CAREER: Practical Compressive Signal Processing

“信号”是人们想要获取的任何数据集，例如，图像、大数据块或音频剪辑。我们可以想象一下，我们需要多快地对音频片段进行采样，以便仅从这些样本中就能准确地恢复音频片段。你需要每纳秒、每毫秒还是每秒钟采样一次吗？压缩信号处理（CSP）表明，可以从比传统认为的更少的样本中获得和恢复许多信号中的重要信息。CSP的应用非常广泛，包括成像（医学、高光谱、显微镜、生物）、模拟-信息转换、雷达、大规模信息合成、地球物理数据分析、计算生物学等等。尽管这些应用是惊人的，但在CSP的理论工作和CSP在实际环境中的使用之间存在脱节。该项目的目标是通过为CSP提供适用于现实世界信号和设置的方法和分析来弥合这一差距。这项工作将缩短MRI的扫描时间，降低计算基础设施的成本和能耗，改进高光谱图像对病虫害作物的检测，提高雷达的精度，并改善许多其他大数据应用的压缩和分析。此外，该项目将涉及各个层次的学生，并向他们介绍严谨的科学研究。PI积极从代表性不足的人群中招募成员，并将继续通过她自己的研究和外展项目促进多样性。早期的CSP模型将信号的类别限制为在非常特定的意义上可压缩的信号（相对于标准正交或非相干基的稀疏）。这个项目的一个目标是放宽这个限制，以允许在实践中实际遇到的信号，例如冗余的、连贯的和高度过完备的字典中的稀疏信号。我们将利用贪婪方法和基于优化的方法，为特定的感兴趣的字典量身定制，以及针对任意基的更通用的方法。此外，该项目将开发自适应CSP采样方案，其中信号的测量是“在飞行中”设计的，因为它们正在进行。传统的测量方案忽略了这些信息，而自适应方案有可能显著减少重建误差、测量次数和计算时间。我们将确定约束和非约束设置的最佳测量策略，并从信息论的角度分析人们实际上可以从适应性中获得多少。该项目还将涉及“一位CSP”的工作，这是CSP的一个令人兴奋的新分支，可以处理极端（通常更现实）的量化。我们将利用亚线性方法，其中大误差自然出现，并且还使用基于优化的技术以及自适应量化阈值将恢复误差降低到非自适应量化的最佳可能以下。通过对这些课题的研究，本研究将弥补CSP理论的巨大空白，并为实践者和研究者提供一个统一的框架。

A "signal" is any data set that one would like to acquire, for example, an image, a large block of data, or an audio clip. One can imagine asking how quickly one would need to sample an audio clip so that from those samples alone, the audio clip could be accurately recovered. Would you need to sample every nanosecond, every millisecond, or every second? Compressive Signal Processing (CSP) shows that the important information in many signals can be obtained and recovered from far fewer samples than traditionally thought. The applications of CSP are widespread and include imaging (medical, hyperspectral, microscopy, biological), analog-to-information conversion, radar, large scale information synthesis, geophysical data analysis, computational biology, and many more. Although these applications are astounding, there has been a disconnect between the theoretical work in CSP and the use of CSP in practical settings. The goals of this project will bridge this gap by providing methods and analysis for CSP that apply to real-world signals and settings. Such work will lead to decreased scan time in MRI, reduced cost and energy consumption in computing infrastructures, improved detection of diseased crops from hyperspectral images, increased accuracy in radar, and improved compression and analysis in many other large-data applications. In addition, this project will involve students at all levels and introduce them to rigorous scientific research. The PI actively recruits members from under-represented populations, and will continue to promote diversity through her own research and outreach programs.Early CSP models restrict the class of signals to those compressible in a very specific sense (sparse with respect to an orthonormal or incoherent basis). One goal of this project is to relax this restriction to allow for signals actually encountered in practice, such as those sparse in redundant, coherent, and highly overcomplete dictionaries. We will utilize both greedy approaches and optimization-based methods, tailored to specific dictionaries of interest, as well as more general methods for arbitrary bases. In addition, this project will develop adaptive CSP sampling schemes, where measurements of the signal are designed "on the fly," as they are being taken. Traditional measurement schemes ignore this information, while adaptive schemes have the potential to significantly reduce reconstruction error, number of measurements, and computation time. We will identify optimal measurement strategies for constrained and unconstrained settings, and analyze how much one can actually gain from adaptivity from an information theoretic point of view. The project will also involve work in "one-bit CSP", a new and exciting branch of CSP which handles extreme (and often more realistic) quantization. We will draw on sub-linear methods, where large errors appear naturally, and also use optimization based techniques along with adaptive quantization thresholds to reduce the recovery error below the best possible for non-adaptive quantization. In studying these topics, the research will bridge a large gap in the theory of CSP and provide a unified framework for both practitioners and researchers.