RaSenQuaSI: Randomized Sensing and Quantization of Signals and Images

RaSenQuaSI：信号和图像的随机传感和量化

• 批准号: 254873217
• 负责人: Professor Dr. Felix Krahmer, Ph.D.
• 金额: --
• 依托单位: Department of Mathematics
• 依托单位国家: 德国
• 项目类别: Independent Junior Research Groups
• 财政年份: 2014
• 资助国家: 德国
• 起止时间: 2013-12-31 至 2019-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/254873217?language=en
• 关键词: RaSenQuaSI; Randomized; Sensing; Quantization; Signals

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.

在最近的数学信号处理工作中，一个常见的范例是随机方法在捕获相关信号信息方面做得很好，而确定性设置产生了相当差的性能。最突出的例子当然是压缩感知。这个年轻的领域源于对稀疏信号的观察，即可以表示为给定表示系统中仅几个元素的线性组合，可以有效地从随机选择的少量线性测量中有效地恢复。迄今为止，这一理论已经在磁共振成像（MRI）、雷达和遥感等领域得到了广泛的应用。随机化方法成功的另一个例子是相位检索问题，这个问题最近受到了广泛的关注。这里只观察到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诸如x射线晶体学等物理应用中。虽然在所有这些应用中，利用自由度可以将有限数量的随机性引入系统，但测量的某些结构通常是由应用程序强加的。以核磁共振成像为例，测量结果可以用傅里叶系数来建模，并且只能随机选择所选的频率。这种带有附加结构的随机系统将在项目中发挥核心作用。一个重要的方面，将是中心的项目是如何纳入模拟到数字转换的过程。也就是说，为了被计算机处理，测量需要被量子化，即由有限字母中的有限数量的符号表示。对于压缩感知，这主要用于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。对于相位检索，这种方法将是全新的。此外，最近观察到结构化测量系统的采样策略依赖于稀疏性诱导表示系统。对于不同的系统，我们将详细研究这种对应关系。最后，对于相位检索，随机化方法的研究才刚刚开始。该项目旨在为开发结构化系统的恢复保证以及算法方面做出贡献。