Single-distance phase retrieval at large phase shifts

X-ray Tomography Applied to Electrochemical Devices and Electrocatalysis

X-ray computed tomography (CT) is a nondestructive three-dimensional (3D) imaging technique used for studying morphological properties of porous and nonporous materials. In the field of electrocatalysis, X-ray CT is mainly used to quantify the morphology of electrodes and extract information such as porosity, tortuosity, pore-size distribution, and other relevant properties. For electrochemical systems such as fuel cells, electrolyzers, and redox flow batteries, X-ray CT gives the ability to study evolution of critical features of interest in ex situ, in situ, and operando environments. These include catalyst degradation, interface evolution under real conditions, formation of new phases (water and oxygen), and dynamics of transport processes. These studies enable more efficient device and electrode designs that will ultimately contribute to widespread decarbonization efforts.

Hierarchically guided in situ nanolaminography for the visualisation of damage nucleation in alloy sheets

Hierarchical guidance is developed for three-dimensional (3D) nanoscale X-ray imaging, enabling identification, refinement, and tracking of regions of interest (ROIs) within specimens considerably exceeding the field of view. This opens up new possibilities for in situ investigations. Experimentally, the approach takes advantage of rapid multiscale measurements based on magnified projection microscopy featuring continuous zoom capabilities. Immediate and continuous feedback on the subsequent experimental progress is enabled by suitable on-the-fly data processing. For this, by theoretical justification and experimental validation, so-called quasi-particle phase-retrieval is generalised to conical-beam conditions, being key for sufficiently fast computation without significant loss of imaging quality and resolution compared to common approaches for holographic microscopy. Exploiting 3D laminography, particularly suited for imaging of ROIs in laterally extended plate-like samples, the potential of hierarchical guidance is demonstrated by the in situ investigation of damage nucleation inside alloy sheets under engineering-relevant boundary conditions, providing novel insight into the nanoscale morphological development of void and particle clusters under mechanical load. Combined with digital volume correlation, we study deformation kinematics with unprecedented spatial resolution. Correlation of mesoscale (i.e. strain fields) and nanoscale (i.e. particle cracking) evolution opens new routes for the understanding of damage nucleation within sheet materials with application-relevant dimensions.

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.

Tofu: a fast, versatile and user-friendly image processing toolkit for computed tomography

Tofu is a toolkit for processing large amounts of images and for tomographic reconstruction. Complex image processing tasks are organized as workflows of individual processing steps. The toolkit is able to reconstruct parallel and cone beam as well as tomographic and laminographic geometries. Many pre- and post-processing algorithms needed for high-quality 3D reconstruction are available, e.g. phase retrieval, ring removal and de-noising. Tofu is optimized for stand-alone GPU workstations on which it achieves reconstruction speed comparable with costly CPU clusters. It automatically utilizes all GPUs in the system and generates 3D reconstruction code with minimal number of instructions given the input geometry (parallel/cone beam, tomography/laminography), hence yielding optimal run-time performance. In order to improve accessibility for researchers with no previous knowledge of programming, tofu contains graphical user interfaces for both optimization of 3D reconstruction parameters and batch processing of data with pre-configured workflows for typical computed tomography reconstruction. The toolkit is open source and extensive documentation is available for both end-users and developers. Thanks to the mentioned features, tofu is suitable for both expert users with specialized image processing needs (e.g. when dealing with data from custom-built computed tomography scanners) and for application-specific end-users who just need to reconstruct their data on off-the-shelf hardware.

Foam-like phantoms for comparing tomography algorithms

Tomographic algorithms are often compared by evaluating them on certain benchmark datasets. For fair comparison, these datasets should ideally (i) be challenging to reconstruct, (ii) be representative of typical tomographic experiments, (iii) be flexible to allow for different acquisition modes, and (iv) include enough samples to allow for comparison of data-driven algorithms. Current approaches often satisfy only some of these requirements, but not all. For example, real-world datasets are typically challenging and representative of a category of experimental examples, but are restricted to the acquisition mode that was used in the experiment and are often limited in the number of samples. Mathematical phantoms are often flexible and can sometimes produce enough samples for data-driven approaches, but can be relatively easy to reconstruct and are often not representative of typical scanned objects. In this paper, we present a family of foam-like mathematical phantoms that aims to satisfy all four requirements simultaneously. The phantoms consist of foam-like structures with more than 100000 features, making them challenging to reconstruct and representative of common tomography samples. Because the phantoms are computer-generated, varying acquisition modes and experimental conditions can be simulated. An effectively unlimited number of random variations of the phantoms can be generated, making them suitable for data-driven approaches. We give a formal mathematical definition of the foam-like phantoms, and explain how they can be generated and used in virtual tomographic experiments in a computationally efficient way. In addition, several 4D extensions of the 3D phantoms are given, enabling comparisons of algorithms for dynamic tomography. Finally, example phantoms and tomographic datasets are given, showing that the phantoms can be effectively used to make fair and informative comparisons between tomography algorithms.

Phase retrieval for arbitrary Fresnel-like linear shift-invariant imaging systems suitable for tomography

We present a generalization of the non-iterative phase retrieval in X-ray phase contrast imaging applicable for an arbitrary linear shift-invariant (LSI) imaging system with a non-negligible amount of free space propagation (termed as Fresnel-like). Our novel approach poses no restrictions on the propagation distance between optical elements of the system. In turn, the requirements are only demanded for the transfer function of the optical elements, which should be approximable by second-order Taylor polynomials. Furthermore, we show that the method can be conveniently used as an initial guess for iterative phase retrieval, resulting in faster convergence. The proposed approach is tested on synthetic and experimentally measured holograms obtained using a Bragg magnifier microscope - a representative of Fresnel-like LSI imaging systems. Finally, the algorithm is applied to a whole micro-tomographic scan of a biological specimen of a tardigrade, revealing morphological details at the spatial resolution of 300 nm - limiting resolution of the actual imaging system.

Fresnel diffractograms from pure-phase wave fields under perfect spatio-temporal coherence: Non-linear/non-local aspects and far-field behavior

Recently, the diffractogram, that is, the Fourier transform of the intensity contrast induced by Fresnel free-space propagation of a given (exit) wave field, was investigated non-perturbatively in the phase-scaling factor S (controlling the strength of phase variation) for the special case of a Gaussian phase of width [Formula: see text]. Surprisingly, an additional low-frequency zero σ_* = σ_*(S, F) >0 emerges critically at small Fresnel number F (σ proportional to square of 2D spatial frequency). Here, we study the S-scaling behavior of the entire diffractogram. We identify a valley of maximum S-scaling linearity in the F - σ plane corresponding to a nearly universal physical frequency ξml = (0:143 ± 0.001)w ^-1/2. Large values of F (near field) are shown to imply S-scaling linearity for low σ but nowhere else (overdamped non-oscillatory). In contrast, small F values (far field) entail distinct, sizable s-bands of good S-scaling linearity (damped oscillatory). These bands also occur in simulated diffractograms induced by a complex phase map (Lena). The transition from damped oscillatory to overdamped non-oscillatory diffractograms is shown to be a critical phenomenon for the Gaussian case. We also give evidence for the occurrence of this transition in an X-ray imaging experiment. Finally, we show that the extreme far-field limit generates a σ-universal diffractogram under certain requirements on the phase map: information on phase shape then is solely encoded in S-scaling behavior.

SYRMEP Tomo Project: a graphical user interface for customizing CT reconstruction workflows

When considering the acquisition of experimental synchrotron radiation (SR) X-ray CT data, the reconstruction workflow cannot be limited to the essential computational steps of flat fielding and filtered back projection (FBP). More refined image processing is often required, usually to compensate artifacts and enhance the quality of the reconstructed images. In principle, it would be desirable to optimize the reconstruction workflow at the facility during the experiment (beamtime). However, several practical factors affect the image reconstruction part of the experiment and users are likely to conclude the beamtime with sub-optimal reconstructed images. Through an example of application, this article presents SYRMEP Tomo Project (STP), an open-source software tool conceived to let users design custom CT reconstruction workflows. STP has been designed for post-beamtime (off-line use) and for a new reconstruction of past archived data at user's home institution where simple computing resources are available. Releases of the software can be downloaded at the Elettra Scientific Computing group GitHub repository https://github.com/ElettraSciComp/STP-Gui.

Towards on-the-fly data post-processing for real-time tomographic imaging at TOMCAT

Sub-second full-field tomographic microscopy at third-generation synchrotron sources is a reality, opening up new possibilities for the study of dynamic systems in different fields. Sustained elevated data rates of multiple GB/s in tomographic experiments will become even more common at diffraction-limited storage rings, coming in operation soon. The computational tools necessary for the post-processing of raw tomographic projections have generally not experienced the same efficiency increase as the experimental facilities, hindering optimal exploitation of this new potential. We present here a fast, flexible, and user-friendly post-processing pipeline overcoming this efficiency mismatch and delivering reconstructed tomographic datasets just few seconds after the data have been acquired, enabling fast parameter and image quality evaluation as well as efficient post-processing of TBs of tomographic data. With this new tool, also able to accept a stream of data directly from a detector, few selected tomographic slices are available in less than half a second, providing advanced previewing capabilities paving the way to new concepts for on-the-fly control of dynamic experiments.

Holographic imaging with a hard x-ray nanoprobe: ptychographic versus conventional phase retrieval

We have performed near-field x-ray imaging with simultaneous object and probe reconstruction. By an advanced ptychographic algorithm based on longitudinal and lateral translations, full-field images of nanoscale objects are reconstructed with quantitative contrast values, along with the extended wavefronts used to illuminate the objects. The imaging scheme makes idealizing assumptions on the probe obsolete, and efficiently disentangles phase shifts related to the object from the imperfections in the illumination. We validate this approach by comparison to the conventional reconstruction scheme without simultaneous probe retrieval, based on the contrast transfer function algorithm. To this end, a set of semiconductor nanowires with controlled chemical composition (InP core, insulating SiO₂ layer, and indium tin oxide cover) is imaged using the quasi-point source illumination realized by the hard x-ray nanofocus (26  nm×39  nm spot size) of the ID16A Nano-Imaging beamline at the European Synchrotron Radiation Facility.

Contextual phase estimation from two-plane intensity measurements

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

Gauging low-dose X-ray phase-contrast imaging at a single and large propagation distance

The interactions of a beam of hard and spatio-temporally coherent X-rays with a soft-matter sample primarily induce a transverse distribution of exit phase variations δϕ (retardations or advancements in pieces of the wave front exiting the object compared to the incoming wave front) whose free-space propagation over a distance z gives rise to intensity contrast gz. For single-distance image detection and |δϕ| ≪ 1 all-order-in-z phase-intensity contrast transfer is linear in δϕ. Here we show that ideal coherence implies a decay of the (shot-)noise-to-signal ratio in gz and of the associated phase noise as z(-1/2) and z(-1), respectively. Limits on X-ray dose thus favor large values of z. We discuss how a phase-scaling symmetry, exact in the limit δϕ → 0 and dynamically unbroken up to |δϕ| ∼ 1, suggests a filtering of gz in Fourier space, preserving non-iterative quasi-linear phase retrieval for phase variations up to order unity if induced by multi-scale objects inducing phase variations δϕ of a broad spatial frequency spectrum. Such an approach continues to be applicable under an assumed phase-attenuation duality. Using synchrotron radiation, ex and in vivo microtomography on frog embryos exemplifies improved resolution compared to a conventional single-distance phase-retrieval algorithm.

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.

High contrast microstructural visualization of natural acellular matrices by means of phase-based x-ray tomography

Acellular scaffolds obtained via decellularization are a key instrument in regenerative medicine both per se and to drive the development of future-generation synthetic scaffolds that could become available off-the-shelf. In this framework, imaging is key to the understanding of the scaffolds’ internal structure as well as their interaction with cells and other organs, including ideally post-implantation. Scaffolds of a wide range of intricate organs (esophagus, lung, liver and small intestine) were imaged with x-ray phase contrast computed tomography (PC-CT). Image quality was sufficiently high to visualize scaffold microarchitecture and to detect major anatomical features, such as the esophageal mucosal-submucosal separation, pulmonary alveoli and intestinal villi. These results are a long-sought step for the field of regenerative medicine; until now, histology and scanning electron microscopy have been the gold standard to study the scaffold structure. However, they are both destructive: hence, they are not suitable for imaging scaffolds prior to transplantation, and have no prospect for post-transplantation use. PC-CT, on the other hand, is non-destructive, 3D and fully quantitative. Importantly, not only do we demonstrate achievement of high image quality at two different synchrotron facilities, but also with commercial x-ray equipment, which makes the method available to any research laboratory.

Visualization of water drying in porous materials by X-ray phase contrast imaging

We present in this study results from X-ray tomographic microscopy with synchrotron radiation performed both in attenuation and phase contrast modes on a limestone sample during two stages of water drying. No contrast agent was used in order to increase the X-ray attenuation by water. We show that only by using the phase contrast mode it is possible to achieve enough water content change resolution to investigate the drying process at the pore-scale. We performed 3D image analysis of the time-differential phase contrast tomogram. We show by the results of such analysis that it is possible to obtain a reliable characterization of the spatial redistribution of water in the resolved pore system in agreement with what expected from the theory of drying in porous media and from measurements performed with other approaches. We thus show the potential of X-ray phase contrast imaging for pore-scale investigations of reactive water transport processes which cannot be imaged by adding a contrast agent for exploiting the standard attenuation contrast imaging mode.

TV-based conjugate gradient method and discrete L-curve for few-view CT reconstruction of X-ray in vivo data

High-resolution, three-dimensional (3D) imaging of soft tissues requires the solution of two inverse problems: phase retrieval and the reconstruction of the 3D image from a tomographic stack of two-dimensional (2D) projections. The number of projections per stack should be small to accommodate fast tomography of rapid processes and to constrain X-ray radiation dose to optimal levels to either increase the duration of in vivo time-lapse series at a given goal for spatial resolution and/or the conservation of structure under X-ray irradiation. In pursuing the 3D reconstruction problem in the sense of compressive sampling theory, we propose to reduce the number of projections by applying an advanced algebraic technique subject to the minimisation of the total variation (TV) in the reconstructed slice. This problem is formulated in a Lagrangian multiplier fashion with the parameter value determined by appealing to a discrete L-curve in conjunction with a conjugate gradient method. The usefulness of this reconstruction modality is demonstrated for simulated and in vivo data, the latter acquired in parallel-beam imaging experiments using synchrotron radiation.

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.

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.

X-ray phase-contrast imaging: from pre-clinical applications towards clinics

Phase-contrast x-ray imaging (PCI) is an innovative method that is sensitive to the refraction of the x-rays in matter. PCI is particularly adapted to visualize weakly absorbing details like those often encountered in biology and medicine. In past years, PCI has become one of the most used imaging methods in laboratory and preclinical studies: its unique characteristics allow high contrast 3D visualization of thick and complex samples even at high spatial resolution. Applications have covered a wide range of pathologies and organs, and are more and more often performed in vivo. Several techniques are now available to exploit and visualize the phase-contrast: propagation- and analyzer-based, crystal and grating interferometry and non-interferometric methods like the coded aperture. In this review, covering the last five years, we will give an overview of the main theoretical and experimental developments and of the important steps performed towards the clinical implementation of PCI.

Criticality in single-distance phase retrieval

We investigate why in free-space propagation single-distance phase retrieval based on a modified contrast-transfer function of linearized Fresnel theory yields good results for moderately strong pure-phase objects. Upscaling phase-variations in the exit plane, the growth of maxima of the modulus of the Fourier transformed intensity contrast dominates the minima. Cutting out small regions around the latter thus keeps information loss due to nonlocal, nonlinear effects negligible. This quasiparticle approach breaks down at a critical upscaling where the positions of the minima start to move rapidly. We apply our results to X-ray data of an early-stage Xenopus (frog) embryo.

Talbot-Lau x-ray interferometry for high energy density plasma diagnostic

High resolution density diagnostics are difficult in high energy density laboratory plasmas (HEDLP) experiments due to the scarcity of probes that can penetrate above solid density plasmas. Hard x-rays are one possible probe for such dense plasmas. We study the possibility of applying an x-ray method recently developed for medical imaging, differential phase-contrast with Talbot-Lau interferometers, for the diagnostic of electron density and small-scale hydrodynamic instabilities in HEDLP experiments. The Talbot method uses micro-periodic gratings to measure the refraction and ultra-small angle scatter of x-rays through an object and is attractive for HEDLP diagnostic due to its capability to work with incoherent and polychromatic x-ray sources such as the laser driven backlighters used for HEDLP radiography. Our paper studies the potential of the Talbot method for HEDLP diagnostic, its adaptation to the HEDLP environment, and its extension of high x-ray energy using micro-periodic mirrors. The analysis is illustrated with experimental results obtained using a laboratory Talbot interferometer.