Chapter 1 initially concerns the introduction of Fourier Phase Retrieval. In many imaging systems, one can only measure the magnitude-square of the Fourier transform of the underlying signal, known as the power spectral density. At a large enough distance from the imaging plane, the measurements are given by the Fourier transform of the image. Thus for capturing images of large distances, optical devices essentially measure the Fourier transform magnitude of the object being imaged. The problem of reconstructing a signal from its Fourier magnitude is known as Fourier phase retrieval. This problem arises in many areas of engineering and applied physics, including optics, X-ray crystallography (determining the atomic and molecular structure of a crystal), astronomical imaging, speech processing, computational biology, etc. Also, in Chapter 1, we introduce the concept of Dimensionality Reduction, which is a tool used in several disciplines, including statistics, data mining, pattern recognition, machine learning, artificial intelligence, and optimization. Dimensionality reduction refers to the act of transforming data from a high-dimensional space to a low-dimensional space. Chapter 2 discusses Fourier Ptychography, which is an imaging technique that involves sample being illuminated at different angles of incidence (effectively shifting the sample's Fourier transform) after which a lens acts as a low-pass filter, thereby effectively providing localized Fourier information about the sample around frequencies dictated by each angle of illumination. Near-Field (Fourier) Ptychography (NFP) occurs when the sample is placed at a short defocus distance having a large Fresnel number. We prove that certain NFP measurements are robustly invertible (up to an unavoidable global phase ambiguity) for specific Point Spread Functions (PSFs) and physical masks which lead to well-conditioned lifted linear systems. We then apply a block phase retrieval algorithm using weighted angular synchronization and prove that the proposed approach accurately recovers the measured sample for these specific PSF and mask pairs. Finally, we also propose using a Wirtinger Flow for NFP problems and numerically evaluate that alternate approach both against our main proposed approach, as well as with NFP measurements for which our main approach does not apply. We now move onto Chapter 3 concerning Blind Ptychography. Far-field Ptychography occurs when there is a large enough defocus distance (when the Fresnel number is much greater than 1) to obtain magnitude-square Fourier transform measurements. In an attempt to remove ambiguities, masks are utilized to ensure unique outputs to any recovery algorithm are unique up to a global phase. In Chapter 3, we assume that both the sample and the mask are unknown, and we apply blind deconvolutional techniques to solve for both. Numerical experiments demonstrate that the technique works well in practice, and is robust under noise. Finally, we have Chapter 4. Let M be a compact d-dimensional submanifold of RN. In this chapter, we prove that a nonlinear function f from RN into Rm exists as a Johnson-Lindenstrauss embedding with low distortion, for all x in M and y in RN. In effect, f not only serves as a bi-Lipschitz function from M into Rm with bi-Lipschitz constants close to one, but also approximately preserves all distances from points not in M to all points in M in its image. Furthermore, the proof is constructive and yields an algorithm which works well in practice. In particular, it is empirically demonstrated herein that such nonlinear functions allow for more accurate compressive nearest neighbor classification than standard linear Johnson-Lindenstrauss embeddings do in practice. Furthermore, it is demonstrated that this approach works when the labelled data consists of NFP measurements.
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