The PIs build on their recently developed techniques that fuse methods from numerical approximation and inverse theory, such as for purposes of reconstruction fidelity, with those that exploit specific application information, such as sparsity. Their methods address both theoretical and practical considerations. Algorithms are proposed for function and/or image recovery, as well as to characterize and extract important information from a data set, such as edges, or other features, without necessarily determining the underlying function. Research objectives include (i) developing novel approximation approaches for functional and/or feature recovery from data that is deficient with respect to one or multiple perspectives, i.e. data is under-sampled or missing, may be noisy, is measured via a dual representation, or is otherwise non-standard with respect to traditional numerical approximation techniques; (ii) developing numerical approximation operators that can be directly applied to practical data sets, or may provide a feedback to practitioners for improving sampling protocols; and (iii) integrating techniques from numerical linear algebra and statistical regularization that are specifically pertinent for obtaining robust but efficient solutions of ill-conditioned problems when handling practical data. This research will provide rigorous analysis of all new algorithms in terms of accuracy, efficiency, and robustness, especially in the presence of of noise, perturbations, or otherwise incomplete data information.
Practical data collection techniques are becoming increasingly more sophisticated. User friendly software packages allow disciplinary scientists to successfully diagnose, predict, model, and determine important characteristics from a plethora of measured data. Yet, recent investigations into various reconstruction algorithms for data collected under modern magnetic resonance imaging (MRI) protocols have clearly demonstrated shortcomings that arise when pragmatic algorithmic modifications are used without considering fundamental mathematical issues regarding accuracy and measurement error. Some currently employed algorithms in fact yield both incorrect diagnoses and additional procedural costs. This project extends the PIs prior research and addresses the development of novel mathematical techniques for handling issues associated with extracting functional and feature information from data acquired by non-standard sampling protocols.
