Phase Retrieval by Alternating Direction Method of Multipliers

This dissertation aims at reconstructing a signal from the magnitude of its Fourier transform, known as phase retrieval. The problem arises in variety of areas such as crystallography, astronomy, optics, voice recognition, and coherent diffraction imaging (CDI). In particular, we focus on two types of phaseless measurements: short-time Fourier transform (STFT) and frequency-resolved optical gating (FROG). STFT takes the Fourier transform when passing a short-time window over a signal. When the window function is given, the problem is referred to as non-blind STFT, while blind STFT means to simultaneously estimate both the signal and the window from the magnitude measurements. FROG is closely related to STFT in such a way that the window function in FROG is just the signal itself. We apply alternating direction method of multipliers (ADMM) to solve all the aforementioned problems: non-blind STFT, blind STFT, and FROG. Specifically for the blind STFT, we discuss three approaches to address the scaling ambiguity. We also consider a special type of signals that has only a few non-zero elements by minimizing the L1 norm to promote sparsity in the objective function. Numerical experiments are provided to demonstrate the proposed algorithms outperform the state-of-the-art in non-blind STFT and FROG. As the blind STFT is one of the first kind, we compare the performance of the three proposed approaches.