RaSenQuaSI: Randomized Sensing and Quantization of Signals and Images

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A common paradigm in many recent works in mathematical signal processing is that randomized approaches do a good job capturing relevant signal information while determinstic setups yield a considerably worse performance. The most prominent example is certainly compressed sensing. This young area arose from the observation that signals which are sparse, i.e., which can be represented as a linear combination of just few elements of a given representation system, can be efficiently recovered efficiently from a small number of linear measurements chosen at random. To date, this theory has found manifold applications such as Magnetic Resonance Imaging (MRI), radar, and remote sensing. Another example for the success of randomized approaches is the problem of phase retrieval, which has received much attention recently. Here only the measurement amplitudes, not the sign or phase information, of the linear measurements are observed. This problem arises particularly in applications in physics such as X-ray crystallography. While in all these applications a limited amount of randomness can be introduced into the system exploiting the degrees of freedom, some structure of the measurements is usually imposed by the application. For MRI, for example, the measurements can be modeled by Fourier coefficient, and only the frequencies selected can be chosen at random. Such randomized systems with additional structure will play a central role in the project.An important aspect that will be central to the project is how to incorporate analog to digital conversion into the process. Namely, in order to be processed by a computer, the measurements need to be quantized, i.e., represented by a finite number of symbols from a finite alphabet. For compressed sensing, this has mainly been done for measurements without imposed structure, so a main goal will be to study application oriented structured scenarios. For phase retrieval, such approaches will be completely new.Also, the sampling strategies for structured measurement systems have recently been observed to depend on the sparsity inducing representation system. This correspondence will be studied in detail for various systems.Lastly, for phase retrieval, research on randomized approaches is only at its beginnings. The project intends to contribute to developing recovery guarantees for structured systems as well as to algorithmic aspects.