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RaSenQuaSI: Randomized Sensing and Quantization of Signals and Images

Subject Area Mathematics
Term from 2014 to 2020
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 254873217
 
Final Report Year 2024

Final Report Abstract

The Emmy Noether Junior Research Group Randomized sensing and quantization of signals and images led by Felix Krahmer investigated the interplay between structure and randomness in signal processing applications. The goal was to understand how random measurement design with appropriate reconstruction algorithms can allow for recovery guarantees. Key feature of the group’s research agenda was a unified viewpoint where random choice of design parameters in line with structural constraints posed by the applications is combined with well-adapted reconstruction algorithms. The three pillars of the agenda revolved around compressive sensing, quantization, and phase retrieval. Compressive sensing studies the reconstruction of signals that admit particularly simple representations from subsampled measurements. Major challenges successfully addressed by the group include the mathematical performance analysis of reconstruction methods for data which is noisy with unknown noise level and reconstruction methods that are efficient enough to deal with very high-dimensional data. The quantization problem incorporates digital representations into the picture and again asks the question how well the signal can be recovered from the representing bits. The group provided the first near-optimal error analysis for Sigma-Delta quantization for a structured random compressive sensing systems, an application-driven recursive quantization scheme and also was among the first to study quantization of randomly sampled signals of limited frequency range. A different challenge arising in nano-scale imaging, is that only intensities can be measured and the phase information is lost. The resulting phase retrieval problem was studied by the group both from a theoretical viewpoint, aiming to understand the general limitations, and in the more practical context of ptychography where a concrete scenario of multiple masked measurements is considered. Results of the group include design and analysis of a method to address limitations arising from conditioning problems and symmetries. From a mathematical point of view, the considered problems gave rise to challenging problems at the interface of many mathematical disciplines. The work of the group combines techniques from probability theory, optimization, Banach space geometry, approximation theory and Fourier analysis, and also established results of independent mathematical interest, for example the characterization of the geometry of random polytopes spanned by heavy-tailed random vectors.

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