High resolution signal and image recovery: Fast algorithms and analysis

In this dissertation we address three issues arising in signal recovery problems: developing fast and efficient algorithms for convex set constrained signal recovery, analyzing resolution limits in signal recovery algorithms, and developing new regularization techniques for reducing the ill effects of noise in signal recovery algorithms.;For convex set constrained signal recovery, we develop an approach that is well-suited for applications with a limited number of measurements. We show that convex set constraints do indeed improve resolution in signal recovery. Next, we develop a quadratically convergent Newton algorithm to compute the reconstruction that is consistent with the convex set constraints and measured data, and that is closest to a known nominal signal. We also suggest suitable modifications to the algorithm in order that it has the desired local and global convergence properties and is computation and memory efficient. We demonstrate the algorithm on several practical applications.;We next investigate the key issue of resolution limits in signal recovery. The classical Rayleigh limit serves only as a lower bound on the achievable resolution. We show that in the ideal situation in which infinitely many noise-free measurements are available, the resolution limit depends only on the intersample spacing and not on the shape and width of the sampling kernel. In the practical situation of finitely many noise-corrupted measurements, we show that details finer than the Rayleigh resolution limit can be recovered by simple linear processing. In the process, we derive an algorithm for high resolution signal recovery (from linear measurements) and show how one can precompute worst-case error bounds and resolution ability of the algorithm. We illustrate the results on one-dimensional and two-dimensional examples.;Most reconstruction algorithms require solution of a linear system of equations that is almost always highly ill-conditioned and, hence, very sensitive to noise. We investigate the reduced rank SVD-based regularization scheme and present a new technique for rank selection in this scheme. The selection is based on maximizing a similarity measure. The similarity measure is constructed by enforcing our belief that signals with large norms are less likely than ones with small norms. The method is tested extensively for the bandlimited extrapolation application with much success.