RaSenQuaSI: Randomized Sensing and Quantization of Signals and Images

在数学信号处理的许多最近的工作中的一个共同的范例是，随机化的方法做了很好的工作，捕获相关的信号信息，而确定性设置产生相当差的性能。最突出的例子当然是压缩感知。这个年轻的区域来自于对稀疏信号的观察，即，其可以被表示为给定表示系统的仅仅几个元素的线性组合，可以从随机选择的少量线性测量中有效地恢复。到目前为止，这一理论已经发现了多方面的应用，如磁共振成像（MRI），雷达和遥感。随机方法成功的另一个例子是相位恢复问题，最近受到了广泛关注。这里，仅观察线性测量的测量幅度，而不是符号或相位信息。这个问题特别出现在物理学的应用中，如X射线晶体学。虽然在所有这些应用中，可以将有限量的随机性引入到利用自由度的系统中，但是测量的某些结构通常由应用强加。例如，对于MRI，可以通过傅立叶系数对测量进行建模，并且仅可以随机选择所选择的频率。这种具有额外结构的随机系统将在该项目中发挥核心作用。该项目的一个重要方面是如何将模数转换纳入过程。即，为了由计算机处理，测量需要被量化，即，由来自有限字母表的有限数目的符号表示。对于压缩感知，这主要是针对没有强加结构的测量，因此主要目标将是研究面向应用的结构化场景。对于相位恢复，这种方法将是全新的。而且，最近已经观察到结构化测量系统的采样策略依赖于稀疏诱导表示系统。最后，对于相位恢复，对随机方法的研究才刚刚开始。该项目旨在为结构化系统的恢复保证以及算法方面做出贡献。

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.