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.
This project took place at two universities. The Princeton participation has involved a variety of contributions, most of which are described in the report by the collaborating PI's. Due to some personnel changes related to this project, we found ourselves with the opportunity to branch out in the study of compression to include the study of compression with security constraints. Several important results were discovered with respect to compression for security. First, information theory can provide guarantees of security for compressed information in the form of a minimal level of distortion that an eavesdropper would incur if they tried to decode it (without access to the secret key or private, secure communication). This is in contrast to the conventional method of security used in the literature, which is quantified by entropy. Second, the distortion approach to secure compression provides not only a deeper understanding of how to compress information but in fact can be shown to contain the entropy approach as a special case. In other words, the theory provided by this work generalizes the theory on the subject in the information theory literature. Third, a new encoder, refered to as the likelihood encoder, which was designed for the perpose of obtaining secure compression, has proven to be a useful encoder for the general theory of compression, even without security concerns. In addition to the above intellectual contributions of this work, the project has funded multiple Ph.D students. Additionally, the PI's have worked with undergraduates on research related to this project.
