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.
