Recovering Signals Sparse in a Frame: Theory and Applications

在过去的二十年里，高维数据传感、表示和恢复的发展取得了爆炸性的增长。核磁共振或CT形式的医学成像、高光谱卫星成像、雷达成像和遥感等问题可能涉及大维数据，在这些数据中，经典设计的方法变得不实用。高维数据的挑战需要压缩感知等技术，它能够提取信号的非线性低维结构。研究人员的目标是进一步推进压缩感知的范围，重点放在由一组原子或一个帧稀疏地表示的信号上。该奖项将支持这一问题的数学技术的发展及其在成像方面的应用，同时提供理论保证。本项目中的分析将广泛应用于信号处理问题，如相位恢复、一位传感，以及一般的许多数据科学问题。压缩感知的目标是从严重欠采样的测量中准确而稳定地恢复稀疏信号。研究人员将进行理论分析，以便从极少的测量中恢复在帧或字典中稀疏合成的信号，并在使用次采样傅立叶或卷积测量的成像应用中进行数值实验。这是由于(1)在MRI、MIMO雷达、层析成像、遥感等领域的大量实际应用，其中利用了稀疏性；(2)在冗余帧中稳健的信号捕获和扩展的好处；以及(3)缺乏关于合成稀疏传感问题的理论工作。该项目的预期成果是建立一个综合稀疏传感问题的框架，分析随机传感的有效性，并通过可证明的保证提高图像恢复质量。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.