CIF: Small: Collaborative Research: Compressed Sensing for High-Resolution Image Inversion

Princeton University, Princeton NJ

Investigators

Abstract

Abstract What physical, chemical, or biological configuration produced the measurements one has made or the images one has formed? This is a question of inverting an image for the field that produced it, and it arises in almost all fields of science and engineering. The emerging methodology of compressed sensing has opened up many applications in imaging science, signal processing, and networking. However, its applicability to high-resolution image inversion is as yet unproven. The objective of this research is to provide a comprehensive analysis of the performance of compressed sensing as an image inversion principle. The program is interdisciplinary, with signal processing forming the bridge between imaging science and mathematics. The theory of compressed sensing suggests that sub-sampling of an image of a physical field has manageable consequences for image inversion, provided that the image is sparse in a known basis. But in reality, no physical field is sparse in a known basis and therefore any presumed basis for sparsity is always mismatched to the actual sparsity basis chosen by the physics of the problem. This is called model mismatch. This research establishes bounds on the sensitivities to model mismatch of compressed sensing and compares its performance to more established principles of image inversion. The goal of the research is to establish quantitative trade-offs between basis over-fitting, compressed sampling rate, and robustness to mismatch. The research develops principles for compressed sensing that preserve the fidelity of inversions, even under conditions of mismatch. It extends the theory of compressed sensing from a first-order theory of modeling to a second-order theory for sparse covariance and frequency-wave-number spectrum estimation.