X-ray phase imaging: Demonstration of extended conditions for homogeneous objects

Fast algorithms for nonlinear and constrained phase retrieval in near-field X-ray holography based on Tikhonov regularization

Based on phase retrieval, lensless coherent imaging and in particular holography offers quantitative phase and amplitude images. This is of particular importance for spectral ranges where suitable lenses are challenging, such as for hard x-rays. Here, we propose a phase retrieval approach for inline x-ray holography based on Tikhonov regularization applied to the full nonlinear forward model of image formation. The approach can be seen as a nonlinear generalization of the well-established contrast transfer function (CTF) reconstruction method. While similar methods have been proposed before, the current work achieves nonlinear, constrained phase retrieval at competitive computation times. We thus enable high-throughput imaging of optically strong objects beyond the scope of CTF. Using different examples of inline holograms obtained from illumination by a x-ray waveguide-source, we demonstrate superior image quality even for samples which do not obey the assumption of a weakly varying phase. Since the presented approach does not rely on linearization, we expect it to be well suited also for other probes such as visible light or electrons, which often exhibit strong phase interaction.

Mixed scale dense convolutional networks for x-ray phase contrast imaging

X-ray in-line phase contrast imaging relies on the measurement of Fresnel diffraction intensity patterns due to the phase shift and the attenuation induced by the object. The recovery of phase and attenuation from one or several diffraction patterns is a nonlinear ill-posed inverse problem. In this work, we propose supervised learning approaches using mixed scale dense (MS-D) convolutional neural networks to simultaneously retrieve the phase and the attenuation from x-ray phase contrast images. This network architecture uses dilated convolutions to capture features at different image scales and densely connects all feature maps. The long range information in images becomes quickly available, and greater receptive field size can be obtained without losing resolution. This network architecture seems to account for the effect of the Fresnel operator very efficiently. We train the networks using simulated data of objects consisting of either homogeneous components, characterized by a fixed ratio of the induced refractive phase shifts and attenuation, or heterogeneous components, consisting of various materials. We also train the networks in the image domain by applying a simple initial reconstruction using the adjoint of the Fréchet derivative. We compare the results obtained with the MS-D network to reconstructions using U-Net, another popular network architecture, as well as to reconstructions using the contrast transfer function method, a direct phase and attenuation retrieval method based on linearization of the direct problem. The networks are evaluated using simulated noisy data as well as images acquired at NanoMAX (MAX IV, Lund, Sweden). In all cases, large improvements of the reconstruction errors are obtained on simulated data compared to the linearized method. Moreover, on experimental data, the networks improve the reconstruction quantitatively, improving the low-frequency behavior and the resolution.

3D virtual histopathology of cardiac tissue from Covid-19 patients based on phase-contrast X-ray tomography

For the first time, we have used phase-contrast X-ray tomography to characterize the three-dimensional (3d) structure of cardiac tissue from patients who succumbed to Covid-19. By extending conventional histopathological examination by a third dimension, the delicate pathological changes of the vascular system of severe Covid-19 progressions can be analyzed, fully quantified and compared to other types of viral myocarditis and controls. To this end, cardiac samples with a cross-section of 3.5mm were scanned at a laboratory setup as well as at a parallel beam setup at a synchrotron radiation facility the synchrotron in a parallel beam configuration. The vascular network was segmented by a deep learning architecture suitable for 3d datasets (V-net), trained by sparse manual annotations. Pathological alterations of vessels, concerning the variation of diameters and the amount of small holes, were observed, indicative of elevated occurrence of intussusceptive angiogenesis, also confirmed by high-resolution cone beam X-ray tomography and scanning electron microscopy. Furthermore, we implemented a fully automated analysis of the tissue structure in the form of shape measures based on the structure tensor. The corresponding distributions show that the histopathology of Covid-19 differs from both influenza and typical coxsackie virus myocarditis.

Adorym: a multi-platform generic X-ray image reconstruction framework based on automatic differentiation

We describe and demonstrate an optimization-based X-ray image reconstruction framework called Adorym. Our framework provides a generic forward model, allowing one code framework to be used for a wide range of imaging methods ranging from near-field holography to fly-scan ptychographic tomography. By using automatic differentiation for optimization, Adorym has the flexibility to refine experimental parameters including probe positions, multiple hologram alignment, and object tilts. It is written with strong support for parallel processing, allowing large datasets to be processed on high-performance computing systems. We demonstrate its use on several experimental datasets to show improved image quality through parameter refinement.

Three-dimensional imaging of xylem at cell wall level through near field nano holotomography

Detailed imaging of the three-dimensionally complex architecture of xylary plants is important for studying biological and mechanical functions of woody plants. Apart from common two-dimensional microscopy, X-ray micro-computed tomography has been established as a three-dimensional (3D) imaging method for studying the hydraulic function of wooden plants. However, this X-ray imaging method can barely reach the resolution needed to see the minute structures (e.g. pit membrane). To complement the xylem structure with 3D views at the nanoscale level, X-ray near-field nano-holotomography (NFH) was applied to analyze the wood species Pinus sylvestris and Fagus sylvatica. The demanded small specimens required focused ion beam (FIB) application. The FIB milling, however, influenced the image quality through gallium implantation on the cell-wall surfaces. The measurements indicated that NFH is appropriate for imaging wood at nanometric resolution. With a 26 nm voxel pitch, the structure of the cell-wall surface in Pinus sylvestris could be visualized in genuine detail. In wood of Fagus sylvatica, the structure of a pit pair, including the pit membrane, between two neighboring fibrous cells could be traced tomographically.

3D virtual histology of human pancreatic tissue by multiscale phase-contrast X-ray tomography

A multiscale three-dimensional (3D) virtual histology approach is presented, based on two configurations of propagation phase-contrast X-ray tomography, which have been implemented in close proximity at the GINIX endstation at the beamline P10/PETRA III (DESY, Hamburg, Germany). This enables the 3D reconstruction of characteristic morphological features of human pancreatic normal and tumor tissue, as obtained from cancer surgery, first in the form of a large-scale overview by parallel-beam illumination, followed by a zoom into a region-of-interest based on zoom tomography using a Kirkpatrick-Baez mirror with additional waveguide optics. To this end 1 mm punch biopsies of the tissue were taken. In the parallel tomography, a volumetric throughput on the order of 0.01 mm3 s-1 was achieved, while maintaining the ability to segment isolated cells. With a continuous rotation during the scan, a total acquisition time of less than 2 min was required for a full tomographic scan. Using the combination of both setups, islets of Langerhans, a three-dimensional cluster of cells in the endocrine part of the pancreas, could be located. Cells in such an islet were segmented and visualized in 3D. Further, morphological alterations of tumorous tissue of the pancreas were characterized. To this end, the anisotropy parameter Ω, based on intensity gradients, was used in order to quantify the presence of collagen fibers within the entire biopsy specimen. This proof-of-concept experiment of the multiscale approach on human pancreatic tissue paves the way for future 3D virtual pathology.

3D virtual pathohistology of lung tissue from Covid-19 patients based on phase contrast X-ray tomography

We present a three-dimensional (3D) approach for virtual histology and histopathology based on multi-scale phase contrast x-ray tomography, and use this to investigate the parenchymal architecture of unstained lung tissue from patients who succumbed to Covid-19. Based on this first proof-of-concept study, we propose multi-scale phase contrast x-ray tomography as a tool to unravel the pathophysiology of Covid-19, extending conventional histology by a third dimension and allowing for full quantification of tissue remodeling. By combining parallel and cone beam geometry, autopsy samples with a maximum cross section of 8 mm are scanned and reconstructed at a resolution and image quality, which allows for the segmentation of individual cells. Using the zoom capability of the cone beam geometry, regions-of-interest are reconstructed with a minimum voxel size of 167 nm. We exemplify the capability of this approach by 3D visualization of diffuse alveolar damage (DAD) with its prominent hyaline membrane formation, by mapping the 3D distribution and density of lymphocytes infiltrating the tissue, and by providing histograms of characteristic distances from tissue interior to the closest air compartment.

A phase-retrieval toolbox for X-ray holography and tomography

Propagation-based phase-contrast X-ray imaging is by now a well established imaging technique, which - as a full-field technique - is particularly useful for tomography applications. Since it can be implemented with synchrotron radiation and at laboratory micro-focus sources, it covers a wide range of applications. A limiting factor in its development has been the phase-retrieval step, which was often performed using methods with a limited regime of applicability, typically based on linearization. In this work, a much larger set of algorithms, which covers a wide range of cases (experimental parameters, objects and constraints), is compiled into a single toolbox - the HoloTomoToolbox - which is made publicly available. Importantly, the unified structure of the implemented phase-retrieval functions facilitates their use and performance test on different experimental data.

Nanoscale x-ray holotomography of human brain tissue with phase retrieval based on multienergy recordings

X-ray cone-beam holotomography of unstained tissue from the human central nervous system reveals details down to subcellular length scales. This visualization of variations in the electron density of the sample is based on phase-contrast techniques using intensities formed by self-interference of the beam between object and detector. Phase retrieval inverts diffraction and overcomes the phase problem by constraints such as several measurements at different Fresnel numbers for a single projection. Therefore, the object-to-detector distance (defocus) can be varied. However, for cone-beam geometry, changing defocus changes magnification, which can be problematic in view of image processing and resolution. Alternatively, the photon energy can be altered (multi-E). Far from absorption edges, multi-E data yield the wavelength-independent electron density. We present the multi-E holotomography at the Göttingen Instrument for Nano-Imaging with X-Rays (GINIX) setup of the P10 beamline at Deutsches Elektronen-Synchrotron. The instrument is based on a combined optics of elliptical mirrors and an x-ray waveguide positioned in the focal plane for further coherence, spatial filtering, and high numerical aperture. Previous results showed the suitability of this instrument for nanoscale tomography of unstained brain tissue. We demonstrate that upon energy variation, the focal spot is stable enough for imaging. To this end, a double-crystal monochromator and automated alignment routines are required. Three tomograms of human brain tissue were recorded and jointly analyzed using phase retrieval based on the contrast transfer function formalism generalized to multiple photon energies. Variations of the electron density of the sample are successfully reconstructed.

Determination of size distributions of non-spherical pores or particles from single x-ray phase contrast images

Although x-ray tomography is commonly used to characterize the three-dimensional structure of materials, sometimes this is impractical due either to limited time for data collection (such as in rapidly-evolving systems) or the need to limit the radiation exposure of the sample. In such situations, it is desirable to extract as much information as possible from a more limited data set. In this paper, we describe how to extract the size distribution of non-spherical pores (or, equivalently, particles) from single x-ray phase contrast imaging (XPCI). Because the pores overlap in projection, interpreting the images and extracting quantitative information about the size distribution is non-trivial. In this paper we extend a previously-developed Fourier-based framework for interpreting the speckle pattern of XPCI images from materials with spherical pores to the more challenging case of non-spherical pores. We develop an analytical expression for the XPCI image from a distribution of randomly-oriented ellipsoidal pores, and show that we can use this expression to extract quantitative information about the size distribution from single images. We discuss three approaches to evaluating this expression, corresponding to different assumptions about the nature of the size distribution, and validate our results with simulated XPCI images and experimental data from Berea sandstone.

Contrast enhancement for visualizing neuronal cytoarchitecture by propagation-based x-ray phase-contrast tomography

Knowledge of the three-dimensional (3d) neuronal cytoarchitecture is an important factor in order to understand the connection between tissue structure and function or to visualize pathological changes in neurodegenerative diseases or tumor development. The gold standard in neuropathology is histology, a technique which provides insights into the cellular organization based on sectioning of the sample. Conventional histology, however, misses the complete 3d information as only individual two-dimensional slices through the object are available. In this work, we use propagation-based phase-contrast x-ray tomography to perform 3d virtual histology on cerebellar tissue from mice. This technique enables us to non-invasively visualize the entire 3d density distribution of the examined samples at isotropic (sub-)cellular resolution. One central challenge, however, of the technique is the fact that contrast for important structural features can be easily lost due to small electron density differences, notably between the cells and surrounding tissue. Here, we evaluate the influence of different embedding media, which are intermediate steps in sample preparation for classical histology, on contrast formation and examine the applicability of the different sample preparations both at a synchrotron-based holotomography setup as well as a laboratory source.

Fast three-dimensional phase retrieval in propagation-based X-ray tomography

The following article describes a method for 3D reconstruction of multi-material objects based on propagation-based X-ray phase-contrast tomography (PB-CT) with phase retrieval using the homogeneous form of the transport of intensity equation (TIE-Hom). Unlike conventional PB-CT algorithms that perform phase retrieval of individual projections, the described post-reconstruction phase-retrieval method is applied in 3D to a localized region of the CT-reconstructed volume. This work demonstrates, via numerical simulations, the accuracy and noise characteristics of the method under a variety of experimental conditions, comparing it with both conventional absorption tomography and 2D TIE-Hom phase retrieval applied to projection images. The results indicate that the 3D post-reconstruction method generally achieves a modest improvement in noise suppression over existing PB-CT methods. It is also shown that potentially large computational gains over projection-based phase retrieval for multi-material samples are possible. In particular, constraining phase retrieval to a localized 3D region of interest reduces the overall computational cost and eliminates the need for multiple CT reconstructions and global 2D phase retrieval operations for each material within the sample.

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.

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.

Label-free quantitative 3D tomographic imaging for partially coherent light microscopy

Crucial benefits provided by partially coherent light microscopy such as improved spatial resolution, optical sectioning and speckle-noise suppression are exploited here to achieve 3D quantitative imaging: reconstruction of the object refractive index (RI). We present a partially coherent optical diffraction tomography technique (PC-ODT) that can be easily implemented in commercially available bright-field microscopes. We show that the high numerical apertures of the objective and condenser lenses, together with optical refocusing, are main issues for achieving fast and successful 3D RI reconstruction of weak objects. In particular, the optical refocusing is performed by a high-speed focus tunable lens mounted in front of the digital camera enabling compatibility with commercial microscopes. The technique is experimentally demonstrated on different examples: diatom cells (biosilica shells), polystyrene micro-spheres and blood cells. The results confirm the straightforward 3D-RI reconstruction of the samples providing valuable quantitative information for their analysis. Thus, the PC-ODT can be considered as an efficient and affordable alternative to coherent ODT which requires specially designed holographic microscopes.

Three-dimensional single-cell imaging with X-ray waveguides in the holographic regime

X-ray tomography at the level of single biological cells is possible in a low-dose regime, based on full-field holographic recordings, with phase contrast originating from free-space wave propagation. Building upon recent progress in cellular imaging based on the illumination by quasi-point sources provided by X-ray waveguides, here this approach is extended in several ways. First, the phase-retrieval algorithms are extended by an optimized deterministic inversion, based on a multi-distance recording. Second, different advanced forms of iterative phase retrieval are used, operational for single-distance and multi-distance recordings. Results are compared for several different preparations of macrophage cells, for different staining and labelling. As a result, it is shown that phase retrieval is no longer a bottleneck for holographic imaging of cells, and how advanced schemes can be implemented to cope also with high noise and inconsistencies in the data.

Partially coherent contrast-transfer-function approximation

The contrast-transfer-function (CTF) approximation, widely used in various phase-contrast imaging techniques, is revisited. CTF validity conditions are extended to a wide class of strongly absorbing and refracting objects, as well as to nonuniform partially coherent incident illumination. Partially coherent free-space propagators, describing amplitude and phase in-line contrast, are introduced and their properties are investigated. The present results are relevant to the design of imaging experiments with partially coherent sources, as well as to the analysis and interpretation of the corresponding images.

Regularized Newton methods for x-ray phase contrast and general imaging problems

Like many other advanced imaging methods, x-ray phase contrast imaging and tomography require mathematical inversion of the observed data to obtain real-space information. While an accurate forward model describing the generally nonlinear image formation from a given object to the observations is often available, explicit inversion formulas are typically not known. Moreover, the measured data might be insufficient for stable image reconstruction, in which case it has to be complemented by suitable a priori information. In this work, regularized Newton methods are presented as a general framework for the solution of such ill-posed nonlinear imaging problems. For a proof of principle, the approach is applied to x-ray phase contrast imaging in the near-field propagation regime. Simultaneous recovery of the phase- and amplitude from a single near-field diffraction pattern without homogeneity constraints is demonstrated for the first time. The presented methods further permit all-at-once phase contrast tomography, i.e. simultaneous phase retrieval and tomographic inversion. We demonstrate the potential of this approach by three-dimensional imaging of a colloidal crystal at 95nm isotropic resolution.

Three-dimensional propagation in near-field tomographic X-ray phase retrieval

An extension of phase retrieval algorithms for near-field X-ray (propagation) imaging to three dimensions is presented, enhancing the quality of the reconstruction by exploiting previously unused three-dimensional consistency constraints.

This paper presents an extension of phase retrieval algorithms for near-field X-ray (propagation) imaging to three dimensions, enhancing the quality of the reconstruction by exploiting previously unused three-dimensional consistency constraints. The approach is based on a novel three-dimensional propagator and is derived for the case of optically weak objects. It can be easily implemented in current phase retrieval architectures, is computationally efficient and reduces the need for restrictive prior assumptions, resulting in superior reconstruction quality.

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.

Compound focusing mirror and X-ray waveguide optics for coherent imaging and nano-diffraction

A compound optical system for coherent focusing and imaging at the nanoscale is reported, realised by high-gain fixed-curvature elliptical mirrors in combination with X-ray waveguide optics or different cleaning apertures. The key optical concepts are illustrated, as implemented at the Göttingen Instrument for Nano-Imaging with X-rays (GINIX), installed at the P10 coherence beamline of the PETRA III storage ring at DESY, Hamburg, and examples for typical applications in biological imaging are given. Characteristic beam configurations with the recently achieved values are also described, meeting the different requirements of the applications, such as spot size, coherence or bandwidth. The emphasis of this work is on the different beam shaping, filtering and characterization methods.

Phase-contrast zoom tomography reveals precise locations of macrophages in mouse lungs

We have performed x-ray phase-contrast tomography on mouse lung tissue. Using a divergent x-ray beam generated by nanoscale focusing, we used zoom tomography to produce three-dimensional reconstructions with selectable magnification, resolution, and field of view. Thus, macroscopic tissue samples extending over several mm can be studied in sub-cellular-level structural detail. The zoom capability and, in particular, the high dose efficiency are enabled by the near-perfect exit wavefront of an optimized x-ray waveguide channel. In combination with suitable phase-retrieval algorithms, challenging radiation-sensitive and low-contrast samples can be reconstructed with minimal artefacts. The dose efficiency of the method is demonstrated by the reconstruction of living macrophages both with and without phagocytized contrast agents. We also used zoom tomography to visualize barium-labelled macrophages in the context of morphological structures in asthmatic and healthy mouse lung tissue one day after intratracheal application. The three-dimensional reconstructions showed that the macrophages predominantly localized to the alveoli, but they were also found in bronchial walls, indicating that these cells might be able to migrate from the lumen of the bronchi through the epithelium.

Quantitative X-ray phase contrast waveguide imaging of bacterial endospores

Quantitative X-ray phase contrast imaging uniquely offers quantitative imaging information in terms of electron density maps allowing for mass and mass density determinations of soft biological samples (‘weighing with light’). Here, it was carried out using coherent X-ray waveguide illumination, yielding values of the mass and mass density of freeze-dried bacterial endospores (Bacillus spp.).

Quantitative waveguide-based X-ray phase contrast imaging has been carried out on the level of single, unstained, unsliced and freeze-dried bacterial cells of Bacillus thuringiensis and Bacillus subtilis using hard X-rays of 7.9 keV photon energy. The cells have been prepared in the metabolically dormant state of an endospore. The quantitative phase maps obtained by iterative phase retrieval using a modified hybrid input–output algorithm allow for mass and mass density determinations on the level of single individual endospores but include also large field of view investigations. Additionally, a direct reconstruction based on the contrast transfer function is investigated, and the two approaches are compared. Depending on the field of view and method, a resolution down to 65 nm was achieved at a maximum applied dose of below 5 × 10⁵ Gy. Masses in the range of about ∼110–190 (20) fg for isolated endospores have been obtained.

Phase imaging for absorptive phase objects using hybrid uniform and structured illumination transport of intensity equation

Transport of intensity equation (TIE) has been a popular and convenient phase imaging method that retrieves phase profile from the measurement of intensity differentials. Conventional 2-shot uniform illumination TIE can give reliable inversion of the phase from intensity in many situations of practical interest; however, it has a null space consisting of fields with non-zero circulation of the Poynting vector. Here, we propose the hybrid illumination TIE method to disambiguate such objects. By comparing the diffraction signals using uniform and structured (sinusoidal) illumination patterns, we obtain a modulation-induced signal that depends solely on the phase gradient. In this way, we also increase signal sensitivity in the low spatial frequency region.

Low-noise phase imaging by hybrid uniform and structured illumination transport of intensity equation

We demonstrate a new approach to the transport of intensity equation (TIE) phase retrieval method which uses structured illumination to improve low-frequency noise performance. In the hybrid scheme, two phase images are acquired: one with uniform and one with sinusoidal grating illumination intensity. The former preserves the high spatial frequency features of the phase best, whereas the latter dramatically increase the response at low spatial frequencies (where traditional TIE notoriously suffers). We then theoretically prove the design of a spectral filter that optimally combines the two phase results while suppressing noise. The combination of uniformly and structured illuminated TIE (hybrid TIE) phase imaging is experimentally demonstrated optically with a calibrated pure phase object.

Simple merging technique for improving resolution in qualitative single image phase contrast tomography

For dynamic samples and/or for simple ease-of-use experiments, single-image phase contrast tomography is a very effective method for the 3D visualization of materials which would otherwise be indiscernible in attenuation based x-ray imaging. With binary samples (e.g. air-material) and monochromatic wavefields a transport-of-intensity (TIE)-based phase retrieval algorithm is known to retrieve accurate quantitative maps of the phase distribution. For mixed material samples and/or white beam radiation the algorithm can still produce useful qualitative tomographic reconstructions with significantly improved area contrast. The stability of the algorithm comes with a recognized associated loss of spatial resolution due to its essential behaviour as a low-pass filter. One possible answer to this is an image fusion technique that merges the slices reconstructed from raw phase contrast images and those after phase retrieval, where the improved contrast may be acquired without the associated loss of high-frequency information. We present this technique as a simple few-parameter Fourier method, which is easily tunable and highly compatible with current reconstruction steps.

Measurement of absolute regional lung air volumes from near-field x-ray speckles

Propagation-based phase contrast x-ray (PBX) imaging yields high contrast images of the lung where airways that overlap in projection coherently scatter the x-rays, giving rise to a speckled intensity due to interference effects. Our previous works have shown that total and regional changes in lung air volumes can be accurately measured from two-dimensional (2D) absorption or phase contrast images when the subject is immersed in a water-filled container. In this paper we demonstrate how the phase contrast speckle patterns can be used to directly measure absolute regional lung air volumes from 2D PBX images without the need for a water-filled container. We justify this technique analytically and via simulation using the transport-of-intensity equation and calibrate the technique using our existing methods for measuring lung air volume. Finally, we show the full capabilities of this technique for measuring regional differences in lung aeration.

Total variation minimization approach in in-line x-ray phase-contrast tomography

The reconstruction problem in in-line X-ray Phase-Contrast Tomography is usually approached by solving two independent linearized sub-problems: phase retrieval and tomographic reconstruction. Both problems are often ill-posed and require the use of regularization techniques that lead to artifacts in the reconstructed image. We present a novel reconstruction approach that solves two coupled linear problems algebraically. Our approach is based on the assumption that the frequency space of the tomogram can be divided into bands that are accurately recovered and bands that are undefined by the observations. This results in an underdetermined linear system of equations. We investigate how this system can be solved using three different algebraic reconstruction algorithms based on Total Variation minimization. These algorithms are compared using both simulated and experimental data. Our results demonstrate that in many cases the proposed algebraic algorithms yield a significantly improved accuracy over the conventional L2-regularized closed-form solution. This work demonstrates that algebraic algorithms may become an important tool in applications where the acquisition time and the delivered radiation dose must be minimized.

Comparison of single distance phase retrieval algorithms by considering different object composition and the effect of statistical and structural noise

Phase retrieval is a technique for extracting quantitative phase information from X-ray propagation-based phase-contrast tomography (PPCT). In this paper, the performance of different single distance phase retrieval algorithms will be investigated. The algorithms are herein called phase-attenuation duality Born Algorithm (PAD-BA), phase-attenuation duality Rytov Algorithm (PAD-RA), phase-attenuation duality Modified Bronnikov Algorithm (PAD-MBA), phase-attenuation duality Paganin algorithm (PAD-PA) and phase-attenuation duality Wu Algorithm (PAD-WA), respectively. They are all based on phase-attenuation duality property and on weak absorption of the sample and they employ only a single distance PPCT data. In this paper, they are investigated via simulated noise-free PPCT data considering the fulfillment of PAD property and weakly absorbing conditions, and with experimental PPCT data of a mixture sample containing absorbing and weakly absorbing materials, and of a polymer sample considering different degrees of statistical and structural noise. The simulation shows all algorithms can quantitatively reconstruct the 3D refractive index of a quasi-homogeneous weakly absorbing object from noise-free PPCT data. When the weakly absorbing condition is violated, the PAD-RA and PAD-PA/WA obtain better result than PAD-BA and PAD-MBA that are shown in both simulation and mixture sample results. When considering the statistical noise, the contrast-to-noise ratio values decreases as the photon number is reduced. The structural noise study shows that the result is progressively corrupted by ring-like artifacts with the increase of structural noise (i.e. phantom thickness). The PAD-RA and PAD-PA/WA gain better density resolution than the PAD-BA and PAD-MBA in both statistical and structural noise study.

Transport of intensity phase reconstruction to solve the twin image problem in holographic x-ray imaging

We have implemented a deterministic method for solving the phase problem in hard x-ray in-line holography which overcomes the twin image problem. The phase distribution in the detector plane is retrieved by using two images with slightly different Fresnel numbers. We then use measured intensities and reconstructed phases in the detection plane to compute the exit wave in the sample plane. No further a priori information like a limited support or the assumption of pure phase objects is necessary so that it can be used for a wide range of complex samples. Using a nano-focused hard x-ray beam half period resolutions better than 30 nm are achieved.

Phase retrieval in in-line x-ray phase contrast imaging based on total variation minimization

State-of-the-art techniques for phase retrieval in propagation based X-ray phase-contrast imaging are aiming to solve an underdetermined linear system of equations. They commonly employ Tikhonov regularization - an L2-norm regularized deconvolution scheme - despite some of its limitations. We present a novel approach to phase retrieval based on Total Variation (TV) minimization. We incorporated TV minimization for deconvolution in phase retrieval using a variety of the most common linear phase-contrast models. The results of our TV minimization was compared with Tikhonov regularized deconvolution on simulated as well as experimental data. The presented method was shown to deliver improved accuracy in reconstructions based on a single distance as well as multiple distance phase-contrast images corrupted by noise and hampered by errors due to nonlinear imaging effects.

In-Line Phase-Contrast X-ray Imaging and Tomography for Materials Science

X-ray phase-contrast imaging and tomography make use of the refraction of X-rays by the sample in image formation. This provides considerable additional information in the image compared to conventional X-ray imaging methods, which rely solely on X-ray absorption by the sample. Phase-contrast imaging highlights edges and internal boundaries of a sample and is thus complementary to absorption contrast, which is more sensitive to the bulk of the sample. Phase-contrast can also be used to image low-density materials, which do not absorb X-rays sufficiently to form a conventional X-ray image. In the context of materials science, X-ray phase-contrast imaging and tomography have particular value in the 2D and 3D characterization of low-density materials, the detection of cracks and voids and the analysis of composites and multiphase materials where the different components have similar X-ray attenuation coefficients. Here we review the use of phase-contrast imaging and tomography for a wide variety of materials science characterization problems using both synchrotron and laboratory sources and further demonstrate the particular benefits of phase contrast in the laboratory setting with a series of case studies.

Multi-wavelength elemental contrast absorption imaging

We report experimental demonstrations of a quantitative technique for elemental mapping. The technique operates in full-field imaging mode and uses three intensity measurements at energies across an absorption edge of an element of interest to obtain its elemental distribution. The experimental results show that the technique can overcome some limitations in the conventional Absorption Edge Contrast Imaging. The technique allows for an accurate determination of the elemental distribution in a compound sample even at a low level of percentage composition. It is also robust to the choice of energy intervals.

Imaging the 3D structure of secondary osteons in human cortical bone using phase-retrieval tomography

By applying a phase-retrieval step before carrying out standard filtered back-projection reconstructions in tomographic imaging, we were able to resolve structures with small differences in density within a densely absorbing sample. This phase-retrieval tomography is particularly suited for the three-dimensional segmentation of secondary osteons (roughly cylindrical structures) which are superimposed upon an existing cortical bone structure through the process of turnover known as remodelling. The resulting images make possible the analysis of the secondary osteon structure and the relationship between an osteon and the surrounding tissue. Our observations have revealed many different and complex 3D structures of osteons that could not be studied using previous methods. This work was carried out using a laboratory-based x-ray source, which makes obtaining these sorts of images readily accessible.

Single-distance phase retrieval at large phase shifts

For coherent X-ray imaging of pure phase objects we study the reliability of linear relations in phase-retrieval algorithms based on a single intensity map after free-space propagation. For large phase changes and/or large propagation distances we propose two venues of working beyond linearity: Projection onto an effective, linear and local model in Fourier space and expansion of intensity contrast in powers of object-detector distance. We apply both algorithms successfully to simulated data.

Phase retrieval in X-ray phase-contrast imaging suitable for tomography

In-line phase-contrast X-ray imaging provides images where both absorption and refraction contribute. For quantitative analysis of these images, the phase needs to be retrieved numerically. There are many phase-retrieval methods available. Those suitable for phase-contrast tomography, i.e., non-iterative phase-retrieval methods that use only one image at each projection angle, all follow the same pattern though derived in different ways. We outline this pattern and use it to compare the methods to each other, considering only phase-retrieval performance and not the additional effects of tomographic reconstruction. We also outline derivations, approximations and assumptions, and show which methods are similar or identical and how they relate to each other. A simple scheme for choosing reconstruction method is presented, and numerical phase-retrieval performed for all methods.

Quantitative phase imaging with polychromatic x-ray sources

We introduce theoretically and demonstrate experimentally a contrast transfer function based phase retrieval algorithm that reconstructs the projected thickness of an homogeneous sample using a polychromatic x-ray source. We show excellent quantitative recovery of test samples in 2D using a synchrotron source with significant harmonic contamination, and in 3D using a laboratory source.

Optimization of image recording distances for quantitative X-ray in-line phase contrast imaging

Compared to phase retrieval from single sample-to-detector distance (SDD) image, phase retrieval with multiple SDD images could improve the precision in quantitative X-ray in-line phase contrast imaging (QXIPCI). Among all the related phase retrieval approaches, the two-SDD-image-based one is the simplest and well compromises between precision and dose. However, how to optimize the recording distances for the two images to achieve highest precision, remains unsolved. In this paper, the problem was investigated systematically based on digital simulation and related experiments. Spectral correlation degree (SCD) is introduced to evaluate the pertinence between the two SDD images. The simulation results show that the highest retrieving precision could be obtained while the SDD of the second image is three times that of the first image. The best retrieval could be achieved when SDD of the first image is selected properly, meanwhile the SCD occurs with a typical damping oscillation. Experiments, carried out at the X-ray imaging beamline of SSRF, demonstrated the simulation results.

Nonlinear phase retrieval from single-distance radiograph

Phase contrast in the object plane of a phase object is retrieved from intensity contrast at a {\sl single} object-detector distance. Expanding intensity contrast and phase shift in the detector plane in powers of object-detector distance, phase retrieval is performed beyond the solution to the linearized transport-of-intensity equation. The expansion coefficients are determined by the entire paraxial wave equation. The Laplacian of the phase shift in the object plane thus is written as a local expression linear in the intensity contrast and nonlinear in the phase shift in the object plane. A perturbative approach to this expression is proposed and tested with simulated phantom data.

Optimisation of phase imaging geometry

Certain phase retrieval methods use knowledge about the free space propagation of a wave to phase a paraxial beam passing through one or more measurement planes. This approach has been widely applied and has been shown to quantitatively retrieve the refractive index profile of a sample. The quality of the phase retrieval will depend on a range of factors including sample feature size, propagation distance, measurement plane separation, wavelength and noise. Here we describe an optimisation study for two-plane phase retrieval using a laboratory-based X-ray source that considers all of these factors. We discuss our results in the context of a three-dimensional reconstruction of a sample refractive index profile.

Regularization of phase retrieval with phase-attenuation duality prior for 3-D holotomography

We consider the phase retrieval problem in 3-D holotomography for strongly absorbing objects. Holotomography combines phase retrieval from Fresnel diffraction patterns with tomographic reconstruction to reconstruct the 3-D refractive index distribution. The main interest is the increase in sensitivity of up to three orders of magnitude compared to standard, absorption based tomography. Most existing algorithms are based upon linearization of the forward problem. This is motivated by the large problem size, since it yields computationally efficient solutions. Here, the mixed approach is used, which allows for both strong absorption and long propagation distances. Previous implementations have shown promising results, but in practice often suffer from strong low frequency artifacts. To address this problem, we introduce a homogeneous object assumption through a regularizing term based upon the absorption image. This allows the homogeneous object assumption to be introduced only in the low frequency range. The proportionality constant between absorption and refractive index is assumed to be known. The regularizing parameter is found using the standard L-curve technique. The benefits of our approach are illustrated using data measured at the European Synchrotron Radiation Facility. Low frequency noise in the reconstruction is alleviated, but the result is only quantitative in the areas of the sample where the homogeneous object assumption is fulfilled.

Use of a complex constraint in coherent diffractive imaging

We demonstrate use of a complex constraint based on the interaction of x-rays with matter for reconstructing images from coherent X-ray diffraction. We show the complementary information provided by the phase and magnitude of the reconstructed wavefield greatly improves the quality of the resulting estimate of the transmission function of an object without the need for a priori information about the object composition.

Relationships between smooth- and small-phase conditions in X-ray phase-contrast imaging

We analyze some relationships between the small- and smooth-phase conditions in propagation-based X-ray phase-contrast imaging. Although these conditions are generally well-understood, our analysis yields the identification of physical conditions under which they are mathematically equivalent. We also demonstrate that the smooth-phase condition depends not only on the imaging system resolution and object-to-detector distance, but also on the topography of the wavefield phase function.

Multispectral intensity diffraction tomography: single material objects with variable densities

A reconstruction theory for multispectral intensity diffraction tomography (I-DT) is established and investigated for use with single material objects whose dispersion characteristics are known a priori. Instead of varying the object-to-detector distance, as prescribed by the original I-DT method and other classic in-line holographic reconstruction methods, the temporal frequency of the illuminating plane wave represents the degree of freedom of the imaging system that is varied to acquire two independent intensity measurements at each tomographic view angle. Unlike previous multispectral I-DT methods, the proposed method does not require a nondispersive assumption. A computer-simulation study is presented to demonstrate and corroborate the method.

Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography

A well-known problem in x-ray microcomputed tomography is low sensitivity. Phase contrast imaging offers an increase of sensitivity of up to a factor of 10(3) in the hard x-ray region, which makes it possible to image soft tissue and small density variations. If a sufficiently coherent x-ray beam, such as that obtained from a third generation synchrotron, is used, phase contrast can be obtained by simply moving the detector downstream of the imaged object. This setup is known as in-line or propagation based phase contrast imaging. A quantitative relationship exists between the phase shift induced by the object and the recorded intensity and inversion of this relationship is called phase retrieval. Since the phase shift is proportional to projections through the three-dimensional refractive index distribution in the object, once the phase is retrieved, the refractive index can be reconstructed by using the phase as input to a tomographic reconstruction algorithm. A comparison between four phase retrieval algorithms is presented. The algorithms are based on the transport of intensity equation (TIE), transport of intensity equation for weak absorption, the contrast transfer function (CTF), and a mixed approach between the CTF and TIE, respectively. The compared methods all rely on linearization of the relationship between phase shift and recorded intensity to yield fast phase retrieval algorithms. The phase retrieval algorithms are compared using both simulated and experimental data, acquired at the European Synchrotron Radiation Facility third generation synchrotron light source. The algorithms are evaluated in terms of two different reconstruction error metrics. While being slightly less computationally effective, the mixed approach shows the best performance in terms of the chosen criteria.

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

We present a method for phase retrieval in propagation-based x-ray imaging, based on the contrast transfer and transport of intensity equation approaches. We show that the contrast transfer model does not coincide with the transport of intensity in the limit of small propagation distances, and we derive a new model that alleviates this problem. Using this model, we devise an algorithm to retrieve the phase from slowly varying samples that is valid beyond the limit of small distances. We show its utility by imaging in three dimensions a biological sample that causes both strong absorption and phase shift.