Conjugate Phase Retrieval in Paley-Wiener Space

Iterative algorithms for phase retrieval from intensity data are compared to gradient search methods. Both the problem of phase retrieval from two intensity measurements (in electron microscopy or wave front sensing) and the problem of phase retrieval from a single intensity measurement plus a non-negativity constraint (in astronomy) are considered, with emphasis on the latter. It is shown that both the error-reduction algorithm for the problem of a single intensity measurement and the Gerchberg-Saxton algorithm for the problem of two intensity measurements converge. The error-reduction algorithm is also shown to be closely related to the steepest-descent method. Other algorithms, including the input-output algorithm and the conjugate-gradient method, are shown to converge in practice much faster than the error-reduction algorithm. Examples are shown.

The classical problem of phase retrieval has found a wide array of applications in optics, imaging and signal processing. In this paper, we consider the phase retrieval problem in a one-bit setting, where the signals are sampled using one-bit analog-to-digital converters (ADCs). A significant advantage of deploying one-bit ADCs in signal processing systems is their superior sampling rates as compared to their high-resolution counterparts. This leads to an enormous amount of one-bit samples gathered at the output of the ADC in a short period of time. We demonstrate that this advantage pays extraordinary dividends when it comes to convex phase retrieval formulations, namely that the often encountered matrix semi-definiteness constraints as well as rank constraints (that are computationally prohibitive to enforce), become redundant for phase retrieval in the face of a growing sample size. Several numerical results are presented to illustrate the effectiveness of the proposed methodologies.

There are several important remote sensing applications where the development of Ground Penetrating Synthetic Aperture Radar (GPENSAR) is the logical approach, e.g., searching for buried military facilities, minefield mapping, survey of underground pipelines. Penetration of sufficient soil depth for useful results require a SAR to operate at VHF/UHF frequencies, e.g., 200 - 300 MHz. At these frequencies a satellite SAR will encounter substantial distortion in the double passage of the SAR signal through the ionosphere. One of the ionospheric distortions is equivalent the phase aberrations caused in imaging through the turbulent atmosphere, and the problem of phase retrieval for the GPENSAR becomes a necessity. For GPENSR there are imaging concepts that exploit dual polarization radiation of the SAR pulse. The phase retrieval problem then becomes one of compensation for the phase aberrations induced in each of the polarization components returned to the satellite receiver. We discuss the use of the two polarizations to cancel the ionospheric phase aberrations. Unfortunately, the resulting signal has only relative phase of the two polarizations. We discuss an algorithm for the retrieval of the absolute phase. The algorithm is based on an optimization approach. Although phase retrieval by optimization is difficult because of local minima, the retrieval of absolute phase in the dual polarization case is substantially less difficult, because the two polarizations constrain the solution sufficiently to eliminate many local minima.

i»?The problem of phase retrieval, i.e., the recovery of a function given the magnitude of its Fourier transform, arises in various fields of science and engineering, including electron microscopy, crystallography, astronomy, and optical imaging. Exploring phase retrieval in optical settings, specifically when the light originates from a laser, is natural since optical detection devices [e.g., charge-coupled device (CCD) cameras, photosensitive films, and the human eye] cannot measure the phase of a light wave. This is because, generally, optical measurement devices that rely on converting photons to electrons (current) do not allow for direct recording of the phase: the electromagnetic field oscillates at rates of ~1015 Hz, which no electronic measurement device can follow. Indeed, optical measurement/detection systems measure the photon flux, which is proportional to the magnitude squared of the field, not the phase. Consequently, measuring the phase of optical waves (electromagnetic fields oscillating at 1015 Hz and higher) involves additional complexity, typically by requiring interference with another known field, in the process of holography.

Signal recovery from the magnitude of the Fourier transform, or equivalently, from the autocorrelation, is a classical problem known as phase retrieval. Due to the absence of phase information, some form of additional information is required in order to be able to uniquely identify the underlying signal. In this work, we consider the problem of phase retrieval using masks. Due to our interest in developing robust algorithms with theoretical guarantees, we explore a convex optimization-based framework. In this work, we show that two specific masks (each mask provides 2n Fourier magnitude measurements) or five specific masks (each mask provides n Fourier magnitude measurements) are sufficient for a convex relaxation of the phase retrieval problem to provably recover almost all signals (up to global phase). We also show that the recovery is stable in the presence of measurement noise. This is a significant improvement over the existing results, which require O(log2 n) random masks (each mask provides n Fourier magnitude measurements) in order to guarantee unique recovery (up to global phase). Numerical experiments complement our theoretical analysis and show interesting trends, which we hope to explain in a future publication.

Phase-Contrast Tomography (PCT) is becoming an important technique for nondestructive, in-situ characterization of soft and hard condensed matter. This thesis sheds light on our progress in developing novel tomographic reconstruction algorithms in combination with in-situ experimental approaches that employ propagation-based PCT. These developments are aimed to improve the spatial and temporal resolutions of in-situ studies of the soft and hard condensed matter. The mechanical properties of materials such as steel are, to a large extent, defined by its microstructure. Propagation-based PCT (i.e. holotomography) can be used to visualize the microstructures associated with small variations in density or composition in the bulk of the specimen. This imaging technique has the advantage of being non-destructive and allows to carry out in-situ, time-resolved observations of miscrostructural changes during dynamic processes such as solid-state phase transformations. Sufficient spatial and temporal resolution can be achieved only when an extremely bright x-ray source is used. In order to test whether a novel table-top x-ray source, the MIRRORCLE-6X, can be used for implementation of a propagation-based PCT system, we have carried out a series of experiments. The characteristics of a propagation-based PCI system that employs the MIRRORCLE-6X were analyzed. In our further research we have focused on the experiments that could be performed at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. We reported on the investigation of the cementite microstructure in carbon steel. In this work we present the reconstruction of the three-dimensional morphology of cementite grains in the bulk of steel using a non-destructive imaging technique – x-ray Phase-Contrast Tomography (PCT). Complementary information about the crystalline structure of the ferrite grains surrounding the cementite is obtained using x-ray Diffraction-Contrast Tomography (DCT). The work was continued by a time-resolved x-ray PCT investigation of the austenite grain growth during the ferrite-to-austenite phase transformation in low-carbon steel. Experimental data allows to visualize the evolution of the three-dimensional morphology of the austenite-ferrite interfaces with a spatial resolution on the order of 1 ?m and a time resolution of less than 9 minutes. The in-situ measurements show that the theory for grain growth during diffusional solid-state phase transformations needs to be extended in order to describe the drastically lower mobility that we observed for flat interfaces compared to curved interfaces. We observe that some flat interfaces are completely immobile, which is not predicted by the state-of-the-art theories and showed - for the first time - that new computational model for grain growth are needed. Quantitative image reconstruction based on PCT data requires solving two inverse problems: phase retrieval and tomographic reconstruction. In most cases, the problem of phase retrieval is ill-posed and requires that some type of a prior knowledge about the reconstructed image (i.e. particular regularization method) is used. We present a novel algebraic approach suitable for phase retrieval using various linear models. In this approach Total Variation (TV) minimization is used for regularization of the linearized inverse problem by promoting the solution with a sparse gradient magnitude (i.e. piece-wise constant solution). In that case prior knowledge about the reconstructed image may allow to (partially) recover the unknown spatial frequencies that are undefined by the experimental data. The problem of tomographic reconstruction based on PCT data can be solved using a similar approach. We conclude our investigation of the image reconstruction algorithms by introducing several algebraic approaches suitable for tomographic reconstruction based on PCT data. In these approaches TV minimization is used to find a regularized solution to an underdetermined linear system based on a linearized representation of the PCT. When the density of the reconstructed object is piece-wise constant (or close to it) a virtually artifact-free solution can be computed for the tomographic problem. The proposed approach can also be used to radically improve the accuracy of the tomographic reconstruction based on incomplete data (e.g. small number of projections) or data with low signal to noise ratio.

The problem of phase retrieval is revisited and studied from a fresh perspective. In particular, we establish a connection between the phase retrieval problem and the sensor network localization problem, which allows us to utilize the vast theoretical and algorithmic literature on the latter to tackle the former. Leveraging this connection, we develop a two-stage algorithm for phase retrieval that can provably recover the desired signal. In both sparse and dense settings, our proposed algorithm improves upon prior approaches simultaneously in the number of required measurements for recovery and the reconstruction time. We present numerical results to corroborate our theory and to demonstrate the efficiency of the proposed algorithm. As a side result, we propose a new form of phase retrieval problem and connect it to the complex rigidity theory proposed by Gortler and Thurston (in: Connelly R, Ivic Weiss A, Whiteley W (eds) Rigidity and symmetry, Springer, New York, pp 131–154, 2014).

We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó–Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

The two-dimensional discrete phase-retrieval problem is concerned with the reconstruction of a signal, or image, x(m, n), from the magnitude (intensity) of its Fourier transform, |X(ω1,ω2)|- Phase retrieval is an important problem that arises in a variety of different applications including x-ray crystallography, astronomy, electron microscopy, optics, and signal processing [1-5]. There are three issues that need to be considered in the solution of the phase retrieval problem: the uniqueness of the solution, the development of phase retrieval algorithms that reconstruct a signal from its Fourier transform intensity and any á priori information that might be available, and the sensitivity of the reconstruction to computational noise and measurements errors. It is now well known that if a two-dimensional signal with finite support has a z-transform that is an irreducible polynomial then the signal is uniquely defined to within a trivial ambiguity by the intensity of its Fourier transform [6]. This result becomes important with the fact that it has been shown that “almost all” discrete two-dimensional signals with finite support have z-transforms that are irreducible [7]. In spite of this uniqueness of the solution, however, the reconstruction of a signal from its Fourier intensity remains a difficult problem.

Two- and higher-dimensional bandlimited functions are almost always non-factorizable, meaning that their zeros form a single analytic curve. In principle one can use this property to separate the product of two bandlimited functions into its respective factors; this is important in Fourier phase retrieval and deconvolution problems. The intersection of this zero structure with the real plane is at points, closed curves or lines stretching to infinity. The location of zero points, curves or lines can only be estimated, in practice, from available noisy data. The estimated locations can be used to write a factorizable approximation to the original function and we explore the consequences of doing this. Of importance is the fact that point zeros can be used to represent a 2D bandlimited function. Hence from intensity data, point zero locations in the intensity can be used directly to estimate the complex spectrum and provide an approximate solution to the phase retrieval problem. Examples will be given and their importance for the general blind deconvolution problem discussed.

In the article, we consider the problem of phase retrieval from magnitudes of STFT (short-time Fourier transform) measurements. The uniqueness of STFT phase retrieval had been established when the signal is one-dimensional function. In this article, we investigate the uniqueness of STFT phase retrieval when the signal is a two-dimensional vector function. And we prove the uniqueness theorem of STFT phase retrieval for bandlimited real-valued and complex-valued vector functions, respectively.

In this paper, we consider the problem of phase retrieval under masked Fourier measurements. Specifically, we focus on a question raised by E Candès et al, namely, to reduce the sampling complexity of such problem in noiseless case. We provide a truncated Wirtinger flow (TWF) algorithm with resampling to solve the problem. This algorithm starts with an initial guess computed by some truncated spectral method, then proceeds by successively refining the estimation via an update rule that bears a strong resemblance to a gradient descent scheme. These careful selection rules provide better initial guess and descent direction. We prove that when the sampling complexity is O(n log n), we get some good initial solution. Furthermore, the sequence generated by the resampled TWF algorithm can linearly converge to the true signal x∈Cn with ϵ~ -accuracy up to a global phase, provided with the sampling complexity O(log(1ϵ~)nlog2⁡n) . To some degree, we improve the results proposed by E Candès et al, which demand O(n log4 n) samples.

The problem of phase retrieval from intensity measurements plays an important role in many fields of physical research, e.g. optics, electron and x-ray microscopy, crystallography, diffraction tomography and others. in practice the recorded images contain information only on the intensity distribution I(x,y) = ψ*ψ*= |A|2 of the imaging wave function ψ = A*exp(-iϕ) and the phase information (ϕ(x,y) is usually lost. in general, the phase problem can be solved either by special holographic/interferometric methods, or by noninterferometric approaches based on intensity measurements in far Fraunhofer zone or in the Fresnel zone at two adjacent planes orthogonal to the optical axis. The latter approach uses the transport-of-intensity equation (TIE) formalism, introduced originally by Teague [1] and developed later in [2]. Applications of TIE to nonmagnetic materials and magnetic inductance mapping were successfully made in [3,4]. However, this approach still needs further improvement both in mathematics and in practical solutions, since the result is very sensitive to many experimental parameters.

Phase retrieval for sub-Gaussian measurements

Due to its appearance in a remarkably wide field of applications, such as audio processing and coherent diffraction imaging, the short-time Fourier transform (STFT) phase retrieval problem has seen a great deal of attention in recent years. A central problem in STFT phase retrieval concerns the question for which window functions g∈L2(Rd) and which sampling sets Λ⊆R2d is every f∈L2(Rd) uniquely determined (up to a global phase factor) by phaseless samples of the form |Vgf(Λ)|=|Vgf(λ)|:λ∈Λ,where Vgf denotes the STFT of f with respect to g. The investigation of this question constitutes a key step towards making the problem computationally tractable. However, it deviates from ordinary sampling tasks in a fundamental and subtle manner: recent results demonstrate that uniqueness is unachievable if Λ is a lattice, i.e Λ=AZ2d,A∈GL(2d,R). Driven by this discretization barrier, the present article centers around the initiation of a novel sampling scheme which allows for unique recovery of any square-integrable function via phaseless STFT-sampling. Specifically, we show that square-root lattices, i.e., sets of the form Λ=AZ2d,Z={±n:n∈N0},guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians