THE RECOVERY OF SIGNALS FROM SATURATED LINEAR MEASUREMENTS

Abstract

This dissertation is committed to the study of two situations where this problem occurs, saturation recovery and phase retrieval. In saturation recovery, the goal is to recover a vector when frame coefficients whose magnitude exceeds a certain threshold λ are clipped. In terms of this problem, characterizations of frames that perform saturation recovery are described, upper and lower bounds on the critical saturation level are examined, and iterative algorithms that implement saturation recovery are presented. In phase retrieval, the goal is to recover a vector when the phase of the frame coefficients is lost. The main result in this area establishes new quantitative bounds on how the stability constant for phase retrieval is affected by a small perturbation of the frame vectors. These bounds are significant in that they are independent of the dimension of the Hilbert space and the number of vectors in the frame.