Robust reconstruction techniques for nonuniformly sampled data

在这项研究中，PI开发并分析了用于从非均匀获取的多维数据进行稳健重建的数学和数值框架。这项工作的一个特别的重点是基于小波的，基于无限维(即模拟/连续)信号和图像模型的稀疏性利用算法。研究目标包括：(I)引入一个全面的数学采样理论，用于非均匀数据在任意基数和任意帧上的稳定、可实现的重建；(Ii)将压缩感知理论和技术扩展到非均匀和非理想数据，以及将压缩感知继续发展到无限维环境；(Iii)实现和分析基于数值线性代数的高效重建算法；以及(Iv)为均匀和非均匀数据的稳定重建建立新的基本障碍。这项研究将提供彻底的数学分析，特别是关于准确性和稳定性的关键问题。在许多不同的领域，包括医学成像、层析成像、地震成像、雷达和天文学，数据收集是不统一的。例如，在医学成像中，非均匀采样几何结构允许快速、更高分辨率的扫描，而对噪声和伪影的敏感性较低。然而，用于从这些数据重建图像的标准算法往往存在严重缺陷，特别是在准确性和对噪声和其他误差的稳健性方面。这可能会导致错误的图像配准，并且在医学成像中会导致误诊。本项目介绍了非标准和非均匀采样数据的新的和改进的算法，特别强调稀疏性开发方法，并解决了从这些数据重建图像的基本数学分析。这项工作的好处包括：(1)发展了更现实的采样和压缩传感理论，更接近和代表实践者的需要，以及(2)引入了重建算法，具有更好的重建质量，更短的数据采集时间和更好的在存在噪声和扰动的情况下的稳健性。

In this research the PI develops and analyzes mathematical and numerical frameworks for robust reconstructions from nonuniformly-acquired, multidimensional data. A particular focus of this work is on wavelet-based, sparsity-exploiting algorithms based on infinite-dimensional (i.e. analog/continuous) signal and image models. Research objectives include (i) introducing a comprehensive mathematical sampling theory for stable, realizable reconstructions in arbitrary bases and frames from nonuniform data, (ii) extending compressed sensing theory and techniques to nonuniform and nonideal data, as well as continuing the development of compressed sensing to the infinite-dimensional setting, (iii) implementing and analyzing efficient algorithms for reconstruction based on numerical linear algebra, and (iv) establishing new fundamental barriers for stable reconstructions from uniform and nonuniform data. The research will provide thorough mathematical analysis, in particular as regards the key issues of accuracy and stability.In many different areas, including medical imaging, tomography, seismic imaging, radar, and astronomy, data is collected nonuniformly. In medical imaging, for example, nonuniform sampling geometries allow for fast, higher-resolution scans with lower susceptibility to noise and artifacts. However, standard algorithms used for image reconstruction from such data often have critical shortcomings, especially as regards accuracy and robustness to noise and other errors. This can lead to incorrect image registration and, in medical imaging, misdiagnosis. This project introduces new and improved algorithms for non-standard and nonuniformly sampled data, with a particular emphasis on sparsity-exploiting methods, and addresses the fundamental mathematical analysis of image reconstruction from such data. The benefits of this work include (i) the development of a more realistic theory of sampling and compressed sensing that is closer to and more representative of the practitioner's needs, and (ii) the introduction of reconstruction algorithms with superior reconstruction quality, lower data acquisition times and improved robustness in the presence of noise and perturbations.