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.
A general question that arises in almost all science and engineering disciplines is this: 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 configuration or field that produced it. The emerging methodologies of compressed sensing and sparse inversion have found applications in imaging science, signal processing, and networking. The intellectual merit of this research project is its comprehensive analysis of the performance of compressed sensing and sparse inversion as useful image inversion principles. The program is interdisciplinary, with signal processing forming the bridge between imaging science and mathematics. The theories of compressed sensing and sparse inversion suggest 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 determined by the physics of the problem. This is called model mismatch. This research project has provided a comprehensive analysis of the performance of compressed sensing (CS) and sparse inversion as image inversion principles. The broad impact of our studies is that compressed sensing brings loss in signal-to-noise ratio, which manifests itself in poorer inversion quality, and that there is a predictable limit to the resolution that can be achieved in image inversion by replacing a nonlinear physical model by a fine-grained linear mathematical model. As a consequence, there is caution to be exercised when designing hardware and software based on the principles of compressed sensing and sparse inversion. Qualitatively, one may say that compressed sensing followed by sparse inversion appears to be a technology that may be successfully applied when the signal-to-noise ratio of measurements is high and the required resolution of the inversion is not high. This project supported or partially supported two faculty in electrical and computer engineering, and mathematics; and three PhD students, two in electrical and computer engineering and one in statistics. Two of these students have graduated and now hold professional appointments at CGG, a geophysical prospecting company, and MD Anderson Cancer Center. The third continues his studies at Colorado State University.
