Phase retrieval from power spectra of masked signals

In diffraction imaging, one is tasked with reconstructing a signal from its power spectrum. To resolve the ambiguity in this inverse problem, one might invoke prior knowledge about the signal, but phase retrieval algorithms in this vein have found limited success. One alternative is to create redundancy in the measurement process by illuminating the signal multiple times, distorting the signal each time with a different mask. Despite several recent advances in phase retrieval, the community has yet to construct an ensemble of masks which uniquely determines all signals and admits an efficient reconstruction algorithm. In this paper, we leverage the recently proposed polarization method to construct such an ensemble. We also present numerical simulations to illustrate the stability of the polarization method in this setting. In comparison to a state-of-the-art phase retrieval algorithm known as PhaseLift, we find that polarization is much faster with comparable stability.Comment: 18 pages, 3 figure

Fourier Phase Retrieval with a Single Mask by Douglas-Rachford Algorithm

Douglas-Rachford (DR) algorithm is analyzed for Fourier phase retrieval with a single random phase mask. Local, geometric convergence to a unique fixed point is proved with numerical demonstration of global convergence

Ab initio compressive phase retrieval

Any object on earth has two fundamental properties: it is finite, and it is made of atoms. Structural information about an object can be obtained from diffraction amplitude measurements that account for either one of these traits. Nyquist-sampling of the Fourier amplitudes is sufficient to image single particles of finite size at any resolution. Atomic resolution data is routinely used to image molecules replicated in a crystal structure. Here we report an algorithm that requires neither information, but uses the fact that an image of a natural object is compressible. Intended applications include tomographic diffractive imaging, crystallography, powder diffraction, small angle x-ray scattering and random Fourier amplitude measurements.Comment: 7 pages, 4 figures, presented at the XXI IUCr Congress, Aug. 2008, Osaka Japa

Phase retrieval from single biomolecule diffraction pattern

In this paper, we propose the SPR (sparse phase retrieval) method, which is a new phase retrieval method for coherent x-ray diffraction imaging (CXDI). Conventional phase retrieval methods effectively solve the problem for high signal-to-noise ratio measurements, but would not be sufficient for single biomolecular imaging which is expected to be realized with femto-second x-ray free electron laser pulses. The SPR method is based on the Bayesian statistics. It does not need to set the object boundary constraint that is required by the commonly used hybrid input-output (HIO) method, instead a prior distribution is defined with an exponential distribution and used for the estimation. Simulation results demonstrate that the proposed method reconstructs the electron density under a noisy condition even some central pixels are masked.Comment: 13 pages, 13 figures, submitted for a journa

Phase Retrieval with Application to Optical Imaging

This review article provides a contemporary overview of phase retrieval in optical imaging, linking the relevant optical physics to the information processing methods and algorithms. Its purpose is to describe the current state of the art in this area, identify challenges, and suggest vision and areas where signal processing methods can have a large impact on optical imaging and on the world of imaging at large, with applications in a variety of fields ranging from biology and chemistry to physics and engineering