Abstract: This Letter presents a novel absolute phase recovery technique with phase coding. Unlike the conventional gray-coding method, the codeword is embedded into the phase and then used to determine the fringe order for absolute phase retrieval. This technique is robust because it uses phase instead of intensity to determine codewords, and it could achieve a faster measurement speed, since three additional images can represent more than 8(23) unique codewords for phase unwrapping. Experimental results will be presented to verify the performance of the proposed technique.

Abstract: X-ray ptychographic computed tomography has recently emerged as a nondestructive characterization tool for samples with representative sizes of several tens of micrometers, yet offering a resolution currently lying in but not limited to the 100-nm range. Here we evaluate the quantitativeness of this technique using a model sample with a known structure and density, and we discuss its sensitivity as a function of resolution. Additionally, we show an example application for the determination of the mass density of individual $2\ensuremath{-}\ensuremath{\mu}$m-sized SiO${}_{2}$ microspheres with a relative error of 2$%$. The accuracy and sensitivity demonstrated in this paper will enable quantitative imaging, segmentation, and identification of different phases in complex materials at the nanoscale.

Abstract: We consider stability and uniqueness in real phase retrieval problems over general input sets. Specifically, we assume the data consists of noisy quadratic measurements of an unknown input x in R^n that lies in a general set T and study conditions under which x can be stably recovered from the measurements. In the noise-free setting we derive a general expression on the number of measurements needed to ensure that a unique solution can be found in a stable way, that depends on the set T through a natural complexity parameter. This parameter can be computed explicitly for many sets T of interest. For example, for k-sparse inputs we show that O(k\log(n/k)) measurements are needed, and when x can be any vector in R^n, O(n) measurements suffice. In the noisy case, we show that if one can find a value for which the empirical risk is bounded by a given, computable constant (that depends on the set T), then the error with respect to the true input is bounded above by an another, closely related complexity parameter of the set. By choosing an appropriate number N of measurements, this bound can be made arbitrarily small, and it decays at a rate faster than N^{-1/2+\delta} for any \delta>0. In particular, for k-sparse vectors stable recovery is possible from O(k\log(n/k)\log k) noisy measurements, and when x can be any vector in R^n, O(n \log n) noisy measurements suffice. We also show that the complexity parameter for the quadratic problem is the same as the one used for analyzing stability in linear measurements under very general conditions. Thus, no substantial price has to be paid in terms of stability if there is no knowledge of the phase.

Abstract: X-ray phase contrast imaging has overcome the limitations of X-ray absorption imaging in many fields. Particular effort has been directed towards developing phase retrieval methods: These reveal quantitative information about a sample, which is a requirement for performing X-ray phase tomography, allows material identification and better distinction between tissue types, etc. Phase retrieval seems impossible with conventional X-ray sources due to their low spatial coherence. In the only previous example where conventional sources have been used, collimators were employed to produce spatially coherent secondary sources. We present a truly incoherent phase retrieval method, which removes the spatial coherence constraints and employs a conventional source without aperturing, collimation, or filtering. This is possible because our technique, based on the pixel edge illumination principle, is neither interferometric nor crystal based. Beams created by an X-ray mask to image the sample are smeared due to the incoherence of the source, yet we show that their displacements can still be measured accurately, obtaining strong phase contrast. Quantitative information is extracted from only two images rather than a sequence as required by several coherent methods. Our technique makes quantitative phase imaging and phase tomography possible in applications where exposure time and radiation dose are critical. The technique employs masks which are currently commercially available with linear dimensions in the tens of centimeters thus allowing for a large field of view. The technique works at high photon energy and thus promises to deliver much safer quantitative phase imaging and phase tomography in the future.

Abstract: The twin-image problem in phase retrieval is characterized by the simultaneous occurrence of features from the original object and its inversion about the origin (twin image). This problem can occur in reconstructions for which the object support is centrosymmetric or loose, and in severe cases it can greatly hinder image quality. In this paper we examine this problem and find that it arises when the retrieved Fourier-domain phase is divided into sets of regions, some of which reconstruct the object while others the twin. We examine sample reconstructions that present the twin-image problem to different extents and find that, even when the twin-image problem is not visually evident, it can exist in small regions of the retrieved Fourier phase. The reduced-support constraint approach is shown to be effective in escaping stagnation caused by the twin-image problem.

Abstract: Random illumination is proposed to enforce absolute uniqueness and resolve all types of ambiguity, trivial or nontrivial, in phase retrieval. Almost sure irreducibility is proved for any complex-valued object whose support set has rank 2. While the new irreducibility result can be viewed as a probabilistic version of the classical result by Bruck, Sodin and Hayes, it provides a novel perspective and an effective method for phase retrieval. In particular, almost sure uniqueness, up to a global phase, is proved for complex-valued objects under general two-point conditions. Under a tight sector constraint absolute uniqueness is proved to hold with probability exponentially close to unity as the object sparsity increases. Under a magnitude constraint with random amplitude illumination, uniqueness modulo global phase is proved to hold with probability exponentially close to unity as object sparsity increases. For general complex-valued objects without any constraint, almost sure uniqueness up to global phase is established with two sets of Fourier magnitude data under two independent illuminations. Numerical experiments suggest that random illumination essentially alleviates most, if not all, numerical problems commonly associated with the standard phasing algorithms. (Some figures may appear in colour only in the online journal)

Abstract: Coherent diffraction imaging (CDI) is high-resolution lensless microscopy that has been applied to image a wide range of specimens using synchrotron radiation, X-ray free electron lasers, high harmonic generation, soft X-ray laser and electrons. Despite these rapid advances, it remains a challenge to reconstruct fine features in weakly scattering objects such as biological specimens from noisy data. Here we present an effective iterative algorithm, termed oversampling smoothness (OSS), for phase retrieval of noisy diffraction intensities. OSS exploits the correlation information among the pixels or voxels in the region outside of a support in real space. By properly applying spatial frequency filters to the pixels or voxels outside the support at different stage of the iterative process (i.e. a smoothness constraint), OSS finds a balance between the hybrid input-output (HIO) and error reduction (ER) algorithms to search for a global minimum in solution space, while reducing the oscillations in the reconstruction. Both our numerical simulations with Poisson noise and experimental data from a biological cell indicate that OSS consistently outperforms the HIO, ER-HIO and noise robust (NR)-HIO algorithms at all noise levels in terms of accuracy and consistency of the reconstructions. We expect OSS to find application in the rapidly growing CDI field as well as other disciplines where phase retrieval from noisy Fourier magnitudes is needed.

Abstract: We propose a new method using coherent diffractive imaging for optical color-image encryption and synthesis in the Fresnel domain. An optical multiple-random-phase-mask encryption system is applied, and a strategy based on lateral translations of a phase-only mask is employed during image encryption. For the decryption, an iterative phase retrieval algorithm is applied to extract high-quality decrypted color images from diffraction intensity maps (i.e., ciphertexts). In addition, optical color-image synthesis is also investigated based on coherent diffractive imaging. Numerical results are presented to demonstrate feasibility and effectiveness of the proposed method. Compared with conventional interference methods, coherent diffractive imaging approach may open up a new research perspective or can provide an effective alternative for optical color-image encryption and synthesis.

Abstract: Using the maximum and the minimum of interference, a novel two-step phase demodulation algorithm is proposed to perform the phase extraction in phase-shifting interferometry. By means of the simulation calculation and the experimental research, it is proved that both the measured phase and the phase shift with high precision can be obtained in the proposed algorithm.

Abstract: We present a sparsity-based method for subwavelength coherent diffractive imaging: an algorithmic approach for reconstruction of subwavelength images from a single intensity measurement of their far-field diffraction pattern.

Abstract: The phase problem is inherent to crystallographic, astronomical and optical imaging where only the intensity of the scattered signal is detected and the phase information is lost and must somehow be recovered to reconstruct the object’s structure. Modern imaging techniques at the molecular scale rely on utilizing novel coherent light sources like X-ray free electron lasers for the ultimate goal of visualizing such objects as individual biomolecules rather than crystals. Here, unlike in the case of crystals where structures can be solved by model building and phase refinement, the phase distribution of the wave scattered by an individual molecule must directly be recovered. There are two well-known solutions to the phase problem: holography and coherent diffraction imaging (CDI). Both techniques have their pros and cons. In holography, the reconstruction of the scattered complex-valued object wave is directly provided by a well-defined reference wave that must cover the entire detector area which often is an experimental challenge. CDI provides the highest possible, only wavelength limited, resolution, but the phase recovery is an iterative process which requires some pre-defined information about the object and whose outcome is not always uniquely-defined. Moreover, the diffraction patterns must be recorded under oversampling conditions, a pre-requisite to be able to solve the phase problem. Here, we report how holography and CDI can be merged into one superior technique: holographic coherent diffraction imaging (HCDI). An inline hologram can be recorded by employing a modified CDI experimental scheme. We demonstrate that the amplitude of the Fourier transform of an inline hologram is related to the complex-valued visibility, thus providing information on both, the amplitude and the phase of the scattered wave in the plane of the diffraction pattern. With the phase information available, the condition of oversampling the diffraction patterns can be relaxed, and the phase problem can be solved in a fast and unambiguous manner. We demonstrate the reconstruction of various diffraction patterns of objects recorded with visible light as well as with low-energy electrons. Although we have demonstrated our HCDI method using laser light and low-energy electrons, it can also be applied to any other coherent radiation such as X-rays or high-energy electrons.

Abstract: We address the problem of phase retrieval, which is frequently encountered in optical imaging. The measured quantity is the magnitude of the Fourier spectrum of a function (in optics, the function is also referred to as an object). The goal is to recover the object based on the magnitude measurements. In doing so, the standard assumptions are that the object is compactly supported and positive. In this paper, we consider objects that admit a sparse representation in some orthonormal basis. We develop a variant of the Fienup algorithm to incorporate the condition of sparsity and to successively estimate and refine the phase starting from the magnitude measurements. We show that the proposed iterative algorithm possesses Cauchy convergence properties. As far as the modality is concerned, we work with measurements obtained using a frequency-domain optical-coherence tomography experimental setup. The experimental results on real measured data show that the proposed technique exhibits good reconstruction performance even with fewer coefficients taken into account for reconstruction. It also suppresses the autocorrelation artifacts to a significant extent since it estimates the phase accurately.

Abstract: A phase retrieval method for microscopy using multiple illumination wavelengths is proposed. A fast algorithm suitable for calculations with high numerical aperture is used for the iterative retrieval of the object wavefront. The advantages and limitations of the technique are systematically analyzed and demonstrated by both simulation and experimental results.

Abstract: An important problem in radar waveform optimization is the synthesis of discrete time constant modulus signals from Fourier magnitude data. Iterative algorithms for solving this problem have been proposed in the literature, but the algorithms are only applicable in limited cases, and the convergent behavior of these algorithms has not been established. We connect waveform design to the well-studied problem of phase retrieval. This is useful for explaining the success of the proposed iterative methods. We generalize and extend the existing algorithms to handle the case in which the dimensions of the time domain waveform and the frequency domain data are unequal, and we provide a convergence analysis. We also relate the phase retrieval problem to the problem of synthesizing discrete time constant modulus signals from power spectral density (PSD) data, which is different and more appropriate for the waveform design problem. We compare the iterative methods to direct search gradient methods for both problems, and establish that the proposed algorithms can provide comparable performance with reduced computational complexity.

Abstract: We present a theoretical and experimental comparison of three X-ray phase-contrast techniques: propagation-based imaging, analyzer-based imaging and grating interferometry. The signal-to-noise ratio and the figure of merit are quantitatively compared for the three techniques on the same phantoms and using the same X-ray source and detector. Principal dependencies of the signal upon the numerous acquisition parameters, the spatial resolution and X-ray energy are discussed in detail. The sensitivity of each technique, in terms of the smallest detectable phase shift, is also evaluated.

Abstract: We present a method for phase retrieval from x-ray Fresnel diffraction patterns for multimaterial objects. Previously, homogeneous object assumptions have been used and have been introduced in the Radon domain. Here, we apply prior knowledge in the object domain, which permits the introduction of multiple materials. This is achieved first by a tomographic reconstruction of an attenuation scan and then introduction of the prior followed by a forward projection step to yield the a priori phase maps. The method is applied to the reconstruction of an object of known composition consisting of both soft and hard materials and is shown to perform better than previously proposed algorithms.

Abstract: There are two well-known solutions to the phase problem: holography and coherent diffraction imaging (CDI). We show how holography and CDI can be merged into one superior technique: holographic coherent diffraction imaging (HCDI).

Abstract: We propose a new phase retrieval algorithm for optical image encryption in three-dimensional (3D) space. The two-dimensional (2D) plaintext is considered as a series of particles distributed in 3D space, and an iterative phase retrieval algorithm is developed to encrypt the series of particles into phase-only masks. The feasibility and effectiveness of the proposed method are demonstrated by a numerical experiment, and the advantages and security of the proposed optical cryptosystems are also analyzed and discussed.

Abstract: The attenuation of x-rays in a material forms the basis of x-ray radiography and tomography. By measuring the transmission of the x-rays over a large amount of raypaths, the three-dimensional (3D) distribution of the x-ray linear attenuation coefficient can be reconstructed in a 3D volume. In x-ray microtomography (μCT), however, the x-ray refraction yields a significant signal in the transmission image and the 3D distribution of the refractive index can be reconstructed in a 3D volume. To do so, several methods exist, on both a hardware and software level. In this paper, we compare two similar software methods, the modified Bronnikov algorithm and the simultaneous phase-and-amplitude retrieval. The first method assumes a pure phase object, whereas the latter assumes a homogeneous object. Although these assumptions seem very restrictive, both methods have proven to yield good results on experimental data.

Abstract: A technique for enhanced deterministic phase retrieval using a partially developed speckle field (PDSF) and a spatial light modulator (SLM) is demonstrated experimentally. A smooth test wavefront impinges on a phase diffuser, forming a PDSF that is directed to a 4f setup. Two defocused speckle intensity measurements are recorded at the output plane corresponding to axially-propagated representations of the PDSF in the input plane. The speckle intensity measurements are then used in a conventional transport of intensity equation (TIE) to reconstruct directly the test wavefront. The PDSF in our technique increases the dynamic range of the axial intensity derivative for smooth phase objects, resulting in a more robust solution to the TIE. The SLM setup enables a fast and accurate recording of speckle intensity. Experimental results are in good agreement with those obtained using the iterative phase retrieval and digital holographic methods of wavefront reconstruction.

Abstract: A new phase-measurement technique is proposed, which utilizes a three-beam interferometer. Three-wave interference in the interferometer generates a uniform lattice of optical vortices, which is distorted by the presence of an object inserted in one arm of the interferometer. The transverse displacement of the vortices is proportional to the phase shift in the object wave. Tracking the vortices permits the phase of the object to be reconstructed. We demonstrate the method experimentally using a simple lens and a more complex object, namely the wing of a common house fly. Since the technique is implemented in real space, it is capable of reconstructing the phase locally.

Abstract: ?Ghosted? fringe patterns simultaneously reflected from both the upper and lower sides of a transparent target in the fringe reflection technique are captured for transparent surface 3D shape measurement, but the phase retrieval from the captured ?ghosted? fringe patterns is still not solved. A novel method is proposed to solve this issue by using two sets of phase-shifted fringe patterns with slightly different frequencies. The nonlinear least-squares method is used to estimate the fringe phase and modulation from both front and rear interfaces. Several simulations are done to show the feasibility of the proposed method. The influence of fringe noise on the algorithm is studied as well, which indicates that the proposed method is able to retrieve the phase from double-sided reflective fringe patterns with fringe noise equivalent to that in practical measurements. The merits and limitations of the method are discussed and recommendations for future studies are made.

Abstract: In this work, we demonstrate an improved method for iterative phase retrieval with application to coherent diffractive imaging. By introducing additional operations inside the support term of existing iterated projection algorithms, we demonstrate improved convergence speed, higher success rate and, in some cases, improved reconstruction quality. New algorithms take a particularly simple form with the introduction of a generalized projection-based reflector. Numerical simulations verify that these new algorithms surpass the current standards without adding complexity to the reconstruction process. Thus the introduction of this new class of algorithms offers a new array of methods for efficiently deconvolving intricate data.

Abstract: We demonstrate optical inspection of a technical component under thermal load by means of phase retrieval. The scheme is based on the determination of the phase distribution of light scattered by an object using phase retrieval from a set of intensity measurements. In contrast to already existing configurations, the employed setup enables a rapid acquisition of the required intensity distributions, thus the slow changes of the object’s surface during the thermal loading can be tolerated in the subsequent evaluation process. The presented system exhibits all the properties of an inspection system based on phase shifting interferometry, yet it does not require a reference wave, which makes it more tolerant against environmental disturbances and therefore a preferred choice for future industrial applications.

Abstract: Simultaneous measurement of multidimensional displacements using digital holographic interferometry involves multi-directional illumination of the deformed object and requires the reliable estimation of the resulting multiple interference phase distributions. The paper introduces an elegant method to simultaneously estimate the desired multiple phases from a single fringe pattern. The proposed method relies on modeling the reconstructed interference field as a piecewise multicomponent polynomial phase signal. Effectively, in a given region or segment, the reconstructed interference field is represented as the sum of different components i.e. complex signals with polynomial phases. The corresponding polynomial coefficients are estimated using the product high-order ambiguity function. To ensure proper matching of the estimated coefficients with the corresponding components, an amplitude based discrimination criterion is used. The main advantage of the proposed method is direct retrieval of multiple phases without the application of spatial carrier based filtering operations.

Abstract: A phase retrieval technique using a spatial light modulator (SLM) and a phase diffuser for a fast reconstruction of smooth wave fronts is demonstrated experimentally. Diffuse illumination of a smooth test object with the aid of a phase diffuser (maximum phase shift, Df = 0.85p) results in a significant diversity in the intensity measurements which, in turn, is beneficial for a non-stagnating iterative phase reconstruction. The use of the SLM enables accurate and fast speckle intensity recording and active correction of misalignments in the setup. The effectiveness of the technique is demonstrated in the optical testing of lenses.

Abstract: Measuring transmission and optical thickness of an object with a single intensity recording is desired in many fields of imaging research. One possibility to achieve this is to employ phase retrieval algorithms. We propose a method to significantly improve the performance of such algorithms in optical imaging. The method relies on introducing a specially designed phase object into the specimen plane during the image recording, which serves as a constraint in the subsequent phase retrieval algorithm. This leads to faster algorithm convergence and improved final accuracy. Quantitative imaging can be performed by a single recording of the resulting diffraction pattern in the camera plane, without using lenses or other optical elements. The method allows effective suppression of the “twin-image”, an artefact that appears when holograms are read out. Results from numerical simulations and experiments confirm a high accuracy which can be comparable to that of phase-stepping interferometry.

Abstract: To simplify the reconstruction optical system, an improved method is developed to reconstruct the three-dimensional (3D) quantitative refractive index of live cells based on a local plane wave approximation. In this method, the cell is illuminated by a convergent beam whose cross section is much larger than the cell, so the beam passing through the cell can be treated as a local plane wave. With the one-dimensional moving of cells in the beam cross section, multi-directional phase projections can be obtained using a Mach–Zehnder interferometer and the Hilbert transform phase retrieval algorithm. An inverse Radon–Radon iterative algorithm is used to reconstruct the 3D refractive index distribution which is verified by a simulation reconstruction. The corresponding set-up is applied to obtain the multi-directional phase projections and reconstruct the 3D refractive index of red blood cells (RBCs). The results show that only with a simple device could the method measure the 3D refractive index of cells with high precision. The reconstruction method could also be applied in opto-fluidic microscopy to fabricate a compact on-chip opto-fluidic tomographic microscope.