Robust reconstruction techniques for nonuniformly sampled data

在这项研究中，PI开发和分析了数学和数值框架，用于从非均匀采集的多维数据中进行鲁棒重建。 这项工作的一个特别重点是基于小波的，稀疏利用算法的基础上无限维（即模拟/连续）的信号和图像模型。 研究目标包括：（i）引入一种全面的数学采样理论，用于从非均匀数据中在任意基和帧中进行稳定的、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，以及继续将压缩感知发展到无限维环境，（iii）实现和分析基于数值线性代数的重建的有效算法，以及（iv）为从均匀和非均匀数据进行稳定重建建立新的基本障碍。 该研究将提供全面的数学分析，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，断层扫描，地震成像，雷达和天文学，数据收集不均匀。 例如，在医学成像中，非均匀采样几何形状允许快速、更高分辨率的扫描，同时对噪声和伪影的敏感性更低。 然而，用于从这样的数据进行图像重建的标准算法通常具有严重的缺点，特别是关于精度和对噪声和其他误差的鲁棒性。 这可能导致不正确的图像配准，并且在医学成像中导致误诊。 该项目介绍了新的和改进的算法，为非标准和非均匀采样数据，特别强调稀疏利用方法，并解决了基本的数学分析，从这些数据的图像重建。 这项工作的好处包括（i）一个更现实的采样和压缩传感理论的发展，更接近和更代表从业者的需求，以及（ii）引入重建算法，具有上级重建质量，降低数据采集时间和改善的鲁棒性，在噪声和扰动的存在。

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.