Whole optical wave field reconstruction from double or multi in-line holograms by phase retrieval algorithm

Off-axis digital lensless holographic microscopy based on spatially multiplexed interferometry

Significance

Digital holographic microscopy (DHM) is a label-free microscopy technique that provides time-resolved quantitative phase imaging (QPI) by measuring the optical path delay of light induced by transparent biological samples. DHM has been utilized for various biomedical applications, such as cancer research and sperm cell assessment, as well as for in vitro drug or toxicity testing. Its lensless version, digital lensless holographic microscopy (DLHM), is an emerging technology that offers size-reduced, lightweight, and cost-effective imaging systems. These features make DLHM applicable, for example, in limited resource laboratories, remote areas, and point-of-care applications.

Aim

In addition to the abovementioned advantages, in-line arrangements for DLHM also include the limitation of the twin-image presence, which can restrict accurate QPI. We therefore propose a compact lensless common-path interferometric off-axis approach that is capable of quantitative imaging of fast-moving biological specimens, such as living cells in flow.

Approach

We suggest lensless spatially multiplexed interferometric microscopy (LESSMIM) as a lens-free variant of the previously reported spatially multiplexed interferometric microscopy (SMIM) concept. LESSMIM comprises a common-path interferometric architecture that is based on a single diffraction grating to achieve digital off-axis holography. From a series of single-shot off-axis holograms, twin-image free and time-resolved QPI is achieved by commonly used methods for Fourier filtering-based reconstruction, aberration compensation, and numerical propagation.

Results

Initially, the LESSMIM concept is experimentally demonstrated by results from a resolution test chart and investigations on temporal stability. Then, the accuracy of QPI and capabilities for imaging of living adherent cell cultures is characterized. Finally, utilizing a microfluidic channel, the cytometry of suspended cells in flow is evaluated.

Conclusions

LESSMIM overcomes several limitations of in-line DLHM and provides fast time-resolved QPI in a compact optical arrangement. In summary, LESSMIM represents a promising technique with potential biomedical applications for fast imaging such as in imaging flow cytometry or sperm cell analysis.

Structured modulation multi-height microscopy for high-resolution imaging

Conventional multi-height microscopy techniques introduce different object-to-detector distances to obtain multiple measurements for phase retrieval. However, surpassing the diffraction limit imposed by the numerical aperture (NA) of the objective lens remains a challenging task. Here, we report a novel structured modulation multi-height microscopy technique for quantitative high-resolution imaging. In our platform, a thin diffuser is placed in between the sample and the objective lens. By translating the diffuser to different axial positions, a sequence of modulated intensity images is captured for reconstruction. The otherwise inaccessible high-resolution object information can thus be encoded into the optical system for detection. In the construction process, we report a ptychographic phase retrieval algorithm to recover the existing wavefront of the complex object. We validate our approach using a resolution target, a phase target, and various biological samples. We demonstrate a ∼4-fold resolution gain over the diffraction limit. We also demonstrate our approach to achieve a 6.5 mm by 4.3 mm field of view and a half-pitch resolution of 1.2 µm. The reported methodology provides a portable, turnkey solution for quantitative high-resolution imaging with potential applications in disease diagnosis, sample screening, and other fields.

Multi-Illumination Single-Holographic-Exposure Lensless Fresnel (MISHELF) Microscopy: Principles and Biomedical Applications

Lensless holographic microscopy (LHM) comes out as a promising label-free technique since it supplies high-quality imaging and adaptive magnification in a lens-free, compact and cost-effective way. Compact sizes and reduced prices of LHMs make them a perfect instrument for point-of-care diagnosis and increase their usability in limited-resource laboratories, remote areas, and poor countries. LHM can provide excellent intensity and phase imaging when the twin image is removed. In that sense, multi-illumination single-holographic-exposure lensless Fresnel (MISHELF) microscopy appears as a single-shot and phase-retrieved imaging technique employing multiple illumination/detection channels and a fast-iterative phase-retrieval algorithm. In this contribution, we review MISHELF microscopy through the description of the principles, the analysis of the performance, the presentation of the microscope prototypes and the inclusion of the main biomedical applications reported so far.

Recent advances in experimental design and data analysis to characterize prokaryotic motility

Bacterial motility plays a key role in important cell processes such as chemotaxis and biofilm formation, but is challenging to quantify due to the small size of the individual microorganisms and the complex interplay of biological and physical factors that influence motility phenotypes. Swimming, the first type of motility described in bacteria, still remains largely unquantified. Light microscopy has enabled qualitative characterization of swimming patterns seen in different strains, such as run and tumble, run-reverse-flick, run and slow, stop and coil, and push and pull, which has allowed for elucidation of the underlying physics. However, quantifying these behaviors (e.g., identifying run distances and speeds, turn angles and behavior by surfaces or cell-cell interactions) remains a challenging task. A qualitative and quantitative understanding of bacterial motility is needed to bridge the gap between experimentation, omics analysis, and bacterial motility theory. In this review, we discuss the strengths and limitations of how phase contrast microscopy, fluorescence microscopy, and digital holographic microscopy have been used to quantify bacterial motility. Approaches to automated software analysis, including cell recognition, tracking, and track analysis, are also discussed with a view to providing a guide for experimenters to setting up the appropriate imaging and analysis system for their needs.

Diffraction-Net: a robust single-shot holography for multi-distance lensless imaging

Digital holography based on lensless imaging is a developing method adopted in microscopy and micro-scale measurement. To retrieve complex-amplitude on the sample surface, multiple images are required for common reconstruction methods. A promising single-shot approach points to deep learning, which has been used in lensless imaging but suffering from the unsatisfied generalization ability and stability. Here, we propose and construct a diffraction network (Diff-Net) to connect diffraction images at different distances, which breaks through the limitations of physical devices. The Diff-Net based single-shot holography is robust as there is no practical errors between the multiple images. An iterative complex-amplitude retrieval approach based on light transfer function through the Diff-Net generated multiple images is used for complex-amplitude recovery. This process indicates a hybrid-driven method including both physical model and deep learning, and the experimental results demonstrate that the Diff-Net possesses qualified generalization ability for samples with significantly different morphologies.

Pixel Super-Resolution Phase Retrieval for Lensless On-Chip Microscopy via Accelerated Wirtinger Flow

Empowered by pixel super-resolution (PSR) and phase retrieval techniques, lensless on-chip microscopy opens up new possibilities for high-throughput biomedical imaging. However, the current PSR phase retrieval approaches are time consuming in terms of both the measurement and reconstruction procedures. In this work, we present a novel computational framework for PSR phase retrieval to address these concerns. Specifically, a sparsity-promoting regularizer is introduced to enhance the well posedness of the nonconvex problem under limited measurements, and Nesterov's momentum is used to accelerate the iterations. The resulting algorithm, termed accelerated Wirtinger flow (AWF), achieves at least an order of magnitude faster rate of convergence and allows a twofold reduction in the measurement number while maintaining competitive reconstruction quality. Furthermore, we provide general guidance for step size selection based on theoretical analyses, facilitating simple implementation without the need for complicated parameter tuning. The proposed AWF algorithm is compatible with most of the existing lensless on-chip microscopes and could help achieve label-free rapid whole slide imaging of dynamic biological activities at subpixel resolution.

Blood-Coated Sensor for High-Throughput Ptychographic Cytometry on a Blu-ray Disc

The Blu-ray drive is an engineering masterpiece that integrates disc rotation, pickup head translation, and three lasers in a compact and portable format. Here, we integrate a blood-coated image sensor with a modified Blu-ray drive for high-throughput cytometric analysis of various biospecimens. In this device, samples are mounted on the rotating Blu-ray disc and illuminated by the built-in lasers from the pickup head. The resulting coherent diffraction patterns are then recorded by the blood-coated image sensor. The rich spatial features of the blood-cell monolayer help down-modulate the object information for sensor detection, thus forming a high-resolution computational biolens with a theoretically unlimited field of view. With the acquired data, we develop a lensless coherent diffraction imaging modality termed rotational ptychography for image reconstruction. We show that our device can resolve the 435 nm line width on the resolution target and has a field of view only limited by the size of the Blu-ray disc. To demonstrate its applications, we perform high-throughput urinalysis by locating disease-related calcium oxalate crystals over the entire microscope slide. We also quantify different types of cells on a blood smear with an acquisition speed of ∼10,000 cells per second. For in vitro experiments, we monitor live bacterial cultures over the entire Petri dish with single-cell resolution. Using biological cells as a computational lens could enable new intriguing imaging devices for point-of-care diagnostics. Modifying a Blu-ray drive with the blood-coated sensor further allows the spread of high-throughput optical microscopy from well-equipped laboratories to citizen scientists worldwide.

A constrained method for lensless coherent imaging of thin samples

Lensless inline holography can produce high-resolution images over a large field of view (FoV). In a previous work [Appl. Opt.60, B38 (2021)APOPAI0003-693510.1364/AO.414976], we showed that (i) the actual FoV can be extrapolated outside of the camera FoV and (ii) the effective resolution of the setup can be several times higher than the resolution of the camera. In this paper, we present a reconstruction method to recover high resolution with an extrapolated FoV image of the phase and the amplitude of a sample from aliased intensity measurements taken at a lower resolution.

Quantitative multi-height phase retrieval via a coded image sensor

Multi-height phase retrieval introduces different object-to-detector distances for obtaining phase diversity measurements. In the acquisition process, the slow-varying phase information, however, cannot be converted to intensity variations for detection. Therefore, the low-frequency contents of the phase profile are lost during acquisition and cannot be properly restored via phase retrieval. Here, we demonstrate the use of a coded image sensor for addressing this challenge in multi-height phase retrieval. In our scheme, we add a coded layer on top of the image sensor for encoding the slow-varying complex wavefronts into intensity variations of the modulated patterns. Inspired by the concept of blind ptychography, we report a reconstruction scheme to jointly recover the complex object and the unknown coded layer using multi-height measurements. With both simulation and experimental results, we show that the recovered phase is quantitative and the slow-varying phase profiles can be properly restored using lensless multi-height measurements. We also show that the image quality using the coded sensor is better than that of a regular image sensor. For demonstrations, we validate the reported scheme with various biospecimens and compare the results to those of regular lensless multi-height phase retrieval. The use of a coded image sensor may enable true quantitative phase imaging for the lensless multi-height, multi-wavelength, and transport-of-intensity equation approaches.

High-throughput lensless whole slide imaging via continuous height-varying modulation of a tilted sensor

We report a new, to the best of our knowledge, lensless microscopy configuration by integrating the concepts of transverse translational ptychography and defocus multi-height phase retrieval. In this approach, we place a tilted image sensor under the specimen for introducing linearly increasing phase modulation along one lateral direction. Similar to the operation of ptychography, we laterally translate the specimen and acquire the diffraction images for reconstruction. Since the axial distance between the specimen and the sensor varies at different lateral positions, laterally translating the specimen effectively introduces defocus multi-height measurements while eliminating axial scanning. Lateral translation further introduces sub-pixel shift for pixel super-resolution imaging and naturally expands the field of view for rapid whole slide imaging. We show that the equivalent height variation can be precisely estimated from the lateral shift of the specimen, thereby addressing the challenge of precise axial positioning in conventional multi-height phase retrieval. Using a sensor with 1.67 µm pixel size, our low-cost and field-portable prototype can resolve the 690 nm linewidth on the resolution target. We show that a whole slide image of a blood smear with a 120mm² field of view can be acquired in 18 s. We also demonstrate accurate automatic white blood cell counting from the recovered image. The reported approach may provide a turnkey solution for addressing point-of-care and telemedicine-related challenges.

Improving axial localization of weak phase particles in digital in-line holography

One shortcoming of digital in-line holography (DIH) is the low axial position accuracy due to the elongated particle traces in the reconstruction field. Here, we propose a method that improves the axial localization of DIH when applying it to track the motion of weak phase particles in dense suspensions. The proposed method detects particle positions based on local intensities in the reconstruction field consisting of scattering and incident waves. We perform both numerical and experimental tests and demonstrate that the proposed method has a higher axial position accuracy than the previous method based on the local intensities in the reconstructed scattered field. We show that the proposed method has an axial position error below 1.5 particle diameters for holograms with a particle concentration of 4700particles/mm³. The proposed method is further validated by tracking the Brownian motion of 1µmparticles in dense suspensions.

Non-propagation fast phase diverse phase retrieval for wavefront measurement with generalized FFT-based basis function

Phase retrieval is an attractive optical testing method with a simple experimental arrangement. The sampling grids wave propagation computation based on the FFT operations is usually involved in each iterative process for the classical phase retrieval model. In this paper, a novel non-propagation optimization phase retrieval technique with the FFT-based basis function is proposed to accelerate wavefront measurement. The sampling grids wave diffraction propagation computation is converted to matrix-vector products that have small dimensions to reduce the computational burden. The diffraction basis function based on generalized numerical orthogonal polynomial and two-step Fresnel propagation is deduced, which is suitable for the generally shaped pupil. This paper provides a universal non-propagation framework to accelerate phase retrieval which is applicable to the arbitrarily shaped wavefront measurement.

Holographic optical field recovery using a regularized untrained deep decoder network

Image reconstruction using minimal measured information has been a long-standing open problem in many computational imaging approaches, in particular in-line holography. Many solutions are devised based on compressive sensing (CS) techniques with handcrafted image priors or supervised deep neural networks (DNN). However, the limited performance of CS methods due to lack of information about the image priors and the requirement of an enormous amount of per-sample-type training resources for DNNs has posed new challenges over the primary problem. In this study, we propose a single-shot lensless in-line holographic reconstruction method using an untrained deep neural network which is incorporated with a physical image formation algorithm. We demonstrate that by modifying a deep decoder network with simple regularizers, a Gabor hologram can be inversely reconstructed via a minimization process that is constrained by a deep image prior. The outcoming model allows to accurately recover the phase and amplitude images without any training dataset, excess measurements, or specific assumptions about the object's or the measurement's characteristics.

Separating twin images in digital holographic microscopy using weak scatterers

When using inline digital holographic microscopy (DHM) and placing the hologram plane within a particle suspension, both real and virtual images come into focus during reconstruction, limiting our ability to resolve three-dimensional (3D) particle distribution. Here, we propose a new method to distinguish between real and virtual images in the 3D reconstruction field. This new method is based on the use of weak scatterers, and the fact that the real and virtual images of weak scatterers display distinct intensity distributions along the optical axis. We experimentally demonstrate this method by localizing and tracking 1 µm particles in a 3D volume with a particle concentration ranging from 200 to 6000particles/mm³. Unlike previous approaches to address the virtual image problem, this method does not require the recording of multiple holograms or the insertion of additional optical components. The proposed method allows the hologram plane to be placed within the sample volume, and extends the capability of DHM to measure the 3D movements of particles in deep samples far away from the optical window.

Whole phase curvature-based particle positioning and size determination by digital holography

We demonstrate the positioning and characterization of a transparent particle with a diameter of 60 µm in sparse particle fields. Particles appear elongated in optical setups with small numerical apertures that are used in digital holography; thus, an accurate method to position them is required. We propose a new optimization method using the whole phase curvature of a reconstructed wave along the optical axis to obtain not only the precise axial position but also the radius and refractive index of a particle. Experimental results show that the axial positions of particles can be detected with a standard deviation of 38.6 µm, corresponding to 66% of the average particle diameter. A radius of 29.3±0.4µm and a refractive index of 1.5910±0.0017 agree well with the manufacturer specifications of particles.

Holography and Coherent Diffraction Imaging with Low-(30-250 eV) and High-(80-300 keV) Energy Electrons: History, Principles, and Recent Trends

In this paper, we present the theoretical background to electron scattering in an atomic potential and the differences between low- and high-energy electrons interacting with matter. We discuss several interferometric techniques that can be realized with low- and high-energy electrons and which can be applied to the imaging of non-crystalline samples and individual macromolecules, including in-line holography, point projection microscopy, off-axis holography, and coherent diffraction imaging. The advantages of using low- and high-energy electrons for particular experiments are examined, and experimental schemes for holography and coherent diffraction imaging are compared.

Three-dimensional measurement of a particle field using phase retrieval digital holography

Digital inline holography (DIH) has long been used to measure the three-dimensional (3D) distribution of micrometer particles in suspensions. However, DIH experiences a virtual image problem that limits the particle density and the placement of the hologram plane relative to the sample volume. Here, we apply virtual-image-free phase retrieval digital holography (PRDH) to detect opaque particles in 3D volumes that exceed $ 2000\;{\rm particles}/{{\rm mm}^3} $2000particles/mm³. PRDH is based on recording two holograms whose planes are displaced along the optical axis, and then reconstructing the complete optical waves estimated by the iterative phase retrieval algorithm. Both numerical and experimental tests are performed, and results show that PRDH recovers the original 3D particle distributions even when the hologram planes are within the particle suspensions. Moreover, compared to single-hologram-based DIH, PRDH is proved to have better particle detection qualities. The uncertainty in the localization of particle centers is reduced to less than one particle diameter.

Information retrieval using overlapping holograms with partial complementarity

An information retrieval technique from superimposed holograms representing 2D and 3D objects using complementary fringes is presented. By adding two different computer generated holograms with quasi-complementarity information is possible to retrieve information at a specific depth.

Iterative phase retrieval for digital holography: tutorial

This paper provides a tutorial of iterative phase retrieval algorithms based on the Gerchberg-Saxton (GS) algorithm applied in digital holography. In addition, a novel GS-based algorithm that allows reconstruction of 3D samples is demonstrated. The GS-based algorithms recover a complex-valued wavefront using wavefront back-and-forth propagation between two planes with constraints superimposed in these two planes. Iterative phase retrieval allows quantitatively correct and twin-image-free reconstructions of object amplitude and phase distributions from its in-line hologram. The present work derives the quantitative criteria on how many holograms are required to reconstruct a complex-valued object distribution, be it a 2D or 3D sample. It is shown that for a sample that can be approximated as a 2D sample, a single-shot in-line hologram is sufficient to reconstruct the absorption and phase distributions of the sample. Previously, the GS-based algorithms have been successfully employed to reconstruct samples that are limited to a 2D plane. However, realistic physical objects always have some finite thickness and therefore are 3D rather than 2D objects. This study demonstrates that 3D samples, including 3D phase objects, can be reconstructed from two or more holograms. It is shown that in principle, two holograms are sufficient to recover the entire wavefront diffracted by a 3D sample distribution. In this method, the reconstruction is performed by applying iterative phase retrieval between the planes where intensity was measured. The recovered complex-valued wavefront is then propagated back to the sample planes, thus reconstructing the 3D distribution of the sample. This method can be applied for 3D samples such as 3D distribution of particles, thick biological samples, and other 3D phase objects. Examples of reconstructions of 3D objects, including phase objects, are provided. Resolution enhancement obtained by iterative extrapolation of holograms is also discussed.

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.

Amplitude and phase retrieval with simultaneous diversity estimation using expectation maximization

Iterative amplitude and phase retrieval algorithms have been proven to accurately reconstruct arbitrary wavefronts from multiple intensity measurements when system parameters are known exactly, given the ability to induce phase diversity between images. Such sets of intensity images with phase diversity can be generated by moving a lens in the optical system, but any position error on the lens will degenerate the reconstruction result. We demonstrate the use of an expectation-maximization algorithm with Kalman smoothing for recovering both the complex field and the lens position from a stack of intensity images. Our method successfully reduces the mean-squared-error of the estimated wavefront in comparison to an approach without position error estimation. We present and discuss the results of using a Kalman smoother and nonlinear least-squares optimization for the estimation of the moving lens position.

High-precision rotation angle measurement method based on a lensless digital holographic microscope

To accurately measure ultrasmall rotation angles, a robust and effective method based on lensless digital holographic microscopy is proposed in this paper. The method combines holographic microscopy, solid geometry, and 3D measurement, including holographic measurement and angle measurement processes. We can calculate the 3D shape by the angular spectrum algorithm and the least-squares phase-unwrapping algorithm in the holographic process. According to the relationship between the surface shape and rotation angles, the real-time rotation angles can be calculated. To validate the feasibility and practicability of the proposed approach, numerical noise simulations and experiments were performed. The measurement precision of rotation angle can reach 0.5″ in the range of 1000″ in this paper's experiments. The holographic method has high measurement precision and good stability. In addition, the compact small volume has great potential in small-angle sensor applications.

Coherence measurements of scattered incoherent light for lensless identification of an object's location and size

In the absence of a lens to form an image, incoherent or partially coherent light scattering off an obstructive or reflective object forms a broad intensity distribution in the far field with only feeble spatial features. We show here that measuring the complex spatial coherence function can help in the identification of the size and location of a one-dimensional object placed in the path of a partially coherent light source. The complex coherence function is measured in the far field through wavefront sampling, which is performed via dynamically reconfigurable slits implemented on a digital micromirror device (DMD). The impact of an object - parameterized by size and location - that either intercepts or reflects incoherent light is studied. The experimental results show that measuring the spatial coherence function as a function of the separation between two slits located symmetrically around the optical axis can identify the object transverse location and angle subtended from the detection plane (the ratio of the object width to the axial distance from the detector). The measurements are in good agreement with numerical simulations of a forward model based on Fresnel propagators. The rapid refresh rate of DMDs may enable real-time operation of such a lensless coherency imaging scheme.

X-ray Phase-Contrast Imaging and Metrology through Unified Modulated Pattern Analysis

We present a method for x-ray phase-contrast imaging and metrology applications based on the sample-induced modulation and subsequent computational demodulation of a random or periodic reference interference pattern. The proposed unified modulated pattern analysis (UMPA) technique is a versatile approach and allows tuning of signal sensitivity, spatial resolution, and scan time. We characterize the method and demonstrate its potential for high-sensitivity, quantitative phase imaging, and metrology to overcome the limitations of existing methods.

Compact, cost-effective and field-portable microscope prototype based on MISHELF microscopy

We report on a reduced cost, portable and compact prototype design of lensless holographic microscope with an illumination/detection scheme based on wavelength multiplexing, working with single hologram acquisition and using a fast convergence algorithm for image processing. All together, MISHELF (initials coming from Multi-Illumination Single-Holographic-Exposure Lensless Fresnel) microscopy allows the recording of three Fresnel domain diffraction patterns in a single camera snap-shot incoming from illuminating the sample with three coherent lights at once. Previous implementations have proposed an illumination/detection procedure based on a tuned (illumination wavelengths centered at the maximum sensitivity of the camera detection channels) configuration but here we report on a detuned (non-centered ones) scheme resulting in prototype miniaturization and cost reduction. Thus, MISHELF microscopy in combination with a novel and fast iterative algorithm allows high-resolution (μm range) phase-retrieved (twin image elimination) quantitative phase imaging of dynamic events (video rate recording speed). The performance of this microscope prototype is validated through experiments using both amplitude (USAF resolution test) and complex (live swine sperm cells and flowing microbeads) samples. The proposed method becomes in an alternative instrument improving some capabilities of existing lensless microscopes.

Phase retrieval method for digital holography with two cameras in particle measurement

This paper presents a phase retrieval method for digital holography with two high-speed cameras in particle measurement. The conditions for recording two holograms are derived theoretically. We focus in particular on the distance between the two holograms. The relative misalignment of the two holograms is also evaluated by numerical simulations. Finally, we experimentally compared settling particles reconstructed by the presented method and the Gabor method.

Resolution and quality enhancement in terahertz in-line holography by sub-pixel sampling with double-distance reconstruction

We demonstrate the enhancement of resolution and image quality in terahertz (THz) lens-free in-line digital holography by sub-pixel sampling with double-distance reconstruction. Multiple sub-pixel shifted low-resolution (LR) holograms recorded by a pyroelectric array detector (100 μm × 100 μm pixel pitch, 124 × 124 pixels) are aligned precisely to synthesize a high-resolution (HR) hologram. By this method, the lateral resolution is no more limited by the pixel pitch, and lateral resolution of 150 μm is obtained, which corresponds to 1.26λ with respect to the illuminating wavelength of 118.8 μm (2.52 THz). Compared with other published works, to date, this is the highest resolution in THz digital holography when considering the illuminating wavelength. In addition, to suppress the twin-image and zero-order artifacts, the complex amplitude distributions of both object and illuminaing background wave fields are reconstructed simultaneously. This is achieved by iterative phase retrieval between the double HR holograms and background images at two recording planes, which does not require any constraints on object plane or a priori knowledge of the sample.

Contextual phase estimation from two-plane intensity measurements

In this work we construct examples of paraxial light fields whose intensities defined at all points in space do not have a corresponding cross-spectrally pure field amplitude reproducing the same set of transported intensities at all transverse planes. Nevertheless, two spatially separated transverse plane intensities as drawn from these examples are shown to have a corresponding cross-spectrally pure field amplitude, which, through paraxial free propagation between these two planes, reproduces the drawn transverse plane intensities. And the phase associated with such a field amplitude at a given transverse plane is found to be contextual, and intrinsically dependent on the pairing plane.

Pixel super-resolution using wavelength scanning

Undersampling and pixelation affect a number of imaging systems, limiting the resolution of the acquired images, which becomes particularly significant for wide-field microscopy applications. Various super-resolution techniques have been implemented to mitigate this resolution loss by utilizing sub-pixel displacements in the imaging system, achieved, for example, by shifting the illumination source, the sensor array and/or the sample, followed by digital synthesis of a smaller effective pixel by merging these sub-pixel-shifted low-resolution images. Herein, we introduce a new pixel super-resolution method that is based on wavelength scanning and demonstrate that as an alternative to physical shifting/displacements, wavelength diversity can be used to boost the resolution of a wide-field imaging system and significantly increase its space-bandwidth product. We confirmed the effectiveness of this new technique by improving the resolution of lens-free as well as lens-based microscopy systems and developed an iterative algorithm to generate high-resolution reconstructions of a specimen using undersampled diffraction patterns recorded at a few wavelengths covering a narrow spectrum (10-30 nm). When combined with a synthetic-aperture-based diffraction imaging technique, this wavelength-scanning super-resolution approach can achieve a half-pitch resolution of 250 nm, corresponding to a numerical aperture of ~1.0, across a large field of view (>20 mm²). We also demonstrated the effectiveness of this approach by imaging various biological samples, including blood and Papanicolaou smears. Compared with displacement-based super-resolution techniques, wavelength scanning brings uniform resolution improvement in all directions across a sensor array and requires significantly fewer measurements. This technique would broadly benefit wide-field imaging applications that demand larger space-bandwidth products.

Propagation phasor approach for holographic image reconstruction

To achieve high-resolution and wide field-of-view, digital holographic imaging techniques need to tackle two major challenges: phase recovery and spatial undersampling. Previously, these challenges were separately addressed using phase retrieval and pixel super-resolution algorithms, which utilize the diversity of different imaging parameters. Although existing holographic imaging methods can achieve large space-bandwidth-products by performing pixel super-resolution and phase retrieval sequentially, they require large amounts of data, which might be a limitation in high-speed or cost-effective imaging applications. Here we report a propagation phasor approach, which for the first time combines phase retrieval and pixel super-resolution into a unified mathematical framework and enables the synthesis of new holographic image reconstruction methods with significantly improved data efficiency. In this approach, twin image and spatial aliasing signals, along with other digital artifacts, are interpreted as noise terms that are modulated by phasors that analytically depend on the lateral displacement between hologram and sensor planes, sample-to-sensor distance, wavelength, and the illumination angle. Compared to previous holographic reconstruction techniques, this new framework results in five- to seven-fold reduced number of raw measurements, while still achieving a competitive resolution and space-bandwidth-product. We also demonstrated the success of this approach by imaging biological specimens including Papanicolaou and blood smears.

In-line digital holographic imaging in volume holographic microscopy

A dual-plane in-line digital holographic imaging method incorporating volume holographic microscopy (VHM) is presented to reconstruct objects in a single shot while eliminating zero-order and twin-image diffracted waves. The proposed imaging method is configured such that information from different axial planes is acquired simultaneously using multiplexed volume holographic imaging gratings, as used in VHM, and recorded as in-line holograms where the corresponding reference beams are generated in the fashion of Gabor's in-line holography. Unlike conventional VHM, which can take axial intensity information only at focal depths, the proposed method digitally reconstructs objects at any axial position. Further, we demonstrate the proposed imaging technique's ability to effectively eliminate zero-order and twin images for single-shot three-dimensional object reconstruction.

Improved quantitative phase imaging in lensless microscopy by single-shot multi-wavelength illumination using a fast convergence algorithm

We report on a novel algorithm for high-resolution quantitative phase imaging in a new concept of lensless holographic microscope based on single-shot multi-wavelength illumination. This new microscope layout, reported by Noom et al. along the past year and named by us as MISHELF (initials incoming from Multi-Illumination Single-Holographic-Exposure Lensless Fresnel) microscopy, rises from the simultaneous illumination and recording of multiple diffraction patterns in the Fresnel domain. In combination with a novel and fast iterative phase retrieval algorithm, MISHELF microscopy is capable of high-resolution (micron range) phase-retrieved (twin image elimination) biological imaging of dynamic events. In this contribution, MISHELF microscopy is demonstrated through qualitative concept description, algorithm implementation, and experimental validation using both a synthetic object (resolution test target) and a biological sample (swine sperm sample) for the case of three (RGB) illumination wavelengths. The proposed method becomes in an alternative instrument improving the capabilities of existing lensless microscopes.

High speed phase retrieval of in-line holograms by the assistance of corresponding off-axis holograms

Retrieving correct phase information from an in-line hologram is difficult as the object wave field and the diffractions of the zero order and the conjugate object term overlap. The existing iterative numerical phase retrieval methods are slow, especially in the case of high Fresnel number systems. Conversely, the reconstruction of the object wave field from an off-axis hologram is simple, but due to the applied spatial frequency filtering the achievable resolution is confined. Here, a new, high-speed algorithm is introduced that efficiently incorporates the data of an auxiliary off-axis hologram in the phase retrieval of the corresponding in-line hologram. The efficiency of the introduced combined phase retrieval method is demonstrated by simulated and measured holograms.

Iterative phase retrieval algorithms. I: optimization

Two modified Gerchberg-Saxton (GS) iterative phase retrieval algorithms are proposed. The first we refer to as the spatial phase perturbation GS algorithm (SPP GSA). The second is a combined GS hybrid input-output algorithm (GS/HIOA). In this paper (Part I), it is demonstrated that the SPP GS and GS/HIO algorithms are both much better at avoiding stagnation during phase retrieval, allowing them to successfully locate superior solutions compared with either the GS or the HIO algorithms. The performances of the SPP GS and GS/HIO algorithms are also compared. Then, the error reduction (ER) algorithm is combined with the HIO algorithm (ER/HIOA) to retrieve the input object image and the phase, given only some knowledge of its extent and the amplitude in the Fourier domain. In Part II, the algorithms developed here are applied to carry out known plaintext and ciphertext attacks on amplitude encoding and phase encoding double random phase encryption systems. Significantly, ER/HIOA is then used to carry out a ciphertext-only attack on AE DRPE systems.

Research on object-plane constraints and hologram expansion in phase retrieval algorithms for continuous-wave terahertz inline digital holography reconstruction

In terahertz inline digital holography, zero-order diffraction light and conjugate images can cause the reconstructed image to be blurred. In this paper, three phase retrieval algorithms are applied to conduct reconstruction based on the same near-field diffraction propagation conditions and image-plane constraints. The impact of different object-plane constraints on CW terahertz inline digital holographic reconstruction is studied. The results show that in the phase retrieval algorithm it is not suitable to impose restriction on the phase when the object is not isolated in the transmission-type CW terahertz inline digital holography. In addition, the effects of zero-padding expansion, boundary replication expansion, and apodization operation on reconstructed images are studied. The results indicate that the conjugate image can be eliminated, and a better reconstructed image can be obtained by adopting an appropriate phase retrieval algorithm after the normalized hologram extending to the minimum area, which meets the applicable range of the angular spectrum reconstruction algorithm by means of boundary replication.

Separating twin images and locating the center of a microparticle in dense suspensions using correlations among reconstructed fields of two parallel holograms

This paper deals with two issues affecting the application of digital holographic microscopy (DHM) for measuring the spatial distribution of particles in a dense suspension, namely discriminating between real and virtual images and accurate detection of the particle center. Previous methods to separate real and virtual fields have involved applications of multiple phase-shifted holograms, combining reconstructed fields of multiple axially displaced holograms, and analysis of intensity distributions of weakly scattering objects. Here, we introduce a simple approach based on simultaneously recording two in-line holograms, whose planes are separated by a short distance from each other. This distance is chosen to be longer than the elongated trace of the particle. During reconstruction, the real images overlap, whereas the virtual images are displaced by twice the distance between hologram planes. Data analysis is based on correlating the spatial intensity distributions of the two reconstructed fields to measure displacement between traces. This method has been implemented for both synthetic particles and a dense suspension of 2 μm particles. The correlation analysis readily discriminates between real and virtual images of a sample containing more than 1300 particles. Consequently, we can now implement DHM for three-dimensional tracking of particles when the hologram plane is located inside the sample volume. Spatial correlations within the same reconstructed field are also used to improve the detection of the axial location of the particle center, extending previously introduced procedures to suspensions of microscopic particles. For each cross section within a particle trace, we sum the correlations among intensity distributions in all planes located symmetrically on both sides of the section. This cumulative correlation has a sharp peak at the particle center. Using both synthetic and recorded particle fields, we show that the uncertainty in localizing the axial location of the center is reduced to about one particle's diameter.

Terahertz in-line digital holography of dragonfly hindwing: amplitude and phase reconstruction at enhanced resolution by extrapolation

We report here on terahertz (THz) digital holography on a biological specimen. A continuous-wave (CW) THz in-line holographic setup was built based on a 2.52 THz CO(2) pumped THz laser and a pyroelectric array detector. We introduced novel statistical method of obtaining true intensity values for the pyroelectric array detector's pixels. Absorption and phase-shifting images of a dragonfly's hindwing were reconstructed simultaneously from single in-line hologram. Furthermore, we applied phase retrieval routines to eliminate twin image and enhanced the resolution of the reconstructions by hologram extrapolation beyond the detector area. The finest observed features are 35 μm width cross veins.

Optical phase imaging using a synthetic aperture phase retrieval technique

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

Lensless phase contrast microscopy based on multiwavelength Fresnel diffraction

We demonstrate a compact, wide-field, quantitative phase contrast microscope that does not require lenses for image formation. High-resolution images are retrieved from Fresnel diffraction patterns recorded at multiple wavelengths, combined with a robust iterative phase retrieval algorithm. Quantitative phase contrast images of living cultured neurons are obtained with a transverse resolution of <2 μm. Our system is well suited for high-resolution live cell imaging and provides a compact, cost-effective alternative to full-sized phase-contrast microscopes.

Field-portable pixel super-resolution colour microscope

Based on partially-coherent digital in-line holography, we report a field-portable microscope that can render lensfree colour images over a wide field-of-view of e.g., >20 mm(2). This computational holographic microscope weighs less than 145 grams with dimensions smaller than 17×6×5 cm, making it especially suitable for field settings and point-of-care use. In this lensfree imaging design, we merged a colorization algorithm with a source shifting based multi-height pixel super-resolution technique to mitigate 'rainbow' like colour artefacts that are typical in holographic imaging. This image processing scheme is based on transforming the colour components of an RGB image into YUV colour space, which separates colour information from brightness component of an image. The resolution of our super-resolution colour microscope was characterized using a USAF test chart to confirm sub-micron spatial resolution, even for reconstructions that employ multi-height phase recovery to handle dense and connected objects. To further demonstrate the performance of this colour microscope Papanicolaou (Pap) smears were also successfully imaged. This field-portable and wide-field computational colour microscope could be useful for tele-medicine applications in resource poor settings.

Wide-field computational color imaging using pixel super-resolved on-chip microscopy

Lens-free holographic on-chip imaging is an emerging approach that offers both wide field-of-view (FOV) and high spatial resolution in a cost-effective and compact design using source shifting based pixel super-resolution. However, color imaging has remained relatively immature for lens-free on-chip imaging, since a 'rainbow' like color artifact appears in reconstructed holographic images. To provide a solution for pixel super-resolved color imaging on a chip, here we introduce and compare the performances of two computational methods based on (1) YUV color space averaging, and (2) Dijkstra's shortest path, both of which eliminate color artifacts in reconstructed images, without compromising the spatial resolution or the wide FOV of lens-free on-chip microscopes. To demonstrate the potential of this lens-free color microscope we imaged stained Papanicolaou (Pap) smears over a wide FOV of ~14 mm(2) with sub-micron spatial resolution.

Phase shifting technique for extended inline holographic microscopy with a pinhole array

Digital inline holographic microscopy using a pinhole for sample illumination allows lensless imaging. To overcome restrictions of the sample size and density in the setup additional reference waves are generated by extending the single pinhole to a regular 2D pinhole array illumination. A technique is presented that uses phase shifting between the pinhole waves. Multiple foci with stable phase differences and a phase error (rms) of 0.027 rad generate pinhole waves which illuminate an undiluted, dense blood smear sample. Amplitude and phase images of the blood sample were successfully reconstructed.

Lensless phase microscopy using phase retrieval with multiple illumination wavelengths

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

Digital holography from shadowgraphic phase estimates

We show experimentally that the recently proposed iterative shadowgraphic method can be applied to in-line digital holography, provided a suitable regularization step is introduced. We show that the method correctly solves the "twin-image" problem using just two samples of the intensity field scattered by a phase object. Field retrieval accuracy is significantly improved when compared to a reconstruction obtained with the defocusing variation algorithms.

Wavefield imaging via iterative retrieval based on phase modulation diversity

We present a fast and robust non-interferomentric wavefield retrieval approach suitable for imaging of both amplitude and phase distributions of scalar coherent beams. It is based on the diversity of the intensity measurements obtained under controlled astigmatism and it can be easily implemented in standard imaging systems. Its application for imaging in microscopy is experimentally studied. Relevant examples illustrate the approach capabilities for image super-resolution, numerical refocusing, quantitative imaging and phase mapping.

Phase retrieval by pinhole scanning

We describe a method where phase and amplitude of a wavefront are obtained by processing a sequence of pattern produced by the interference between the light transmitted by a scanning pinhole (which is sequentially shifted) and a reference pinhole. Simulations and experimental results are presented.

Phase and amplitude imaging from noisy images by Kalman filtering

We propose and demonstrate a computational method for complex-field imaging from many noisy intensity images with varying defocus, using an extended complex Kalman filter. The technique offers dynamic smoothing of noisy measurements and is recursive rather than iterative, so is suitable for adaptive measurements. The Kalman filter provides near-optimal results in very low-light situations and may be adapted to propagation through turbulent, scattering, or nonlinear media.

Quantitative phase microscopy using defocusing by means of a spatial light modulator

A new method for recovery the quantitative phase information of microscopic samples is presented. It is based on a spatial light modulator (SLM) and digital image processing as key elements to extract the sample's phase distribution. By displaying a set of lenses with different focal power, the SLM produces a set of defocused images of the input sample at the CCD plane. Such recorded images are then numerically processed to retrieve phase information. This iterative process is based on the wave propagation equation and leads on a complex amplitude image containing information of both amplitude and phase distributions of the input sample diffracted wave front. The proposed configuration is a non-interferometric architecture (conventional transmission imaging mode) where no moving elements are included. Experimental results perfectly correlate with the results obtained by conventional digital holographic microscopy (DHM).