Twin-Image-Free Holography: A Compressive Sensing Approach

Single-shot lensless masked imaging with enhanced self-calibrated phase retrieval

Single-shot lensless imaging with a binary amplitude mask enables a low-cost and miniaturized configuration for wave field recovery. However, the mask only allows a part of the wave field to be captured, and thus the inverse decoding process becomes a highly ill-posed problem. Here we propose an enhanced self-calibrated phase retrieval (eSCPR) method to realize single-shot joint recovery of mask distribution and the sample's wavefront. In our method, a sparse regularized phase retrieval (SrPR) algorithm is designed to calibrate the mask distribution. Then, a denoising regularized phase retrieval (DrPR) algorithm is constructed to reconstruct the wavefront of the sample. Compared to conventional single-shot methods, our method shows robust and flexible image recovery. Experimental results of different samples are given to demonstrate the superiority of our method.

Quantitative phase imaging based on holography: trends and new perspectives

In 1948, Dennis Gabor proposed the concept of holography, providing a pioneering solution to a quantitative description of the optical wavefront. After 75 years of development, holographic imaging has become a powerful tool for optical wavefront measurement and quantitative phase imaging. The emergence of this technology has given fresh energy to physics, biology, and materials science. Digital holography (DH) possesses the quantitative advantages of wide-field, non-contact, precise, and dynamic measurement capability for complex-waves. DH has unique capabilities for the propagation of optical fields by measuring light scattering with phase information. It offers quantitative visualization of the refractive index and thickness distribution of weak absorption samples, which plays a vital role in the pathophysiology of various diseases and the characterization of various materials. It provides a possibility to bridge the gap between the imaging and scattering disciplines. The propagation of wavefront is described by the complex amplitude. The complex-value in the complex-domain is reconstructed from the intensity-value measurement by camera in the real-domain. Here, we regard the process of holographic recording and reconstruction as a transformation between complex-domain and real-domain, and discuss the mathematics and physical principles of reconstruction. We review the DH in underlying principles, technical approaches, and the breadth of applications. We conclude with emerging challenges and opportunities based on combining holographic imaging with other methodologies that expand the scope and utility of holographic imaging even further. The multidisciplinary nature brings technology and application experts together in label-free cell biology, analytical chemistry, clinical sciences, wavefront sensing, and semiconductor production.

Analytical solution for single-pixel ptychography through linear modeling

Amplitude-modulated single-pixel ptychography (SPP) enables non-interferometric complex-field imaging of objects. However, the conventional iterative and nondeterministic reconstruction methods, based on the ptychography algorithm, pose challenges in fully understanding the role of critical optical parameters. In response, this paper introduces an innovative analytical approach that establishes a theoretical foundation for the uniqueness of SPP reconstruction results. The proposed method conceptualizes SPP as a system of linear equations in the frequency domain, involving both object and modulated illumination. Solving this equation system reveals a determined solution for the complex object, providing an alternative to iterative and nondeterministic techniques. Through a series of simulations, this approach not only validates the uniqueness of SPP reconstruction, but also explores key properties influencing accuracy.

Fresnel incoherent compressive holography toward 3D videography via dual-channel simultaneous phase-shifting interferometry

Fresnel incoherent correlation holography (FINCH) enables high-resolution 3D imaging of objects from several 2D holograms under incoherent light and has many attractive applications in motionless 3D fluorescence imaging. However, FINCH has difficulty implementing 3D imaging of dynamic scenes since multiple phase-shifting holograms need to be recorded for removing the bias term and twin image in the reconstructed scene, which requires the object to remain static during this progress. Here, we propose a dual-channel Fresnel noncoherent compressive holography method. First, a pair of holograms with π phase shifts obtained in a single shot are used for removing the bias term noise. Then, a physic-driven compressive sensing (CS) algorithm is used to achieve twin-image-free reconstruction. In addition, we analyze the reconstruction effect and suitability of the CS algorithm and two-step phase-shift filtering algorithm for objects with different complexities. The experimental results show that the proposed method can record hologram videos of 3D dynamic objects and scenes without sacrificing the imaging field of view or resolution. Moreover, the system refocuses images at arbitrary depth positions via computation, hence providing a new method for fast high-throughput incoherent 3D imaging.

Single-shot inline holography using a physics-aware diffusion model

Among holographic imaging configurations, inline holography excels in its compact design and portability, making it the preferred choice for on-site or field applications with unique imaging requirements. However, effectively holographic reconstruction from a single-shot measurement remains a challenge. While several approaches have been proposed, our novel unsupervised algorithm, the physics-aware diffusion model for digital holographic reconstruction (PadDH), offers distinct advantages. By seamlessly integrating physical information with a pre-trained diffusion model, PadDH overcomes the need for a holographic training dataset and significantly reduces the number of parameters involved. Through comprehensive experiments using both synthetic and experimental data, we validate the capabilities of PadDH in reducing twin-image contamination and generating high-quality reconstructions. Our work represents significant advancements in unsupervised holographic imaging by harnessing the full potential of the pre-trained diffusion prior.

Dual-constrained physics-enhanced untrained neural network for lensless imaging

An untrained neural network (UNN) paves a new way to realize lensless imaging from single-frame intensity data. Based on the physics engine, such methods utilize the smoothness property of a convolutional kernel and provide an iterative self-supervised learning framework to release the needs of an end-to-end training scheme with a large dataset. However, the intrinsic overfitting problem of UNN is a challenging issue for stable and robust reconstruction. To address it, we model the phase retrieval problem into a dual-constrained untrained network, in which a phase-amplitude alternating optimization framework is designed to split the intensity-to-phase problem into two tasks: phase and amplitude optimization. In the process of phase optimization, we combine a deep image prior with a total variation prior to retrain the loss function for the phase update. In the process of amplitude optimization, a total variation denoising-based Wirtinger gradient descent method is constructed to form an amplitude constraint. Alternative iterations of the two tasks result in high-performance wavefield reconstruction. Experimental results demonstrate the superiority of our method.

Physics-driven universal twin-image removal network for digital in-line holographic microscopy

Digital in-line holographic microscopy (DIHM) enables efficient and cost-effective computational quantitative phase imaging with a large field of view, making it valuable for studying cell motility, migration, and bio-microfluidics. However, the quality of DIHM reconstructions is compromised by twin-image noise, posing a significant challenge. Conventional methods for mitigating this noise involve complex hardware setups or time-consuming algorithms with often limited effectiveness. In this work, we propose UTIRnet, a deep learning solution for fast, robust, and universally applicable twin-image suppression, trained exclusively on numerically generated datasets. The availability of open-source UTIRnet codes facilitates its implementation in various DIHM systems without the need for extensive experimental training data. Notably, our network ensures the consistency of reconstruction results with input holograms, imparting a physics-based foundation and enhancing reliability compared to conventional deep learning approaches. Experimental verification was conducted among others on live neural glial cell culture migration sensing, which is crucial for neurodegenerative disease research.

Randomness assisted in-line holography with deep learning

We propose and demonstrate a holographic imaging scheme exploiting random illuminations for recording hologram and then applying numerical reconstruction and twin image removal. We use an in-line holographic geometry to record the hologram in terms of the second-order correlation and apply the numerical approach to reconstruct the recorded hologram. This strategy helps to reconstruct high-quality quantitative images in comparison to the conventional holography where the hologram is recorded in the intensity rather than the second-order intensity correlation. The twin image issue of the in-line holographic scheme is resolved by an unsupervised deep learning based method using an auto-encoder scheme. Proposed learning technique leverages the main characteristic of autoencoders to perform blind single-shot hologram reconstruction, and this does not require a dataset of samples with available ground truth for training and can reconstruct the hologram solely from the captured sample. Experimental results are presented for two objects, and a comparison of the reconstruction quality is given between the conventional inline holography and the one obtained with the proposed technique.

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.

Light People: Professor Liangcai Cao

EDITORIAL

Holography utilizes the principles of wave interference and diffraction to record and reconstruct images, which can highly restore the three-dimensional features of objects and provide an immersive visual experience. Dennis Gabor proposed the concept of holography in 1947 and was awarded the Nobel Prize in Physics in 1971. Holography has gradually developed into two major research directions: digital holography (DH) and computer-generated holography (CGH). Holography has empowered the development of fields such as 6G communication, intelligent healthcare, and commercial MR headsets. In recent years, the general solution to optical inverse problems contained in holography also provides theoretical support for its wide integration with computational lithography, optical metamaterials, optical neural networks, orbital angular momentum (OAM), and other areas. This demonstrates its enormous potential for research and application. We are delighted to invite Professor Liangcai Cao from Tsinghua University, a leading scientist in the field of holography, to give us a profound interpretation of the opportunities and challenges of holography. In the interview, Prof. Cao will take us on a journey through the history of holography, share fascinating stories from his academic visits and exchanges, and shed light on the mentor and tutor culture in teaching. Through this episode of "Light People," we will have the privilege of getting to know Prof. Cao on a deeper level.

Compressive holographic sensing simplifies quantitative phase imaging

Quantitative phase imaging (QPI) has emerged as method for investigating biological specimen and technical objects. However, conventional methods often suffer from shortcomings in image quality, such as the twin image artifact. A novel computational framework for QPI is presented with high quality inline holographic imaging from a single intensity image. This paradigm shift is promising for advanced QPI of cells and tissues.

Autofocusing of Fresnel zone aperture lensless imaging for QR code recognition

Fresnel zone aperture (FZA) lensless imaging encodes the incident light into a hologram-like pattern, so that the scene image can be numerically focused at a long imaging range by the back propagation method. However, the target distance is uncertain. The inaccurate distance causes blurs and artifacts in the reconstructed images. This brings difficulties for the target recognition applications, such as quick response code scanning. We propose an autofocusing method for FZA lensless imaging. By incorporating the image sharpness metrics into the back propagation reconstruction process, the method can acquire the desired focusing distance and reconstruct noise-free high-contrast images. By combining the Tamura of the gradient metrics and nuclear norm of gradient, the relative error of estimated object distance is only 0.95% in the experiment. The proposed reconstruction method significantly improves the mean recognition rate of QR code from 4.06% to 90.00%. It paves the way for designing intelligent integrated sensors.

Wave-optics-based image synthesis for super resolution reconstruction of a FZA lensless camera

A Fresnel Zone Aperture (FZA) mask for a lensless camera, an ultra-thin and functional computational imaging system, is beneficial because the FZA pattern makes it easy to model the imaging process and reconstruct captured images through a simple and fast deconvolution. However, diffraction causes a mismatch between the forward model used in the reconstruction and the actual imaging process, which affects the recovered image's resolution. This work theoretically analyzes the wave-optics imaging model of an FZA lensless camera and focuses on the zero points caused by diffraction in the frequency response. We propose a novel idea of image synthesis to compensate for the zero points through two different realizations based on the linear least-mean-square-error (LMSE) estimation. Results from computer simulation and optical experiments verify a nearly two-fold improvement in spatial resolution from the proposed methods compared with the conventional geometrical-optics-based method.

Hough transform-based multi-object autofocusing compressive holography

Reconstruction of multiple objects from one hologram can be affected by the focus metric judgment of autofocusing. Various segmentation algorithms are applied to obtain a single object in the hologram. Each object is unambiguously reconstructed to acquire its focal position, which produces complicated calculations. Herein, Hough transform (HT)-based multi-object autofocusing compressive holography is presented. The sharpness of each reconstructed image is computed by using a focus metric such as entropy or variance. According to the characteristics of the object, the standard HT is further used for calibration to remove redundant extreme points. The compressive holographic imaging framework with a filter layer can eliminate the inherent noise in in-line reconstruction including cross talk noise of different depth layers, two-order noise, and twin image noise. The proposed method can effectively obtain 3D information on multiple objects and achieve noise elimination by only reconstructing from one hologram.

DH-GAN: a physics-driven untrained generative adversarial network for holographic imaging

Digital holography is a 3D imaging technique by emitting a laser beam with a plane wavefront to an object and measuring the intensity of the diffracted waveform, called holograms. The object's 3D shape can be obtained by numerical analysis of the captured holograms and recovering the incurred phase. Recently, deep learning (DL) methods have been used for more accurate holographic processing. However, most supervised methods require large datasets to train the model, which is rarely available in most DH applications due to the scarcity of samples or privacy concerns. A few one-shot DL-based recovery methods exist with no reliance on large datasets of paired images. Still, most of these methods often neglect the underlying physics law that governs wave propagation. These methods offer a black-box operation, which is not explainable, generalizable, and transferrable to other samples and applications. In this work, we propose a new DL architecture based on generative adversarial networks that uses a discriminative network for realizing a semantic measure for reconstruction quality while using a generative network as a function approximator to model the inverse of hologram formation. We impose smoothness on the background part of the recovered image using a progressive masking module powered by simulated annealing to enhance the reconstruction quality. The proposed method exhibits high transferability to similar samples, which facilitates its fast deployment in time-sensitive applications without the need for retraining the network from scratch. The results show a considerable improvement to competitor methods in reconstruction quality (about 5 dB PSNR gain) and robustness to noise (about 50% reduction in PSNR vs noise increase rate).

Complex wave and phase retrieval from a single off-axis interferogram

Single-frame off-axis holographic reconstruction is promising for quantitative phase imaging. However, reconstruction accuracy and contrast are degraded by noise, frequency spectrum overlap of the interferogram, severe phase distortion, etc. In this work, we propose an iterative single-frame complex wave retrieval based on an explicit model of object and reference waves. We also develop a phase restoration algorithm that does not resort to phase unwrapping. Both simulation and real experiments demonstrate higher accuracy and robustness compared to state-of-the-art methods, for both complex wave estimation and phase reconstruction. Importantly, the allowed bandwidth for the object wave is significantly improved in realistic experimental conditions (similar amplitudes for object and reference waves), which makes it attractive for large field-of-view and high-resolution imaging applications.

Physics-enhanced neural network for phase retrieval from two diffraction patterns

In this work, we propose a physics-enhanced two-to-one Y-neural network (two inputs and one output) for phase retrieval of complex wavefronts from two diffraction patterns. The learnable parameters of the Y-net are optimized by minimizing a hybrid loss function, which evaluates the root-mean-square error and normalized Pearson correlated coefficient on the two diffraction planes. An angular spectrum method network is designed for self-supervised training on the Y-net. Amplitudes and phases of wavefronts diffracted by a USAF-1951 resolution target, a phase grating of 200 lp/mm, and a skeletal muscle cell were retrieved using a Y-net with 100 learning iterations. Fast reconstructions could be realized without constraints or a priori knowledge of the samples.

CoCoCs: co-optimized compressive imaging driven by high-level vision

Compressive imaging senses optically encoded high-dimensional scene data with far fewer measurements and then performs reconstruction via appropriate algorithms. In this paper, we present a novel noniterative end-to-end deep learning-based framework for compressive imaging, dubbed CoCoCs. In comparison to existing approaches, we extend the pipeline by co-optimizing the recovery algorithm with optical coding as well as cascaded high-level computer vision tasks to boost the quality of the reconstruction. We demonstrate the proposed framework on two typical compressive imaging systems, i.e., single pixel imaging and snapshot video compressive imaging. Extensive results, including conventional image quality criteria, mean opinion scores, and accuracy in image classification and motion recognition, confirm that CoCoCs can yield realistic images and videos, which are friendly to both human viewing and computer vision. We hope CoCoCs will give impetus to bridge the gap between compressive imagers and computer vision and the perception of human.

Advances in Digital Holographic Interferometry

Holographic interferometry is a well-established field of science and optical engineering. It has a half-century history of successful implementation as the solution to numerous technical tasks and problems. However, fast progress in digital and computer holography has promoted it to a new level of possibilities and has opened brand new fields of its application. In this review paper, we consider some such new techniques and applications.

Lensfree on-chip microscopy based on single-plane phase retrieval

We propose a novel single-plane phase retrieval method to realize high-quality sample reconstruction for lensfree on-chip microscopy. In our method, complex wavefield reconstruction is modeled as a quadratic minimization problem, where total variation and joint denoising regularization are designed to keep a balance of artifact removal and resolution enhancement. In experiment, we built a 3D-printed field-portable platform to validate the imaging performance of our method, where resolution chart, dynamic target, transparent cell, polystyrene beads, and stained tissue sections are employed for the imaging test. Compared to state-of-the-art methods, our method eliminates image degradation and obtains a higher imaging resolution. Different from multi-wavelength or multi-height phase retrieval methods, our method only utilizes a single-frame intensity data record to accomplish high-fidelity reconstruction of different samples, which contributes a simple, robust, and data-efficient solution to design a resource-limited lensfree on-chip microscope. We believe that it will become a useful tool for telemedicine and point-of-care application.

Resolution and Contrast Enhancement for Lensless Digital Holographic Microscopy and Its Application in Biomedicine

An important imaging technique in biomedicine, the conventional optical microscopy relies on relatively complicated and bulky lens and alignment mechanics. Based on the Gabor holography, the lensless digital holographic microscopy has the advantages of light weight and low cost. It has developed rapidly and received attention in many fields. However, the finite pixel size at the sensor plane limits the spatial resolution. In this study, we first review the principle of lensless digital holography, then go over some methods to improve image contrast and discuss the methods to enhance the image resolution of the lensless holographic image. Moreover, the applications of lensless digital holographic microscopy in biomedicine are reviewed. Finally, we look forward to the future development and prospect of lensless digital holographic technology.

Arbitrary amplitude and phase control in visible by dielectric metasurface

Metasurfaces have been widely studied for arbitrary manipulation of the amplitude, phase and polarization of a field at the sub-wavelength scale. However, realizing a high efficiency metasurface with simultaneous and independent control of the amplitude and phase in visible remains a challenge. In this work, an ultrathin single-cell dielectric metasurface which can modulate arbitrary complex amplitude in transmission mode is proposed. The amplitude is controlled by adjusting the dipoles and quadrupoles by tuning the geometric size, while the phase is manipulated based on the Pancharatnam-Berry phase by rotating the meta-atom. Complex amplitude fields for generating holographic images and structure light are utilized to verify the reliability of the proposed structure. It has been experimentally demonstrated that the quality of holographic image of complex-amplitude hologram encoded on the proposed metasurface is better than that of phase-only holograms and verified by simulation that complex structure light can be generated by the proposed structure. Our work expands the superior limits of various applications, including arbitrary beam shaping, 3D biological imaging, optical computing, and optics-on-chip devices.

Scanning-free digital holography for decoupling the refractive index and thickness via a constrained equation of higher degree

Digital holography is one of the most popular quantitative phase imaging techniques, but the refractive index and the thickness are always coupled in the phase. To solve the decoupling problem, multiple scanning methods such as tomography and total reflection are usually used, which is time-consuming. To increase the imaging speed and reduce the system cost, it is urgent to seek the decoupling method of scanning-free digital holography. In this paper, we find that the decoupling method of scanning-free digital holography can be transformed into a problem of solving constrained higher order equations. By introducing the Fresnel reflection formula, a six-degree equation about refractive index is constructed and the corresponding algorithm for solving the equation is given. By using the algorithm, the refractive index and thickness can be decoupled successfully. A series of results show that the proposed method is effective and has high anti-noise performance. This method provides a mathematical possibility for scanning-free digital holography to decouple the refractive index and complex pixel stepped thickness distributions. Therefore, it may provide a theoretical basis for the subsequent development of a real scanning-free digital holography system, which may have potential applications in the measurement of optical devices produced by the modern film deposition process and etching process.

Compressive propagation with coherence

In this Letter, we present wave propagation models of spatially partially coherent (or spatially incoherent) light to compress the computational load of forward and back propagations in inverse problems. In our model, partially coherent light is approximated as a set of random or plane wavefronts passing through spatial bandpass filters, which corresponds to an illumination pupil, and each wave coherently propagates onto a sensor plane through object space. We show that our models reduce the number of coherent propagations in inverse problems, which are essential in optical control and sensing, such as computer-generated holography (CGH) and quantitative phase imaging. We verify the proposed models by numerical and experimental demonstrations of CGH incorporating spatially partially coherent light.

Efficient compressive holographic reconstruction based on hologram segmentation

Compressive holography can successfully reconstruct a three-dimensional layered object from a two-dimensional hologram. However, the extremely time-consuming reconstruction limits its range of applications. We propose a dimension reduction of measurement matrix (DRMM) method to accelerate compressive holographic reconstruction. The calculation time is substantially reduced while the reconstruction quality is improved by DRMM, which is implemented by a hologram segmentation approach and a parallel computing technique. Holograms of specific target objects are segmented from the hologram of a three-dimensional layered object, and the reconstruction can be implemented in parallel using multicore processors. We present both simulation and experimental results to show the feasibility and effectiveness of the proposed method.

Further improvements to iterative off-axis digital holography

In order to break through the limitation of off-axis holography in the field of measuring rough or strong scattering objects, a new iterative algorithm based on the concept of wavefront-coding was proposed. The reference wave is regarded as a wave modulator and it starts with random guess freed from the result of traditional off-axis holography. The full frequency spectrum could be retrieved iteratively after taking full advantage of the space-bandwidth production of the detector. As one form of coherent diffractive imaging, the theoretical resolution is diffraction limitation. According to the simulations and experiments with random phase plate, when the object fails to be reconstructed by traditional off-axis holography and other iterative off-axis holography algorithm due to the frequency spectrum of object is too wide, the proposed algorithm works well. It could be a general algorithm to prominently improve the capability of off-axis holography to measure rough or strong scattering objects.

Dynamical machine learning volumetric reconstruction of objects' interiors from limited angular views

Limited-angle tomography of an interior volume is a challenging, highly ill-posed problem with practical implications in medical and biological imaging, manufacturing, automation, and environmental and food security. Regularizing priors are necessary to reduce artifacts by improving the condition of such problems. Recently, it was shown that one effective way to learn the priors for strongly scattering yet highly structured 3D objects, e.g. layered and Manhattan, is by a static neural network [Goy et al. Proc. Natl. Acad. Sci. 116, 19848-19856 (2019)]. Here, we present a radically different approach where the collection of raw images from multiple angles is viewed analogously to a dynamical system driven by the object-dependent forward scattering operator. The sequence index in the angle of illumination plays the role of discrete time in the dynamical system analogy. Thus, the imaging problem turns into a problem of nonlinear system identification, which also suggests dynamical learning as a better fit to regularize the reconstructions. We devised a Recurrent Neural Network (RNN) architecture with a novel Separable-Convolution Gated Recurrent Unit (SC-GRU) as the fundamental building block. Through a comprehensive comparison of several quantitative metrics, we show that the dynamic method is suitable for a generic interior-volumetric reconstruction under a limited-angle scheme. We show that this approach accurately reconstructs volume interiors under two conditions: weak scattering, when the Radon transform approximation is applicable and the forward operator well defined; and strong scattering, which is nonlinear with respect to the 3D refractive index distribution and includes uncertainty in the forward operator.

Shifted band-extended angular spectrum method for off-axis diffraction calculation

The shifted band-extended angular spectrum method (Shift-BEASM) is proposed to calculate free-space diffraction between two parallel planes with an off-axis offset. Off-axis numerical propagation is useful for simulating non-paraxial and large-scale fields. The proposed Shift-BEASM allow us to calculate the off-axis diffraction in a wide propagation range by extending the effective bandwidth using the nonuniform fast Fourier transform. The calculation accuracy is higher than that of existing techniques, such as the shifted-Fresnel method and shifted band-limited angular spectrum method, not only in the near field but also in the far field. Numerical examples and accuracy as well as theoretical formulation are presented to confirm validity of the proposed method.

Influence of sparse constraint functions on compressive holographic tomography

In this paper, we quantified and analyzed the impact of the l₁ norm and total variation (TV) norm sparse constraints on the reconstruction quality under different interlayer spacings, sampling rates, and signal-to-noise ratios. For high-quality holograms, the results of compressive-sensing reconstruction using l₁ norm achieved higher quality than those by the TV norm. In contrast, for low-quality holograms, the quality of TV-norm-based reconstruction results was relatively stable and better than that of l₁ norm. In addition, we explained why interlayer spacing cannot be smaller and recommend the use of axial resolution of the digital holography system as the interlayer spacing. The conclusions are valuable in the choice of sparse constraints in compressive holographic tomography.

Axial resolution analysis in compressive digital holographic microscopy

Digital holographic microscopy with compressive sensing (CDHM) has successfully achieved tomography and has been applied in many fields. However, the enhancement of axial resolution in CDHM remains to be elucidated. By deducing accurate formulas for the lateral and axial resolutions without paraxial approximation, we quantized the elongation effect of a digital holography (DH) system in this study. Thus, we revealed that the elongation effect, which is affected only by the system's numerical aperture (NA), is an inherent property of DH systems. We present a detailed analysis herein on the physical significance of the coherence parameter, which is the ratio of a system's limit axial resolution to the interlayer spacing more thoroughly than in previous research. Further, we achieved the tomography of a fiber by using a DH system with a 10 × microscope, with CS to eliminate the elongation effect, and experimentally validated our theoretical results. By applying these theoretical guidelines, we distinguished crossed fibers at distances of 36.4 μm and 48.5 μm, respectively, using the same experimental setup. There would be potential applications of this theory in tomography and observation of microscale objects in the areas of biological and fluid.

Analysis of numerical diffraction calculation methods: from the perspective of phase space optics and the sampling theorem

Diffraction calculations are widely used in applications that require numerical simulation of optical wave propagation. Different numerical diffraction calculation methods have their own transform and sampling properties. In this study, we provide a unified analysis where five popular fast diffraction calculation methods are analyzed from the perspective of phase space optics and the sampling theorem: single fast Fourier transform-based Fresnel transform, Fresnel transfer function approach, Fresnel impulse response approach, angular spectrum method, and Rayleigh-Sommerfeld convolution. The evolutions of an input signal's space-bandwidth product (SBP) during wave propagation are illustrated with the help of a phase space diagram (PSD) and an ABCD matrix. It is demonstrated that all of the above methods cannot make full use of the SBP of the input signal after diffraction; and some transform properties have been ignored. Each method has its own restrictions and applicable range. The reason why different methods have different applicable ranges is explained with physical models. After comprehensively studying and comparing the effect on the SBP and sampling properties of these methods, suggestions are given for choosing the proper method for different applications and overcoming the restrictions of corresponding methods. The PSD and ABCD matrix are used to illustrate the properties of these methods intuitively. Numerical results are presented to verify the analysis, and potential ways to develop new diffraction calculation methods are also discussed.

All-dielectric bifocal isotropic metalens for a single-shot hologram generation device

Planar metalenses are regarded as promising functional nanodevices because of their lightweight, nano-resolution properties, and, therefore, they can serve as versatile platforms for imaging and Fourier transforming. Here, we demonstrate a meta-device that functions as an isotropic bifocal all-dielectric Huygens' metalens to realize nanoscale real-time coaxial digital hologram generation. We design an isotropic bifocal metalens for micro/nano hologram recording, and the metalens utilizes the complete region compared to a previously reported interleaved multifocal metalens scheme. In addition, the hologram generation does not depend on complex polarization conversion, thereby improving the practical efficiency. For high-fidelity reconstruction, compressive reconstruction is utilized to remove twin-image and zero-order items and to suppress noise. Such concept would be extended to white-light achromatic meta-holography and three-dimensional micro/nano in vivo incoherent super-resolution imaging under subwavelength modulation.

Single-shot lensless imaging with fresnel zone aperture and incoherent illumination

Lensless imaging eliminates the need for geometric isomorphism between a scene and an image while allowing the construction of compact, lightweight imaging systems. However, a challenging inverse problem remains due to the low reconstructed signal-to-noise ratio. Current implementations require multiple masks or multiple shots to denoise the reconstruction. We propose single-shot lensless imaging with a Fresnel zone aperture and incoherent illumination. By using the Fresnel zone aperture to encode the incoherent rays in wavefront-like form, the captured pattern has the same form as the inline hologram. Since conventional backpropagation reconstruction is troubled by the twin-image problem, we show that the compressive sensing algorithm is effective in removing this twin-image artifact due to the sparsity in natural scenes. The reconstruction with a significantly improved signal-to-noise ratio from a single-shot image promotes a camera architecture that is flat and reliable in its structure and free of the need for strict calibration.

Noise suppression for ballistic-photons based on compressive in-line holographic imaging through an inhomogeneous medium

Noise suppression is one of the most important tasks in imaging through inhomogeneous mediums. Here, we proposed a denoising approach based on compressive in-line holography for imaging through an inhomogeneous medium. A reference-beam-free system with a low-cost continuous-wave laser is presented. The suppression against the noise, which is brought by the scattering photons, is presented in simulations using the proposed algorithm. The noise immunity is demonstrated in lensless imaging behind a random phase mask with an optical depth of 1.42 by single exposure, as well as behind a ground glass with an optical depth of 6.38 by multiple exposures.

Megahertz-rate shock-wave distortion cancellation via phase conjugate digital in-line holography

Holography is a powerful tool for three-dimensional imaging. However, in explosive, supersonic, hypersonic, cavitating, or ionizing environments, shock-waves and density gradients impart phase distortions that obscure objects in the field-of-view. Capturing time-resolved information in these environments also requires ultra-high-speed acquisition. To reduce phase distortions and increase imaging rates, we introduce an ultra-high-speed phase conjugate digital in-line holography (PCDIH) technique. In this concept, a coherent beam passes through the shock-wave distortion, reflects off a phase conjugate mirror, and propagates back through the shock-wave, thereby minimizing imaging distortions from phase delays. By implementing the method using a pulse-burst laser setup at up to 5 million-frames-per-second, time-resolved holograms of ultra-fast events are now possible. This technique is applied for holographic imaging through laser-spark plasma-generated shock-waves and to enable three-dimensional tracking of explosively generated hypersonic fragments. Simulations further advance our understanding of physical processes and experiments demonstrate ultra-high-speed PCDIH techniques for capturing dynamics.

Shock-waves in explosive, supersonic or ionizing environments impart phase distortions to holographic imaging. Here, the authors report an ultra-high-speed phase conjugate digital in-line holography technique where a laser passes through the shock-wave and is reflected back through the phase distortion, thus correcting phase delays.

Design of diffractive optical element projector for a pseudorandom dot array by an improved encoding method

Here we achieved the structured light patterns of a pseudorandom dot array by a single diffractive optical element. The dot array can be applied to achieve three-dimensional imaging. First, the pseudorandom dot array was generated by the proposed improved encoding methods, which are an improved formula-method-based encoding algorithm and an improved enumeration-method-based encoding algorithm. Second, diffractive optical elements were designed as dot projectors to generate pseudorandom dots by the Gerchberg-Saxton algorithm. Pseudorandom dot arrays with different sizes were generated to validate the proposed encoding methods. A pseudorandom dot array with a maximal size of 713×449 was experimentally achieved. By analyzing the intensity distribution of the projecting pattern, the projected dots have a unique window of 7×7, and the dot array is distortion free. The proposed encoding methods, optimization algorithm, and applied fabrication technology have potential applications in three-dimensional imaging, three-dimensional sensing, shape measurement, and deformation measurement with high decoding speed.

Lensfree on-chip microscopy based on dual-plane phase retrieval

In lensfree on-chip microscopy, the iterative phase retrieval with defocused images easily enables a high-resolution and whole field reconstruction. However, on the reconstruction of the dense sample, conventional methods suffer from the stagnation problem and noise affection under two intensity measurements, which gives rise to a remarkable loss of the image contrast and resolution. Here we propose a novel dual-plane phase retrieval algorithm to perform a stable and versatile lensless reconstruction. A weighted feedback constraint was utilized to speed up the convergence. Then, a gradient descent minimization based on total variation metric was proposed to suppress the noise affection. With these two object constraints, a smoothed but resolution-preserving result can be achieved. Numerical simulations of Gaussian and Poisson noise were given to prove the noise-robustness of our method. The experiments of USAF resolution target, H&E stained pathological slide, and label-free microglia cell demonstrated the superior performance of our approach compared to other state-of-the-art methods.

Intensity-only inverse scattering with MUSIC

We present a method for inverse scattering that relies on intensity-only measurements of the scattered field on a single measurement plane. By collecting measurements from a suite of experiments in which the sample is illuminated using different incident fields, we create sufficient data diversity to overcome the limitations of the intensity-only measurements. We give an explicit procedure that uses an algebraic relation called the polarization identity to convert intensity measurements of scattered fields to interferometric measurements in which one of the scattered fields serves as the reference. By adjusting the multiple signal classification (MUSIC) method for these interferometric data, we effectively recover the location and shapes of multiple objects contained in the imaging region. This method is effective and robust to noise as long as there is sufficiently high data diversity and the fractional volume of the scattering objects is not too high. We present image reconstructions for several three-dimensional examples with simulated data computed using the Method of Fundamental Solutions that demonstrate the effectiveness of this imaging method.

Digital holography free of 2π ambiguity, using coherence modulation

In this Letter, a quantitative measurement method with an extended axial range in low-coherence light digital holography is presented. Based on the characteristics of the light source, the degree of coherence and phase values are obtained. Because the degree of coherence is modulated with respect to the optical path difference, it can be used to remove the 2π ambiguity of the phase, without the use of numerical or dual-wavelength methods. The mathematical procedures from three phase-shifting holograms are numerically described. From experimental results, the accurate measurements of a sample with high step are presented to confirm the effectiveness.

Alignment-tolerant single-shot digital holographic microscopy based on computer-controlled telecentricity

An alignment-tolerant telecentric digital holographic microscopy (AT-T-DHM) system based on computer-controlled telecentricity is proposed. It consists of a three-step process-optical recording, computational compensation, and retrieving processes. With a tube-lens-based two-beam interferometer, phase information of the object is recorded on the hologram, where another optical quadratic phase error (O-QPE) due to the misalignment of the tube lens happens to be added. In the computational compensation process, this phase error can be estimated, by which the O-QPE is balanced out from the recorded hologram. Then, only the phase information of the object can be retrieved from the O-QPE-compensated hologram. This computational compensation process makes the proposed system virtually operate in a telecentric imaging mode, which enables implementing a practical AT-T-DHM. Wave-optical analysis and experiments with a test object confirm the feasibility of the proposed system.

Digital holographic phase imaging based on phase iteratively enhanced compressive sensing

Digital holography has been widely applied in quantitative phase imaging (QPI) for monolayer objects within a limited depth. For multilayer imaging, compressive sensing is employed to eliminate defocused images but with missing phase information. A phase iteratively enhanced compressive sensing (PIE-CS) algorithm is proposed to achieve phase imaging and eliminate defocused images simultaneously. Linear filtering is first applied to the off-axis hologram in Fourier domain, and an intermediate reconstructed complex image is obtained. A periodic phase mask is then superimposed on the intermediate reconstructed image to iteratively eliminate the defocused images and recover the object with phase information. The experimental recovery of amplitude and phase of a two-layer sample with as little as 7% random measurement is demonstrated. The average phase error of the PIE-CS algorithm is analyzed, and the results show the feasibility for QPI.

Resolution priority holographic stereogram based on integral imaging with enhanced depth range

Conventional holographic stereogram (HS) can be generated through fast Fourier transforming parallax images into hogels. Conventional HS uses multiple plane waves to reconstruct 3D images with low resolution and is similar to the principle of depth priority integral imaging (II). We proposed the concept of resolution priority HS for the first time, which is based on the principle of resolution priority II, by adding a quadratic phase term on the conventional Fourier transform. In the proposed resolution priority HS, the resolution of reconstructed 3D images is much better than conventional HS, but the depth range is limited. To enhance the depth range, a multi-plane technique was used to present multiple central depth planes simultaneously. The proposed resolution priority HS with high resolution and enhanced depth range was verified by both simulation and optical experiment.

Sparsity-based continuous wave terahertz lens-free on-chip holography with sub-wavelength resolution

We demonstrate terahertz (THz) lens-free in-line holography on a chip in order to achieve 40 μm spatial resolution corresponding to ~0.7λ with a numerical aperture of ~0.87. We believe that this is the first time that sub-wavelength resolution in THz holography and the 40 μm resolution were both far better than what was already reported. The setup is based on a self-developed high-power continuous wave THz laser at 5.24 THz (λ = 57.25 μm) and a high-resolution microbolometer detector array (640 × 512 pixels) with a pitch of 17 μm. This on-chip in-line holography, however, suffers from the twin-image artifacts which obfuscate the reconstruction. To address this problem, we propose an iterative optimization framework, where the conventional object constraint and the L₁ sparsity constraint can be combined to efficiently reconstruct the complex amplitude distribution of the sample. Note that the proposed framework and the sparsity-based algorithm can be applied to holography in other wavebands without limitation of wavelength. We demonstrate the success of this sparsity-based on-chip holography by imaging biological samples (i.e., a dragonfly wing and a bauhinia leaf).