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We report our study on magnetic structural evolution of artificially patterned micron and submicron magnetic arrays as a function of applied field using in situ electron microscopy. To understand magnetic dynamics and switching behavior we employ our newly developed phase retrieval method, based on Lorentz phase microscopy, to map local induction distribution at nanometric scale. We outline the principle of the new method and discuss its advantages and drawbacks in comparison with off-axis electron holography.

In this paper, we consider the noisy phase retrieval problem which occurs in many different areas of science and physics. The PhaseMax algorithm is an efficient convex method to tackle with phase retrieval problem. On the basis of this algorithm, we propose two kinds of extended formulations of the PhaseMax algorithm, namely, PhaseMax with bounded and non-negative noise and PhaseMax with outliers to deal with the phase retrieval problem under different noise corruptions. Then we prove that these extended algorithms can stably recover real signals from independent sub-Gaussian measurements under optimal sample complexity. Specially, such results remain valid in noiseless case. As we can see, these results guarantee that a broad range of random measurements such as Bernoulli measurements with erasures can be applied to reconstruct the original signals by these extended PhaseMax algorithms. Finally, we demonstrate the effectiveness of our extended PhaseMax algorithm through numerical simulations. We find that with the same initialization, extended PhaseMax algorithm outperforms Truncated Wirtinger Flow method, and recovers the signal with corrupted measurements robustly.

We introduce the ICQPhase algorithm for iteratively consistent quantized phase retrieval. ICQPhase addresses the problem of recovering a signal $x\in {𝔽}^d={ℝ}^d$ or ${ℂ}^d$ from one-bit quantized phaseless measurements $y_j=Q(\parallel P_{j}x\parallel )$, $1\le j\le m$, where $P_j:{𝔽}^d\to {𝔽}^d$ are orthogonal projections and where $Q:ℝ\to \{0,1\}$ is a one-bit scalar quantizer. We numerically validate the performance of ICQPhase and demonstrate that it experimentally achieves mean squared error of order $1/m^2$ in a variety of settings. We provide a theoretical analysis of the algorithm for the case of rank-one projections of signals in two-dimensional real space and prove that the mean squared error is of order $1/m^2$.

In this paper, we study the problem of phase retrieval with short-time linear canonical transform (STLCT). The relation between signal, window and their STLCT is provided through Fourier transform. Based on this theorem, a uniqueness result is established for all square integrable functions. For nonseparable real continuous signal, we prove the uniqueness theorems under some weaker conditions. In complex bandlimited and cardinal $B$-spline spaces, uniqueness results are provided with magnitude-only STLCT.

We address the problem of phase retrieval (PR) from quantized measurements. The goal is to reconstruct a signal from quadratic measurements encoded with a finite precision, which is indeed the case in practical applications. We develop an iterative projected-gradient-type algorithm that recovers the signal subject to ensuring consistency with the measurement, meaning that the recovered signal, when encoded, must yield the same set of measurements that one started with. The algorithm involves rank-1 projection, which stems from the idea of lifting, originally proposed in the context of PhaseLift. The consistency criterion is enforced using a one-sided quadratic cost. We also determine the probability with which different vectors lead to the same set of quantized measurements, which makes it impossible to resolve them. Naturally, this probability depends on how correlated such vectors are, and how coarsely/finely the measurements are quantized. The proposed algorithm is also capable of incorporating a sparsity constraint on the signal. An analysis of the cost function reveals that it is bounded probabilistically, both above and below, by functions that are dependent on how well-correlated the estimate is with the ground-truth. We also derive the Cramér-Rao lower bound (CRB) on the achievable reconstruction accuracy. A comparison with the state-of-the-art algorithms shows that the proposed algorithm has a higher reconstruction accuracy and is about 2 to 3 dB away from the CRB. The edge, in terms of the reconstruction signal-to-noise ratio, over the competing algorithms is higher (about 5 to 6 dB) when the quantization is coarse, thereby making the proposed scheme particularly attractive in such scenarios. We also demonstrate a concrete application of the proposed method to frequency-domain optical-coherence tomography (FDOCT).

In this paper, the ${\ell }_1$-analysis model for the phase retrieval problem of sparse unknown signals in the redundant dictionary is extended to the ${\ell }_p$-analysis model, where $0<p\le 1$. It’s shown that if the measurement matrix $A$ satisfies the strong restricted isometry property adapted to D (S-DRIP) condition, the unknown signal $x$ can be stably recovered by analyzing the ${\ell }_p$$(0<p\le 1)$ minimization model.

In this paper, we will present several new results in finite and countable dimensional separable real Hilbert space phase retrieval and norm retrieval by fusion frames. We will characterize of norm retrieval for fusion frames similar norm retrieval for vectors and we will show that only one direction holds for fusion frames. In similar vector case, we will show that every tight fusion frame can do norm retrieval. Also we will show that the unitary operators preserve phase (norm) retrievability of fusion frames. We will make a detailed study of when hyperplanes do norm retrieval and show a general result about it. We will provide numerous examples to show that our results are best possible.

The phase retrieval (PR) problem is to reconstruct real/complex functions from the magnitudes of their Fourier/frame measurements in classical computational imaging. In this paper, we consider phase retrieval of complex vectors/images from the magnitudes of short-time fractional Fourier transform (STFrFT). In our setting, the above problem is solved by minimizing a least square ReLu loss function and a novel algorithm by alternating direction method of multipliers (ADMM) is presented. As shown in the numerical simulations on complex signals/images, our proposed PR-ADMM algorithm from STFrFT has a better recovery performance with flexible window functions and appropriate fractional orders. It demonstrates to have satisfactory performance from mixed phaseless STFrFT measurements. Compared with several six other main algorithms, the proposed algorithm explicitly recovers the phase of image with higher the peak signal-to-noise ratio. Meanwhile, the proposed algorithm is robust to noise. These also generalize some of about phase retrievals with Fourier measurements.

Focusing light though scattering media beyond the ballistic regime is a challenging task in biomedical optical imaging. This challenge can be overcome by wavefront shaping technique, in which a time-reversed (TR) wavefront of scattered light is generated to suppress the scattering. In previous TR optical focusing experiments, a phase-only spatial light modulator (SLM) has been typically used to control the wavefront of incident light. Unfortunately, although the phase information is reconstructed by the phase-only SLM, the amplitude information is lost, resulting in decreased peak-to-background ratio (PBR) of optical focusing in the TR wavefront reconstruction. A new method of TR optical focusing through scattering media is proposed here, which numerically reconstructs the full phase and amplitude of a simulated scattered light field by using a single phase-only SLM. Simulation results and the proposed optical setup show that the time-reversal of a fully developed speckle field can be digitally implemented with both phase and amplitude recovery, affording a way to improve the performance of light focusing through scattering media.

X-ray phase contrast imaging (XPCi) provides a much higher visibility of low-absorbing details than conventional, attenuation-based radiography. This is due to the fact that image contrast is determined by the unit decrement of the real part of the complex refractive index of an object rather than by its imaginary part (the absorption coefficient), which can be up to 1000 times larger for energies in the X-ray regime. This finds applications in many areas, including medicine, biology, material testing, and homeland security. Until lately, XPCi has been restricted to synchrotron facilities due to its demanding coherence requirements on the radiation source. However, edge illumination XPCi, first developed by one of the authors at the ELETTRA Synchrotron in Italy, substantially relaxes these requirements and therefore provides options to overcome this problem. Our group has built a prototype scanner that adapts the edge-illumination concept to standard laboratory conditions and extends it to large fields of view. This is based on X-ray sources and detectors available off the shelf, and its use has led to impressive results in mammography, cartilage imaging, testing of composite materials and security inspection. This article presents the method and the scanner prototype, and reviews its applications in selected biomedical and non-medical disciplines.

We demonstrate a method, sub-intensity pattern stitching, to extend the amplitude measurement range of wavefront in phase retrieval. An algorithm based on least square to stitch is presented. An experimental setup used to measure a spherical mirror with 100mm diameter and 650mm radius using this stitching method is arranged. The stitching results and corresponding interferometer results are in excellent agreement, which promises the validity of this method.

The aim of this paper is to study the recovery performance of the ℓ0 minimization for the sparse problem in the real phase retrieval setting. Though this problem is closely related to the compressive sensing problem, lacking of phase information makes this harder. As cost functions are often at the heart of the problem, we here first study the iterative hard algorithms that are minimizing the ℓ0 penalised cost functions. Herein we propose a new IHT algorithm for the problem of one dimensional (1D) phase retrieval, namely, recovery of 1D sparse signal from the magnitude of its measurement. And we give the theoretical results of error bound and convergence of the algorithm. Simulation results indicate that the proposed algorithm is accurate and more faster than existing techniques.

X-ray free-electron lasers provide a greatly increased peak intensity on femtosecond time scales, opening up new opportunities to study the structure and dynamics of nanoparticles, viruses and biological macromolecules. These opportunities are being realized with new crystallography techniques to study nanoscale periodic samples and with coherent diffractive imaging techniques to study non-periodic samples. This chapter provides an introduction to coherent diffractive imaging techniques, a review of its applications and development at synchrotron sources, and finally a survey of the first imaging and crystallography results from X-ray laser experiments.