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Abstract

Compressed sensing is a new sampling paradigm of mathematical signal processing which, under certain assumptions, allows signal recovery from highly undersampled measurements. The extension of the mathematical theory and the analysis and development of new applications in many fields are the subject of numerous international research activities. In this thesis an industrial problem from experimental fluid dynamics is consider, exemplarily. The current state of the art methodology solves the problem in two independent stages: First it recovers particle images by nonstandard tomography, and secondly it estimates the motion between two given time points. This motivates the problem of joint signal and motion estimation while raising theoretical questions in compressed sensing related to the recovery of sparse time-varying signals. In particular, two different approaches are presented for recovering a time-varying signal and its motion from undersampled linear measurements taken at two different points in time. The first approach formulates a problem at hand as optimal transport between two indirectly observed densities with a physical constraint. Several methods are proposed to integrate the projection constraints into the convex optimization framework of Benamou and Brenier. In the second approach, the signal is modeled as if observed by the real sensor specified by the experimental setup and an additional virtual sensor due to motion. The combination of these two sensors is called compressed motion sensor and its properties are examined from the viewpoint of compressed sensing. It is shown that in compressed motion sensing (CMS), besides sparsity, a sufficient change of signal leads to recovery guarantees and it is demonstrated that the compressed motion sensor at least doubles the performance of the real sensor. Moreover, for certain sparsity levels the signal motion can be established, too.