Quantitative phase microscopy via optimized inversion of the phase optical transfer function

Partially coherent broadband 3D optical transfer functions with arbitrary temporal and angular power spectra

Optical diffraction tomography is a powerful technique to produce 3D volumetric images of biological samples using contrast produced by variations in the index of refraction in an unlabeled specimen. While this is typically performed with coherent illumination from a variety of angles, interest has grown in partially coherent methods due to the simplicity of the illumination and the computation-free axial sectioning provided by the coherence window of the source. However, such methods rely on the symmetry or discretization of a source to facilitate quantitative analysis and are unable to efficiently handle arbitrary illumination that may vary asymmetrically in angle and continuously in the spectrum, such as diffusely scattered or thermal sources. A general broadband theory may expand the scope of illumination methods available for quantitative analysis, as partially coherent sources are commonly available and may benefit from the effects of spatial and temporal incoherence. In this work, we investigate partially coherent tomographic phase microscopy from arbitrary sources regardless of angular distribution and spectrum by unifying the effects of spatial and temporal coherence into a single formulation. This approach further yields a method for efficient computation of the overall systems' optical transfer function, which scales with O(n ³), down from O(mn ⁴) for existing convolutional methods, where n ³ is the number of spatial voxels in 3D space and m is the number of discrete wavelengths in the illumination spectrum. This work has important implications for enabling partially coherent 3D quantitative phase microscopy and refractive index tomography in virtually any transmission or epi-illumination microscope.

Research on partially coherent spatial light interference microscopy

Based on partial coherence theory, this study rigorously deduces the principle of spatial light interference microscopy (SLIM) and improves the calculation method of SLIM. The main problem we found with SLIM is that it simply defaults the phase of the direct light to 0. To address this problem, we propose and experimentally demonstrate a double four-step phase shift method. Simulation results show that this method can reduce the relative error of oil-immersed microsphere reconstruction to about 3.7%, and for red blood cell reconstruction, the relative error can be reduced to about 13%.

Time-lapse imaging using dual-color coded quantitative differential phase contrast microscopy

SIGNIFICANCE

Quantitative differential phase contrast (qDPC) microscopy enhances phase contrast by asymmetric illumination using partially coherent light and multiple intensity measurements. However, for live cell imaging, motion artifacts and image acquisition time are important issues. For live cell imaging, a large number of intensity measurements can limit the imaging quality and speed. The minimum number of intensity measurements in qDPC can greatly enhance performance for live imaging.

AIM

To obtain high-contrast, isotropic qDPC images with two intensity measurements and perform time-lapse imaging of biological samples.

APPROACH

Based on the color-coded design, a dual-color linear-gradient pupil is proposed to achieve isotropic phase contrast response with two intensity measurements. In our method, the purpose of designing a dual-color coded pupil is twofold: first, to obtain a linear amplitude gradient for asymmetric illumination, which is required to get a circular symmetry of transfer function, and second, to reduce the required number of frames for phase retrieval.

RESULTS

To demonstrate the imaging performance of our system, standard microlens arrays were used as samples. We performed time-lapse quantitative phase imaging of rat astrocytes under a low-oxygen environment. Detailed morphology and dynamic changes such as the apoptosis process and migration of cells were observed.

CONCLUSIONS

It is shown that dual-color linear-gradient pupils in qDPC can outperform half-circle and vortex pupils, and isotropic phase transfer function can be achieved with only two-axis measurements. The reduced number of frames helps in achieving faster imaging speed as compared to the typical qDPC system. The imaging performance of our system is evaluated by time-lapse imaging of rat astrocytes. Different morphological changes in cells during their life cycle were observed in terms of quantitative phase change values.

Simultaneous 3D deconvolution and halo removal for spatial light interference microscopy through a two-edge apodized Wiener filter

As one of the most sensitive quantitative phase microscopy techniques, spatial light interference microscopy (SLIM) has undergone rapid development in the past decade and has seen wide application in both basic science and clinical studies. However, as with any other traditional microscope, the axial resolution is the worst among the three dimensions. This leads to lower contrast in the thicker regions of cell samples. Another common foe in the phase contrast image is the halo artifact, which can block underlying structures, in particular when high resolution is desired. Current solutions focus on either halo removal or contrast enhancement alone, and thus need two processing steps to create both high contrast and halo-free phase images. Further, raw images often suffer from artifacts that are both bright and slowly varying, dubbed here as cloud-like artifacts. After deconvolution, these cloud-like artifacts often dominate the image and obscure high-frequency information, which is typically of greatest interest. In this paper, we first analyzed the unique characteristics of the phase transfer function associated with SLIM to find the root of the cloud-like artifacts and halo artifacts. Then we designed a two-edge apodized deconvolution scheme as a counter measure. We show that even with a simple Wiener filter, the two-edge apodization (TEA) can effectively improve the contrast while suppressing the halo and cloud-like artifacts. Our algorithm, named TEA-Weiner, is non-iterative and thus can be implemented in real time. For low-contrast structures inside the cell such as the endoplasmic reticulum (ER), where ringing artifacts are more likely, we show that two-edge apodization can be combined with additional constraints such as total variation so that their contrast can be enhanced simultaneously with other bright structures inside the cell. Comparing our method with other state-of-the-art algorithms, our method has two advantages: First, deconvolution and halo removal are accomplished simultaneously; second, the image quality is highest using TEA-Weiner filtering.

Quantifying the data quality of focal series for inline electron holography

Inline electron holography, the recovery of amplitude and phase of an electron wave function having passed through a thin specimen from a focal series recorded in a transmission electron microscope is being applied in many labs worldwide. At medium range magnification (i.e. typically ≥0.8 nm where the lattice of small unit cell crystals such as silicon is not resolved), where the defocus needs to be varied over a rather large range of several hundred nm or even μm, the retrieval of low spatial frequency information is severely affected by the choice of experimental parameters as well as the way of data normalization. Methods to quantitatively analyze the reliability of phase maps obtained by inline electron holography are presented, and data recorded and processed in different ways are compared. While, even under optimized conditions, the phase reconstructed from an experimental focal series still lacks very low spatial frequency components, regularization schemes exist and are demonstrated to effectively hide artifacts associated with this lack of information.

Accurate quantitative phase imaging by the transport of intensity equation: a mixed-transfer-function approach

As a well-established deterministic phase retrieval approach, the transport of intensity equation (TIE) is able to recover the quantitative phase of a sample under coherent or partially coherent illumination with its through-focus intensity measurements. Nevertheless, the inherent paraxial approximation limits its validity to low-numerical-aperture imaging and slowly varying objects, precluding its application to high-resolution quantitative phase imaging (QPI). Alternatively, QPI can be achieved by phase deconvolution approaches based on the coherent contrast transfer function or partially coherent weak object transfer function (WOTF) without invoking paraxial approximation. But these methods are generally appropriate for "weakly scattering" samples in which the total phase delay induced by the object should be small. Consequently, high-resolution high-accuracy QPI of "nonweak" phase objects with fine details and large phase excursions remains a great challenge. In this Letter, we propose a mixed-transfer-function (MTF) approach to address the dilemma between measurement accuracy and imaging resolution. By effectively merging the phases reconstructed by TIE and WOTF in the frequency domain, the high-accuracy low-frequency phase "global" profile can be secured, and high-resolution high-frequency features can be well preserved simultaneously. Simulations and experimental results on a microlens array and unstained biological cells demonstrate the effectiveness of MTF.

Phase sensitivity in differential phase contrast microscopy: limits and strategies to improve it

The phase sensitivity limit of Differential Phase Contrast (DPC) with partially coherent light is analyzed in details. The parameters to tune phase sensitivity, such as the diameter of illumination, the numerical aperture of the objective, and the noise of the camera are taken into account to determine the minimum phase contrast that can be detected. We found that a priori information about the sample can be used to fine-tune these parameters to increase phase contrast. Based on this information, we propose a simple algorithm to predict phase sensitivity of a DPC setup, which can be performed before the setup is built. Experiments confirm the theoretical findings.

Revealing architectural order with quantitative label-free imaging and deep learning

We report quantitative label-free imaging with phase and polarization (QLIPP) for simultaneous measurement of density, anisotropy, and orientation of structures in unlabeled live cells and tissue slices. We combine QLIPP with deep neural networks to predict fluorescence images of diverse cell and tissue structures. QLIPP images reveal anatomical regions and axon tract orientation in prenatal human brain tissue sections that are not visible using brightfield imaging. We report a variant of U-Net architecture, multi-channel 2.5D U-Net, for computationally efficient prediction of fluorescence images in three dimensions and over large fields of view. Further, we develop data normalization methods for accurate prediction of myelin distribution over large brain regions. We show that experimental defects in labeling the human tissue can be rescued with quantitative label-free imaging and neural network model. We anticipate that the proposed method will enable new studies of architectural order at spatial scales ranging from organelles to tissue.

Two improved defocus quantitative phase imaging methods: discussion

Multifilter phase imaging with partially coherent light (MFPI-PC) and phase optical transfer function recovery (POTFR) are two viable defocus-based, two-dimensional quantitative phase imaging (QPI) methods. While both methods use transfer function inversion, MFPI-PC is based on the in-focus intensity derivative, while POTFR is based on the intensity difference between symmetrically defocused images. This paper compares and contrasts MFPI-PC and POTFR. Six disadvantages (five in MFPI-PC and one in POTFR) are identified. Improvement strategies to overcome each of the six shortcomings are identified and implemented, and both methods are shown to be clearly improved. The revised MFPI-PC is shown to be more accurate than the original MFPI-PC and generally more accurate than the revised POTFR. The revised POTFR is shown to be inherently faster than the original POTFR and also slightly faster than the revised MFPI-PC.

Weighted-least-squares multi-filter phase imaging with partially coherent light: characteristics of annular illumination

Multi-filter phase imaging with partially coherent light (MFPI-PC) is a promising microscopic quantitative phase imaging (QPI) method that measures the phase of a transparent object. In the present work, a weighted-least-squares version is developed and applied to the important case of annular illumination. The resulting improved algorithms have largely solved the noise magnification problem associated with the original MFPI-PC in annular illumination. Simulation and microlens experiments are used to validate the new QPI method for the case of annular illumination.

Quantitative phase imaging with increased spatial coherence based on Fourier filtering

This Letter reports a method for tuning the degree of spatial coherence in an optical microscope. The method employs an amplitude spatial light modulator (aSLM) placed in the Fourier plane of the microscope. The aSLM displays a set of binary filters that blocks specific frequencies of the Fourier spectrum of the complex object field. It is shown that numerical processing of these filtered fields provides a final intensity with increased spatial coherence. Coherence tuning is used in a new hybrid phase reconstruction algorithm that employs the transport of intensity equation and an iterative phase retrieval technique. It is validated experimentally that this hybrid approach is able to retrieve the phase information with high resolution, and effective artifact noise suppression when employing at minimum two symmetrical defocused intensities and large sources.

Partially coherent microscope in phase space

Explicit expressions are presented for different phase-space representations (mutual intensity, Wigner distribution function, and ambiguity function) of the partially coherent image wave field in a microscope system. These are separated into system- and object-dependent parts. The partially coherent image in phase space can be described in terms of different 6D system-dependent kernels, all Fourier transforms of the system mutual spectrum, the region of overlap of two displaced objective pupils and the effective source. The image intensity can be expressed in terms of a 4D kernel, the convolution in spatial frequency of the source, and the Wigner distribution function of the objective pupil, given by a marginal of, or a section through, the respective phase-space kernels.

Isotropic differential phase contrast microscopy for quantitative phase bio-imaging

Quantitative phase imaging (QPI) has been investigated to retrieve optical phase information of an object and applied to biological microscopy and related medical studies. In recent examples, differential phase contrast (DPC) microscopy can recover phase image of thin sample under multi-axis intensity measurements in wide-field scheme. Unlike conventional DPC, based on theoretical approach under partially coherent condition, we propose a new method to achieve isotropic differential phase contrast (iDPC) with high accuracy and stability for phase recovery in simple and high-speed fashion. The iDPC is simply implemented with a partially coherent microscopy and a programmable thin-film transistor (TFT) shield to digitally modulate structured illumination patterns for QPI. In this article, simulation results show consistency of our theoretical approach for iDPC under partial coherence. In addition, we further demonstrate experiments of quantitative phase images of a standard micro-lens array, as well as label-free live human cell samples.

Three-dimensional tomographic microscopy technique with multi-frequency combination with partially coherent illuminations

We demonstrate a three-dimensional (3D) optical diffraction tomographic technique with multi-frequency combination (MFC-ODT) for the 3D quantitative phase imaging of unlabeled specimens. Three sets of through-focus intensity images are captured under an annular aperture and two circular apertures with different coherence parameters. The 3D phase optical transfer functions (POTF) corresponding to different illumination apertures are combined to obtain a synthesized frequency response, achieving high-quality, low-noise 3D reconstructions with imaging resolution up to the incoherent diffraction limit. Besides, the expression of 3D POTF for arbitrary illumination pupils is derived and analyzed, and the 3D imaging performance of annular illumination is explored. It is shown that the phase-contrast washout effect in high-NA circular apertures can be effectively addressed by introducing a complementary annular aperture, which strongly boosts the phase contrast and improves the imaging resolution. By incorporating high-NA illumination as well as high-NA detection, MFC-ODT can achieve a theoretical transverse resolution up to 200 nm and an axial resolution of 645 nm. To test the feasibility of the proposed MFC-ODT technique, the 3D refractive index reconstruction results are based on a simulated 3D resolution target and experimental investigations of micro polystyrene bead and unstained biological samples are presented. Due to its capability for high-resolution 3D phase imaging as well as the compatibility with a widely available commercial microscope, the MFC-ODT is expected to find versatile applications in biological and biomedical research.

Multi-focus TIE algorithm including partial spatial coherence and overlapping filters

The transport of intensity equation (TIE) relates the variation of intensity of a wave-front along its mean direction of propagation with its phase. In experimental application, characteristic artefacts may affect the retrieved phase. These originate from inadequacies in estimating the axial derivative and the amplification of noise in the inversion of the TIE. To tackle these issues, images recorded at multiple planes of focus can be integrated into a multi-focus TIE (MFTIE) solution. This methodology relies on the efficient sampling of phase information in the spatial-frequency domain, typically by the definition of band pass filters implemented as a progression of sharp spatial frequency cut-offs. We present a convenient MFTIE implementation which avoids the need for recording images at very specific planes of focus and applies overlapping cut-offs, greatly simplifying the experimental application. This new approach additionally also accounts for partial spatial coherence in a flux-preserving framework. Using simulated data as well as experimental data from optical microscopy and electron microscopy we show that the frequency response of this MFTIE algorithm recovers efficiently a wide range of spatial frequencies of the phase that can be further extended by simple iterative refinement.

Clarification and unification of the obliquity factor in diffraction and scattering theories: discussion

Two-dimensional (2D) and three-dimensional (3D) diffraction theories form the underlying basis of quantitative phase imaging. This paper reviews how 2D and 3D diffraction theories are developed based on thin and thick object requirements. However, some previously reported work has mixed 2D and 3D theories. This discussion shows that it is possible to enable consistent mixed use of 2D and 3D theories by applying appropriate obliquity factor (OF) modifications. The discussion is concluded with an overall unifying representation for the usage of the OF modifications in 2D and 3D diffraction theories as applied to both thin and thick objects.

Efficient quantitative phase microscopy using programmable annular LED illumination

In this work, we present an efficient quantitative phase imaging (QPI) approach using programmable annular LED illumination. As a new type of coded light source, the LED array provides flexible illumination control for noninterferometric QPI based on a traditional microscopic configurations. The proposed method modulates the transfer function of system by changing the LED illumination pattern, which provides noise-robust response of transfer function and achieves twice resolution limit of objective NA. The quantitative phase can be recovered from slightly defocused intensity images through inversion of transfer function. Moreover, the weak object transfer function (WOTF) of axis-symmetric oblique source is derived, and the noise-free and noisy simulation results validate the predicted theory. Finally, we experimentally confirm accurate and repeatable performance of our method by imaging calibrated phase samples and cellular specimens with different NA objectives.

Refractive Index Imaging of Cells with Variable-Angle Near-Total Internal Reflection (TIR) Microscopy

The refractive index in the interior of single cells affects the evanescent field depth in quantitative studies using total internal reflection (TIR) fluorescence, but often that index is not well known. We here present method to measure and spatially map the absolute index of refraction in a microscopic sample, by imaging a collimated light beam reflected from the substrate/buffer/cell interference at variable angles of incidence. Above the TIR critical angle (which is a strong function of refractive index), the reflection is 100%, but in the immediate sub-critical angle zone, the reflection intensity is a very strong ascending function of incidence angle. By analyzing the angular position of that edge at each location in the field of view, the local refractive index can be estimated. In addition, by analyzing the steepness of the edge, the distance-to-substrate can be determined. We apply the technique to liquid calibration samples, silica beads, cultured Chinese hamster ovary cells, and primary culture chromaffin cells. The optical technique suffers from decremented lateral resolution, scattering, and interference artifacts. However, it still provides reasonable results for both refractive index (~1.38) and for distance-to-substrate (~150 nm) for the cells, as well as a lateral resolution to about 1 µm.

High-resolution transport-of-intensity quantitative phase microscopy with annular illumination

For quantitative phase imaging (QPI) based on transport-of-intensity equation (TIE), partially coherent illumination provides speckle-free imaging, compatibility with brightfield microscopy, and transverse resolution beyond coherent diffraction limit. Unfortunately, in a conventional microscope with circular illumination aperture, partial coherence tends to diminish the phase contrast, exacerbating the inherent noise-to-resolution tradeoff in TIE imaging, resulting in strong low-frequency artifacts and compromised imaging resolution. Here, we demonstrate how these issues can be effectively addressed by replacing the conventional circular illumination aperture with an annular one. The matched annular illumination not only strongly boosts the phase contrast for low spatial frequencies, but significantly improves the practical imaging resolution to near the incoherent diffraction limit. By incorporating high-numerical aperture (NA) illumination as well as high-NA objective, it is shown, for the first time, that TIE phase imaging can achieve a transverse resolution up to 208 nm, corresponding to an effective NA of 2.66. Time-lapse imaging of in vitro Hela cells revealing cellular morphology and subcellular dynamics during cells mitosis and apoptosis is exemplified. Given its capability for high-resolution QPI as well as the compatibility with widely available brightfield microscopy hardware, the proposed approach is expected to be adopted by the wider biology and medicine community.

Single-exposure quantitative phase imaging in color-coded LED microscopy

We demonstrate single-shot quantitative phase imaging (QPI) in a platform of color-coded LED microscopy (cLEDscope). The light source in a conventional microscope is replaced by a circular LED pattern that is trisected into subregions with equal area, assigned to red, green, and blue colors. Image acquisition with a color image sensor and subsequent computation based on weak object transfer functions allow for the QPI of a transparent specimen. We also provide a correction method for color-leakage, which may be encountered in implementing our method with consumer-grade LEDs and image sensors. Most commercially available LEDs and image sensors do not provide spectrally isolated emissions and pixel responses, generating significant error in phase estimation in our method. We describe the correction scheme for this color-leakage issue, and demonstrate improved phase measurement accuracy. The computational model and single-exposure QPI capability of our method are presented by showing images of calibrated phase samples and cellular specimens.

Quantitative phase imaging method based on an analytical nonparaxial partially coherent phase optical transfer function

Multifilter phase imaging with partially coherent light (MFPI-PC) is a promising new quantitative phase imaging method. However, the existing MFPI-PC method is based on the paraxial approximation. In the present work, an analytical nonparaxial partially coherent phase optical transfer function is derived. This enables the MFPI-PC to be extended to the realistic nonparaxial case. Simulations over a wide range of test phase objects as well as experimental measurements on a microlens array verify higher levels of imaging accuracy compared to the paraxial method. Unlike the paraxial version, the nonparaxial MFPI-PC with obliquity factor correction exhibits no systematic error. In addition, due to its analytical expression, the increase in computation time compared to the paraxial version is negligible.

Quantitative phase imaging of weakly scattering objects using partially coherent illumination

In this paper, we extend our recent work on partially coherent quantitative phase imaging (pcQPI) from two-dimensional (2D) to three-dimensional (3D) imaging of weakly scattering samples. Due to the mathematical complexity, most theoretical modeling of quantitative phase image formation under partial coherence has focused on thin, well-focused samples. It is unclear how these abberations are affected by defocusing. Also, as 3D QPI techniques continue to develop, a better model needs to be developed to understand and quantify these aberrations when imaging thicker samples. Here, using the first order Born's approximation, we derived a mathematical framework that provides an intuitive model of image formation under varying degrees of coherence. Our description provides a clear connection between the halo effect and phase underestimation, defocusing and the 3D structure of the sample under investigation. Our results agree very well with the experiments and show that the microscope objective defines the sectioning ability of the imaging system while the condenser lens is responsible for the halo effect.

Three-dimensional quantitative phase imaging via tomographic deconvolution phase microscopy

The field of three-dimensional quantitative phase imaging (3D QPI) is expanding rapidly with applications in biological, medical, and industrial research, development, diagnostics, and metrology. Much of this research has centered on developing optical diffraction tomography (ODT) for biomedical applications. In addition to technical difficulties associated with coherent noise, ODT is not congruous with optical microscopy utilizing partially coherent light, which is used in most biomedical laboratories. Thus, ODT solutions have, for the most part, been limited to customized optomechanical systems which would be relatively expensive to implement on a wide scale. In the present work, a new phase reconstruction method, called tomographic deconvolution phase microscopy (TDPM), is described which makes use of commercial microscopy hardware in realizing 3D QPI. TDPM is analogous to methods used in deconvolution microscopy which improve spatial resolution and 3D-localization accuracy of fluorescence micrographs by combining multiple through-focal scans which are deconvolved by the system point spread function. TDPM is based on the 3D weak object transfer function theory which is shown here to be capable of imaging "nonweak" phase objects with large phase excursions. TDPM requires no phase unwrapping and recovers the entire object spectrum via object rotation, mitigating the need to fill in the "missing cone" of spatial frequencies algorithmically as in limited-angle ODT. In the present work, TDPM is demonstrated using optical fibers, including single-mode, polarization-maintaining, and photonic-crystal fibers as well as an azimuthally varying CO2-laser-induced long-period fiber grating period as test phase objects.