  In this work we consider the problem of reconstruction of a signal from the
magnitude of its Fourier transform, also known as phase retrieval. The problem
arises in many areas of astronomy, crystallography, optics, and coherent
diffraction imaging (CDI). Our main goal is to develop an efficient
reconstruction method based on continuous optimization techniques. Unlike
current reconstruction methods, which are based on alternating projections, our
approach leads to a much faster and more robust method. However, all previous
attempts to employ continuous optimization methods, such as Newton-type
algorithms, to the phase retrieval problem failed. In this work we provide an
explanation for this failure, and based on this explanation we devise a
sufficient condition that allows development of new reconstruction
methods---approximately known Fourier phase. We demonstrate that a rough (up to
$\pi/2$ radians) Fourier phase estimate practically guarantees successful
reconstruction by any reasonable method. We also present a new reconstruction
method whose reconstruction time is orders of magnitude faster than that of the
current method-of-choice in phase retrieval---Hybrid Input-Output (HIO).
Moreover, our method is capable of successful reconstruction even in the
situations where HIO is known to fail. We also extended our method to other
applications: Fourier domain holography, and interferometry. Additionally we
developed a new sparsity-based method for sub-wavelength CDI. Using this method
we demonstrated experimental resolution exceeding several times the physical
limit imposed by the diffraction light properties (so called diffraction
limit).
