Mixed transfer function and transport of intensity approach for phase retrieval in the Fresnel region

Quantitative phase contrast X-ray tomography of aluminium metal matrix composite

The quantitative assessment of micro-structure and load-induced damages in Al-SiC metal matrix composites (MMC) is important for its design optimization, performance evaluation and structure-property correlation. X-ray Phase contrast micro-tomography is potentially used for evaluation of its 3 dimensional micro-structure manifested in the form of voids, cracks, embedded particles, and load-induced damages. However, the contrast between Al matrix and SiC particles is insufficient for their clear morphological identification and quantitative assessment. In the present study, we have proposed and applied single image-based phase retrieval as a pre-processing step to micro-tomography reconstruction for improved assessment of micro-structure and cohesion-induced damages in Al-SiC MMC. The advantages of applying different phase retrieval techniques in the enhancement of image quality and morphological quantification of SiC particles, pores and cohesion damages are discussed. It is observed that the Paganin method offers the best improvement in contrast to noise ratio for the measurement of SiC particles embedded in the Al matrix.

Structured illumination contrast transfer function for high resolution quantitative phase imaging

We report a sub-diffraction resolution imaging of non-fluorescent samples through quantitative phase imaging. This is achieved through a novel application of structured illumination microscopy (SIM), a super-resolution imaging technique established primarily for fluorescence microscopy. Utilizing our contrast transfer function formalism with SIM, we extract the high spatial frequency components of the phase profile from the defocused intensity images, enabling the reconstruction of a quantitative phase image with a frequency spectrum that surpasses the diffraction limit imposed by the imaging system. Our approach offers several advantages including a deterministic, phase-unwrapping-free algorithm and an easily implementable, non-interferometric setup. We validate the proposed technique for high-resolution phase imaging through both simulation and experimental results, demonstrating a two-fold enhancement in resolution. A lateral resolution of 0.814 µm is achieved for the phase imaging of human cheek cells using a 0.42 NA objective lens and an illumination wavelength of 660 nm, highlighting the efficacy of our approach for high-resolution quantitative phase imaging.

Three-dimensional contrast-transfer-function approach in phase-contrast tomography

A new method is developed for 3D reconstruction of multimaterial objects using propagation-based x-ray phase-contrast tomography (PB-CT) with phase retrieval via contrast-transfer-function (CTF) formalism. The approach differs from conventional PB-CT algorithms, which apply phase retrieval to individual 2D projections. Instead, this method involves performing phase retrieval to the CT-reconstructed volume in 3D. The CTF formalism is further extended to the cases of partially coherent illumination and strongly absorbing samples. Simulated results demonstrate that the proposed post-reconstruction CTF method provides fast and stable phase retrieval, producing results equivalent to conventional pre-reconstruction 2D CTF phase retrieval. Moreover, it is shown that application can be highly localized to isolated objects of interest, without a significant loss of quality, thus leading to increased computational efficiency. Combined with the extended validity of the CTF to greater propagation distances, this method provides additional advantages over approaches based on the transport-of-intensity equation.

Investigating the robustness of a deep learning-based method for quantitative phase retrieval from propagation-based x-ray phase contrast measurements under laboratory conditions

Objective.Quantitative phase retrieval (QPR) in propagation-based x-ray phase contrast imaging of heterogeneous and structurally complicated objects is challenging under laboratory conditions due to partial spatial coherence and polychromaticity. A deep learning-based method (DLBM) provides a nonlinear approach to this problem while not being constrained by restrictive assumptions about object properties and beam coherence. The objective of this work is to assess a DLBM for its applicability under practical scenarios by evaluating its robustness and generalizability under typical experimental variations.Approach.Towards this end, an end-to-end DLBM was employed for QPR under laboratory conditions and its robustness was investigated across various system and object conditions. The robustness of the method was tested via varying propagation distances and its generalizability with respect to object structure and experimental data was also tested.Main results.Although the end-to-end DLBM was stable under the studied variations, its successful deployment was found to be affected by choices pertaining to data pre-processing, network training considerations and system modeling.Significance.To our knowledge, we demonstrated for the first time, the potential applicability of an end-to-end learning-based QPR method, trained on simulated data, to experimental propagation-based x-ray phase contrast measurements acquired under laboratory conditions with a commercial x-ray source and a conventional detector. We considered conditions of polychromaticity, partial spatial coherence, and high noise levels, typical to laboratory conditions. This work further explored the robustness of this method to practical variations in propagation distances and object structure with the goal of assessing its potential for experimental use. Such an exploration of any DLBM (irrespective of its network architecture) before practical deployment provides an understanding of its potential behavior under experimental settings.

Oscillation of the orientation of cardiomyocyte aggregates in human left ventricle free wall

Orientation of local cardiomyocyte aggregates in the human left ventricle free wall experiences an oscillation in the laminar structure regions, besides its gradual change trend. We described this oscillation using five transmural samples imaged at the European Synchrotron Radiation Facility with an isotropic voxel size of 3.5 × 3.5 × 3.5 μm³ . In the reconstructed volume of each sample, we manually selected a region containing a regular laminar structure as the region of interest and measured the distribution of the orientation of local cardiomyocyte aggregates inside using a Fourier-based method. Then, we extracted the gradual change part of the orientation of cardiomyocyte aggregates with a three-dimensional centered Gaussian filter and measured the angle between the original orientation vector of local cardiomyocyte aggregates and its gradual change part. Further, we assessed the measured angles in different local coordinates. The results indicate that the oscillation amplitude of the orientation of cardiomyocyte aggregates is regional in the left ventricle wall, which may promote our understanding of the rearrangement mechanism of the cardiomyocyte aggregates and provide a new biomarker to study the heart physiological status.

Deep Gauss-Newton for phase retrieval

We propose the deep Gauss-Newton (DGN) algorithm. The DGN allows one to take into account the knowledge of the forward model in a deep neural network by unrolling a Gauss-Newton optimization method. No regularization or step size needs to be chosen; they are learned through convolutional neural networks. The proposed algorithm does not require an initial reconstruction and is able to retrieve simultaneously the phase and absorption from a single-distance diffraction pattern. The DGN method was applied to both simulated and experimental data and permitted large improvements of the reconstruction error and of the resolution compared with a state-of-the-art iterative method and another neural-network-based reconstruction algorithm.

Nonlinear primal-dual algorithm for the phase and absorption retrieval from a single phase contrast image

We propose a nonlinear primal-dual algorithm for the retrieval of phase shift and absorption from a single x ray in-line phase contrast, or Fresnel diffraction, image. The algorithm permits us to regularize phase and absorption separately. We demonstrate that taking into account the nonlinearity in the reconstruction improves reconstruction compared with linear methods. We also demonstrate that choosing different regularizers for absorption and phase can improve the reconstructions. The use of the total variation and its generalization in a primal-dual approach allows us to exploit the sparsity of the investigated sample. On both simulated and real datasets, the proposed nonlinear primal-dual hybrid gradient (NL-PDHG) method yields reconstructions with considerably fewer artifacts and improved the normalized mean squared error compared with its linearized version.

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.

Characterization of internal fatigue cracks in aluminum alloys by simulation of phase contrast tomography

Synchrotron Radiation Computed Tomography (SRCT) allows a better detection of fatigue cracks in metals than laboratory CT due to the existence of phase contrast. However the presence in reconstructed images of fringes at the edges of objects generated by Fresnel diffraction makes it difficult to identify and analyze the cracks quantitatively. Simulations of phase contrast synchrotron tomography images containing cracks with different sizes and shapes are obtained by using GATE software. Analyzing the simulation results, firstly, we confirmed that the bright parts with strong contrast in SRCT image are streak artifacts; secondly, we found that the gray scale values within the cracks in SRCT images are related to the crack size; these simulation results are used to analyse SRCT images of internal fatigue cracks in a cast Al alloy, providing a clearer visualisation of damage.

Measurement of local orientation of cardiomyocyte aggregates in human left ventricle free wall samples using X-ray phase-contrast microtomography

Most cardiomyocytes in the left ventricle wall are grouped in aggregates of four to five units that are quasi-parallel to each other. When one or more "cardiomyocyte aggregates" are delimited by two cleavage planes, this defines a "sheetlet" that can be considered as a "work unit" that contributes to the thickening of the wall during the cardiac cycle. In this paper, we introduce the skeleton method to measure the local three-dimensional (3D) orientation of cardiomyocyte aggregates in the sheetlets in three steps: data segmentation; extraction of the skeleton of the sheetlets; and calculation of the local orientation of the cardiomyocyte aggregates inside the sheetlets. These data include a series of virtual tissue volumes and five transmural human left ventricle free wall samples, imaged with 3D synchrotron radiation phase-contrast microtomography, and reconstructed with a 3.5×3.5×3.5μm³ voxel size. We computed the local orientation of the cardiomyocyte aggregates inside the sheetlets with a working window of 112×112×112μm³ in size. These data demonstrate that the skeleton method can provide accurate 3D measurements and reliable screening of the 3D evolution of the orientation of cardiomyocyte aggregates within the sheetlets. We showed that in regions that contain one population of quasi-parallel sheetlets, the orientation of the cardiomyocyte aggregates undergo "oscillations" along the perpendicular direction of the sheetlets. In regions that contain two populations of sheetlets with a different angular range, we demonstrate some discontinuity of the helix angle of the cardiomyocyte aggregates at the interface between the two populations.

Dual-energy phase retrieval algorithm for inline phase sensitive x-ray imaging system

Phase retrieval is vital for quantitative x-ray phase contrast imaging. This work presents an iterative method to simultaneously retrieve the x-ray absorption and phase images from a single x-ray exposure. The proposed approach uses the photon-counting detectors' energy-resolving capability in providing multiple spectrally resolved phase contrast images from a single x-ray exposure. The retrieval method is derived, presented, and experimentally tested with a multi-material phantom in an inline phase contrast imaging setup. By separating the contributions of photoelectric absorption and Compton scattering to the attenuation, the authors divide the phase contrast image into two portions, the attenuation map arises from photoelectric absorption and a pseudo phase contrast image generated by electron density. This way one can apply the Phase Attenuation Dualiby (PAD) algorithm and Fresnel propagation for the iteration. The retrieval results from the experimental images show that this iterative method is fast, accurate, robust against noise, and thus yields noticeable enhancement in contrast to noise ratios.

PyPhase - a Python package for X-ray phase imaging

X-ray propagation-based imaging techniques are well established at synchrotron radiation and laboratory sources. However, most reconstruction algorithms for such image modalities, also known as phase-retrieval algorithms, have been developed specifically for one instrument by and for experts, making the development and diffusion of such techniques difficult. Here, PyPhase, a free and open-source package for propagation-based near-field phase reconstructions, which is distributed under the CeCILL license, is presented. PyPhase implements some of the most popular phase-retrieval algorithms in a highly modular framework supporting its deployment on large-scale computing facilities. This makes the integration, the development of new phase-retrieval algorithms, and the deployment on different computing infrastructures straightforward. Its capabilities and simplicity are presented by application to data acquired at the synchrotron source MAX IV (Lund, Sweden).

Evaluation of the Weighted Mean X-ray Energy for an Imaging System Via Propagation-Based Phase-Contrast Imaging

For imaging events of extremely short duration, like shock waves or explosions, it is necessary to be able to image the object with a single-shot exposure. A suitable setup is given by a laser-induced X-ray source such as the one that can be found at GSI (Helmholtzzentrum für Schwerionenforschung GmbH) in Darmstadt (Society for Heavy Ion Research), Germany. There, it is possible to direct a pulse from the high-energy laser Petawatt High Energy Laser for Heavy Ion eXperiments (PHELIX) on a tungsten wire to generate a picosecond polychromatic X-ray pulse, called backlighter. For grating-based single-shot phase-contrast imaging of shock waves or exploding wires, it is important to know the weighted mean energy of the X-ray spectrum for choosing a suitable setup. In propagation-based phase-contrast imaging the knowledge of the weighted mean energy is necessary to be able to reconstruct quantitative phase images of unknown objects. Hence, we developed a method to evaluate the weighted mean energy of the X-ray backlighter spectrum using propagation-based phase-contrast images. In a first step wave-field simulations are performed to verify the results. Furthermore, our evaluation is cross-checked with monochromatic synchrotron measurements with known energy at Diamond Light Source (DLS, Didcot, UK) for proof of concepts.

Modeling classical wavefront sensors

We present an image formation model for deterministic phase retrieval in propagation-based wavefront sensing, unifying analysis for classical wavefront sensors such as Shack-Hartmann (slopes tracking) and curvature sensors (based on Transport-of-Intensity Equation). We show how this model generalizes commonly seen formulas, including Transport-of-Intensity Equation, from small distances and beyond. Using this model, we analyze theoretically achievable lateral wavefront resolution in propagation-based deterministic wavefront sensing. Finally, via a prototype masked wavefront sensor, we show simultaneous bright field and phase imaging numerically recovered in real-time from a single-shot measurement.

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.

Forward model for propagation-based x-ray phase contrast imaging in parallel- and cone-beam geometry

We demonstrate a fast, flexible, and accurate paraxial wave propagation model to serve as a forward model for propagation-based X-ray phase contrast imaging (XPCI) in parallel-beam or cone-beam geometry. This model incorporates geometric cone-beam effects into the multi-slice beam propagation method. It enables rapid prototyping and is well suited to serve as a forward model for propagation-based X-ray phase contrast tomographic reconstructions. Furthermore, it is capable of modeling arbitrary objects, including those that are strongly or multi-scattering. Simulation studies were conducted to compare our model to other forward models in the X-ray regime, such as the Mie and full-wave Rytov solutions.

Unsupervised solution for in-line holography phase retrieval using Bayesian inference

In propagation based phase contrast imaging, intensity patterns are recorded on a x-ray detector at one or multiple propagation distances, called in-line holograms. They form the input of an inversion algorithm that aims at retrieving the phase shift induced by the object. The problem of phase retrieval in in-line holography is an ill-posed inverse problem. Consequently an adequate solution requires some form of regularization with the most commonly applied being the classical Tikhonov regularization. While generally satisfying this method suffers from a few issues such as the choice of the regularization parameter. Here, we offer an alternative to the established method by applying the principles of Bayesian inference. We construct an iterative optimization algorithm capable of both retrieving the unknown phase and determining a multi-dimensional regularization parameter. In the end, we highlight the advantages of the introduced algorithm, chief among them being the unsupervised determination of the regularization parameter(s). The proposed approach is tested on both simulated and experimental data and is found to provide robust solutions, with improved response to typical issues like low frequency noise and the twin-image problem.

3D Imaging Based on Depth Measurement Technologies

Three-dimensional (3D) imaging has attracted more and more interest because of its widespread applications, especially in information and life science. These techniques can be broadly divided into two types: ray-based and wavefront-based 3D imaging. Issues such as imaging quality and system complexity of these techniques limit the applications significantly, and therefore many investigations have focused on 3D imaging from depth measurements. This paper presents an overview of 3D imaging from depth measurements, and provides a summary of the connection between the ray-based and wavefront-based 3D imaging techniques.

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.

Improved phase-attenuation duality method with space-frequency joint domain iterative regularization

PURPOSE

A common problem of in-line phase contrast imaging systems based on laboratory source and detector is the negative effects of finite source size, limited spatial resolution, and system noise. These negative effects swamp the fine phase contrast fringes and impede the precise retrieval of phase maps. This study aims to develop a novel phase retrieval method to restore phase information that is lost due to an imperfect system.

METHODS

An improved phase-attenuation duality (PAD) method based on space-frequency joint domain iterative regularization (JDIR) is proposed to overcome the problems of the analytical PAD method and the spatial-domain iterative regularization (SDIR) based PAD method. These problems include noise robustness and optical transfer function compensation. The proposed method was compared with the two former PAD methods through computer simulations and experiments for validation.

RESULTS

Results reveal that JDIR method outperforms the other two methods in terms of improving the visibility of structures in the retrieved phase maps. Among all the phase retrieval algorithms, the TV-norm-based JDIR method performed the best in considering the contrast and noise performance.

CONCLUSIONS

This paper provides a new method to investigate quantitative phase-contrast imaging when considering the negative effects of an imperfect system.

Evaluation of phase retrieval approaches in magnified X-ray phase nano computerized tomography applied to bone tissue

X-ray phase contrast imaging offers higher sensitivity compared to conventional X-ray attenuation imaging and can be simply implemented by propagation when using a partially coherent synchrotron beam. We address the phase retrieval in in-line phase nano-CT using multiple propagation distances. We derive a method which extends Paganin's single distance method and compare it to the contrast transfer function (CTF) approach in the case of a homogeneous object. The methods are applied to phase nano-CT data acquired at the voxel size of 30 nm (ID16A, ESRF, Grenoble, France). Our results show a gain in image quality in terms of the signal-to-noise ratio and spatial resolution when using four distances instead of one. The extended Paganin's method followed by an iterative refinement step provides the best reconstructions while the homogeneous CTF method delivers quasi comparable results for our data, even without refinement step.

Optical convolution for quantitative phase retrieval using the transport of intensity equation

Propagation-based phase imaging using the transport of intensity equation (TIE) allows rapid, deterministic phase retrieval from defocused images. However, computational solutions to the TIE suffer from significant low-frequency noise artifacts and are unique up to the application of boundary conditions on the phase. We demonstrate that quantitative phase can be imaged directly at the detector for a class of pure-phase samples by appropriately patterning the illumination to solve the TIE through an optical convolution with the source. This can reduce noise artifacts, obviates the need for user-supplied boundary conditions and is demonstrated via simulation and experiment.

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.

Robust phase-retrieval-based X-ray tomography for morphological assessment of early hepatic echinococcosis infection in rats

Propagation-based phase-contrast computed micro-tomography (PPCT) dominates the non-destructive, three-dimensional inner-structure measurement in synchrotron-based biomedical research due to its simple experimental setup. To quantitatively visualize tiny density variations in soft tissues and organs closely related to early pathological morphology, an experimental study of synchrotron-based X-ray PPCT combined with generalized phase and attenuation duality (PAD) phase retrieval was implemented with the hepatic echinococcosis (HE) infection rat model at different stages. We quantitatively analyzed and evaluated the different pathological characterizations of hepatic echinococcosis during the development of this disease via our PAD-based PPCT and especially provided evidence that hepatic alveolar echinococcosis invades the liver tissue and spreads through blood flow systems with abundant blood supply in the early stage. Additionally, the infiltration of tiny vesicles in HE lesions can be clearly observed by our PAD-PPCT technique due to the striking contrast-to-noise ratio (CNR) and mass density resolution, which cannot be found by the medical imaging techniques, such as magnetic resonance imaging (MRI), computed tomography (CT), and ultrasound, in hospitals. The results demonstrated that our PAD-PPCT technique has a great potential for indicating the subtle structural information of pathological changes in soft biomedical specimens, especially helpful for the research of early micro-morphology of diseases.

Registration of phase-contrast images in propagation-based X-ray phase tomography

X-ray phase tomography aims at reconstructing the 3D electron density distribution of an object. It offers enhanced sensitivity compared to attenuation-based X-ray absorption tomography. In propagation-based methods, phase contrast is achieved by letting the beam propagate after interaction with the object. The phase shift is then retrieved at each projection angle, and subsequently used in tomographic reconstruction to obtain the refractive index decrement distribution, which is proportional to the electron density. Accurate phase retrieval is achieved by combining images at different propagation distances. For reconstructions of good quality, the phase-contrast images recorded at different distances need to be accurately aligned. In this work, we characterise the artefacts related to misalignment of the phase-contrast images, and investigate the use of different registration algorithms for aligning in-line phase-contrast images. The characterisation of artefacts is done by a simulation study and comparison with experimental data. Loss in resolution due to vibrations is found to be comparable to attenuation-based computed tomography. Further, it is shown that registration of phase-contrast images is nontrivial due to the difference in contrast between the different images, and the often periodical artefacts present in the phase-contrast images if multilayer X-ray optics are used. To address this, we compared two registration algorithms for aligning phase-contrast images acquired by magnified X-ray nanotomography: one based on cross-correlation and one based on mutual information. We found that the mutual information-based registration algorithm was more robust than a correlation-based method.

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.

Source diversity for transport of intensity phase imaging

The transport of intensity equation (TIE) is a phase retrieval method that relies on measurements of the intensity of a paraxial field under propagation between two or more closely spaced planes. A limitation of TIE is its susceptibility to low frequency noise artifacts in the reconstructed phase. Under Köhler illumination, when both illumination power and exposure time are limited, the use of larger sources can improve low-frequency performance although it introduces blurring. Appropriately combining intensity measurements taken with a diversity of source sizes can improve both low- and high-frequency performance in phase reconstruction.

Contrast-transfer-function phase retrieval based on compressed sensing

We report on a new contrast-transfer-function (CTF) phase-retrieval method based on the alternating direction method of multipliers (ADMMs), which allows us to exploit any compressed sensing regularization scheme reflecting the sparsity of the investigated object. The proposed iterative algorithm retrieves accurate phase maps from highly noisy single-distance projection microscopy data and is characterized by a stable convergence, not bounded to the prior knowledge of the object support or to the initialization strategy. Experiments on simulated and real datasets show that ADMM-CTF yields reconstructions with a substantial lower amount of artifacts and enhanced signal-to-noise ratio compared to the standard analytical inversion.

Extraction of the 3D local orientation of myocytes in human cardiac tissue using X-ray phase-contrast micro-tomography and multi-scale analysis

This paper presents a methodology to access the 3D local myocyte arrangements in fresh human post-mortem heart samples. We investigated the cardiac micro-structure at a high and isotropic resolution of 3.5 µm in three dimensions using X-ray phase micro-tomography at the European Synchrotron Radiation Facility. We then processed the reconstructed volumes to extract the 3D local orientation of the myocytes using a multi-scale approach with no segmentation. We created a simplified 3D model of tissue sample made of simulated myocytes with known size and orientations, to evaluate our orientation extraction method. Afterwards, we applied it to 2D histological cuts and to eight 3D left ventricular (LV) cardiac tissue samples. Then, the variation of the helix angles, from the endocardium to the epicardium, was computed at several spatial resolutions ranging from 3.6³ mm³ to 112³ µm³. We measure an increased range of 20° to 30° from the coarsest resolution level to the finest level in the experimental samples. This result is in line with the higher values measured from histology. The displayed tractography demonstrates a rather smooth evolution of the transmural helix angle in six LV samples and a sudden discontinuity of the helix angle in two septum samples. These measurements bring a new vision of the human heart architecture from macro- to micro-scale.

Visualization and Pathological Characteristics of Hepatic Alveolar Echinococcosis with Synchrotron-based X-ray Phase Sensitive Micro-tomography

Propagation-based phase-contrast computed tomography (PPCT) utilizes highly sensitive phase-contrast technology applied to X-ray micro-tomography, especially with the extensive use of synchrotron radiation (SR). Performing phase retrieval (PR) on the acquired angular projections can enhance image contrast and enable quantitative imaging. We employed the combination of SR-PPCT and PR for the histopathological evaluation of hepatic alveolar echinococcosis (HAE) disease and demonstrated the validity and superiority of PR-based SR-PPCT. A high-resolution angular projection data set of a human postoperative specimen of HAE disease was acquired, which was processed by graded ethanol concentration fixation (GECF). The reconstructed images from both approaches, with the projection data directly used and preprocessed by PR for tomographic reconstruction, were compared in terms of the tissue contrast-to-noise ratio and density spatial resolution. The PR-based SR-PPCT was selected for microscale measurement and the 3D visualization of HAE disease. Our experimental results demonstrated that the PR-based SR-PPCT technique is greatly suitable for the discrimination of pathological tissues and the characterization of HAE. In addition, this new technique is superior to conventional hospital CT and microscopy for the three-dimensional, non-destructive microscale measurement of HAE. This PR-based SR-PPCT technique has great potential for in situmicroscale histopathological analysis and diagnosis, especially for applications involving soft tissues and organs.

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.

Technical Note: Synchrotron-based high-energy x-ray phase sensitive microtomography for biomedical research

PURPOSE

Propagation-based phase-contrast CT (PPCT) utilizes highly sensitive phase-contrast technology applied to x-ray microtomography. Performing phase retrieval on the acquired angular projections can enhance image contrast and enable quantitative imaging. In this work, the authors demonstrate the validity and advantages of a novel technique for high-resolution PPCT by using the generalized phase-attenuation duality (PAD) method of phase retrieval.

METHODS

A high-resolution angular projection data set of a fish head specimen was acquired with a monochromatic 60-keV x-ray beam. In one approach, the projection data were directly used for tomographic reconstruction. In two other approaches, the projection data were preprocessed by phase retrieval based on either the linearized PAD method or the generalized PAD method. The reconstructed images from all three approaches were then compared in terms of tissue contrast-to-noise ratio and spatial resolution.

RESULTS

The authors' experimental results demonstrated the validity of the PPCT technique based on the generalized PAD-based method. In addition, the results show that the authors' technique is superior to the direct PPCT technique as well as the linearized PAD-based PPCT technique in terms of their relative capabilities for tissue discrimination and characterization.

CONCLUSIONS

This novel PPCT technique demonstrates great potential for biomedical imaging, especially for applications that require high spatial resolution and limited radiation exposure.

Using sparsity information for iterative phase retrieval in x-ray propagation imaging

For iterative phase retrieval algorithms in near field x-ray propagation imaging experiments with a single distance measurement, it is indispensable to have a strong constraint based on a priori information about the specimen; for example, information about the specimen's support. Recently, Loock and Plonka proposed to use the a priori information that the exit wave is sparsely represented in a certain directional representation system, a so-called shearlet system. In this work, we extend this approach to complex-valued signals by applying the new shearlet constraint to amplitude and phase separately. Further, we demonstrate its applicability to experimental data.

Contrast transfer functions for Zernike phase contrast in full-field transmission hard X-ray microscopy

Full-field transmission hard X-ray microscopy (TXM) has been widely applied to study morphology and structures with high spatial precision and to dynamic processes. Zernike phase contrast (ZPC) in hard X-ray TXM is often utilized to get an in-line phase contrast enhancement for weak-absorbing materials with little contrast differences. Here, following forward image formation, we derive and simplify the contrast transfer functions (CTFs) of the Zernike phase imaging system in TXM based on a linear space-shift-invariant imaging mode under certain approximations. The CTFs in ZPC in their simplified forms show a high similarity to the one in free-space propagation X-ray imaging systems.

Signal-to-noise criterion for free-propagation imaging techniques at free-electron lasers and synchrotrons

We propose a signal-to-noise criterion which predicts whether a feature of a given size and scattering strength, placed inside a larger object, can be retrieved with two common X-ray imaging techniques: coherent diffraction imaging and projection microscopy. This criterion, based on how efficiently these techniques detect the scattered photons and validated through simulations, shows in general that projection microscopy can resolve smaller phase differences and features than coherent diffraction imaging. Our criterion can be used to design optimized imaging experiments and perform feasibility studies for sensitive biological materials in free-electron lasers, where the number of photons per pulse is limited, or in synchrotron experiments, for both techniques.

3D X-ray ultra-microscopy of bone tissue

We review the current X-ray techniques with 3D imaging capability at the nano-scale: transmission X-ray microscopy, ptychography and in-line phase nano-tomography. We further review the different ultra-structural features that have so far been resolved: the lacuno-canalicular network, collagen orientation, nano-scale mineralization and their use as basis for mechanical simulations. X-ray computed tomography at the micro-metric scale is increasingly considered as the reference technique in imaging of bone micro-structure. The trend has been to push towards increasingly higher resolution. Due to the difficulty of realizing optics in the hard X-ray regime, the magnification has mainly been due to the use of visible light optics and indirect detection of the X-rays, which limits the attainable resolution with respect to the wavelength of the visible light used in detection. Recent developments in X-ray optics and instrumentation have allowed to implement several types of methods that achieve imaging that is limited in resolution by the X-ray wavelength, thus enabling computed tomography at the nano-scale. We review here the X-ray techniques with 3D imaging capability at the nano-scale: transmission X-ray microscopy, ptychography and in-line phase nano-tomography. Further, we review the different ultra-structural features that have so far been resolved and the applications that have been reported: imaging of the lacuno-canalicular network, direct analysis of collagen orientation, analysis of mineralization on the nano-scale and use of 3D images at the nano-scale to drive mechanical simulations. Finally, we discuss the issue of going beyond qualitative description to quantification of ultra-structural features.

Variational Phase Imaging Using the Transport-of-Intensity Equation

We introduce a variational phase retrieval algorithm for the imaging of transparent objects. Our formalism is based on the transport-of-intensity equation (TIE), which relates the phase of an optical field to the variation of its intensity along the direction of propagation. TIE practically requires one to record a set of defocus images to measure the variation of intensity. We first investigate the effect of the defocus distance on the retrieved phase map. Based on our analysis, we propose a weighted phase reconstruction algorithm yielding a phase map that minimizes a convex functional. The method is nonlinear and combines different ranges of spatial frequencies - depending on the defocus value of the measurements - in a regularized fashion. The minimization task is solved iteratively via the alternating-direction method of multipliers. Our simulations outperform commonly used linear and nonlinear TIE solvers. We also illustrate and validate our method on real microscopy data of HeLa cells.

Correcting multi material artifacts from single material phase retrieved holo-tomograms with a simple 3D Fourier method

Here we present a method for the removal of multi-material artifacts which occur during the application of a single material phase retrieval procedure to X-ray tomographic data sets. For the phase retrieval we chose the most common method which is the single material filter. The correction method which we describe in the following has been designed for samples consisting of three distinct materials, hence effectively two different material interfaces. Furthermore the material phase with the strongest X-ray interaction needs to show sufficient absorption in order to allow for segmenting this phase through application of a grey value threshold. If these conditions are fulfilled the method is easy to apply through post processing as is shown for the volume images of two sample types.

Iterative optimum frequency combination method for high efficiency phase imaging of absorptive objects based on phase transfer function

In this work, an optimum frequency combination (OFC) method is proposed to reconstruct high quality phase information of the complex light field, which is really valuable for many objects such as optical elements and cells. It is shown that the difference image between two symmetrical separated, larger defocused planes contains a lot of lower frequency components of the phase distribution and the higher frequency components can be easily observed in the difference image between two nearly focused planes. Based on the phase transfer function (PTF), our method combines different frequency components with high Signal-to-Noise Ratio (SNR) together to estimate a more accurate frequency spectrum of the object's phase distribution without any complicated linear or nonlinear regression. Then, we can directly reconstruct a high-quality phase map through inverse Fourier transform. What's more, in order to compensate the phase discrepancy resulted from strong absorption in the intensity, an iterative compensation algorithm is proposed. Both the simulation and experimental results demonstrate that our iterative OFC (IOFC) method can give a computationally efficient and noise-robust phase reconstruction for absorptive phase objects with higher accuracy and fewer defocus planes.

Quantitative phase retrieval with arbitrary pupil and illumination

We present a general algorithm for combining measurements taken under various illumination and imaging conditions to quantitatively extract the amplitude and phase of an object wave. The algorithm uses the weak object transfer function, which incorporates arbitrary pupil functions and partially coherent illumination. The approach is extended beyond the weak object regime using an iterative algorithm. We demonstrate the method on measurements of Extreme Ultraviolet Lithography (EUV) multilayer mask defects taken in an EUV zone plate microscope with both a standard zone plate lens and a zone plate implementing Zernike phase contrast.

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

Although the field of quantitative phase imaging (QPI) has wide-ranging biomedical applicability, many QPI methods are not well-suited for such applications due to their reliance on coherent illumination and specialized hardware. By contrast, methods utilizing partially coherent illumination have the potential to promote the widespread adoption of QPI due to their compatibility with microscopy, which is ubiquitous in the biomedical community. Described herein is a new defocus-based reconstruction method that utilizes a small number of efficiently sampled micrographs to optimally invert the partially coherent phase optical transfer function under assumptions of weak absorption and slowly varying phase. Simulation results are provided that compare the performance of this method with similar algorithms and demonstrate compatibility with large phase objects. The accuracy of the method is validated experimentally using a microlens array as a test phase object. Lastly, time-lapse images of live adherent cells are obtained with an off-the-shelf microscope, thus demonstrating the new method's potential for extending QPI capability widely in the biomedical community.

Optimization of reconstructed quality of hard x-ray phase microtomography

For applications of hard x-ray propagation-based phase-contrast computed microtomography (PPCT) in high-resolution biological research, high spatial resolution and high contrast-to-noise ratio are simultaneously required for tiny structural discrimination and characterization. Most existing micro-CT techniques to improve image quality are limited by high cost, physical limitations, and complexity of the experimental hardware and setup. In this work, a novel PPCT technique, which combines a wavelet-transform-based modulation transform function compensation algorithm and a generalized phase-retrieval algorithm, is proposed to optimize the reconstruction quality of tomographic slices. Our experimental results, which compared the spatial resolutions and contrast-to-noise ratios of reconstructed images, demonstrated the validity of the proposed generalized PPCT technique. The experimental results showed that the proposed generalized PPCT technique is superior to the direct PPCT and the linearized phase-retrieval PPCT techniques. This novel PPCT technique demonstrates great potential for biological imaging, especially for applications that require high spatial resolution and limit radiation exposure.

Hybrid single-beam reconstruction technique for slow and fast varying wave fields

An iterative single-beam wave field reconstruction technique that employs both non-paraxial, wave propagation based and paraxial deterministic phase retrieval techniques is presented. This approach overcomes two major obstacles that exist in the current state of the art techniques: iterative methods do not reconstruct slowly varying wave fields due to slow convergence and stagnation, and deterministic methods have paraxial limits, making the reconstructions of quickly varying object features impossible. In this work, a hybrid approach is reported that uses paraxial wave field corrections within iterative phase retrieval solvers. This technique is suitable for cases ranging from slow to fast varying wave fields, and unlike the currently available methods, can also reconstruct measurement objects with different regions of both slowly and quickly varying object features. It is further shown that this technique gives a higher accuracy than current single-beam phase retrieval techniques, and in comparison to the iterative methods, has a higher convergence speed.

Partially coherent phase imaging with simultaneous source recovery

We propose a new method for phase retrieval that uses partially coherent illumination created by any arbitrary source shape in Köhler geometry. Using a stack of defocused intensity images, we recover not only the phase and amplitude of the sample, but also an estimate of the unknown source shape, which describes the spatial coherence of the illumination. Our algorithm uses a Kalman filtering approach which is fast, accurate and robust to noise. The method is experimentally simple and flexible, so should find use in optical, electron, X-ray and other phase imaging systems which employ partially coherent light. We provide an experimental demonstration in an optical microscope with various condenser apertures.

Contrast enhancement in X-ray phase contrast tomography

We demonstrate phase contrast enhancement of X-ray computed tomography derived from propagation based imaging. In this method, the absorption and phase components are assumed to be correlated, allowing for phase retrieval from a single image. Experimental results are shown for liquid samples. Signal-to-noise ratio is greatly enhanced relative to pure attenuation based imaging.

Transport of Intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes

We propose an alternative method for solving the Transport of Intensity equation (TIE) from a stack of through-focus intensity images taken by a microscope or lensless imager. Our method enables quantitative phase and amplitude imaging with improved accuracy and reduced data capture, while also being computationally efficient and robust to noise. We use prior knowledge of how intensity varies with propagation in the spatial frequency domain in order to constrain a fitting algorithm [Gaussian process (GP) regression] for estimating the axial intensity derivative. Solving the problem in the frequency domain inspires an efficient measurement scheme which captures images at exponentially spaced focal steps, significantly reducing the number of images required. Low-frequency artifacts that plague traditional TIE methods can be suppressed without an excessive number of captured images. We validate our technique experimentally by recovering the phase of human cheek cells in a brightfield microscope.

Priors for X-ray in-line phase tomography of heterogeneous objects

We present a new prior for phase retrieval from X-ray Fresnel diffraction patterns. Fresnel diffraction patterns are achieved by letting a highly coherent X-ray beam propagate in free space after interaction with an object. Previously, either homogeneous or multi-material object assumptions have been used. The advantage of the homogeneous object assumption is that the prior can be introduced in the Radon domain. Heterogeneous object priors, on the other hand, have to be applied in the object domain. Here, we let the relationship between attenuation and refractive index vary as a function of the measured attenuation index. The method is evaluated using images acquired at beamline ID19 (ESRF, Grenoble, France) of a phantom where the prior is calculated by linear interpolation and of a healing bone obtained from a rat osteotomy model. It is shown that the ratio between attenuation and refractive index in bone for different levels of mineralization follows a power law. Reconstruction was performed using the mixed approach but is compatible with other, more advanced models. We achieve more precise reconstructions than previously reported in literature. We believe that the proposed method will find application in biomedical imaging problems where the object is strongly heterogeneous, such as bone healing and biomaterials engineering.