1-D and 2-D phase retrieval by solving linear systems of equations and by using the wavelet transform.

Abstract

The phase retrieval problem arises when a signal must be reconstructed from only the magnitude of its Fourier transform; if the phase information were also available, the signal could simply be synthesized using the inverse Fourier transform. The phase retrieval problem occurs in several areas of engineering and applied physics such as antenna array design, optical astronomy, electron microscopy, and x-ray crystallography. Although a unique relationship exists between a signal and its Fourier transform, the same cannot be said for a signal and the intensity of its Fourier transform. Given certain a priori information about the signal, the magnitude and phase of its Fourier transform are no longer independent functions. Compact support, real-valuedness, and non-negativity are some of the constraints that are often used to reconstruct the signal. In this thesis we describe our research concerning the solution of one- and two-dimensional, discrete and continuous time phase retrieval problems. In discrete phase retrieval we formulate the problem as a linear system of equations; our methods do not require polynomial rooting, tracking zero curves of algebraic functions, or any sort of iteration like previous methods. In continuous phase retrieval, most previous solutions rely on simply discretizing the problem and then employing an iterative algorithm. We avoid this approximation by using wavelet expansions to transform this uncountably infinite problem into a linear system of equations. The wavelet bases provide the following advantages: they easily allow incorporation of the a priori signal information; they provide a structured system of equations which permits a fast algorithm; and, they represent signals which are self-similar across scales (e.g. fractals) efficiently. Our solutions obviate the stagnation problems associated with iterative algorithms and our solutions are computationally simpler and more stable than alternative non-iterative algorithms. Moreover, our algorithms can accommodate noisy Fourier magnitude information and their performance with signals corrupted by noise is explored.