Structured compressed sensing algorithms: design, analysis and applications

科学和工程中的许多问题需要从测量集合中重建对象--例如，图像、信号或高维函数。由于时间、成本或其他方面的限制，人们往往受到可收集的数据量的严重限制，这严重影响了人们准确恢复未知对象的能力。本研究项目涉及基于压缩感知(CS)理论和技术的针对该问题的新算法的开发、分析和应用。相关应用的例子包括医学成像、显微镜、物理系统中的不确定性量化、机器学习和偏微分方程组的数值解。它的首要目标是为这类应用引入新的、计算高效的数值优化技术，这些技术既具有更高的精度，又具有较低的获取时间和成本。*具体目标*1)设计和实施新一代基于CS的成像算法，该算法在采样和恢复过程中都包含额外的结构。通过利用这种结构，这项工作有望带来对当前最先进算法的实质性改进，在磁共振成像、X射线CT以及电子和荧光显微镜等关键成像技术方面产生切实的好处。*2)开发和研究基于CS的高维近似的新方法，该方法利用结构平滑-稀疏先验和随机采样技术来提高精度。随着数据收集变得更容易和更广泛，科学和工程中迫切需要通过逼近高维函数来理解日益复杂的现象。这项工作的成果将是这一问题的改进技术，为重要的实际任务带来好处，如物理(例如生物、机械或流体)系统中的不确定性量化。*3)调查科学计算中基于采样的算法的稳定性和准确性的限制，并设计基于保角映射的新计算方法，以达到这些限制。基于采样的算法在科学计算中有多种用途，包括曲面重建、偏微分方程组的数值方法和数值软件。这项工作将通过更好地理解任何算法对此问题可达到的理论极限来增强知识，其好处将是基于这些极限的一套改进的方法。*总的来说，本研究计划的重点是为挑战面向数据的问题开发新的算法。它的目的是在国家需要的科学、工程和医学的关键领域的一系列应用中带来好处。这项研究还将通过HQP的培训，促进学术界和工业界对这些领域技能的迫切需求。

Many problems in science and engineering require the reconstruction of an object - an image, signal or high-dimensional function, for example - from a collection of measurements. Due to time, cost or other constraints, one is often severely limited by the amount of data that can be collected, which significantly affects ones ability to recover the unknown object accurately. This research program involves the development, analysis and application of new algorithms for this problem, based on the theory and techniques of compressed sensing (CS). Examples of relevant applications include medical imaging, microscopy, uncertainty quantification in physical systems, machine learning and the numerical solution of PDEs. Its overarching objective is to introduce new, computationally-efficient numerical optimization techniques for such applications that possess both better accuracy and lower acquisition time and cost. ****Specific objectives***1) To design and implement a new generation of CS-based algorithms for imaging that incorporate additional structure in both the sampling and recovery process. By leveraging such structure, this work is expected to bring substantial improvements over current state-of-the-art algorithms, yielding tangible benefits in key imaging technologies such as MRI, X-ray CT and electron and fluorescence microscopy.***2) To develop and study new CS-based methods for high-dimensional approximation that exploit structured smoothness-sparsity priors and randomized sampling techniques to enhance accuracy. As data collection becomes easier and more widespread, there is a pressing need in science and engineering to understand increasingly complex phenomena by approximating high-dimensional functions. The outcomes of this work will be improved techniques for this problem, bringing benefits to important practical tasks such as uncertainty quantification in physical (e.g. biological, mechanical or fluid) systems.***3) To investigate the limits of stability and accuracy for sampling-based algorithms in scientific computing, and to design new computational methods based on conformal mappings that attain such limits. Sampling-based algorithms have a variety of uses in scientific computing, including surface reconstruction, numerical methods for PDEs and numerical software. This work will enhance knowledge through a better understanding of the theoretical limits achievable by any algorithm for this problem, and its benefit will be an improved set of methods based on such limits.***Overall, the focus of this research program is the development of new algorithms for challenging data-oriented problems. It aims to bring benefits in a range of applications in key areas of national need in science, engineering and medicine. This research will also contribute to the pressing need for skills in these areas, both in academia and industry, through the training of HQP.**