Two-dimensional phase unwrapping using the transport of intensity equation

High-resolution cell imaging using white light phase shifting interferometry and iterative phase deconvolution

An optimization algorithm is presented for the deconvolution of a complex field to improve the resolution and accuracy of quantitative phase imaging (QPI). A high-resolution phase map can be recovered by solving a constrained optimization problem of deconvolution using a complex gradient operator. The method is demonstrated on phase measurements of samples using a white light based phase shifting interferometry (WLPSI) method. The application of the algorithm on real and simulated objects shows a significant resolution and contrast improvement. Experiments performed on Escherichia coli bacterium have revealed its sub-cellular structures that were not visible in the raw WLPSI images obtained using a five phase shifting method. These features can give valuable insights into the structures and functioning of biological cells. The algorithm is simple in implementation and can be incorporated into other QPI modalities .

Carrier-frequency estimation for digital holograms of phase objects

Accurate estimation of carrier fringe frequency is essential for the demodulation of off-axis digital holograms. The fringe frequency is often associated with the amplitude peak of the cross-term in the two-dimensional Fourier transform of a digital hologram. We point out that this definition of carrier frequency is not valid in general for holograms associated with phase objects. We examine the carrier-envelope representation for digital holograms from the viewpoint of Mandel's criterion [J. Opt. Soc. Am.57, 613 (1967)10.1364/JOSA.57.000613]. An appropriate definition of carrier frequency is observed to be the centroid of the power spectrum associated with the cross term. This definition is shown to apply uniformly to holograms associated with phase objects, is robust to noise, and leads to the smoothest (or least fluctuating) envelope representation for the demodulated object wave. The proposed definition is illustrated with simulated as well as experimentally recorded off-axis holograms.

Dielectric Metasurface Enabled Compact, Single-Shot Digital Holography for Quantitative Phase Imaging

Quantitative phase imaging (QPI) enables nondestructive, real-time, label-free imaging of transparent specimens and can reveal information about their fundamental properties such as cell size and morphology, mass density, particle dynamics, and cellular fluctuations. Development of high-performance and low-cost quantitative phase imaging systems is thus required in many fields, including on-site biomedical imaging and industrial inspection. Here, we propose an ultracompact, highly stable interferometer based on a single-layer dielectric metasurface for common path off-axis digital holography and experimentally demonstrate quantitative phase imaging. The interferometric imaging system leveraging an ultrathin multifunctional metasurface captures image plane holograms in a single shot and provides quantitative phase information on the test samples for extraction of its physical properties. With the benefits of planar engineering and high integrability, the proposed metasurface-based method establishes a stable miniaturized QPI system for reliable and cost-effective point-of-care devices, live cell imaging, 3D topography, and edge detection for optical computing.

Quantification of pollen viability in Lantana camara by digital holographic microscopy

Pollen grains represent the male gametes of seed plants and their viability is critical for sexual reproduction in the plant life cycle. Palynology and viability studies have traditionally been used to address a range of botanical, ecological and geological questions, but recent work has revealed the importance of pollen viability in invasion biology as well. Here, we report an efficient visual method for assessing the viability of pollen using digital holographic microscopy (DHM). Imaging data reveal that quantitative phase information provided by the technique can be correlated with viability as indicated by the outcome of the colorimetric test. We successfully test this method on pollen grains of Lantana camara, a well-known alien invasive plant in the tropical world. Our results show that pollen viability may be assessed accurately without the usual staining procedure and suggest potential applications of the DHM methodology to a number of emerging areas in plant science.

Single-shot multispectral quantitative phase imaging of biological samples using deep learning

Multispectral quantitative phase imaging (MS-QPI) is a high-contrast label-free technique for morphological imaging of the specimens. The aim of the present study is to extract spectral dependent quantitative information in single-shot using a highly spatially sensitive digital holographic microscope assisted by a deep neural network. There are three different wavelengths used in our method: λ=532, 633, and 808 nm. The first step is to get the interferometric data for each wavelength. The acquired datasets are used to train a generative adversarial network to generate multispectral (MS) quantitative phase maps from a single input interferogram. The network was trained and validated on two different samples: the optical waveguide and MG63 osteosarcoma cells. Validation of the present approach is performed by comparing the predicted MS phase maps with numerically reconstructed (F T+T I E) phase maps and quantifying with different image quality assessment metrices.

Two-dimensional phase unwrapping based on Fourier transforms and the Yukawa potential spectrum

The two-dimensional phase unwrapping problem (PHUP) has been solved with discrete Fourier transforms (FTs) and many other techniques traditionally. Nevertheless, a formal way of solving the continuous Poisson equation for the PHUP, with the use of continuous FT and based on distribution theory, has not been reported yet, to our knowledge. The well-known specific solution of this equation is given in general by a convolution of a continuous Laplacian estimate with a particular Green function, whose FT does not exist mathematically. However, an alternative Green function called the Yukawa potential, with a guaranteed Fourier spectrum, can be considered for solving an approximated Poisson equation, inducing a standard procedure of a FT-based unwrapping algorithm. Thus, the general steps for this approach are described in this work by considering some reconstructions with synthetic and real data.

Robust phase unwrapping via non-local regularization

Phase unwrapping is an indispensable step in recovering the true phase from a modulo-2π phase. Conventional phase unwrapping methods suffer from error propagation under severe noise. In this Letter, we propose an iterative framework for robust phase unwrapping with high fidelity. The proposed method utilizes the transport-of-intensity equation to solve the phase unwrapping problem with high computational efficiency. To further improve reconstruction accuracy, we take advantage of non-local structural similarity using low-rank regularization. Meanwhile, we use an adaptive iteration strategy that dynamically and automatically updates the denoising parameter to avoid over-smoothing and preserve image details. A set of simulation and experimental results validates the proposed method, which can provide satisfying results under severe noise conditions, and outperform existing state-of-the-art phase unwrapping methods with at least 6 dB higher peak SNR (PSNR).

Discontinuous phase unwrapping based on the minimization of Zernike gradient polynomial residual

In many certain optical metrology cases, the pupil is usually divided into multiple connected domains by secondary mirror spiders, thus producing segment piston errors and leaving a false phase unwrapping result. In this paper, a method based on minimization of Zernike gradient polynomial residual (MZGR) is proposed to estimate segment piston errors and correct erroneous phase unwrapping results. Simulations and experiments demonstrated that this method can obtain the segment piston errors precisely under complex aberration forms and varied obscurations, indicating reliable practicality. Comparison to the 4D commercial solution, the RMS (root-mean-square) of the residual decreased from 0.154 λ to 0.020 λ.

2π ambiguity-free digital holography method for stepped phase imaging

It is known that phase ambiguity is always an inherent problem in digital holography. In this paper, a 2π ambiguity-free digital holography method is proposed. The method naturally avoids phase ambiguity by a quasianalytic method. This quasianalytic method accurately calculates the true phase by constructing an equation and solving the solution of the equation. Thus, the inherent wrapping problem in digital holography is eliminated. For example, our experimental result shows that the true phase of the stepped specimen with the phase distributed in [0, 16π] can be obtained unambiguously. Since the proposed method naturally avoids the phase ambiguity problem, it may be beneficial to enlarge the application potential of the digital holography. The effectiveness and accuracy of the proposed method are verified by both numerical simulations and experimental results.

Topography stitching in the spatial frequency domain for the representation of mid-spatial frequency errors

Sub-aperture fabrication techniques such as diamond turning, ion beam figuring, and bonnet polishing are indispensable tools in today's optical fabrication chain. Each of these tools addresses different figure and roughness imperfections corresponding to a broad spatial frequency range. Their individual effects, however, cannot be regarded as completely independent from each other due to the concurrent formation of form and finish errors, particularly in the mid-spatial frequency (MSF) region. Deterministic Zernike polynomials and statistical power spectral density (PSD) functions are often used to represent form and finish errors, respectively. Typically, both types of surface errors are treated separately when their impact on optical performance is considered: (i) wave aberrations caused by figure errors and (ii) stray light resulting from surface roughness. To fill the gap between deterministic and statistical descriptions, a generalized surface description is of great importance for bringing versatility to the entire optical fabrication chain by enabling easy and quick exchange of surface topography data between three disciplines: optical design, manufacturing, and characterization. In this work, we present a surface description by stitching the amplitude and unwrapped phase spectra of several surface topography measurements at different magnifications. An alternative representation of surface errors at different regimes is proposed, allowing us to bridge the gap between figure and finish as well as to describe the well-known MSF errors.

Roadmap on digital holography [Invited]

This Roadmap article on digital holography provides an overview of a vast array of research activities in the field of digital holography. The paper consists of a series of 25 sections from the prominent experts in digital holography presenting various aspects of the field on sensing, 3D imaging and displays, virtual and augmented reality, microscopy, cell identification, tomography, label-free live cell imaging, and other applications. Each section represents the vision of its author to describe the significant progress, potential impact, important developments, and challenging issues in the field of digital holography.

Fast and accurate phase-unwrapping algorithm based on the transport of intensity equation: comment

In [Appl. Opt.56, 7079 (2017)APOPAI0003-693510.1364/AO.56.007079], an attractive phase unwrapping algorithm based on the transport of intensity equation is proposed. Although the effectiveness is experimentally evaluated, the derivation of the angular spectrum based expression is ambiguous. In this paper, the ambiguity is clarified with the Helmholtz equation.

Robust phase unwrapping algorithm for noisy and segmented phase measurements

This paper proposes a robust phase unwrapping algorithm (RPUA) for phase unwrapping in the presence of noise and segmented phase. The RPUA method presents a new model of phase derivatives combined with error-correction iterations to achieve an anti-noise effect. Moreover, it bridges the phase islands in the spatial domain using numerical carrier frequency and fringe extrapolation thus eliminating height faults to enable solving segmented phase unwrapping. Numerical simulation and comparison with three conventional methods were performed, proving the high robustness and efficiency of the RPUA. Further, three experiments demonstrated that the RPUA can obtain the unwrapped phase under different noise accurately and possesses the capability to process segmented phases, indicating reliable practicality.

Single-shot higher-order transport-of-intensity quantitative phase imaging based on computer-generated holography

The imaging quality of quantitative phase imaging (QPI) based on the transport of intensity equation (TIE) can be improved using a higher-order approximation for defocused intensity distributions. However, this requires mechanically scanning an image sensor or object along the optical axis, which in turn requires a precisely aligned optical setup. To overcome this problem, a computer-generated hologram (CGH) technique is introduced to TIE-based QPI. A CGH generating defocused point spread function is inserted in the Fourier plane of an object. The CGH acts as a lens and grating with various focal lengths and orientations, allowing multiple defocused intensity distributions to be simultaneously detected on an image sensor plane. The results of a numerical simulation and optical experiment demonstrated the feasibility of the proposed method.

Performance analysis of phase retrieval using transport of intensity with digital holography [Invited]

The performance of direct and unwrapped phase retrieval, which combines digital holography with the transport of intensity, is examined in detail in this paper. In this technique, digital holography is used to numerically reconstruct the intensities at different planes around the image plane, and phase retrieval is achieved by the transport of intensity. Digital holography with transport of intensity is examined for inline and off-axis geometries. The effect of twin images in the inline case is evaluated. Phase-shifting digital holography with transport of intensity is introduced. The performance of digital holography with transport of intensity is compared with traditional off-axis single- and dual-wavelength techniques, which employ standard phase unwrapping algorithms. Simulations and experiments are performed to determine and compare the accuracy of phase retrieval through a mean-squared-error figure of merit as well as the computational speeds of the various methods.

High space-bandwidth in quantitative phase imaging using partially spatially coherent digital holographic microscopy and a deep neural network

Quantitative phase microscopy (QPM) is a label-free technique that enables monitoring of morphological changes at the subcellular level. The performance of the QPM system in terms of spatial sensitivity and resolution depends on the coherence properties of the light source and the numerical aperture (NA) of objective lenses. Here, we propose high space-bandwidth quantitative phase imaging using partially spatially coherent digital holographic microscopy (PSC-DHM) assisted with a deep neural network. The PSC source synthesized to improve the spatial sensitivity of the reconstructed phase map from the interferometric images. Further, compatible generative adversarial network (GAN) is used and trained with paired low-resolution (LR) and high-resolution (HR) datasets acquired from the PSC-DHM system. The training of the network is performed on two different types of samples, i.e. mostly homogenous human red blood cells (RBC), and on highly heterogeneous macrophages. The performance is evaluated by predicting the HR images from the datasets captured with a low NA lens and compared with the actual HR phase images. An improvement of 9× in the space-bandwidth product is demonstrated for both RBC and macrophages datasets. We believe that the PSC-DHM + GAN approach would be applicable in single-shot label free tissue imaging, disease classification and other high-resolution tomography applications by utilizing the longitudinal spatial coherence properties of the light source.

Deep Convolutional Neural Network Phase Unwrapping for Fringe Projection 3D Imaging

Phase unwrapping is a very important step in fringe projection 3D imaging. In this paper, we propose a new neural network for accurate phase unwrapping to address the special needs in fringe projection 3D imaging. Instead of labeling the wrapped phase with integers directly, a two-step training process with the same network configuration is proposed. In the first step, the network (network I) is trained to label only four key features in the wrapped phase. In the second step, another network with same configuration (network II) is trained to label the wrapped phase segments. The advantages are that the dimension of the wrapped phase can be much larger from that of the training data, and the phase with serious Gaussian noise can be correctly unwrapped. We demonstrate the performance and key features of the neural network trained with the simulation data for the experimental data.

A practical criterion for focusing of unstained cell samples using a digital holographic microscope

Digital holographic microscopy (DHM) is an important technique that may be used for quantitative phase imaging of unstained biological cell samples. Since the DHM technology is not commonly used in clinics or bioscience research labs, at present there is no well-accepted focusing criterion for unstained samples that users can follow while recording image plane digital holograms of cells. The usual sharpness metrics that are useful for auto-focusing of stained cells do not work well for unstained cells as there is no amplitude contrast. In this work, we report a practical method for estimating the best focus plane for unstained cells in the digital hologram domain. The method is based on an interesting observation that for the best focus plane the fringe pattern associated with individual unstained cells predominantly shows phase modulation effect in the form of bending of fringes and minimal amplitude modulation. This criterion when applied to unstained red blood cells shows that the central dip in the doughnut-like phase profile of cells is maximal in this plane. The proposed methodology is helpful for standardizing the usage of DHM technology across different users and application development efforts. LAY DESCRIPTION: Digital holographic microscopy (DHM) is slowly but steadily becoming an important microscopy modality and gaining acceptability for basic bio-science research as well as clinical usage. One of the important features of DHM is that it allows users to perform quantitative imaging of unstained transparent cells. Instead of using dyes or fluorescent labelling, DHM systems use quantitative phase as a contrast mechanism which depends on the natural refractive index variation within the cell samples. Since minimal wet lab processing is required in order to image cell samples with a DHM, cells can be imaged in their natural state. While DHM is gaining popularity among users, the imaging protocols across the labs or users need to be standardized in order to make sure that the same quantitative phase parameters are used for tasks such as quantitative phased based cell classification. One of the important operational tasks for any microscopy work is to focus the sample under study. While focusing comes naturally to users of brightfield microscopes based on image contrast, the focusing is not straightforward when samples are unstained so that they do not offer any amplitude contrast. When performing quantitative phase imaging, defocus can actually change the phase profile of the cell due to near-zone (Fresnel) diffraction effects. So unless a standardized focusing methodology is used, it will be difficult for multiple DHM users (potentially at different sites) to agree on quantitative results out of their phase images. DHM literature has prior works which perform numerical focusing of recovered complex wave-field in the hologram plane to find the best focus plane. However such methods are not user friendly and do not allow user the same focusing experience as in a brightfield microscope. The numerical focusing is therefore a reasonably good method for an optics researcher but not necessarily so for a microscopy technician looking at cell samples with a DHM system in a clinical setting. The present work provides a simple focusing criterion for unstained samples that works directly in the hologram domain. The technique is based on an interesting observation that the when an unstained cell sample is in the best-focus plane, its corresponding hologram (or fringe pattern) predominantly shows phase modulation manifested by bending of fringes at the location of the cell. This criterion can be converted into a simple numerical method as we have used to find the best-focus plane using a stack of through focus holograms. We believe that the technique can be used manually by visually observing the holograms or can be converted to an auto-focus algorithm for a motorized DHM system.

Phase information extraction for moiré fringes based on multiresolution analysis

In this paper, the multiresolution analysis (MRA) method is used to preprocess moiré fringes, which can reduce the number of data points and increase computation speeds. To discuss the applicability of the method, a candle combustion flow field is chosen as an example for experiment by moiré deflectometry. First, moiré fringes are preprocessed by the MRA method. Then, phase information extraction and refractive index reconstruction are performed on the three-level low-frequency approximation components. Finally, the involved results prove that the calculation time required for phase information extraction and refractive index reconstruction is greatly reduced based on the moiré fringes preprocessed by MRA method. The relative error could be accepted if the suitable approximation level is applied.

Unsupervised organization of cervical cells using bright-field and single-shot digital holographic microscopy

We report results on unsupervised organization of cervical cells using microscopy of Pap-smear samples in brightfield (3-channel color) as well as high-resolution quantitative phase imaging modalities. A number of morphological parameters are measured for each of the 1450 cell nuclei (from 10 woman subjects) imaged in this study. The principal component analysis (PCA) methodology applied to this data shows that the cell image clustering performance improves significantly when brightfield as well as phase information is utilized for PCA as compared to when brightfield-only information is used. The results point to the feasibility of an image-based tool that will be able to mark suspicious cells for further examination by the pathologist. More importantly, our results suggest that the information in quantitative phase images of cells that is typically not used in clinical practice is valuable for automated cell classification applications in general.

Phase unwrapping in optical metrology via denoised and convolutional segmentation networks

The interferometry technique is commonly used to obtain the phase information of an object in optical metrology. The obtained wrapped phase is subject to a 2π ambiguity. To remove the ambiguity and obtain the correct phase, phase unwrapping is essential. Conventional phase unwrapping approaches are time-consuming and noise sensitive. To address those issues, we propose a new approach, where we transfer the task of phase unwrapping into a multi-class classification problem and introduce an efficient segmentation network to identify classes. Moreover, a noise-to-noise denoised network is integrated to preprocess noisy wrapped phase. We have demonstrated the proposed method with simulated data and in a real interferometric system.

Fourier-based solving approach for the transport-of-intensity equation with reduced restrictions

The transport-of-intensity equation (TIE) has been proven as a standard approach for phase retrieval. Some high efficiency solving methods for the TIE, extensively used in many works, is based on a Fourier transform (FT). However, several assumptions have to be made to solve the TIE by these methods. A common assumption is that there are no zero values for the intensity distribution allowed. The two most widespread Fourier-based approaches have further restrictions. One of these requires the uniformity of the intensity distribution and the other assumes the parallelism of the intensity and phase gradients. In this paper, we present an approach, which does not need any of these assumptions and consequently extends the application domain of the TIE.

Defect detection based on a lensless reflective point diffraction interferometer

We propose a defect detection system to identify phase defects on optics based on a lensless reflective point diffraction interferometer (LRPDI). The optics under test are illuminated by a collimated beam to produce a signal wavefront carrying the defect information, and then the signal wavefront is recorded in a high carrier interferogram using the LRPDI. By lensless imaging, amplitude and phase defects, as well as the accurate phase of a phase defect, can be identified. The simulation and experiment demonstrate the success of the proposed system in detecting phase defects, and its high-accuracy and high-resolution dynamic detection abilities are verified.

Fast and accurate phase-unwrapping algorithm based on the transport of intensity equation

The phase information of a complex field is routinely obtained using coherent measurement techniques as, e.g., interferometry or holography. The obtained measurement result is subject to a 2π ambiguity and is often referred to as wrapped phase. Phase-unwrapping algorithms (PUAs) are commonly employed to remove this ambiguity and, hence, obtain the absolute phase. However, implementing PUAs can be computationally intensive, and the accuracy of those algorithms may be low. Recently, the transport of intensity equation (TIE) has been proposed as a simple and practical alternative for obtaining the absolute phase map. Nevertheless, an efficient implementation of this technique has not yet been made. In this work, we propose an accurate solution for the TIE-based PUA that does not require the use of wave-propagation techniques, as previously reported TIE-based approaches. The proposed method calculates directly the axial derivative of the intensity from the wrapped phase when considering the correct propagation method. This is done in order to bypass the time-consuming wave-propagation techniques employed in similar methods. The analytical evaluation of this parameter allows obtaining an accurate solution when unwrapping the phase map with low computational effort. This work further introduces the use of the iterative TIE-PUA that, in a few steps, improves significantly the accuracy of the final absolute phase map, even in the presence of noise or aliasing of the wrapped data. The high accuracy and utility of the developed TIE-PUA technique is proven by both numerical simulations and experiments for various objects.

Weighted least-squares phase unwrapping algorithm based on a non-interfering image of an object

Conventional quality maps for weighted least-squares phase unwrapping are not appropriate in regions of abnormal fringes. In this paper, the use of a non-interfering image of an object for a reliability map is proposed for robust phase unwrapping. First, the conventional weighted least-squares algorithm is applied, and the unreliable region is detected. Then, the unreliable region is unwrapped iteratively using the non-interfering image of the object. Finally, both phase maps are combined and smoothed by a continuity operation. Experimental results show that the proposed algorithm is appropriate for unwrapping phase maps of abnormal fringes.