Structured compressed sensing algorithms: design, analysis and applications

科学和工程中的许多问题需要从一系列测量数据中重建一个对象——例如图像、信号或高维函数。由于时间、成本或其他限制，可以收集的数据量往往受到严重限制，这严重影响了人们准确恢复未知物体的能力。本研究计划涉及基于压缩感知（CS）理论和技术的新算法的开发、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度量化、机器学习和偏微分方程的数值解。它的首要目标是为这些应用引入新的、计算效率高的数值优化技术，这些技术具有更好的精度和更低的采集时间和成本。****具体目标***1)设计和实施新一代基于cs的成像算法，在采样和恢复过程中都包含额外的结构。利用这种结构，这项工作有望对当前最先进的算法进行实质性改进，在核磁共振成像、x射线CT、电子和荧光显微镜等关键成像技术上产生切实的好处。***2)开发和研究新的基于cs的高维近似方法，利用结构化平滑稀疏先验和随机抽样技术来提高精度。随着数据收集变得越来越容易和广泛，在科学和工程中迫切需要通过近似高维函数来理解日益复杂的现象。这项工作的成果将改进该问题的技术，为物理（如生物、机械或流体）系统中的不确定性量化等重要的实际任务带来好处。***3)研究科学计算中基于采样的算法的稳定性和准确性的极限，并设计新的基于保角映射的计算方法来达到这一极限。基于采样的算法在科学计算中有多种用途，包括表面重建、偏微分方程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可以实现的理论极限来增强知识，它的好处将是基于这些极限的一组改进的方法。总的来说，这个研究项目的重点是开发新的算法来解决具有挑战性的面向数据的问题。它的目标是在科学、工程和医学等国家需要的关键领域的一系列应用中带来效益。通过HQP的培训，这项研究也将有助于满足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.**