The detection, localisation and characterisation of stationary and singular points in the phase of an X-ray wavefield is a challenge, particularly given a time-evolving field. In this paper, the associated difficulties are met by the single-grid, single-exposure X-ray phase contrast imaging technique, enabling direct measurement of phase maxima, minima, saddle points and vortices, in both slowly varying fields and as a means to visualise weakly-attenuating samples that introduce detailed phase variations to the X-ray wavefield. We examine how these high-resolution vector measurements can be visualised, using branch cuts in the phase gradient angle to characterise phase features. The phase gradient angle is proposed as a useful modality for the localisation and tracking of sample features and the magnitude of the phase gradient for improved visualization of samples in projection, capturing edges and bulk structure while avoiding a directional bias. In addition, we describe an advanced two-stage approach to single-grid phase retrieval.
© 2016 Optical Society of America
Since the discovery of x-rays, their ability to penetrate materials has been used in imaging, in particular to visualize highly-attenuating structures like bones. In the case of weakly-attenuating samples, like soft tissue structures, a more sensitive method of measurement is required to provide contrast. In recent years, new methods of phase contrast x-ray imaging (PCXI) have been developed to address this . By detecting variations in the phase of the x-ray wavefield, scientists can resolve soft tissue structures for medical purposes and weakly absorbing samples in materials/industrial applications. Phase contrast x-ray imaging can also be used in metrology, characterising optics and experimental set-ups.
In this paper, we directly measure the gradient of the phase, so that stationary and singular points can be identified in the x-ray wavefield. This is useful in metrology and also provides an avenue to visualise samples that alter the x-ray beam. We show how the phase gradient angle can be used to pinpoint sample features that produce stationary points, to then track those features in a time sequence. In addition, we use the magnitude of the phase gradient to provide improved sample visualization, particularly for dynamical imaging. In this experiment, both transverse directions of the phase gradient are collected simultaneously by using a relatively new method called single-grid, single-exposure phase contrast imaging [2–4].
There are several methods by which x-ray phase contrast can be achieved. The simplest method is propagation-based PCXI, where a moderately coherent source is used and a distance of several centimeters or meters is introduced between the sample and the detector. This means that when the sample introduces strong variations in phase (e.g. at sample edges), the wavefield will self-interfere with propagation and produce bright/dark fringes, providing edge contrast or edge-enhancement. To reconstruct the phase of the wavefield, multiple images are required (e.g. taken at different sample-to-detector distances  or different energies) or a single-material approximation may be used. If the edge-enhanced images are sufficient or a single-image phase retrieval approach is valid , fast imaging can capture dynamic processes. Because propagation-based PCXI derives contrast primarily from the second spatial derivative of the x-ray phase, the phase-retrieved locations of phase maxima, minima and saddle points can be easily influenced by low-frequency artefacts resulting from the phase retrieval process (e.g. Fig. 3 d/e from , comparing propagation-based imaging to theory, also noting that the third profile in the plot comes from an imaging system that matches the system used in this manuscript, simply with a different reference object).
PCXI methods that directly measure the first spatial derivative of the x-ray phase can more precisely locate stationary and singular points. Of these techniques, the most widely-used are analyser-based imaging, edge-illumination and Talbot grating interferometry , all of which typically measure the derivative in one direction (the axis aligned to the crystal analyser or perpendicular to the grating lines). This means that information is lost in the perpendicular direction and the direction of the maximum phase gradient cannot be calculated. These techniques can be extended to measure phase gradients in both transverse x and y directions, but usually at the cost of additional exposures. This can be done by rotating the set-up or sample by 90 degrees and capturing a second image sensitive to the perpendicular transverse direction . In the methods that use a grating, a grid type structure can be used instead of a line grating structure and stepped over several exposures [9,10] to achieve the same result. This means it is difficult to measure a moving or changing sample with these techniques, unless the changes are over time scales that enable multiple exposures/set-ups.
X-ray phase singularities and stationary points may be observed from a sample of interest, from specially designed optics or from the random interference of x-ray waves. One particularly interesting singular point is an x-ray vortex , identified by a wrapping of 2πn in phase around a single point, with zero intensity observed at this point (where n is an integer). In previous work, x-ray vortices have been created using a spiral phase plate , a spiral Fresnel zone plate , a spiral photon sieve  and diffractive optical elements . The interference of several x-ray beams, whether random  or specifically designed , can also create vortices. Alternatively, the x-ray source itself can be arranged to produce photons with quantized orbital angular momentum, for example by using a helical undulator set-up [18,19], or by manipulating the electron beam in an x-ray free electron laser (XFEL) . One of the main difficulties associated with imaging x-ray vortices is the size of the vortex. Because a vortex can be one of the smallest features in a wavefield, the associated intensity minimum in an image and/or a vortex itself can be easily lost as a result of smoothing effects from an extended source, the detector system and scattered photons [12,21].
Nevertheless, X-ray vortices have been detected using a range of methods. One approach is to scan a phase edge across the field so that the propagation-based fringes from that edge are distorted when they overlap with the x-ray vortex . The wavefield can also be reconstructed using ptychography [13,22], three propagation distances , shearing interferometry  or diffraction through an aperture [24,25]. These techniques are not particularly compatible with fast imaging, so are best suited for stationary samples or static wavefields. In visible light imaging, a Shack-Hartmann sensor has been used to capture vortices , and this kind of approach is used in the work presented here.
Stationary points in the x-ray phase are also interesting topological features, particularly in the transverse phase gradient image. Figure 1 shows the most basic singular and stationary phase features including vortices, local phase maxima (or focus points), local phase minima (or inverted focus/anti-focus points) and a saddle point. As seen in the first panel, vortices are characterised by a wrapping of 2π in phase around a point (or n2π, where n = 1, 2, 3...), forming a singular point. In contrast, the precise location of maxima, minima and saddle stationary points can be unclear in slowly-varying phase maps. Given that many of the PCXI methods discussed earlier measure the phase derivative, not the phase itself, we can differentiate the phase distribution to visualize the phase gradients as a vector field, observed in the third row of Fig. 1. It is apparent here that the direction of the maximum phase gradient (shown by the vector direction) is pointing towards a phase maximum, away from a phase minimum and circles a vortex. A visualization of the gradient direction in this vector field is given in the second row of Fig. 1, with ‘0’ (gray) corresponding to a vector pointing in the positive x direction. All phase features are now easily located, but it is more difficult to separate the different kinds of stationary/singular points, since for most of these topological features, the vectors rotate in a clockwise direction if the feature is traversed in a clockwise direction. In other words, where π corresponds to white and −π corresponds to black, the phase gradient angle progresses from white to black in a clockwise direction around the features. The only exception is the saddle point, where the phase gradient angle rotates in the opposite direction (black to white for a clockwise traversal). This visualization in the second row of Fig. 1 provides better localisation of features than the vector map and is higher resolution than a vector field, which is particularly useful if there are many features within the image. In Fig. 1 it is also interesting to look at the direction from which a −π/π boundary in the phase gradient angle approaches each stationary or singular point. This boundary is also known as a branch cut and with our convention corresponds to a phase gradient vector pointing in the negative x direction. This vector would be found on the right of a phase maximum, pointing towards the stationary point and on the left of a phase minimum, pointing away from the stationary point. Similarly, it would be found below an anti-vortex and above a vortex, as the field circulates. Therefore the location of the phase gradient vector that points in the negative x direction, or equivalently, the location and orientation of the −π/π boundary in the phase gradient angle, can be used to infer the identity of the topological feature. With our convention, this boundary will be vertical for vortices (entering from above for a vortex and entering from below for an anti-vortex) and horizontal for maxima/minima (entering from the right for a maximum and the left for a minimum). However, in an experimental wavefield it is possible that the end of this −π/π boundary closest to the stationary point changes direction very locally and hence if the size of the features approaches the resolution of the image, this classification may be somewhat ambiguous. If the reference direction or labelling convention is rotated such that the branch cut corresponds to say, a vector pointing in the negative y direction, the interpretation of the branch cut approach will simply be rotated in the same way (for the negative y vector case mentioned as an alternative convention, the branch cut would approach a vortex from the left, an anti-vortex from the right, a phase maximum from above etc.). While a branch cut in phase does not convey information and is arbitrary, a branch cut in the phase gradient angle contains information.
These phase singularities and stationary points can be classified using the measures of topological charge and topological index. The topological charge describes the quantised change in phase when transversing a simple closed curve anti-clockwise around the phase feature . The topological charge is therefore positive for the anti-vortex and negative for the vortex and zero for the remaining features shown in Fig. 1. In the case of the vortices shown in Fig. 1, the topological charge is +1 and −1, but with multiple windings of 2π around a single point, the topological charge n of a vortex can be higher, although typically these will quickly decay into a vortex of charge ±1 . For several phase singularities/stationary points to disappear altogether, the sum of their topological charges must equal zero. The topological index is measured by performing the same kind of analysis on the angle made by the phase gradient with respect to a given direction (e.g. the second row of Fig. 1 instead of the first row) . As explored in the previous paragraph, all features are associated with an increase in phase gradient angle circling the feature in the same direction, with the exception of the saddle point, hence only the saddle point has negative topological index. Following a similar conservation rule to the topological charge, if, for example, a maximum is created, a saddle point must also be created. Typically, there will be roughly the same number of saddle points as singularities/stationary points in a given field . An extended stationary point, for example resulting from a cylinder of uniform thickness, would experimentally have some very slight variations and hence appear as an alternating sequence of maximum points and saddle points.
In this paper, we used single-grid, single-exposure phase contrast x-ray imaging [3,4] to simultaneously capture phase gradients in both x and y directions to locate and characterise phase singularities and stationary points. This single-exposure approach is compatible with dynamical imaging and is simple to implement. The single-grid PCXI set-up places a grid upstream of the detector (see Fig. 2), creating a reference pattern at the detector. This reference pattern can be created by an absorption grid [2,3], a more efficient phase grid , or even a randomly scattering object like a piece of paper or sandpaper [7,30] (effectively a disordered phase and attenuation grid). A sample is then introduced and sample-induced distortions to the reference pattern are directly resolved at the detector. These distortions are then decoded in software and can be rendered quantitative using the set-up parameters [3,4]. Because the method is sensitive to the first order transverse derivative of the phase , it is more sensitive than propagation-based PCXI to weak/slowly-changing features. In addition, because there is sensitivity to transverse phase variations in both perpendicular directions, it can more accurately localise phase singularities and stationary points than PCXI methods that only detect gradients along a single transverse axis. The limits of the single-grid, single-exposure approach are primarily related to the selection of a grid that is fine enough to populate and hence resolve the sample features, but remains visible over sample-induced intensity variations. This visibility requirement becomes more difficult as the grid period approaches the pixel size and hence the point spread function of the detector system.
The samples used in this experiment produced large transverse phase gradients across large areas and also spatially small phase features, hence we extended the algorithm used to recover the phase gradients from the distorted reference image. If the reference object is well-described by a single high spatial frequency, the differential phase image can be recovered using Fourier analysis [2,32]. If there are multiple spatial frequencies present (which can be useful in creating a well-defined rectangular grid profile that is more easily tracked in the presence of noise), a spatial method of analysis can produce more accurate results . This spatial method performs a local cross-correlation between corresponding regions of the grid-only reference image and the grid-and-sample image, with a given window size and range of movement . The range of movement of the cross-correlation window is set to the size of the largest observed distortions. The size of these distortions can be tuned by adjusting the propagation distance. Initially, the width of the cross-correlation window is chosen to capture at least one grid period or feature in the reference pattern. If the window fits between grid lines, then there will be no feature to track, but if the window covers several grid lines, the distortions of only one of these grid lines will be difficult to detect. In the case of a noisy image, due to either low counts or background intensity variations, the width of the window may be increased to reduce errors. A manual iteration of different window sizes, beginning at the grid period, is therefore undertaken to discover the best reconstruction. A larger window size will result in a smoother reconstructed differential phase image, with reduced resolution. Therefore the size of the window selects the scale for features of interest. Indeed, a fixed window size for correlation analysis is akin to convolution with a top-hat filter. In computer vision based pattern recognition, feature extraction begins with a selection of a scale of interest, typically dictated by image convolution with a Gaussian filter of an appropriate width. Automated scale-selection, to detect local features across many scales, is achieved by creating a ‘scale space’ of varying Gaussian filters, using which locally optimal scales can be determined at different points in an image . In the context of our work, similar schemes could be employed by using multiple window sizes.
The extended method of analysis reported here performs two successive sets of local cross-correlations. First, the images are analysed using a large cross-correlation window and large range of movement to determine the low-spatial-frequency or large-scale phase gradients in the image. This means that the analysis can be performed at a fewer number of pixels, for example, only every 20th pixel, reducing computation time. The resulting shift images in the x and y directions are then used as an initial estimate to perform a second set of local cross-correlations. This second analysis uses a much smaller window (centred at the new estimated shift position), so as to not smooth out small features, and a smaller range of movement, for example, only 2 pixels. In a given experiment, a longer propagation distance may sometimes be chosen to increase the sensitivity to weak phase features by increasing the size of the local transverse shift observed at the detector. If there are strong phase features in the same field of view as these weak features, they will produce a large local reference pattern shift, increasing the probability of spurious phase wrapping or ambiguity in determining whether a periodic feature has shifted from the right or the left. This new approach to the analysis means that both strong and weak phase shifts are resolved at both long and short length scales, reducing the likelihood of phase wrapping artefacts (especially compared to the Fourier method , see  for example).
At each position in the image, the peak of the cross-correlation can be fitted to extract sub-pixel shifts in the reference pattern . An image of the local reference pattern shifts has the appearance of a differential phase image, and can be made quantitative according to the scaling described in Morgan et al. . In this manuscript, differential phase contrast images are shown with a grayscale that describes the number of pixels by which the reference pattern has been shifted.
We use the single-grid imaging method and this two-scale analysis to analyse x-ray wavefield phase features created by random phase screens (Fig. 3), verifying that the orientation of the branch cut in the phase gradient angle indicates the nature of the stationary or singular point. We then apply this methodology to visualize samples, and look at the merits of resolving both transverse directions of the phase gradient (Figs. 5 and 6). The utility of the phase gradient angle in locating sample features and the improved sample visualization that comes with plotting the magnitude of the phase gradient are both shown. A number of different grid periods, detector resolutions and propagation distances are used, as specified in the figure captions.
3. X-ray wavefield phase features
3.1. Local maxima, minima and saddle points
The images in Fig. 3 show the wavefield observed when random variations are introduced to the x-ray beam by a piece of fibrous micropore surgical tape (3M) situated 254 cm upstream of the detector. A diffraction pattern from the fibers can be seen in Fig. 3(a), and the phase of the wavefront is visualized in Fig. 3(f). The phase is captured by comparing the phase-grid-and-tape image [Fig. 3b] to the phase-grid-only image [Fig. 3(c)] to extract differential phase in the vertical [Fig. 3(d)] and horizontal [Fig. 3(e)] directions, then integrating these together using the Fourier derivative theorem (as performed in ). Alternatively, the vertical and horizontal phase gradients can be visualized as a magnitude [Fig. 3(g)] and angle [Fig. 3(h)]. In the phase gradient angle image [Fig. 3(h)], the singularities correspond to the phase singularities and stationary points, and are easily located, highlighting the precise position of these topological features. Where the phase gradient angle increases (black-to-white) in a clockwise direction (negative topological index), a red dot has been placed (following the convention in Fig. 1) identifying a saddle point in the phase derivative. Where the phase gradient angle decreases (white-to-black, positive topological index) in a clockwise direction, a yellow dot has been placed, indicating a local maximum (+) or a minimum (−) in the phase, or possibly an optical vortex. Feature identification is achieved by examining the phase gradient vector field, Fig. 3(i). The pairs of saddle points and maxima/minima demonstrate that there are roughly equal numbers of saddle points and maxima/minima, as predicted [28, 34]. In addition, we see that the phase maxima typically have a −π/π boundary entering from the right and phase minima typically have a −π/π boundary entering from the left of the stationary point. This means that feature identification is possible using the phase gradient image only [Fig. 3(h)].
If the locations of these maxima and minima are compared to the contours in Fig. 3(f), it can be seen that there are some low frequency artefacts introduced in the integration process that introduce slight deviations in position, but otherwise the points match up with the expected contours. The differences in the positions of these features highlights the value of directly detecting the phase gradient to accurately locate phase features. We can also see that along the ‘ridgelines’ of the projected thickness, there is an alternating sequence of maximum points and saddle points, as expected (e.g. imagine the image is the face of a clock and walk from ‘12 o’clock’ to the centre of the clock, then out to ‘three o’clock’).
We can also look at sub-classifications of the stationary and singular points. For example, saddle points can be categorised using the lines of constant phase. Possibilities include an open saddle (where the lines of constant phase form a cross shape and do not meet again), a closed saddle (where the lines of constant phase appear as a figure eight) or a loop saddle (where two arms form a loop and the other two are open) . Studying Fig. 3(f) and tracing between the contour lines, we see that the saddle points closest to the top and bottom of the image are loop saddles and the left and right most saddle points are most likely closed saddles.
The comprehensive phase gradient measurements enable further vector field calculations, following the example of visible light experiments. For example, we can extract the contours of zero first or second derivatives in x and y [Fig. 4(a) and 4(b)], as per . From the complex wavefield Ψ, we can also calculate measures like the longitudinal orbital angular momentum density, Lz =Im[Ψ(x∂yΨ − y∂xΨ)] as per , and shown in Fig. 4(c), or trace the lines where Lz is close to 0, as per , and shown in Fig. 4(d).
The slowly-varying field we have studied in Fig. 3 shows stationary points in the form of phase maxima, minima and saddle points, but no vortical structures are present. It is likely that the phase variations introduced by the tape are not strong enough to ‘tear’ the wavefronts and/or vortices have disappeared with propagation as the features become broader and weaker, or any vortices present are too small to be resolved .
The critical propagation distance Dc to obtain spontaneously-induced phase vortices can be estimated by considering the requirements for three waves to interfere and form a vortex . If the sample has transverse variations in projected phase of ϕ separated by characteristic transverse length scale L, we can estimate the characteristic phase gradients introduced to the wavefield as ϕ/L. A phase gradient of ϕ/L will result in a change in the direction of x-ray wave propagation of θ, according to tanθ ≈ θ = ϕ/(kL), (for a small angle, as is typical of x-rays). In order for these refracted beams to interfere with each other and form a vortex they must propagate a distance of Dc metres. With each ray travelling at an angle of θ, they will meet given they each shift transversely by L/2 with a propagation of Dc, and this can be described by tanθ ≈ θ = L/2D. Combining these two expressions to eliminate θ, we can estimate that the critical distance for interference, Dc, is:
Estimating these parameters from Fig. 3, k = 2π/0.5 × 10−10 m−1, L = 100 × 10−6 m and ϕ = 3.4 radians, this gives Dc ∼ 200 meters (to one significant figure). Leaving aside the feasibility of such a large propagation distance, with a source width of 300 μm and the sample placed approximately 200 m from the source (the parameters of the beamline used for these experiments), the penumbral blurring a further 200 metres downstream at a detector would be 300 μm, larger than the characteristic transverse length scale of the features and hence likely destroying any vortices, even at one of the most coherent x-ray imaging synchrotron beamlines in the world (BL20XU, SPring-8).
While we tested several more strongly-refracting set-ups, in all cases the vortices were either insufficiently spatially separated at this high energy (e.g. three beam interference  with perspex prims ) or the precise refractive optic was difficult to manufacture at this high resolution (e.g. a spiral phase plate ). Nevertheless, it is clear that the single-grid imaging technique would also be able to detect phase vortices, in that the reference grid would simply be displaced in a vortical fashion.
4. Visualizing samples
Full measurement of the phase gradient is also useful in visualizing weakly-absorbing samples. Imaging sugar crystals, we can see that magnitude of the phase gradient is a useful and intuitive way to visualize a sample. As shown in Fig. 5(a)–5(c), this modality retains high spatial frequency components to visualize small features (in contrast to the projected phase depth, Fig. 5(d)–5(f)) and information from both transverse directions of the phase gradient (in contrast to a differential image in a single direction, e.g. Fig. 5(g)–5(k), which is what is usually produced with analyser-based PCXI or Talbot interferometry). The resulting images are reminiscent of Schlieren images , but without a directional bias. In addition, for visualising edges, the magnitude of the phase gradient is more robust to noise than the visibility signal extracted by looking at the magnitude of the cross-correlation peak  [Figs. 5(m)–5(n)]. Note that the reduction in visibility signal may however provide useful information on structures that are below the resolution of the imaging system.
The magnitude of the phase gradient has no directional bias, in the same way as a propagation-based phase contrast x-ray image. This lack of directional bias is a major advantage of propagation-based PCXI, and this advantage is now achieved here with sensitivity to the first spatial derivative of the phase, instead of the second spatial derivative like in propagation-based PCXI. The first spatial derivative is more sensitive to slowly changing sample thickness/composition than the second, and hence provides better visualization of the entire sample shape and not just the sample edges. Since the phase gradient magnitude has contributions from both horizontal and vertical phase gradients, there is increased total contrast. For example, study the lower-left sugar crystal in Fig. 5(a), where the contrast on the side surface comes from Fig. 5(g), the horizontal phase gradient and the contrast on the top surface comes from Fig. 5(j), the vertical phase gradient.
The second column of Fig. 5 demonstrates that it is easier to identify individual grains in the image that shows the magnitude of the phase gradient than in any other modality (compare Fig. 5(b) with 5(e), 5(h), 5(k) and 5(n)). The propagation-based PCXI image shown in Fig. 5(o) is the next most useful, but the overlapping of diffraction fringes can make it difficult to differentiate individual grains. The angle of the phase gradient [Fig. 5(l)] can also assist in identifying, locating or counting particles, with a stationary point (topological index = +1) observed at each particle.
This imaging technique and visualization is also beneficial in biomedical research applications. While the speed and sensitivity of single-grid PCXI has already been utilised to visualize the airway surface interfaces and assess the efficacy of treatments , this application does not make full use of the 2D sensitivity of the technique.
One example that benefits from sensitivity to both transverse phase gradient directions is the tracking of inhaled particles along the airway surface , shown in Fig. 6. The speed at which inhaled particles move away from the lungs provides a measure of airway health, but there is large variability in the speed and trajectory of individual particles. Therefore, to generate sufficient statistics, these measurements benefit from automated tracking of many beads in software. In Fig. 6, we see that (c) the magnitude and (d) the angle of the phase gradient show the particles (e) with a more unique signature than the differential phase, and this is an advantage in image recognition and automated tracking. In the phase gradient direction images, Fig. 6(d), we see that each particle (see Fig. 6(i)) provides a localised maximum in phase (−π to π interface entering from the right), with a circulation of 2π providing a high contrast feature that can be easily extracted from an image or used in a cross-correlation , for example in velocimetry . Of particular advantage in this application is the stability of topological features (i.e. phase maxima in the phase gradient image) with respect to small continuous perturbation. Non-topological features, which are not in general stable with respect to perturbation, are much more likely to disappear between consecutive frames (such non-topological ‘disappearance’ events can be termed ‘non-topological reactions’). Topological features, however, are by definition ‘topologically protected’ and can only disappear in topological reactions that preserve both topological charge and topological index; such topological reactions are typically far less likely than the corresponding non-topological reactions described above.
In addition, the phase gradient direction information is vital in ridge detection and edge linking algorithms, which are useful for automated image segmentation . Alternatively, the phase gradient magnitude image in Fig. 6(c) is called a ‘Sobel’ map in image processing, and this forms the first step for feature highlighting schemes such as Canny edge detection . In our approach, the ‘Sobel’ map is directly measured, so has a higher signal to noise ratio than a numerical calculation on a measured thickness image. It is interesting to note that the two-stage single-grid reconstruction process and cross correlation window size selection are reminiscent of selective pre-smoothing before application of a Sobel filter, in both cases optimising the analysis procedure to reveal features of a certain size in the presence of noise.
The second biomedical research example shown in Fig. 6 is the visualization of the lung periphery of a mouse during the breath cycle, also requiring a fast imaging technique to avoid image blur . A sequence of these images over several breaths is provided in Visualization 1. The magnitude of the phase gradient again provides the most intuitive image [Fig. 6(g)], although this is complicated where there are many overlying alveoli. In addition, each air-filled alveolus creates a phase minimum, with the −π to π interface, or branch cut, entering from the left (compare Fig. 6 (l) to Fig. 6(k)). To avoid the visual complexity of marking all points of singular phase gradient, we have left the saddle points unmarked and only labelled these local minima [Fig. 6(i)]. The phase gradient angle [Fig. 6(h)] also provides a clearer visualization for small scale features like these than the vector map [Fig. 6(j)], demonstrating the value of this kind of visualization. The movement of the lungs during breathing can provide a sensitive measure of lung health , and the local movement and expansion of individual alveoli could be extracted from a sequence of these images [ Visualization 1]. Alveolar movement could be monitored by tracking the stationary points in the phase gradient angle images [Fig. 6(h)] and expansion would be seen in the phase gradient magnitude images [Fig. 6(g)]. Note particularly the precise localisation and topologically-protected stability of the phase minima in the phase gradient angle images in Visualization 1, compared to the lower-frequency features in the other modalities. Phase gradient deformations are expected in time sequence imaging and the persistence of key features that are robust to smooth perturbations, such as topological features, is vital for correlating local structural dynamics.
5. Further applications
These biomedical examples indicate the usefulness of the approach for dynamical imaging, which is increasingly possible with bright new x-ray sources that use a liquid metal jet  or inverse Compton scattering .
This imaging approach can also be used in metrology, as in Berujon et al. [50,51], for example, resolving the slowly varying phase from a focusing lens or locating small defects in a mirror. Single-grid imaging can be used to analyse any system where vortices or other topological features are intentionally produced as a part of the imaging system. One example of this is the interference of beams to create a vortex/topological feature lattice as a reference pattern [52,53]. Another possibility is to trace vortex cores with propagation, enabling 3D mapping of the cores. A turbulent wavefield that changes with time could also be captured, monitoring the appearance and disappearance of phase features in accordance with the conservation laws for topological charge and index. The visualization of the amplitude of the transverse phase gradient also links to the idea of the transverse Poynting vector, which is proportional to I∇⊥ϕ, and describes the directional energy flux density in a wavefield of intensity I and phase ϕ.
The stationary and singular points in the x-ray phase, as extracted here, can be used as a skeleton from which to reconstruct a wavefield . This presents possibilities in terms of data compression, of particular interest in situations where a rapid time-sequence is acquired. For example, x-ray velocimetry combines time sequencing of phase contrast images with local cross-correlation based feature tracking. The sparsely distributed stationary and singular points studied here are key features for such optical flow detection.
Here we have focused on zero-dimensional phase gradient singularities to measure topological fiducial points such as maxima, minima and saddles. Extended stationary curves in the phase can also be measured, such as edges and ridges, which are useful for feature identification and image segmentation in particular. Edges can be detected and linked using Canny’s algorithm  involving the first order Sobel operator and, importantly, the phase gradient direction, which are both directly measured in our experiments. Ridge-like features in the phase require second order gradients computed from a Hessian matrix . Whilst high order numerical derivatives are sensitive to noise, our experiments directly measure phase gradients, hence the numerical operators required for phase ridge detection will be first order only. As seen in Fig. 3, any experimental image will have slight variations along a stationary curve, resulting in a sequence of maxima and saddle points.
We have imaged topological features in x-ray wavefields using the single-grid, single-exposure method of x-ray imaging to capture phase gradients in both transverse directions. These two differential phase images enable full characterisation of the wavefield phase, with stationary and singular points easily located and identified by plotting the angle of the phase gradient. In the case of imaging a weakly-attenuating sample, the singularities in the angle of the phase gradient enable accurate location and identification of phase features, and the tracking of objects/features in an image sequence. Visualizing the magnitude of the phase gradient provides a single image in which large and small phase variations are rendered with no directional bias, which is particularly useful in time-dependent applications where tomography is not possible and a sequence of projection images is instead acquired.
Australian Research Council DECRA (DE120102571); Veski VPRF; German Excellence Initiative; European Union Seventh Framework Programme; (n$^\circ$ 291763); MS McLeod Postdoctoral Fellowship; Cure4CF Foundation; Australian Synchrotron International Synchrotron Access Program (ISAP).
KM was supported by an ARC DECRA, a Veski VPRF, and completed this work with the support of the Technische Universität München Institute for Advanced Study, funded by the German Excellence Initiative and the European Union Seventh Framework Programme and co-funded by the European Union. MD is supported by a MS McLeod Postdoctoral Fellowship and DWP, MD and NF are supported in part by funding from the Cure4CF Foundation. We acknowledge the Japan Synchrotron Radiation Research Institute (JASRI), where imaging was conducted under proposals 2013A1352 [Fig. 5], 2013B1764 [Fig. 3] and 2015A1325 [Fig. 6]. Travel was supported by the Australian Synchrotron International Synchrotron Access Program (ISAP). The ISAP is an initiative of the Australian Government being conducted as a part of the NCRIS. Animal imaging was conducted with permission from the Japan Synchrotron Radiation Institute (JASRI) and the Women’s and Children’s Health Network (AE805c and AE980).
We thank Christian David, Simon Rutishauser and Vitaliy A. Guzenko for creating the phase gratings. We also thank beamline scientists Yoshio Suzuki, Akihisa Takeuchi, Kentaro Uesugi and Naoto Yagi for their assistance with the experiments at the SPring-8 synchrotron, and Isobel Aloisio and David Macindoe for their help during imaging.
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