In this dissertation, we present and study two novel approaches to phase retrieval -- an inverse problem in which one attempts to reconstruct a complex-valued function (or vector) from phaseless (or magnitude-only) measurements. Phase retrieval arises in several scientific areas including bio-chemistry, optics, astronomy, quantum mechanics, and speech signal processing. Early solutions to phase retrieval, although practical, lacked robustness guarantees. To this day, practitioners in scientific imaging are still seeking demonstrably stable and robust recovery algorithms. Ptychography is a form of coherent diffractive imaging where diffraction patterns are processed by algorithms to recover an image of a specimen. More specifically, small regions of a specimen are illuminated one-at-a-time, and a detector captures the intensities of the resulting diffraction patterns. As such, the measurements are local and phaseless. In this work, we present two algorithms to recover signals from ptychographic measurements. The first algorithm aims to recover a discrete one-dimensional signal from discrete spectrogram measurements via a modified Wigner distribution deconvolution (WDD) method. While the method is known to practitioners of scientific imaging, robustness and recovery guarantees are lacking, if not absent; our contribution is to supply such guarantees. The second algorithm aims to approximately recover a compactly supported function from continuous spectrogram measurements via lifting and angular synchronization. This setup can be interpreted as the infinite-dimensional equivalent of discrete ptychographic imaging. Our contribution is a model which assumes infinite-dimensional signals and measurements ab initio, as opposed to most recent algorithms in which discrete models are a necessity. Finally, we consider the worst-case noise robustness of any phase retrieval algorithm which aims to reconstruct all nonvanishing vectors from the magnitudes of an arbitrary collection of local correlation measurements. The robustness results provided therein apply to a wide range of ptychographic imaging scenarios. In particular, our contribution is to show that stable recovery of high-resolution images of extremely large samples is likely to require a vast number of measurements, independent of the recovery algorithm employed. The first chapter introduces the phase retrieval problem and presents historical context, as well as applications in which phase retrieval manifests. In addition, we introduce ptychography, discuss existing WDD formulations, and compare these to our contribution in the discrete setting. Chapter 2 provides recovery guarantees for using aliased WDD methods to solve the phase retrieval problem in a discrete setting with sub-sampled measurements. In Chapter 3 we provide lower Lipschitz bounds for generic phase retrieval algorithms from locally supported measurements. Finally, Chapter 4 presents a numerical method to recover compactly supported functions from local measurements via lifting and angular synchronization.
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