynomial interpolation theory developed for a node set known as Padua points. With appropriately defined polynomial spaces, we will show that the node points of non-degenerate Lissajous curves allow unique interpolation and can be used for quadrature rules in the bivariate setting. An explicit formula for the Lagrange polynomials allows to compute the interpolating polynomial with a simple algorithmic scheme. Compared to the already established schemes of the Padua and Xu points, the numerical results for the proposed scheme show similar approximation errors and a similar growth of the Lebesgue constant.