Sparse super resolution in microscopy: Condition, diffraction limit and trigonometric approximations

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Title: Sparse super resolution in microscopy: Condition, diffraction limit and trigonometric approximations
Authors: Hockmann, Mathias
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Thesis advisor: Prof. Dr. Stefan Kunis
Thesis referee: Prof. Dr. Helmut Bölcskei
Abstract: This thesis concerns the development of two ideas to explain the diffraction limit rigidly and it reveals that a deterministic study of the condition shows the same transition behaviour around the classical Rayleigh diffraction limit as a statistical approach using the Cramer-Rao lower bound. Surprisingly, the analysis of both approaches bases on the construction of a certain minorant function with optimal support conditions. After explaining implications towards the analysis of singular values of Vandermonde matrices with clustering nodes, it deals with approaches to solve the super resolution problem through approximation by polynomials and rational functions. For this, estimates for the rate of weak convergence in the 1-Wasserstein distance and for pointwise convergence are shown. In the last part, it is demonstrated how the deterministic and the statistical point of view allow to explain the increased resolution of microscopy techniques like stochastic optical reconstruction microscopy (STORM) and structured illumination microscopy (SIM).
Subject Keywords: Super resolution; Condition; Resolution limit; Rayleigh limit; Cramer-Rao lower bound; Minorant function; Vandermonde matrix; Christoffel function; STORM; SIM
Issue Date: 23-Feb-2024
License name: Attribution 3.0 Germany
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Type of publication: Dissertation oder Habilitation [doctoralThesis]
Appears in Collections:FB06 - E-Dissertationen

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