Recovering Signals Sparse in a Frame: Theory and Applications

在过去的二十年中，高维数据感知、表示和恢复的发展呈爆炸式增长。诸如MRI或CT形式的医学成像、高光谱卫星成像、雷达成像和遥感等问题可能涉及大维度数据，传统设计的方法变得不切实际。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。研究者的目标是进一步推进压缩传感的范围，强调由原子集合或框架稀疏表示的信号。该奖项将支持该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。本项目的分析将广泛应用于信号处理问题，如相位检索、位传感和许多数据科学问题。压缩感知旨在从严重欠采样测量中准确、稳定地恢复稀疏信号。研究者将对从很少的测量中恢复在帧或字典中稀疏合成的信号进行理论分析，并对使用次采样傅里叶或卷积测量的成像应用进行数值实验。这是由以下因素推动的：(1)在MRI， MIMO雷达，断层扫描，遥感等领域的大量实际应用，其中利用了稀疏性；(2)在冗余帧中进行鲁棒信号采集和扩展的好处；(3)合成-稀疏感知问题的理论研究不足。本项目的预期成果是构建该合成-稀疏感知问题的框架，分析随机感知的有效性，提高具有可证明保证的图像恢复质量。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.