Recovering Signals Sparse in a Frame: Theory and Applications

恢复帧中稀疏信号：理论与应用

• 批准号: 1908880
• 负责人: Xuemei Chen
• 金额: $ 12.35万
• 依托单位: New Mexico State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-08-15 至 2020-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1908880&HistoricalAwards=false
• 关键词: Recovering; Signals; Sparse; Frame; Theory

During the last two decades, there has been an explosive growth of the development of high dimensional data sensing, representation, and recovery. Problems such as medical imaging in the form of MRI or CT, hyperspectral satellite imaging, radar imaging, and remote sensing can involve large dimensional data where classically designed methods become impractical. The challenges of high dimensional data require techniques such as compressed sensing, which is able to extract the nonlinear low-dimensional structure of the signal. The investigator aims to further advance the scope of compressed sensing with an emphasis on signals that are sparsely represented by a collection of atoms, or a frame. This award will support the development of mathematical techniques for this problem as well as its applications on imaging while providing theoretical guarantees. The analysis in this project will apply broadly to signal processing problems like phase retrieval, one-bit sensing, and many data science problems in general. Compressed sensing aims to accurately and stably recover sparse signals from drastically undersampled measurements. The investigator will perform a theoretical analysis for recovering signals that are sparsely synthesized in a frame or dictionary from very few measurements and conduct numerical experiments on imaging applications where subsampled Fourier or convolution measurements are used. This is motivated by (1) numerous practical applications in MRI, MIMO radar, tomography, remote sensing, etc., where sparsity is exploited; (2) the benefit of robust signal acquisition and expansion in redundant frames; and (3) the lack of theoretical work on the synthesis-sparse sensing problem. The expected outcomes of this project are building a framework of this synthesis-sparse sensing problem, analyzing the effectiveness of randomized sensing, and improving image restoration quality with provable guarantees.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在过去的二十年中，高维数据感应，表示和恢复的发展爆炸性增长。诸如MRI或CT形式的医学成像，高光谱卫星成像，雷达成像和遥感的问题可能涉及大量的数据，而经典设计的方法变得不切实际。高维数据的挑战需要压缩传感等技术，该技术能够提取信号的非线性低维结构。研究人员旨在进一步推进压缩感测的范围，重点是由原子集合或框架占用的信号。该奖项将支持针对此问题的数学技术的发展，并在提供理论保证的同时，在成像上的应用。该项目中的分析将广泛用于信号处理问题，例如相位检索，一位感应和许多数据科学问题。 压缩传感旨在准确，稳定地从巨大的采样测量中恢复稀疏信号。研究者将对恢复在框架或字典中稀疏合成的信号进行理论分析，并从极少数测量值中进行，并在使用中采样的傅立叶或卷积测量值的成像应用上进行数值实验。这是由（1）在利用稀疏性的MRI，MIMO雷达，层析成像，遥感等中进行的众多实际应用； （2）在冗余框架中稳健的信号采集和扩展的好处； （3）缺乏关于综合传感问题的理论工作。该项目的预期结果是建立了该综合传感问题的框架，分析随机感应的有效性，并通过可证明的保证来改善图像恢复质量。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力和更广泛影响的评估来通过评估来获得支持的值得的。