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It has been argued that in atomic-resolution transmission electron microscopy (TEM) of sparse weakly scattering structures, such as small biological molecules, multiple electron scattering usually has only a small effect, while the in-molecule Fresnel diffraction can be significant due to the intrinsically shallow depth of focus. These facts suggest that the three-dimensional reconstruction of such structures from defocus image series collected at multiple rotational orientations of a molecule can be effectively performed for each atom separately, using the incoherent first Born approximation. The corresponding reconstruction method, termed here Differential Holographic Tomography, is developed theoretically and demonstrated computationally on several numerical models of biological molecules. It is shown that the method is capable of accurate reconstruction of the locations of atoms in a molecule from TEM data collected at a small number of random orientations of the molecule, with one or more defocus images per orientation. Possible applications to cryogenic electron microscopy and other areas are briefly discussed., 32 pages, 8 figures, version 5c (a few typos have been found in the previous version and fixed in the current version)

A method for extracting the dark-field signal in propagation-based phase-contrast imaging is proposed. In the case of objects consisting predominantly of a single material, or several different materials with similar ratios of the real decrement to the imaginary part of the complex refractive index, the proposed method requires a single image for extraction of the dark-field signal in two-dimensional projection imaging. In the case of three-dimensional tomographic imaging, the method needs only one image to be collected at each projection angle. Initial examples using simulated and experimental data indicate that this method can improve visualization of small sharp features inside a larger object, e.g. the visualization of microcalcifications in propagation-based x-ray breast cancer imaging. It is suggested that the proposed approach may be useful in other forms of biomedical imaging, where it can help one to obtain additional small-angle scattering information without increasing the radiation dose to the sample.

The deterioration of both the signal-to-noise ratio and the spatial resolution in the electron-density distribution reconstructed from diffraction intensities collected at different orientations of a sample is analysed theoretically with respect to the radiation damage to the sample and the variations in the X-ray intensities illuminating different copies of the sample. The simple analytical expressions and numerical estimates obtained for models of radiation damage and incident X-ray pulses may be helpful in planning X-ray free-electron laser (XFEL) imaging experiments and in analysis of experimental data. This approach to the analysis of partially coherent X-ray imaging configurations can potentially be used for analysis of other forms of imaging where the temporal behaviour of the sample and the incident intensity during exposure may affect the inverse problem of sample reconstruction.

The purpose of this article is to review different x-ray phase-contrast breast imaging techniques and their potential application in clinical settings.Phase-contrast imaging depicts not only the absorption contrast but also the refraction contrast of the transmitted x-ray beam. Early data suggest that this new modality may overcome some of the diagnostic limitations associated with current clinically available mammography systems and that it has potential for improving breast cancer detection.

IUCrJ 7(6), 1114 - 1123 (2020). doi:10.1107/S2052252520013019, Resolving the electronic structure of single biological molecules in their native state was among the primary motivations behind X-ray free-electron lasers. The ultra-short pulses they produce can outrun the atomic motion induced by radiation damage, but the electronic structure of the sample is still significantly modified from its original state. This paper explores the decoherence of the scattered signal induced by temporal evolution of the electronic structure in the sample molecule. It is shown that the undamaged electron density of a single-molecule sample can often be retrieved using only the two most occupied modes from the coherent mode decomposition of the partially coherent diffraction fluence., Published by Chester

In this study we demonstrate the first direct comparison between synchrotron x-ray propagation-based CT (PB-CT) and cone-beam breast-CT (CB-CT) on human mastectomy specimens (N = 12) including different benign and malignant lesions. The image quality and diagnostic power of the obtained data sets were compared and judged by two independent expert radiologists. Two cases are presented in detail in this paper including a comparison with the corresponding histological evaluation. Results indicate that with PB-CT it is possible to increase the level of contrast-to-noise ratio (CNR) keeping the same level of dose used for the CB-CT or achieve the same level of CNR reached by CB-CT at a lower level of dose. In other words, PB-CT can achieve a higher diagnostic potential compared to the commercial breast-CT system while also delivering a considerably lower mean glandular dose. Therefore, we believe that PB-CT technique, if translated to a clinical setting, could have a significant impact in improving breast cancer diagnosis.

Phase-contrast X-ray imaging can improve the visibility of weakly absorbing objects (e.g. soft tissues) by an order of magnitude or more compared to conventional radiographs. Combining phase retrieval with computed tomography (CT) can increase the signal-to-noise ratio (SNR) by up to two orders of magnitude over conventional CT at the same radiation dose, without loss of image quality. Our experiments reveal that as the radiation dose decreases, the relative improvement in SNR increases. We show that this enhancement can be traded for a reduction in dose greater than the square of the gain in SNR. Upon reducing the dose 300 fold, the phase-retrieved SNR was still up to 9.6 ± 0.2 times larger than the absorption contrast data with spatial resolution in the tens of microns. We show that this theoretically reveals the potential for dose reduction factors in the tens of thousands without loss in image quality, which would have a profound impact on medical and industrial imaging applications.

The effectiveness of reconstructive imaging using the Homogeneous Transport of Intensity equation may be regarded as "unreasonable", because it has been shown to significantly increase signal-to-noise ratio while preserving spatial resolution, compared to equivalent conventional absorption-based imaging techniques at the same photon fluence. We reconcile this surprising behaviour by analysing the propagation of noise in typical in-line holography experiments. This analysis indicates that novel imaging techniques may be designed which produce high signal-to-noise images at low radiation doses without sacrificing spatial resolution.

Micro-CT scanners find applications in many areas ranging from biomedical research to material sciences. In order to provide spatial resolution on a micron scale, these scanners are usually equipped with micro-focus, low-power x-ray sources and hence require long scanning times to produce high resolution 3D images of the object with acceptable contrast-to-noise.Propagation-based phase-contrast tomography (PB-PCT) has the potential to significantly improve the contrast-to-noise ratio (CNR) or, alternatively, reduce the image acquisition time while preserving the CNR and the spatial resolution. We propose a general approach for the optimisation of the PB-PCT imaging system. When applied to an imaging system with fixed parameters of the source and detector this approach requires optimisation of only two independent geometrical parameters of the imaging system, i.e. the source-to-object distance R1 and geometrical magnification M, in order to produce the best spatial resolution and CNR. If, in addition to R1 and M, the system parameter space also includes the source size and the anode potential this approach allows one to find a unique configuration of the imaging system that produces the required spatial resolution and the best CNR.

We treat the problems of phase-contrast image formation, deterministic phase retrieval and aberration balancing, in the imaging of weak objects using two-dimensional shift-invariant linear imaging systems. Three classes of model sample are considered: weak phase objects, weak phase-amplitude objects and single-material weak phase-amplitude objects. For each class of sample we show how the various aberration coefficients, which characterise a given imaging system, contribute to the structure of the associated phase-contrast image. The corresponding inverse problem, of obtaining a closed-form expression for the input wave-field given one or more aberrated phase-contrast images of the same, is then examined. Two sample applications are considered: analyser-crystal phase-contrast imaging of weak objects using hard X-rays, and Zernike-type phase-contrast imaging. We close with a discussion of how coherent and incoherent aberrations may be “balanced” against one another, briefly mentioning the applications of this idea to both “deblur by defocus” and proximity-corrected X-ray lithography.

We present the outcomes of combined feasibility studies carried out at Elettra and Australian Synchrotron to evaluate novel protocols for threedimensional (3D) mammographic phase contrast imaging. A custom designed plastic phantom and some tissue samples have been studied at diverse resolution scales and experimental conditions. Several computed tomography (CT) reconstruction algorithms with different pre-processing and post-processing steps have been considered. Special attention was paid to the effect of phase retrieval on the diagnostic value of the reconstructed images. The images were quantitatively evaluated using objective quality indices in comparison with subjective assessments performed by three experienced radiologists and one pathologist. We show that the propagation-based phase-contrast imaging (PBI) leads to substantial improvement to the contrast-to-noise and to the intrinsic quality of the reconstructed CT images compared with conventional techniques as well as to an important reduction of the delivered doses, thus opening the way to clinical implementations.

Analytical expressions for the spatial distribution of the spectral density and intensity of projection images are derived for a broad class of object transmission functions in the case of partially coherent Schell-model-type incident illumination. The expressions are linear with respect to the phase distribution in the transmitted wave. Associated methods for phase retrieval are discussed with the emphasis on a technique that allows simultaneous “automatic” phase retrieval and deconvolution of projection images of homogeneous objects by means of mutual cancellation at a specified defocus distance of the Fresnel diffraction effects and the image blurring due to the point-spread function of the imaging system.

A theoretical analysis of the combined analyser-based/propagation-based x-ray phase-contrast imaging of weak objects is presented in the case of partially coherent illumination. It takes into account the finite size and divergence of a polychromatic x-ray source as well as the detector point-spread function. The proposed formalism describes the diffraction of x-rays in the monochromator and analyser crystals and free-space propagations between various elements of the imaging system. As in the case of a monochromatic plane wave incident on the object, the intensity in the image is expressed in terms of the amplitude and phase transfer functions. Analytical expressions for the new 'polychromatic' transfer functions are obtained and compared with the corresponding 'monochromatic' transfer functions by numerical simulations. The crucial role of the monochromator is elucidated in the formation of the analyser-based phase-contrast images with polychromatic illumination.

We have developed a new imaging technique that unifies analyser-based and propagation-based X-ray phase-contrast imaging. A novel theoretical approach, utilising the theory of linear shift-invariant systems, was employed to solve the phase-retrieval problem for the case of large Fresnel/Takagi numbers. In October 2003, we performed some preliminary experiments at BL20B2 at SPring-8 to illustrate this new imaging technique.

Theoretical aspects of quantitative diffraction-enhanced imaging of weak objects are considered using the Fourier optics approach. The amplitude and phase transfer functions are introduced by analogy with the well-known case of in-line (holographic) imaging. The inverse problem of the reconstruction of the phase and amplitude of the incident wave from recorded images is solved in the case of non-absorbing objects and objects consisting of a single material and in the general case of objects with uncorrelated refraction and absorption characteristics. A comparison is given between the solutions to the inverse problem obtained using the new formalism and the geometric-optics approximation.

Linear methods for solving the problem of optical phase retrieval from intensity measurements have the advantage of being deterministic and computationally more efficient compared to most alternative non-linear techniques. In the present work the relationship between two popular linear phase retrieval methods based on the Transport of Intensity equation and the first Born approximation, respectively, is clarified using a unified theoretical approach. It is also shown that a new phase retrieval method combining the two techniques has a broader validity range, higher accuracy and stability compared to the same techniques used separately. The new method is tested on simulated images and is demonstrated to be efficient for phase retrieval in the “near-Fresnel” region.

In free-space propagation of optical beams, the linear contribution of the object-plane phase to the intensity distribution in the Fresnel region is described by the Transport of Intensity equation (TIE). In the near-Fresnel region, i.e. where the Fresnel number is larger than unity, the linear term can often be dominant. Therefore in this region the TIE can provide a good initial approximation which can then be refined by other techniques, e.g. by Gerchberg–Saxton–Fienup (GSF) type algorithms. Multigrid methods with the TIE applied on coarser grids and GSF algorithms applied on finer grids are shown to be particularly effective in this context.

Since R€ os discovery of X-rays just over a century ago the vast majority of radiographs have been collected and interpreted on the basis of absorption contrast and geometrical (ray) optics. Recently the possibility of obtaining new and complementary information in X-ray images by utilizing phase-contrast effects has received considerable attention, both in the laboratory context and at synchrotron sources (where much of this activity is a consequence of the highly coherent X-ray beams which can be produced). Phase-contrast X-ray imaging is capable of providing improved information from weakly absorbing features in a sample, together with improved edge definition. Four different experimental arrangements for achieving phase contrast in the hard X-ray regime, for the purpose of non-destructive characterization of materials, will be described. Two of these, demonstrated at ESRF in France and AR in Japan, are based on parallel-beam geometry; the other two, demonstrated at PLS in Korea and APS in USA, are based on spherical-beam geometry. In each case quite different X-ray optical arrangements were used. Some image simulations will be employed to demonstrate salient features of hard X-ray phase-contrast imaging and examples of results from each of the experiments will be shown. 2002 Elsevier Science B.V. All rights reserved.

Summary We outline a new approach to X-ray projection microscopy in a scanning electron microscope (SEM), which exploits phase contrast to boost the quality and information content of images. These developments have been made possible by the combination of a high-brightness field-emission gun (FEG)-based SEM, direct detection CCD technology and new phase retrieval algorithms. Using this approach we have been able to obtain spatial resolution of

The triple-crystal synchrotron X-ray diffractometry data described by Nikulin, Stevenson, Hashizume, Wilkins, Cookson, Foran & Garrett [J. Appl. Cryst. (1995), 28, 57–60] has been analyzed to map out two-dimensional (two-dimensional) lattice distortions in silicon (111) crystals implanted with B+ ions of 100 keV energy through a periodic SiO2 strip pattern. The lateral periodic structure produced a series of satellite reflections associated with the 111 Bragg peak. The 2D reconstruction incorporates the use of the Petrashen–Chukhovskii method, which retrieves the phases of the Bragg waves for these satellite reflections, together with that for the fundamental. The finite Fourier series is then synthesized with the relative phases determined. Localized distortions perpendicular to the surface arising from deposited B+ ions in near-surface layers of the crystal are clearly displayed with spatial resolutions of 0.016 and 0.265 μm in the depth and lateral directions, respectively. For a sample with the oxide layer removed from the surface, two equally plausible strain maps have been obtained by assigning relative phases to 11 satellites using a sequential trial method and a minimum-energy method. Failed map reconstructions for the oxide-covered sample are discussed in terms of the non-unique solutions of the Petrashen–Chukhovskii phase-recovery algorithm and the ambiguous phases determined for the satellites.

A software system has been developed for high-performance Computed Tomography (CT) reconstruction, simulation and other X-ray image processing tasks utilizing remote computer clusters optionally equipped with multiple Graphics Processing Units (GPUs). The system has a streamlined Graphical User Interface for interaction with the cluster. Apart from extensive functionality related to X-ray CT in plane-wave and cone-beam forms, the software includes multiple functions for X-ray phase retrieval and simulation of phase-contrast imaging (propagation-based, analyzer crystal based and Talbot interferometry). Other features include several methods for image deconvolution, simulation of various phase-contrast microscopy modes (Zernike, Schlieren, Nomarski, dark-field, interferometry, etc.) and a large number of conventional image processing operations (such as FFT, algebraic and geometrical transformations, pixel value manipulations, simulated image noise, various filters, etc.). The architectural design of the system is described, as well as the two-level parallelization of the most computationally-intensive modules utilizing both the multiple CPU cores and multiple GPUs available in a local PC or a remote computer cluster. Finally, some results about the current system performance are presented. This system can potentially serve as a basis for a flexible toolbox for X-ray image analysis and simulation, that can efficiently utilize modern multi-processor hardware for advanced scientific computations.

In-line phase contrast enables weakly absorbing specimens to be imaged successfully with x-rays, and greatly enhances the visibility of fine scale structure in more strongly absorbing specimens. This type of phase contrast requires a spatially coherent beam, a condition that can be met by a microfocus x-ray source. We have developed an x-ray microscope, based on such a source, which is capable of high resolution phase-contrast imaging and tomography. Phase retrieval enables quantitative information to be recovered from phase-contrast microscope images of homogeneous samples of known composition and density, and improves the quality of tomographic reconstructions.

We discuss theoretical, experimental and numerical aspects of several new techniques for quantitative phase-contrast tomography using, for example, unfiltered radiation from a polychromatic X-ray microfocus source. The proposed algorithms allow one to reconstruct the three-dimensional distribution of complex refractive index in a sample consisting of one or more constituent materials, given one or more projection images per view angle. If the sample is weakly absorbing or consists predominantly of a single material, these reconstruction algorithms can be simplified and fewer projections may be required for an unambiguous quantitative reconstruction of the spatial distribution of the refractive index. In the case of weakly absorbing samples, the reconstruction algorithm is shown to be achromatic and stable with respect to high-spatial-frequency noise, in contrast to conventional tomography. A variation of the algorithm exploits the natural combination of binary tomography with a phase-retrieval method that makes explicit use of the single-material nature of the sample. Such consistent use of a priori knowledge dramatically reduces the number of required projections, implying significantly reduced dose and scanning time when compared to most alternative phase-contrast tomography methods. Experimental demonstrations are also given, using data from a point-projection X-ray microscope. The refractive index distribution, in test samples of both a polymer fibre scaffold and an adult mouse, is accurately reconstructed from polychromatic phase-contrast data. Applications of the new techniques to rapid non-destructive testing in materials science and biomedical imaging are considered.

In X-ray computed tomography (CT) increased information requirements (e.g. increased resolution) typically lead to a concurrent increase in the required number of viewing angles, scanning time and delivered dose. We demonstrate that using phase-contrast imaging it is possible to "dissect" two- and three-material objects into their component materials, which in combination with binary tomographic techniques allows us to satisfy increased information requirements without taking the usual images at additional viewing angles. This imaging scheme reduces the scanning time and dose delivered to samples by at least an order of magnitude when compared to conventional X-ray CT. The effects of noise on our reconstruction scheme are investigated for simulated data. Finally, a slice through a glass tube filled with silica and water is reconstructed from 18 projection images taken on an X-ray ultra Microscope (XuM).

A method is presented for quantitative phase-contrast tomography using unfiltered radiation from a small polychromatic source. The three-dimensional distribution of complex refractive index in a monomorphous object is reconstructed given a single projection image per view angle. The reconstruction algorithm is achromatic and stable with respect to high-spatial-frequency noise, in contrast to conventional tomography. The density distribution in a test sample was accurately reconstructed from polychromatic phase-contrast data collected with a point-projection x-ray microscope.

Phase-contrast tomography (PCT) allows three-dimensional imaging of objects that display insufficient contrast for conventional absorption-based tomography. We prove that PCT is stable with respect to high-frequency noise in experimental phase-contrast data, unlike conventional tomography, which is known to be mildly unstable. We use known properties of the three-dimensional x-ray transform and transport-of-intensity equation to construct a matrix representation of the forward PCT operator. We then invert this formula to show that, under natural boundary conditions, the PCT reconstruction operator exists and leads to a unique solution. We show that the singular values s(n) of the reconstruction operator have asymptotic behavior s(n)=O(n(-3/2)), guaranteeing the mathematical stability of the reconstruction process.

X-ray Microtomography bridges the 3D analysis gap between conventional x-ray tomography and TEM tomography. The use of a laboratory-based microfocus source opens up the opportunity to gain additional benefits from in-line phase contrast for enhancing the visibility of fine features, cracks, voids and boundaries in individual views. Coupled with phase retrieval methods, such images can be used as input to conventional reconstruction algorithms for three dimensional visualization. Working at high resolution brings challenges of physical stability of the system. Software approaches to overcoming these difficulties have enabled submicron resolution 3D reconstructions.

Summary The science of wave-field phase retrieval and phase measurement is sufficiently mature to permit the routine reconstruction, over a given plane, of the complex wave-function associated with certain coherent forward-propagating scalar wave-fields. This reconstruction gives total knowledge of the information that has been encoded in the complex wave-field by passage through a sample of interest. Such total knowledge is powerful, because it permits the emulation in software of the subsequent action of an infinite variety of coherent imaging systems. Such ‘virtual optics’, in which software forms a natural extension of the ‘hardware optics’ in an imaging system, may be useful in contexts such as quantitative atom and X-ray imaging, in which optical elements such as beam-splitters and lenses can be realized in software rather than optical hardware. Here, we develop the requisite theory to describe such hybrid virtual-physical imaging systems, which we term ‘omni optics’ because of their infinite flexibility. We then give an experimental demonstration of these ideas by showing that a lensless X-ray point projection microscope can, when equipped with the appropriate software, emulate an infinite variety of optical imaging systems including those which yield interferograms, Zernike phase contrast, Schlieren imaging and diffraction-enhanced imaging.

We demonstrate simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object. Subject to the assumptions explicitly stated in the derivation, the algorithm solves the twin-image problem of in-line holography and is capable of analysing data obtained using X-ray microscopy, electron microscopy, neutron microscopy or visible-light microscopy, especially as they relate to defocus and point projection methods. Our simple, robust, non-iterative and computationally efficient method is applied to data obtained using an X-ray phase contrast ultramicroscope.

Since Rontgen’s discovery of X rays in 1895 the vast majority of radiographs have been collected and interpreted on the basis of absorption contrast and geometrical (ray) optics. In recent years the possibility of utilizing phase-contrast effects has received considerable attention, much of this activity stemming from the possibility of producing X-ray beams of suitably high coherence at synchrotron sources (1). A description of a phase-contrast imaging (PCI) technique which utilizes a laboratory-based microfocus X-ray source to achieve the required spatial coherence, and does not rely on there being a high degree of chromatic coherence, will be described (2). The method employs a relatively large object-to-image distance to allow wave interference (Fresnel diffraction) to occur and manifest itself as contrast formation in the intensity distribution recorded at the image plane. PCI is particularly relevant to the nondestructive characterization of materials and this will be demonstrated by way of some exa...

The quantitative imaging of phase object using 16 keV x-rays is reported. The theoretical basis of the techniques is presented along with its implementation using a synchrotron x-ray source. We find that our phase image is in quantitative agreement with independent measurements of the object.© (1997) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

A dedicated micro-CT beamline is planned for the Australian Synchrotron which will extend the synchrotron's imaging and tomography capability down to the smaller scale, incorporating phase-contrast and absorption-contrast, and an additional focussing-based mode for high-resolution. The beamline will use multi-layer mirror monochromators for enhanced flux, and will focus particularly on dynamic and high throughput studies in both monochromatic and pink-beam mode. Together with the existing Imaging and Medical beamline, this beamline will produce numerous large datasets of 10 GB or more, providing a significant data-processing challenge. The Remote-CT project addresses this by combining the MASSIVE supercomputing GPU cluster with XLI / X-TRACT software, developed at CSIRO. This software has extensive functionality for both processing and simulation of absorption and phase-contrast tomography data and has now been modified for parallel operation on a GPU cluster to take maximum advantage of the speed-up this enables.

This paper presents a summary and overview of heterogeneous algorithms and applications developed by the Commonwealth Scientific and Industrial Research Organisation CSIRO for solving practical and challenging science problems such as: 1 high-content analysis of brain cell images for medical research and drug discovery; 2 genetic analysis of complex experimental designs for crop breeding; 3 solving computational fluid dynamics problems; 4 deconvolving 3D images from microscopy and medical imaging; 5 reconstructing large 3D computed tomography CT images from medical and materials science; 6 quantifying uncertainty in complex environmental models. Applications discussed utilise GPUs and multicore CPUs on a scale ranging from single desktop workstations through to large GPU clusters. Results demonstrate that both types of system can successfully accelerate a variety of practical science applications. We have seen significant gains in productivity and opportunity as a result of leveraging GPUs, tackling computational problems in which execution time was previously infeasible.

A method is presented for quantitative polychromatic cone-beam phase-contrast tomographic imaging of a single-material object from few projections. This algorithm exploits the natural combination of binary tomography with a phase-retrieval method that makes explicit use of the single-material nature of the sample. Such consistent use of a priori knowledge reduces the number of required projections, implying significantly reduced dose and scanning time when compared to existing phase-contrast tomography methods. Reconstructions from simulated data sets are used to investigate the effects of noise and establish a minimum required number of projections. An experimental demonstration is then given, using data from a point-projection X-ray microscope. Here, the complex distribution of refractive index in a sample containing several nylon fibers with diameters between 100 microm and 420 microm is reconstructed at a spatial resolution of approximately 4 microm from 20 polychromatic phase-contrast projection images with a mean photon energy of 8.4 keV.

The first Born and Rytov approximations of scattering theory are introduced in their less familiar near-field versions. Two algorithms for phase retrieval based on these approximations are then described. It is shown theoretically and by numerical simulations that, despite the differences in their formulation, the two algorithms deliver fairly similar results when used for optical phase retrieval in the near and intermediate fields. The algorithms are applied to derive explicit solutions to four phase-retrieval problems of practical relevance to quantitative phase-contrast imaging and tomography. An example of successful phase reconstruction by use of the Born-type algorithm with an experimental x-ray image is presented.

Conventional X-ray radiography is based on absorption contrast and assumes geome trical optics. An “in-line” method and theory of hard X-ray phase-contrast ima ging were introduced. The differences between the conventional X-ray absorption contrast image and the phase contrast image were distinguished by using radiogra phs of biological samples. The effect of X-ray source size on the image contrast was investigated. Moreover, the key parameters of absorption contrast image and phase contrast image were simulated theoretically and discussed.

The quantitative imaging of a phase object using 16 keV x rays is reported. The theoretical basis of the techniques is presented along with its implementation using a synchrotron x-ray source. We find that our phase image is in quantitative agreement with independent measurements of the object.

A new method for extracting quantitative information from phase-contrast x-ray images obtained with microfocus x-ray sources is presented. The proposed technique allows rapid noninvasive characterization of the internal structure of thick optically opaque organic samples. The method does not generally involve any sample preparation and does not need any x-ray optical elements (such as monochromators, zone plates, or interferometers). As a consequence, samples can be imaged in vivo or in vitro, and the images are free from optical aberrations. While alternative techniques of x-ray phase-contrast imaging usually require expensive synchrotron radiation sources, our method can be implemented with conventional, albeit microfocus, x-ray tubes, which greatly enhances its practicality. In the present work, we develop the theoretical framework, perform numerical simulations, and present the first experimental results, demonstrating the viability of the proposed approach. We believe that this method should find wide-ranging applications in clinical radiology and medical research.