Fourier Intensity Research Articles

An approach for predicting surface strong motion using borehole seismometers

South Korea has implemented borehole-type seismometers for reliable earthquake observations and earthquake early-warning systems, with approximately 85% of seismometers being replaced by borehole-type seismometers after the Gyeongju earthquake. Although these seismometers are more effective at detecting earthquakes owing to the reduced artificial ambient noise, they do not record surface-level shaking. Therefore, it is necessary to estimate ground surface shaking directly associated with potential damage when using borehole-type seismometers without surface sensors. This study investigated and compared various methods, including the stochastic point-source ground-motion model, transfer function based on ambient noise, and one-dimensional site response, to estimate horizontal seismograms of the ground surface. We assessed the accuracy of these methods by comparing the waveforms generated in event cases (magnitude from 2.5 to 5.8, with epicentral distances spanning 22 km–209 km) in terms of Fourier spectra, intensity, and spectral acceleration. Among the methods assessed, the transfer function approach, which does not account for the geophysical characteristics such as VS30, proved to be the most appropriate for correcting ground-surface effects.

Towards practical single-shot phase retrieval with physics-driven deep neural network.

Phase retrieval (PR), a long-established challenge for recovering a complex-valued signal from its Fourier intensity-only measurements, has attracted considerable attention due to its widespread applications in optical imaging. Recently, deep learning-based approaches were developed and allowed single-shot PR. However, due to the substantial disparity between the input and output domains of the PR problems, the performance of these approaches using vanilla deep neural networks (DNN) still has much room to improve. To increase the reconstruction accuracy, physics-informed approaches were suggested to incorporate the Fourier intensity measurements into an iterative estimation procedure. Since the approach is iterative, they require a lengthy computation process, and the accuracy is still not satisfactory for images with complex structures. Besides, many of these approaches work on simulation data that ignore some common problems such as saturation and quantization errors in practical optical PR systems. In this paper, a novel physics-driven multi-scale DNN structure dubbed PPRNet is proposed. Similar to other deep learning-based PR methods, PPRNet requires only a single Fourier intensity measurement. It is physics-driven that the network is guided to follow the Fourier intensity measurement at different scales to enhance the reconstruction accuracy. PPRNet has a feedforward structure and can be end-to-end trained. Thus, it is much faster and more accurate than the traditional physics-driven PR approaches. Extensive simulations and experiments on an optical platform were conducted. The results demonstrate the superiority and practicality of the proposed PPRNet over the traditional learning-based PR methods.

Phase retrieval with only a nonnegativity constraint.

We show that it may be possible to reconstruct a real-valued, nonnegative 2D object from the magnitude of its Fourier transform using only a nonnegativity constraint without the usual support constraint, even when significant noise is present in the Fourier intensity data.

Adaptive Optics Compensation for Orbital Angular Momentum Optical Wireless Communications

Adaptive optics (AO) can efficiently compensate for turbulence-induced distortion in orbital angular momentum (OAM)-based optical wireless communication (OWC) systems. In this paper, we design a modified phase diversity algorithm (MPDA)-based wavefront sensor to enhance the reconstruction accuracy of distorted OAM wavefront information. Aiming to further strike a compelling trade-off between AO system complexity and compensation accuracy, we first construct a novel AO system that applies a quickly and electronically controlled focus-tunable lens (FTL). It decontaminates distorted OAM signaling beams while having a low systemic complexity and superior convergence performance. Furthermore, we propose the 3-modified phase diversity algorithm (3-MPDA) AO scheme relying upon a Fourier intensity and two defocused intensities as the prior information, which beneficially balances the compensation effect and the number of defocused intensities and exhibits good noise robustness against charge-coupled device (CCD) detectors. In summary, this paper provides new insight for designing AO schemes with high compensation performance in communication links.

SiSPRNet: end-to-end learning for single-shot phase retrieval.

With the success of deep learning methods in many image processing tasks, deep learning approaches have also been introduced to the phase retrieval problem recently. These approaches are different from the traditional iterative optimization methods in that they usually require only one intensity measurement and can reconstruct phase images in real-time. However, because of tremendous domain discrepancy, the quality of the reconstructed images given by these approaches still has much room to improve to meet the general application requirements. In this paper, we design a novel deep neural network structure named SiSPRNet for phase retrieval based on a single Fourier intensity measurement. To effectively utilize the spectral information of the measurements, we propose a new feature extraction unit using the Multi-Layer Perceptron (MLP) as the front end. It allows all pixels of the input intensity image to be considered together for exploring their global representation. The size of the MLP is carefully designed to facilitate the extraction of the representative features while reducing noises and outliers. A dropout layer is also equipped to mitigate the possible overfitting problem in training the MLP. To promote the global correlation in the reconstructed images, a self-attention mechanism is introduced to the Up-sampling and Reconstruction (UR) blocks of the proposed SiSPRNet. These UR blocks are inserted into a residual learning structure to prevent the weak information flow and vanishing gradient problems due to their complex layer structure. Extensive evaluations of the proposed model are performed using different testing datasets of phase-only images and images with linearly related magnitude and phase. Experiments were conducted on an optical experimentation platform (with defocusing to reduce the saturation problem) to understand the performance of different deep learning methods when working in a practical environment. The results demonstrate that the proposed approach consistently outperforms other deep learning methods in single-shot maskless phase retrieval. The source codes of the proposed method have been released in Github [see references].

Extended depth of field reconstruction with complex field estimation

The contradiction between high resolution and large depth of field (DOF) of microscopic imaging limits its application in imaging 3D distributed samples. Here, we propose a new technique to achieve extended depth of field reconstruction in bright-field microscopy through complex field estimation. The complex field is estimated by a phase retrieval algorithm with the four illumination modulation patterns and the corresponding captured Fourier intensities as dual constraints. Then, the digital refocusing strategy based on Fresnel propagator is integrated into reconstruction framework to achieve the reconstruction of high-resolution complex-valued information for each part of the sample, which ensures an effective extension of the DOF. The feasibility of the proposed method to maintain high resolution over extended DOF was verified in actual experiments with a tilted USAF target, biological sample and microlens array (MLA). The extended DOF reconstruction provides a robust method for imaging in both industrial inspection and biomedical research.

Photoinduced interlayer dynamics in Td-MoTe2: A broadband pump-probe study

We report on time-resolved broadband transient reflectivity (tr-bb-TR) measurements performed on a bulk single crystal of Td-MoTe2 as a function of the incident pump fluence (F). Tr-bb-TR data unveil photoinduced electronic changes progressing on the sub-picosecond timescale as well as the dynamics of the coherent low-frequency 1A1 interlayer shear phonon. Our results indicate a gradual evolution of both the TR and the 1A1 Fourier intensity spectra as a function of F, ruling out the threshold-like change that has been associated with the ultrafast photoinduced Td → 1T ′ phase transition. We also observe a large redshift of the 1A1 Fourier spectral features, which implies that large renormalization effects are taking place on the interband transitions that are dielectrically susceptible to the 1A1 interlayer shear phonon displacement.

Robust Phase Retrieval with Complexity-Guidance for Coherent X-Ray Imaging

Reconstruction of a stable and reliable solution from noisy and incomplete Fourier intensity data is a challenging problem for iterative phase retrieval algorithms. The typical methodology employed in the coherent X-ray imaging (CXI) literature involves thousands of iterations of well-known phase retrieval algorithms, e.g., hybrid input-output (HIO) or relaxed averaged alternating reflections (RAAR), that are concluded with a smaller number of error reduction (ER) iterations. Since the single run of this methodology may not provide a reliable solution, hundreds of trial solutions are first obtained by initializing the phase retrieval algorithm with independent random guesses. The resulting trial solutions are then averaged with appropriate phase adjustment, and resolution of the averaged reconstruction is assessed by plotting the phase retrieval transfer function (PRTF). In this work, we examine this commonly used RAAR-ER methodology from the perspective of the complexity parameter introduced by us in recent years. It is observed that the single run of the RAAR-ER algorithm provides a solution with undesirable grainy artifacts that persist to some extent even after averaging the multiple trial solutions. The grainy features are spurious in the sense that they are smaller in size compared to the resolution predicted by the PRTF curve. This inconsistency can be addressed by a novel methodology that we refer to as complexity-guided RAAR (CG-RAAR). The methodology is demonstrated with simulations and experimental data sets from the CXIDB database. In addition to providing consistent solution, CG-RAAR is also observed to require reduced number of independent trials for averaging.

Complexity-guided Fourier phase retrieval from noisy data

Reconstruction of a stable and good quality solution from noisy single-shot Fourier intensity data is a challenging problem for phase retrieval algorithms. We examine behavior of the solution provided by the hybrid input-output (HIO) algorithm for noisy data, from the perspective of the complexity guidance methodology that was introduced by us in an earlier paper [J. Opt. Soc. Am. A36, 202 (2019)JOAOD60740-323210.1364/JOSAA.36.000202]. We find that for noisy data, the complexity of the solution outside the support keeps increasing as the HIO iterations progress. Based on this observation, a strategy for controlling the solution complexity within and outside the support during the HIO iterations is proposed and tested. In particular, we actively track and control the growth of complexity of the solution outside the support region with iterations. This in turn provides us with guidance regarding the level to which the complexity of the solution within the support region needs to be adjusted, such that the total solution complexity is equal to that estimated from raw Fourier intensity data. In our studies, Poisson noise with mean photon counts per pixel in the Fourier intensity data ranges over four orders of magnitude. We observe that the performance of the proposed strategy is noise robust in the sense that with increasing noise, the quality of the phase solution degrades gradually. For higher noise levels, the solution loses textural details while retaining the main object features. Our numerical experiments show that the proposed strategy can uniformly handle pure phase objects, mixed amplitude-phase objects, and the case of dc blocked Fourier intensity data. The results may find a number of applications where single-shot Fourier phase retrieval is critical to the success of corresponding applications.

Dynamic sampling iterative phase retrieval for holographic data storage.

A dynamic sampling iterative phase retrieval method, which dynamically samples the Fourier intensity distribution of the reconstruction beam captured by the detector, is proposed to shorten the iterative number and decrease the phase error rate of phase retrieval in the phase-modulated holographic data storage. By the dynamic sampling method, that keeping relatively low frequency component of Fourier intensity spectrum at the beginning of iteration and gradually releasing more high frequency component at the subsequent iterations, we shortened the iterative number by 2 times and decreased the phase error rate to some extent because our method provided a better convergent path to the phase retrieval. We also believe the thought of our method can be used in more image retrieval fields.

Pendulum beams: optical modes that simulate the quantum pendulum

The wave equation of electromagnetism, the Helmholtz equation, has the same form as the Schrödinger equation, and so optical waves can be used to study quantum mechanical problems. The electromagnetic wave solutions for non-diffracting beams lead to the two-dimensional Helmholtz equation. When expressed in elliptical coordinates the solution of the angular part is the same as the Schrödinger equation for the simple pendulum. The resulting optical eigenmodes, Mathieu modes, have an optical Fourier transform with a spatial intensity distribution that is proportional to the quantum mechanical probability for the pendulum. Comparison of Fourier intensities of eigenmodes are in excellent agreement with calculated quantum mechanical probabilities of pendulum stationary states. We further investigate wave-packet superpositions of a few modes and show that they mimic the libration and the nonlinear rotation of the classical pendulum, including revivals due to the quantized nature of superpositions. The ability to ‘dial a wavefunction’ with the optical modes allows the exploration of important aspects of quantum wave-mechanics and the pendulum that may not be possible with other physical systems.

Phase retrieval exact solution based on structured window modulation without direct reference waves

Arising in coherent diffraction imaging (CDI), the phase retrieval problem is to reconstruct a complex object by measuring only diffraction intensity patterns. Existing approaches, such as iterative Fourier algorithm (IFA), transport intensity equation (TIE) methods and matrix optimisation, result in near-optimal solutions rather than exact solutions. This paper proposes an analytic method to get the exact phase retrieval solution without relying on direct reference waves. Based on structured template windows, the object far-field phase is solved analytically from the recorded Fourier intensities, from which the exact object reconstruction can be achieved. Sufficient conditions for obtaining exact phase retrieval solution are analysed rigorously and verified by numerical simulations, which also show the robustness of the method through simulations under noises, dynamic range, nonlinear response and bad alignment. A simple experimental result is provided to demonstrate the practical feasibility of the method. It achieves analytic phase retrieval for arbitrary complex-valued objects and may be applicable to quick phase imaging.

Phase retrieval with complexity guidance.

Iterative phase retrieval methods based on the Gerchberg-Saxton (GS) or Fienup algorithm typically show stagnation artifacts even after a large number of iterations. We introduce a complexity parameter ζ that can be computed directly from the Fourier magnitude data and provides a measure of fluctuations in the desired phase retrieval solution. It is observed that when initiated with a constant or a uniformly random phase map, the complexity of the Fienup solution containing stagnation artifacts stabilizes at a numerical value that is higher than ζ. We propose a modified Fienup algorithm that uses a controlled sparsity-enhancing step such that in every iteration the complexity of the resulting guess solution is explicitly made close to ζ. This approach, which we refer to as complexity-guided phase retrieval, is seen to provide an artifact-free phase retrieval solution within a few hundred iterations. Numerical illustrations are provided for both amplitude as well as phase objects with and without Poisson noise introduced in the Fourier intensity data. The complexity-guidance concept may potentially be combined with a variety of phase retrieval algorithms and can enable several practical applications.

Hyperuniformity and anti-hyperuniformity in one-dimensional substitution tilings.

This work considers the scaling properties characterizing the hyperuniformity (or anti-hyperuniformity) of long-wavelength fluctuations in a broad class of one-dimensional substitution tilings. A simple argument is presented which predicts the exponent α governing the scaling of Fourier intensities at small wavenumbers, tilings with α > 0 being hyperuniform, and numerical computations confirm that the predictions are accurate for quasiperiodic tilings, tilings with singular continuous spectra and limit-periodic tilings. Quasiperiodic or singular continuous cases can be constructed with α arbitrarily close to any given value between -1 and 3. Limit-periodic tilings can be constructed with α between -1 and 1 or with Fourier intensities that approach zero faster than any power law.

The Geometry of Ambiguity in One-Dimensional Phase Retrieval

We consider the geometry associated to the ambiguities of the one-dimensional Fourier phase retrieval problem for vectors in $\mathbb{C}^{N+1}$. Our first result states that the space of signals has a finite covering (which we call the root covering) where any two signals in the covering space with the same Fourier intensity function differ by a trivial covering ambiguity. Next we use the root covering to study how the nontrivial ambiguities of a signal vary as the signal varies. This is done by describing the incidence variety of pairs of signals with the same Fourier intensity function modulo global phase. As an application, we give a criterion for a real subvariety of the space of signals to admit generic phase retrieval. The extension of this result to multivectors played an important role in the author's work with Bendory and Eldar on blind phaseless short-time Fourier transform recovery.

Sparse Phase Retrieval of One-Dimensional Signals by Prony's Method

In this paper, we show that sparse signals f representable as a linear combination of a finite number N of spikes at arbitrary real locations or as a finite linear combination of B-splines of order m with arbitrary real knots can be almost surely recovered from O(N^2) Fourier intensity measurements up to trivial ambiguities. The constructive proof consists of two steps, where in the first step the Prony method is applied to recover all parameters of the autocorrelation function and in the second step the parameters of f are derived. Moreover, we present an algorithm to evaluate f from its Fourier intensities and illustrate it at different numerical examples.

Enforcing uniqueness in one-dimensional phase retrieval by additional signal information in time domain

Considering the ambiguousness of the discrete-time phase retrieval problem to recover a signal from its Fourier intensities, one can ask the question: what additional information about the unknown signal do we need to select the correct solution within the large solution set? Based on a characterization of the occurring ambiguities, we investigate different a priori conditions in order to reduce the number of ambiguities or even to receive a unique solution. Particularly, if we have access to additional magnitudes of the unknown signal in the time domain, we can show that almost all signals with finite support can be uniquely recovered. Moreover, we prove that an analogous result can be obtained by exploiting additional phase information.

One-Dimensional Phase Retrieval with Additional Interference Intensity Measurements

The one-dimensional phase retrieval problem consists in the recovery of a complex-valued signal from its Fourier intensity. Due to the well-known ambiguousness of this problem, the determination of the original signal within the extensive solution set is challenging and can only be done under suitable a priori assumption or additional information about the unknown signal. Depending on the application, one has sometimes access to further interference measurements between the unknown signal and a reference signal. Beginning with the reconstruction in the discrete-time setting, we show that each signal can be uniquely recovered from its Fourier intensity and two further interference measurements between the unknown signal and a modulation of the signal itself. Afterwards, we consider the continuous-time problem, where we obtain an equivalent result. Moreover, the unique recovery of a continuous-time signal can also be ensured by using interference measurements with a known or an unknown reference which is unrelated to the unknown signal.

Reconstruction of an object from diffraction intensities averaged over multiple object clusters

A projection operator is derived for use in iterative phase retrieval algorithms when the Fourier intensity data is an average over the intensity from multiple clusters of identical objects. The projection operator is a generalization of the magnitude projection for conventional phase retrieval for a single object, and is applicable when the relative orientations and positions of the objects within the clusters are known. Simulations demonstrate that an iterative projection algorithm equipped with this new projection operator can successfully reconstruct an object from the averaged Fourier intensities from multiple clusters, each containing multiple copies of the object.

Quantitative differentiation of normal and scarred tissues using second-harmonic generation microscopy.

The aim of this study was to differentiate normal and scarred hamster cheek pouch samples by applying a quantitative image analysis technique for determining collagen fiber direction and density in second-harmonic generation microscopy images. This paper presents a collagen tissue analysis of scarred cheek pouches of four adult male Golden Syrian hamsters as an animal model for vocal fold scarring. One cheek pouch was scarred using an electrocautery unit and the other cheek was used as a control for each hamster. A home-built upright microscope and a compact ultrafast fiber laser were used to acquire depth resolved epi-collected second-harmonic generation images of collagen fibers. To quantify the average fiber direction and fiber density in each image, we applied two-dimensional Fourier analysis and intensity thresholding at five different locations for each control and scarred tissue sample, respectively. The resultant depth-resolved average fiber direction variance for scarred hamster cheek pouches (0.61 ± 0.03) was significantly lower (p < 0.05) than control tissue (0.73 ± 0.04), indicating increased fiber alignment within the scar. Depth-resolved average voxel density measurements indicated scarred tissues contained greater (p < 0.005) fiber density (0.72 ± 0.09) compared to controls (0.18 ± 0.03). In the present study, image analysis of both fiber alignment and density from depth-resolved second-harmonic generation images in epi-detection mode enabled the quantification of the increased collagen fiber deposition and alignment typically observed in fibrosis. The epi-detection geometry is the only viable method for in vivo imaging as well as imaging thick turbid tissues. These quantitative endpoints, clearly differentiating between control and scarred hamster cheek pouches, provide an objective means to characterize the extent of vocal fold scarring in vivo in preclinical and clinical research. In particular, this non-invasive method offers advantages for monitoring scar treatments in live animals and following the effects of scarring-related treatments such as application of steroids or drugs targeting pathways involved in fibrosis. SCANNING 38:684-693, 2016. © 2016 Wiley Periodicals, Inc.