Abstract: This note shows that we can recover any complex vector $\boldsymbol {x}_{0} \in \mathbb {C}^{n}$ exactly from on the order of n quadratic equations of the form |?a i ,x 0?|2=b i , i=1,?,m, by using a semidefinite program known as PhaseLift. This improves upon earlier bounds in Candes et al. (Commun. Pure Appl. Math. 66:1241---1274, 2013), which required the number of equations to be at least on the order of nlogn. Further, we show that exact recovery holds for all input vectors simultaneously, and also demonstrate optimal recovery results from noisy quadratic measurements; these results are much sharper than previously known results.

Abstract: In this work we analyze the problem of phase retrieval from Fourier measurements with random diffraction patterns. To this end, we consider the recently introduced PhaseLift algorithm, which expresses the problem in the language of convex optimization. We provide recovery guarantees which require O(log^2 d) different diffraction patterns, thus improving on recent results by Candes et al. [arXiv:1310.3240], which require O(log^4 d) different patterns.

Abstract: Ptychography is an imaging method whereby a coherent beam is scanned across an object, and an image is obtained by iterative phasing of the set of diffraction patterns. It is able to be used to image extended objects at a resolution limited by scattering strength of the object and detector geometry, rather than at an optics-imposed limit. As technical advances allow larger fields to be imaged, computational challenges arise for reconstructing the correspondingly larger data volumes, yet at the same time there is also a need to deliver reconstructed images immediately so that one can evaluate the next steps to take in an experiment. Here we present a parallel method for real-time ptychographic phase retrieval. It uses a hybrid parallel strategy to divide the computation between multiple graphics processing units (GPUs) and then employs novel techniques to merge sub-datasets into a single complex phase and amplitude image. Results are shown on a simulated specimen and a real dataset from an X-ray experiment conducted at a synchrotron light source.

Abstract: Recently Fourier Ptychography (FP) has attracted great attention, due to its marked effectiveness in leveraging snapshot numbers for spatial resolution in large field-of-view imaging. To acquire high signal-to-noise-ratio (SNR) images under angularly varying illuminations for subsequent reconstruction, FP requires long exposure time, which largely limits its practical applications. In this paper, based on the recently reported Wirtinger flow algorithm, we propose an iterative optimization framework incorporating phase retrieval and noise relaxation together, to realize FP reconstruction using low SNR images captured under short exposure time. Experiments on both synthetic and real captured data validate the effectiveness of the proposed reconstruction method. Specifically, the proposed technique could save around 80% exposure time to achieve similar retrieval accuracy compared to the conventional FP. Besides, we have released our source code for non-commercial use.

Abstract: We present a method of quantitatively acquiring a large complex field, containing not only amplitude information but also phase information, based on single-shot phase imaging with a coded aperture (SPICA). In SPICA, the propagating field from an object illuminated by partially coherent visible light is sieved by a coded mask, and the sieved field propagates to an image sensor, where it is captured. The sieved field is recovered from the single captured intensity image via a phase retrieval algorithm with an amplitude support constraint using the mask pattern, and then the object’s complex field is reconstructed from the recovered sieved field by an algorithm employing a sparsity constraint based on compressive sensing. The system model and the theoretical bounds of SPICA are derived. We also verified the concept with numerical demonstrations.

Abstract: We propose a method for fully phase image encryption based on double random-structured phase mask encoding in the gyrator transform (GT) domain. The security of the system is strengthened by parameters used in the construction of a structured phase mask (SPM) based on a devil’s vortex Fresnel lens (DVFL). The input image is recovered using the correct parameters of the SPMs, transform orders of the GT, and conjugate of the random phase masks. The use of a DVFL-based SPM enhances security by increasing the key space for encryption, and also overcomes the problem of axis alignment associated with an optical setup. The proposed scheme can also be implemented optically. The computed values of mean squared error between the retrieved and the original image show the efficacy of the proposed scheme. We have also investigated the scheme’s sensitivity to the encryption parameters, and robustness against occlusion and multiplicative Gaussian noise attacks.

Abstract: Lensless imaging is an approach to microscopy in which a high-resolution image of an object is reconstructed from one or more measured diffraction patterns, providing a solution in situations where the use of imaging optics is not possible. However, current lensless imaging methods are typically limited by the need for a light source with a narrow, stable and accurately known spectrum. We have developed a general approach to lensless imaging without spectral bandwidth limitations or sample requirements. We use two time-delayed coherent light pulses and show that scanning the pulse-to-pulse time delay allows the reconstruction of diffraction-limited images for all the spectral components in the pulse. In addition, we introduce an iterative phase retrieval algorithm that uses these spectrally resolved Fresnel diffraction patterns to obtain high-resolution images of complex extended objects. We demonstrate this two-pulse imaging method with octave-spanning visible light sources, in both transmission and reflection geometries, and with broadband extreme-ultraviolet radiation from a high-harmonic generation source. Our approach enables effective use of low-flux ultra-broadband sources, such as table-top high-harmonic generation systems, for high-resolution imaging. Researchers in The Netherlands have overcome the restriction of monochromatic illumination when performing lensless imaging. Stefan Witte and co-workers from LaserLAB Amsterdam have developed a lensless scheme that employs two coherent time-delayed pulses and is compatible with broadband sources. Lensless imaging — whereby diffraction patterns are interpreted to reconstruct an image of a sample — is popular in the X-ray and extreme-ultraviolet regimes, where high-quality lenses for performing conventional imaging are not available. Because this approach has traditionally been limited to narrowband coherent radiation, scientists have been eager to make it compatible with broadband sources such as tabletop high-harmonic generation. This new broadband technique involves scanning the pulse-to-pulse time delay and then applying a phase-retrieval algorithm to produce high-resolution images of complex objects.

Abstract: In this paper our interest is in investigating properties and numerical solutions of Proximal Split feasibility Problems. First, we consider the problem of finding a point which minimizes a convex function \(f\) such that its image under a bounded linear operator \(A\) minimizes another convex function \(g\). Based on an idea introduced in Lopez (Inverse Probl 28:085004, 2012), we propose a split proximal algorithm with a way of selecting the step-sizes such that its implementation does not need any prior information about the operator norm. Because the calculation or at least an estimate of the operator norm \(\Vert A\Vert \) is not an easy task. Secondly, we investigate the case where one of the two involved functions is prox-regular, the novelty of this approach is that the associated proximal mapping is not nonexpansive any longer. Such situation is encountered, for instance, in numerical solution to phase retrieval problem in crystallography, astronomy and inverse scattering Luke (SIAM Rev 44:169–224, 2002) and is therefore of great practical interest.

Abstract: We consider the classical 1D phase retrieval problem. In order to overcome the difficulties associated with phase retrieval from measurements of the Fourier magnitude, we treat recovery from the magnitude of the short-time Fourier transform (STFT). We first show that the redundancy offered by the STFT enables unique recovery for arbitrary nonvanishing inputs, under mild conditions. An efficient algorithm for recovery of a sparse input from the STFT magnitude is then suggested, based on an adaptation of the recently proposed GESPAR algorithm. We demonstrate through simulations that using the STFT leads to improved performance over recovery from the oversampled Fourier magnitude with the same number of measurements.

Abstract: Partially coherent light provides promising advantages for imaging applications. In contrast to its completely coherent counterpart, it prevents image degradation due to speckle noise and decreases cross-talk among the imaged objects. These facts make attractive the partially coherent illumination for accurate quantitative imaging in microscopy. In this work, we present a non-interferometric technique and system for quantitative phase imaging with simultaneous determination of the spatial coherence properties of the sample illumination. Its performance is experimentally demonstrated in several examples underlining the benefits of partial coherence for practical imagining applications. The programmable optical setup comprises an electrically tunable lens and sCMOS camera that allows for high-speed measurement in the millisecond range.

Abstract: We present a new approach for obtaining significant speedup in the digital processing of extracting unwrapped phase profiles from off-axis digital holograms. The new technique digitally multiplexes two orthogonal off-axis holograms, where the digital reconstruction, including spatial filtering and two-dimensional phase unwrapping on a decreased number of pixels, can be performed on both holograms together, without redundant operations. Using this technique, we were able to reconstruct, for the first time to our knowledge, unwrapped phase profiles from off-axis holograms with 1 megapixel in more than 30 frames per second using a standard single-core personal computer on a MATLAB platform, without using graphic-processing-unit programming or parallel computing. This new technique is important for real-time quantitative visualization and measurements of highly dynamic samples and is applicable for a wide range of applications, including rapid biological cell imaging and real-time nondestructive testing. After comparing the speedups obtained by the new technique for holograms of various sizes, we present experimental results of real-time quantitative phase visualization of cells flowing rapidly through a microchannel.

Abstract: We report here on terahertz (THz) digital holography on a biological specimen. A continuous-wave (CW) THz in-line holographic setup was built based on a 2.52 THz CO2 pumped THz laser and a pyroelectric array detector. We introduced novel statistical method of obtaining true intensity values for the pyroelectric array detector's pixels. Absorption and phase-shifting images of a dragonfly's hind wing were reconstructed simultaneously from single in-line hologram. Furthermore, we applied phase retrieval routines to eliminate twin image and enhanced the resolution of the reconstructions by hologram extrapolation beyond the detector area. The finest observed features are 35 {\mu}m width cross veins.

Abstract: In this Letter, we propose a method for retrieving the phase of a wavefront from the diffraction patterns recorded when the object is sequentially illuminated by spatially modulated light. For wavefronts having a smooth phase, the retrieval is achieved by using a deterministic method. When the phase has discontinuities, an iterative process is used for the retrieval and enhancement of the spatial resolution. Both the deterministic and iterative phase reconstructions are demonstrated by experiments.

Abstract: Boundary conditions play a crucial role in the solution of the transport of intensity equation (TIE). If not appropriately handled, they can create significant boundary artifacts across the reconstruction result. In a previous paper [Opt. Express 22, 9220 (2014)], we presented a new boundary-artifact-free TIE phase retrieval method with use of discrete cosine transform (DCT). Here we report its experimental investigations with applications to the micro-optics characterization. The experimental setup is based on a tunable lens based 4f system attached to a non-modified inverted bright-field microscope. We establish inhomogeneous Neumann boundary values by placing a rectangular aperture in the intermediate image plane of the microscope. Then the boundary values are applied to solve the TIE with our DCT-based TIE solver. Experimental results on microlenses highlight the importance of boundary conditions that often overlooked in simplified models, and confirm that our approach effectively avoid the boundary error even when objects are located at the image borders. It is further demonstrated that our technique is non-interferometric, accurate, fast, full-field, and flexible, rendering it a promising metrological tool for the micro-optics inspection.

Abstract: We consider the problem of recovering a complex signal x ∊ Cn from m intensity measurements of the form |a i x|, 1 ≤ i ≤ m, where a i is a measurement row vector. Our main focus is on the case where the measurement vectors are unconstrained, and where x is exactly K-sparse, or the so-called general compressive phase-retrieval problem. We introduce PhaseCode, a novel family of fast and efficient algorithms (that includes Unicolor PhaseCode and Multicolor PhaseCode) that are based on a sparse-graph coding framework. As one instance, our Unicolor PhaseCode algorithm can provably recover, with high probability, all but a tiny 10−7 fraction of the significant signal components, using at most m = 14K measurements, which is a small constant factor from the fundamental limit, with an optimal O(K) decoding time and an optimal O(K) memory complexity. We provide extensive simulation results that validate the practical power of our proposed algorithms. A key contribution of our work is the novel use of coding-theoretic tools like density evolution methods for the design and analysis of fast and efficient algorithms for compressive phase-retrieval problems. This contrasts and complements popular approaches to the phase retrieval problem based on alternating-minimization, convex-relaxation, and semi-definite programming.

Abstract: Phase retrieval has become a very active area of research. We will classify when phase retrieval by Parseval frames passes to the Naimark complement and when phase retrieval by projections passes to the orthogonal complements. We introduce a new concept we call norm retrieval and show that this is what is necessary for passing phase retrieval to complements. This leads to a detailed study of norm retrieval and its relationship to phase retrieval. One fundamental result: a frame $\{\varphi_i\}_{i=1}^M$ yields phase retrieval if and only if $\{T\varphi_i\}_{i=1}^M$ yields norm retrieval for every invertible operator $T$.

Abstract: A simultaneous phase-shifting dual-wavelength interferometry based on two-step demodulation algorithm is proposed in this Letter First, two lasers with different wavelengths go through the same inline phase-shifting interference system simultaneously, and a sequence of five frames of simultaneous phase-shifting dual-wavelength interferograms (SPSDWIs) with the special phase shifts are captured by a monochrome CCD Subsequently, using the subtraction between the first SPSDWI and the other SPSDWI, each wavelength of two frames of single-wavelength interference images (SWIIs) without the background can be achieved Finally, using two-step demodulation algorithm, the wrapped phase of each single-wavelength can be determined easily and quickly with high accuracy

Abstract: From three interferograms with unknown phase shifts, an innovative phase retrieval approach based on the normalized difference maps is proposed. Using the subtraction operation between interferograms, two difference maps without background can be achieved. To eliminate the amplitude inequality of difference maps, normalization process is employed so that two normalized difference maps are obtained. Finally, combining two normalized difference maps and two-step phase retrieval algorithm, the measured phase with high precision can be retrieved rapidly. Comparing with the conventional two-step phase retrieval algorithm with high-pass filtering, the accuracy and processing time of the proposed approach are greatly improved. Importantly, when the phase shift is close to π, almost all two-step algorithms become invalid, but the proposed approach still performs well. That is, the proposed normalized difference maps approach is suitable for the phase retrieval with arbitrary phase shifts.

Abstract: We consider the problem of phase retrieval in the Fresnel regime. In recent years, several techniques have been used to solve this problem applying different a priori assumptions on the two-dimensional object in space, such as positivity, finite support and amplitude constraints. In this paper, we propose a new constraint, namely the assumption that the object possesses a sparse representation in a shearlet frame. We show how a shearlet soft-thresholding procedure can be used for phase reconstruction with Fresnel data. As it turns out, the shearlet sparsity constraint yields reconstruction results that are far superior to the support constraint, and comparable to the support plus positivity constraint.

Abstract: This paper describes a novel algorithm to encrypt double color images into a single undistinguishable image in quaternion gyrator domain. By using an iterative phase retrieval algorithm, the phase masks used for encryption are obtained. Subsequently, the encrypted image is generated via cascaded quaternion gyrator transforms with different rotation angles. The parameters in quaternion gyrator transforms and phases serve as encryption keys. By knowing these keys, the original color images can be fully restituted. Numerical simulations have demonstrated the validity of the proposed encryption system as well as its robustness against loss of data and additive Gaussian noise.

Abstract: Recent advances in multiwavelength interferometry techniques [Appl. Opt.52, 5758 (2013)] give new insights to phase unwrapping problems and allow the fringe order information contained in the measured phase to be extracted with low computational effort. This work introduces an algebraic solution to the phase unwrapping problem that allows the direct calculation of the unknown integer fringe order. The procedure resembles beat-wavelength approaches, but provides greater flexibility in choosing the measurement wavelengths, a larger measurement range, and a higher robustness against noise, due to the ability to correct for errors during the calculation.

Abstract: We present a novel image hiding method based on phase retrieval algorithm under the framework of nonlinear double random phase encoding in fractional Fourier domain. Two phase-only masks (POMs) are efficiently determined by using the phase retrieval algorithm, in which two cascaded phase-truncated fractional Fourier transforms (FrFTs) are involved. No undesired information disclosure, post-processing of the POMs or digital inverse computation appears in our proposed method. In order to achieve the reduction in key transmission, a modified image hiding method based on the modified phase retrieval algorithm and logistic map is further proposed in this paper, in which the fractional orders and the parameters with respect to the logistic map are regarded as encryption keys. Numerical results have demonstrated the feasibility and effectiveness of the proposed algorithms.

Abstract: Optical phase imaging enables visualization of transparent samples, numerical refocusing, and other computational processing. Typically phase is measured quantitatively using interferometric techniques such as digital holography. Researchers have demonstrated image enhancement by synthetic aperture imaging based on digital holography. In this work we introduce a novel imaging technique that implements synthetic aperture imaging using phase retrieval, a non-interferometric technique. Unlike digital holography, phase retrieval obviates the need for a reference arm and provides a more compact, less expensive, and more stable experimental setup. We call this technique synthetic aperture phase retrieval.

Abstract: The present work is concerned with the development and application of a novel fringe analysis technique based on the principles of the windowed-Fourier-transform (WFT) for the determination of temperature and concentration fields from interferometric images for a range of heat and mass transfer applications. Based on the extent of the noise level associated with the experimental data, the technique has been coupled with two different phase unwrapping methods: the Itoh algorithm and the quality guided phase unwrapping technique for phase extraction. In order to generate the experimental data, a range of experiments have been carried out which include cooling of a vertical flat plate in free convection conditions, combustion of mono-propellant flames, and growth of organic as well as inorganic crystals from their aqueous solutions. The flat plate and combustion experiments are modeled as heat transfer applications wherein the interest is to determine the whole-field temperature distribution. Aqueous-solution-based crystal growth experiments are performed to simulate the mass transfer phenomena and the interest is to determine the two-dimensional solute concentration field around the growing crystal. A Mach–Zehnder interferometer has been employed to record the path-integrated quantity of interest (temperature and/or concentration) in the form of interferometric images in the experiments. The potential of the WFT method has also been demonstrated on numerically simulated phase data for varying noise levels, and the accuracy in phase extraction have been quantified in terms of the root mean square errors. Three levels of noise, i.e., 0%, 10%, and 20% have been considered. Results of the present study show that the WFT technique allows an accurate extraction of phase values that can subsequently be converted into two-dimensional temperature and/or concentration distribution fields. Moreover, since WFT is a local processing technique, speckle patterns and the inherent noise in the interferometric data do not affect the resultant phase values. Brief comparisons of the accuracy of the WFT with other standard techniques such as conventional Fourier-filtering methods are also presented.

Abstract: This paper proposes a modified Shack–Hartmann wavefront sensor for phase retrieval. The sensor is revamped by placing a detector at a defocused plane before the focal plane of the lenslet array of the Shack–Hartmann sensor. The algorithm for phase retrieval is an optimization with initial Zernike coefficients calculated by the conventional phase reconstruction of the Shack–Hartmann sensor. Numerical simulations show that the proposed sensor permits sensitive, accurate phase retrieval. Furthermore, experiments tested the feasibility of phase retrieval using the proposed sensor. The surface irregularity for a flat mirror was measured by the proposed method and a Veeco interferometer, respectively. The irregularity for the mirror measured by the proposed method is in very good agreement with that measured using the Veeco interferometer.