Complex-valued Object Research Articles

Reconstruction of a non-negative complex-valued object without a support constraint

Reconstruction of a non-negative complex-valued object without a support constraint

Adaptive support constraint based on pixel difference distribution for twin-image removal in in-line holography

A tight support domain can enhance the reconstruction effect of phase retrieval algorithms using a support domain constraint. Hence, an adaptive support method based on pixel difference distribution to calculate a more accurate support domain is proposed in this paper. The support domain is continuously updated at each iteration of the phase retrieval algorithm, resulting in a support domain that closely corresponds to the shape and size of the object. Simulations are conducted to investigate the reconstruction effect of complex-valued objects as well as the influence of varying the binarization threshold and the difference window size on the convergence performance in the absence of noise. The changing dynamics of the support domain during the iterative process are also showcased. The experiments demonstrate that the adaptive support method can improve the reconstruction image quality and image contrast under actual conditions. Both simulations and experiments affirm the feasibility of this approach.

Phase retrieval for arbitrary complex-valued objects using structured illumination.

A method to solve of the phase retrieval problem in a non-convex formulation for complex-valued objects with a support constraint is proposed. It is shown that two coded diffraction patterns (CDPs) obtained in the same Fresnel or Fraunhofer diffraction plane by masking an object with two, direct and inverse, random binary amplitude masks, are sufficient to reconstruct an arbitrary complex-valued object up to the global phase. The general solution of the problem was found as the sum of two mutually phase-consistent partial solutions obtained by applying the modified error-reduction or hybrid input-output algorithm to each of two "mask+CDP" pairs. The results of model experiments confirmed the possibility of noise-resistant and high-accuracy retrieval of complex-valued objects of various types with the oversampling ratio σ ≥ 2 making use of a small number of iterations. The method is applicable to coherent radiation of any kind.

Adaptive constraints by morphological operations for single-shot digital holography

Digital holography provides access to quantitative measurement of the entire complex field, which is indispensable for the investigation of wave-matter interactions. The emerging iterative phase retrieval approach enables to solve the inverse imaging problem only from the given intensity measurements and physical constraints. However, enforcing imprecise constraints limits the reconstruction accuracy and convergence speed. Here, we propose an advanced iterative phase retrieval framework for single-shot in-line digital holography that incorporates adaptive constraints, which achieves optimized convergence behavior, high-fidelity and twin-image-free reconstruction. In conjunction with morphological operations which can extract the object structure while eliminating the irrelevant part such as artifacts and noise, adaptive constraints allow the support region to be accurately estimated and automatically updated at each iteration. Numerical reconstruction of complex-valued objects and the capability of noise immunity are investigated. The improved reconstruction performance of this approach is experimentally validated. Such flexible and versatile framework has promising applications in biomedicine, X-ray coherent diffractive imaging and wavefront sensing.

Extension of nonnegative constraints to difference-map algorithm on both real and imaginary parts of real space in phase retrieval

The difference-map (DM) algorithm is an iterative method to retrieve the image of an object from its diffraction pattern. Our study proposes the use of nonnegative constraints, reducing the number of unknown variables by half, on both the real and imaginary parts of the object space to reconstruct the image of a complex-valued object in the iterative process of the DM algorithm. The feasibility of the algorithm in biological cell applications was demonstrated using both simulations and optical laser coherent diffraction experiments. The results show that a more accurate image of the object can be retrieved using a loose support by nonnegative constraints than by the same support alone. The comparison indicates that the DM algorithm is superior to the hybrid input–output algorithm in the presence of nonnegative constraints. Therefore, during the execution of a DM algorithm, a loose support allows for more accurate reconstruction than a tight support, which can hardly be found in most cases because of the noise blurring of the retrieved image, and considerable reduction in the reconstruction errors.

Practical phase retrieval using double deep image priors

Phase retrieval (PR) consists of recovering complex-valued objects from their oversampled Fourier magnitudes and takes a central place in scientific imaging. A critical issue around PR is the typical nonconvexity in natural formulations and the associated bad local minimizers. The issue is exacerbated when the support of the object is not precisely known and hence must be overspecified in practice. Practical methods for PR hence involve convolved algorithms, e.g., multiple cycles of hybrid input-output (HIO) + error reduction (ER), to avoid the bad local minimizers and attain reasonable speed, and heuristics to refine the support of the object, e.g., the famous shrinkwrap trick. Overall, the convolved algorithms and the support-refinement heuristics induce multiple algorithm hyperparameters, to which the recovery quality is often sensitive. In this work, we propose a novel PR method by parameterizing the object as the output of a learnable neural network, i.e., deep image prior (DIP). For complex-valued objects in PR, we can flexibly parametrize the magnitude and phase, or the real and imaginary parts separately by two DIPs. We show that this simple idea, free from multi-hyperparameter tuning and support-refinement heuristics, can obtain superior performance than gold-standard PR methods. For the session: Computational Imaging using Fourier Ptychography and Phase Retrieval.

Coherent modulation imaging using a physics-driven neural network.

Coherent modulation imaging (CMI) is a lessness diffraction imaging technique, which uses an iterative algorithm to reconstruct a complex field from a single intensity diffraction pattern. Deep learning as a powerful optimization method can be used to solve highly ill-conditioned problems, including complex field phase retrieval. In this study, a physics-driven neural network for CMI is developed, termed CMINet, to reconstruct the complex-valued object from a single diffraction pattern. The developed approach optimizes the network's weights by a customized physical-model-based loss function, instead of using any ground truth of the reconstructed object for training beforehand. Simulation experiment results show that the developed CMINet has a high reconstruction quality with less noise and robustness to physical parameters. Besides, a trained CMINet can be used to reconstruct a dynamic process with a fast speed instead of iterations frame-by-frame. The biological experiment results show that CMINet can reconstruct high-quality amplitude and phase images with more sharp details, which is practical for biological imaging applications.

Holography with higher-order Stokes correlation

We present and experimentally demonstrate a holography technique with a higher-order Stokes correlation of the random light field. The proposed technique is capable to reconstruct the complex-valued object from the random light field, and overcome twin image issues in the higher-order correlation holography without angular shift, phase shifting, and iterations. A basic principle of the proposed technique is presented and its application is demonstrated by establishing a connection between the higher-order Stokes fluctuations correlations and the far-field complex coherence function of the random field in a lensless configuration. The viability of our technique is demonstrated by considering incoherent reconstruction of the hologram and reconstructing the complex-valued information encoded into the hologram as a distribution of complex coherence function.

Pixel Resolution Imaging in Parallel Phase-Shifting Digital Holography

Parallel phase-shifting digital holography (PPSDH) employing a polarization image sensor can suppress zero-order and twin-image noise through a single exposure, achieve instantaneous measurement of complex-valued dynamic objects, and have broad applications in the areas of biomedicine, etc. To improve the imaging resolution of PPSDH, we propose an oversampled super-pixel image reconstruction method, which can be expressed as the implementation of nearest-neighbor interpolation to replace blank pixels in sparse sub-phase-shift holograms. We found experimentally that the maximum spatial lateral resolution of the reconstructed image based on the existing super-pixel method, B-spline, bicubic, bilinear, and the proposed nearest-neighbor interpolation was 12.4 µm, 11.4 µm, 9.8 µm, 8.8 µm, and 7.8 µm, respectively. The main reason for not reaching the ideal value of 6.9 µm was the inherent residual zero-order and twin-image noise, which needs to be removed in the future.

Single-Shot On-Axis Fizeau Polarization Phase-Shifting Digital Holography for Complex-Valued Dynamic Object Imaging

Digital holography assisted with inline phase-shifting methods has the benefit of a large field of view and a high resolution, but it is limited in dynamic imaging due to sequential detection of multiple holograms. Here we propose and experimentally demonstrate a single-shot phase-shifting digital holography system based on a highly stable on-axis Fizeau-type polarization interferometry. The compact on-axis design of the system with the capability of instantaneous recording of multiple phase-shifted holograms and with robust stability features makes the technique a novel tool for the imaging of complex-valued dynamic objects. The efficacy of the approach is demonstrated experimentally by complex field imaging of various kinds of reflecting-type static and dynamic objects. Moreover, a quantitative analysis on the robust phase stability and sensitivity of the technique is evaluated by comparing the approach with conventional phase-shifting methods. The high phase stability and dynamic imaging potential of the technique are expected to make the system an ideal tool for quantitative phase imaging and real-time imaging of dynamic samples.

Dynamic coherent diffractive imaging with a physics-driven untrained learning method.

Reconstruction of a complex field from one single diffraction measurement remains a challenging task among the community of coherent diffraction imaging (CDI). Conventional iterative algorithms are time-consuming and struggle to converge to a feasible solution because of the inherent ambiguities. Recently, deep-learning-based methods have shown considerable success in computational imaging, but they require large amounts of training data that in many cases are difficult to obtain. Here, we introduce a physics-driven untrained learning method, termed Deep CDI, which addresses the above problem and can image a dynamic process with high confidence and fast reconstruction. Without any labeled data for pretraining, the Deep CDI can reconstruct a complex-valued object from a single diffraction pattern by combining a conventional artificial neural network with a real-world physical imaging model. To our knowledge, we are the first to demonstrate that the support region constraint, which is widely used in the iteration-algorithm-based method, can be utilized for loss calculation. The loss calculated from support constraint and free propagation constraint are summed up to optimize the network's weights. As a proof of principle, numerical simulations and optical experiments on a static sample are carried out to demonstrate the feasibility of our method. We then continuously collect 3600 diffraction patterns and demonstrate that our method can predict the dynamic process with an average reconstruction speed of 228 frames per second (FPS) using only a fraction of the diffraction data to train the weights.

Phase retrieval of complex-valued objects via a randomized Kaczmarz method

Abstract This paper investigates the convergence of the randomized Kaczmarz algorithm for the problem of phase retrieval of complex-valued objects. Although this algorithm has been studied for the real-valued case in [ 28], its generalization to the complex-valued case is nontrivial and has been left as a conjecture. This paper applies a different approach by establishing the connection between the convergence of the algorithm and the convexity of an objective function. Based on the connection, it demonstrates that when the sensing vectors are sampled uniformly from a unit sphere in ${\mathcal{C}}^n$ and the number of sensing vectors $m$ satisfies $m>O(n\log n)$ as $n, m\rightarrow \infty $, then this algorithm with a good initialization achieves linear convergence to the solution with high probability. The method can be applied to other statistical models of sensing vectors as well. A similar convergence result is established for the unitary model, where the sensing vectors are from the columns of random orthogonal matrices. 2000 Math Subject Classification: 68W20, 68W27, 92D25.

Quantitative reconstruction of the complex-valued object based on complementary phase modulations

In the coherent diffraction imaging (CDI) techniques, a key point is to reconstruct the complex-valued object from the far-field intensity measurements, i.e., solving the phase retrieval problem. However, due to this ill-posed problem, traditional phase retrieval algorithms often encounter some problems associated with the iteration convergence. In this work, complementary phase modulations (CPM) are introduced to generate different far-field intensity measurements. The namely CPM-based method aims to find out the global optimal solution by imposing multi-dimensional constraints, including the diverse intensity images at the Fourier plane and the CPM at the object plane. It is proved by the numerical simulations and the optical experiments that the convergence speed and the recovery accuracy could be greatly improved. Furthermore, the shifting complementary phase modulations (SCPM)-based method is proposed by introducing more CPMs. The reconstruction performance is further improved even when the phase range is larger, and the support constraints are not required. In addition, the SCPM-based method is more robust to the Poisson noise. With the outstanding reconstruction performance, the CPM-based methods may be helpful to phase imaging in the application of visible-light microscopy and X-ray imaging.

Non-interferometric accurate phase imaging via linear-convergence iterative optimization

This paper reported a general non-interferometric high-accuracy quantitative phase imaging (QPI) method for arbitrary complex-valued objects. Given by a typical 4-f optical configuration as the imaging system, three frames of small-window phase modulation are applied on the object's Fourier spectrum so that redistributed intensity patterns are produced on the image plane, in which the object phase emerges at different degree. Then, an algebraic relationship that connects the object phase with the output intensity is established to provide us with an approximate closed-form phase recovery. Further, an efficient iterative optimization strategy is developed to turn that approximate solution into an accurate one. Due to the linear convergence property of the iteration, a high-accuracy phase recovery is achieved without requiring heavy iterations. The feasibility and accuracy of the proposed method are verified by both numerical simulations and experiments on diverse phase objects including biomedical tissues.

Mask-modulated lensless imaging via translated structured illumination.

Lensless microscopy technique enables high-resolution image recovery over a large field of view. By integrating the concept of phase retrieval, it can also retrieve the lost phase information from intensity-only measurements. Here we report a mask-modulated lensless imaging platform based on translated structured illumination. In the reported platform, we sandwich the object in-between a coded mask and a naked image sensor for lensless data acquisition. An LED array is used to provide angle-varied illumination for projecting a translated structured pattern without involving mechanical scanning. For different LED elements, we acquire the lensless intensity data for recovering the complex-valued object. In the reconstruction process, we employ the regularized ptychographic iterative engine and implement an up-sampling process in the reciprocal space. As demonstrated by experimental results, the reported platform is able to recover complex-valued object images with higher resolution and better quality than previous implementations. Our approach may provide a cost-effective solution for high-resolution and wide field-of-view ptychographic imaging without involving mechanical scanning.

Hyperspectral phase retrieval: spectral–spatial data processing with sparsity-based complex domain cube filter

Hyperspectral (HS) imaging retrieves information from data obtained across broadband spectral channels. Information to retrieve is a 3D cube, where two coordinates are spatial and the third one is spectral. This cube is complex-valued with varying amplitude and phase. We consider shearography optical setup, in which two phase-shifted broadband copies of the object projections are interfering at a sensor. Registered observations are intensities summarized over spectral channels. For phase reconstruction, the variational setting of the phase retrieval problem is used to derive the iterative algorithm, which includes the original proximity spectral analysis operator and the sparsity modeling of the complex-valued object 3D cube. We resolve the HS phase retrieval problem without random phase coding of wavefronts typical for the most conventional phase retrieval techniques. We show the performance of the algorithm for object phase and thickness imaging in simulation and experimental tests.

Review of partially coherent diffraction imaging

Coherent diffraction imaging (CDI), a type of lensless imaging method, relies on the use of light source with high-degree coherence to compute highly resolved complex-valued objects. The coherence of light source consists of temporal coherence and spatial coherence. In practice, it is difficult to obtain a fully coherent source. Spatial decoherence can be generated in the following three scenarios: no synchronization mechanism for the whole radiation source, a finite (non-zero) point spread function of the detector, and the sample variation within exposure time. Partial temporal coherence means that the beam is not quasi-monochromatic, behaving as the energy spread of the illumination. The consequence of reduced degree of temporal and/or spatial coherence in CDI is the decrease of visibility in the measured diffraction intensity. A fundamental assumption of CDI is the full temporal and spatial coherence, and even a relatively small deviation from full coherence can prevent the phase retrieval algorithm from converging accurately. It is necessary to break the barrier of limited coherence by improving the experimental setups directly or optimizing the phase retrieval algorithms to mitigate decoherence. Based on the Wolf’s model of coherence-mode of light and the framework of CDI using partially coherent light proposed by Nugent et al., various methods have been proposed to solve the problems induced by low coherence. Those methods generally experience a similar development process, that is, from the requirement for measuring the spatial (coherent length or complex coherent factor) or temporal (spectrum distribution) coherence properties to without the need for such priori knowledge. Here in this work, the principles of partial coherent CDI, and the major progress of CDI with partial spatial- and temporal-coherent light are reviewed.

Ghost diffraction holographic microscopy

Unconventional systems that adopt the concept of ghost schemes have led to advancements in some imaging applications. However, their application in quantitative phase imaging remains a challenge. Here, we introduce a basis for quantitative phase imaging with ghost diffraction and demonstrate ghost diffraction holographic microscopy for complex-valued imaging. We achieve this by introducing an off-axis holography approach in the modified ghost diffraction system. We also realize a correlation hologram in the cross-correlation of intensities from two detectors in the modified ghost diffraction setup and digitally process the correlation hologram to image complex-valued objects. To generate experiment results, we use a modified interferometer setup and exploit the spatial statistics of the scattered field from a time-frozen pseudothermal light source. Finally, we evaluate the efficacy of the approach by simulation and follow that with experiments that demonstrate imaging of pure phase objects, planar transparencies, and resolution test targets, among others.

Continuous-wave terahertz reflective ptychography by oblique illumination.

Massive usage scenarios prompt the prosperity of terahertz (THz) reflective imaging methods. In this Letter, we apply ptychography to continuous-wave THz reflective imaging. Our scheme has a compact lensless layout and uses a full-field oblique-illumination recording mode. Diffraction patterns are corrected through tilted plane correction. This method can be used to retrieve the complex-valued object function and to suppress the negative effect of non-uniform illumination. The feasibility is investigated using two metal samples.

High-resolution (diffraction limited) single-shot multiplexed coded-aperture ptychography

We propose and numerically explore a method that upgrades a conventional single-shot microscope into a single-shot ptychographic microscope, without impairing its optical performance. It is based on splitting the microscope’s intermediate image plane into multiple replicas, and detecting a set of their coded Fourier transform magnitudes, using a different sensor for each replica. To code each beam, it is transmitted through a distinct coding mask. A ptychographic phase-retrieval algorithm is used to reconstruct the imaged object’s complex-valued transmission function. The proposed method also enables ultrahigh-speed imaging by using a pulse-burst as illumination and applying a multi-state reconstruction algorithm. This approach paves the way to both single-frame and ultrahigh-speed, diffraction-limited microscopy of complex-valued objects.