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Stability of Moment Problems and Super-Resolution Imaging

Forschungskooperation Niedersachsen-Israel (Nds. Vorab), 2020-2023, jointly with Dmitry Batenkov (Department of Applied Mathematics, School of Mathematical Sciences, Tel-Aviv University, Israel).

Algebraic techniques have proven useful in different imaging tasks such as spikereconstruction (single molecule microscopy), phase retrieval (X-ray crystallogra-phy), and contour reconstruction (natural images). The available data typicallyconsists of (trigonometric) moments of low to moderate order and one asks forthe reconstruction of fine details modeled by zero- or positive-dimensional alge-braic varieties. Often, such reconstruction problems have a generically uniquesolution when the number of data is larger than the degrees of freedom in themodel. Beyond that, we concentrate on simple a-priori conditions to guaranteethat the reconstruction problem is well or only mildly ill conditioned. For the re-construction of points on the complex torus, popular results ask the order of themoments to be larger than the inverse minimal distance of the points. Moreover,simple and efficient eigenvalue based methods achieve this stability numericallyin specific settings. Recently, the situations involving clustered points, points withmultiplicities, and positive-dimensional algebraic varieties are starting to gain in-terest, and these shall be studied in detail within this project.