CAREER: Super-Resolution and Subwavelength Imaging

研究者的目的是将信号估计的超分辨率现象放在定量的基础上，并探讨可靠的超分辨率在层析成像中的后果。当信号的精细结构在小于频带宽度逆的长度尺度上恢复时（即所谓的香农-奈奎斯特缩放），根据其在频带中的傅里叶变换的知识对信号进行估计就有资格获得超分辨。这是一个很难理解的，但却是真实的，绝对是非线性的数值现象。稀疏性被认为是可靠地产生超分辨率的原理。然而，最近对凸规划的分析表明，第二个原则，符号兼容性，需要与稀疏性一起解锁积极的结果。在此基础上，研究者寻求一种新的定量保证，以成功地将信号恢复到理想的分辨率水平，这是任何方法都无法比拟的。这样的结果将推广压缩感知理论在非常相干字典的制度，没有限制等距类型的性质可以预期持有。研究者还研究了基于平移不变性的算法，如矩阵铅笔法、修改匹配追踪、预校正滤波和高维推广。他探讨了层析成像模式，如地震和合成孔径雷达在亚波长范围内的积极和消极的影响。该项目旨在设计先进的非线性处理方法，在数据频率较差的情况下，例如在模糊的情况下，提供更高的图像分辨率和解释水平。对超分辨率提供的可能性的系统理解最终可能会改变从大量反问题数据中提取信息的方法，从医学成像（核磁共振成像，荧光显微镜）到地球物理成像（勘探地震学）和国防应用。该项目还为学生提供了在计算数学、现代数据处理和层析成像交叉领域获得跨学科训练的机会。研究生课程通常不能以令人满意的方式涵盖这些不断发展的材料，尽管工业界对具有定量思维的专家有很大的需求，这些专家可以在所有这些方面都发挥专家的作用。通过组织研究生暑期学校，管理面向少数民族的实习项目，以及通过麻省理工学院的在线教育平台OCW和EdX进行远程教学，研究者参与了几项倡议，以广泛介绍这一新的研究领域。

Demanet1255203 The investigator aims to put the super-resolution phenomenon of signal estimation on a quantitative footing, and to explore the consequences of reliable super-resolution in tomographic imaging. The estimation of a signal from the knowledge of its Fourier transform in a band qualifies as super-resolved when the fine structure of the signal is recovered on a length scale smaller than the inverse of the band's width --- the so-called Shannon-Nyquist scaling. It is a poorly understood, yet real and decidedly nonlinear numerical phenomenon. Sparsity has been suggested as a principle according to which super-resolution reliably occurs. However, recent insights from analysis of convex programs indicate that a second principle, sign compatibility, is needed in tandem with sparsity to unlock positive results. Building on this, the investigator seeks quantitative guarantees of a new kind for successful signal recovery up to the ideal resolution level that no method can beat. Such results would generalize the theory of compressed sensing in the regime of very coherent dictionaries, where no restricted isometry-type property can be expected to hold. The investigator also studies algorithms based on translation-invariance such as the matrix pencil method, modified matching pursuits, precorrective filtering, and generalizations to higher dimensions. He explores implications, both positive and negative, for tomographic imaging modalities such as seismic and synthetic aperture radar in the subwavelength regime. The project aims to design advanced, nonlinear processing methods to deliver higher image resolution and interpretation levels in situations where the data come with poor frequency content, such as in the case of a blurring. A methodical understanding of the possibilities offered by super-resolution could eventually transform the methods by which information is extracted from data in a host of inverse problems, ranging from medical imaging (MRI, fluorescence microscopy) to geophysical imaging (exploration seismology) and defense applications. The project also offers an opportunity for students to acquire strong interdisciplinary training at the intersection of computational mathematics, modern data processing, and tomographic imaging. Graduate curricula typically fall short of covering this evolving material in a satisfactory way, even though there is a great demand in industry for quantitative-minded specialists who can operate as experts on all these fronts. The investigator is involved in several initiatives to introduce this new research area widely, through the organization of graduate summer schools, the administration of an internship program geared toward minorities, and remote teaching via MIT's online educational platforms OCW and EdX.