# Quantifying the x-ray dark-field signal in single-grid imaging

Ying Ying How  1, * and Kaye Morgan1  1School of Physics and Astronomy, Monash
University, Clayton, VIC, 3800, Australia<EMAIL_ADDRESS>
††journal: oe††articletype: Research Article

X-ray dark-field imaging reveals the sample microstructure that is unresolved
when using conventional methods of x-ray imaging. In this paper, we derive a
new method to extract and quantify the x-ray dark-field signal collected using
a single-grid imaging set-up, and relate the signal strength to the number of
sample microstructures, $N$. This was achieved by modelling sample-induced
changes to the shadow of the upstream grid, and fitting experimental data to
this model. Our results suggested that the dark-field scattering angle from
our spherical microstructures is proportional to ${}^{2.19}\sqrt{N}$, which
deviated from the theoretical model of $\sqrt{N}$, but was not inconsistent
with results from other experimental methods. We believe the approach outlined
here can equip quantitative dark-field imaging of small samples, particularly
in cases where only one sample exposure is possible, either due to sample
movement or radiation dose limitations. Future directions include an extension
into directional dark-field imaging.

##  1 Introduction

X-ray imaging has become a widely-adopted technique used to reveal the
internal structure of a sample through a non-invasive approach [1]. It has
been widely used in clinical practice, material science and security screening
[2]. The image contrast in conventional x-ray imaging originates from
differences in the amount of attenuation introduced by each material in the
sample. For a sample made up of highly attenuating materials, such as bone or
metal, conventional x-ray imaging can provide images of high quality. However,
when the sample is made up of weakly attenuating materials such as soft
biological tissues or plastics, the image contrast will be degraded
significantly. This limitation can be addressed by phase contrast x-ray
imaging (PCXI), which is a technique that converts the phase shift experienced
by the x-ray wavefield as it passes through the sample into intensity
modulations to reveal details that are invisible when using conventional x-ray
imaging [3]. There are a number of experimental set-ups by which this phase
contrast signal can be captured.

In addition to the attenuation and phase signals, the dark-field signal can
also be captured using PCXI set-ups, including analyser-based [4], grating
interferometers [5], edge-illumination [6, 7], single-grating/speckle-based
[8, 9, 10] and propagation-based [11]. The dark-field signal originates from
small-angle or ultra-small-angle x-ray scattering (SAXS/USAXS) from unresolved
sample features and provides a complementary picture of the sample properties,
often showing up features that are missing in conventional and phase contrast
x-ray images (e.g. [12]). The dark-field signal is useful since it reveals
information about sub-micron features in a sample without having to fully
resolve those features, which makes it a dose-efficient modality as a detector
with large pixels (relative to the sample features) can be used. The dark-
field contrast modality can also address the limitation that any measurement
of the size of a sample feature will be restricted by the pixel size and the
spatial resolution of the imaging system. Therefore, dark-field signal has the
potential to play an important role in various fields, especially for medical
diagnosis [13, 14, 15], material science [16, 17] and airport screening [18],
with particular benefit seen from the fact it is non-invasive. While dark-
field images provide significant value in a qualitative sense, it can also be
useful to consider quantifying the dark-field signal to extract more
information about the sample.

The dark-field signal has been successfully quantified for several dark-field
imaging techniques, including measuring changes in the width of the rocking
curve with analyser-based imaging (ABI) [19, 20, 4, 21] or equivalently in
edge-illumination (EI) [22] and measuring a relative change in the visibility
of a reference pattern in grating interferometry (GI) [5, 23, 24, 25]. The
dark-field signal extracted from GI is also related to the sample
microstructure size via the linear diffusion coefficient in computed
tomography [24, 25] and the correlation length [26, 27, 28] for specific set-
ups. Higher-order statistical measures of the scattering distribution, e.g.
the kurtosis, can also depend on microstructure size [29]. The dark-field
signal has been related to the sample microstructure properties in a few
experimental tests and applications using these techniques [30, 26, 28, 27,
29, 14]. The dark-field signal described by angular spread can be related to
the number of scattering interfaces via von Nardroff’s equation [31]. Note
that when material microstructure is elongated and aligned, resulting in an
asymmetrical dark-field signal, this directional dark-field signal has been
extracted by capturing the dark-field signal at different orientations of the
sample relative to the gratings [30].

In this paper, we will focus on single-grid imaging, which is an emerging
dark-field imaging technique where the dark-field signal has been extracted
qualitatively [32, 8, 9], but not yet quantitatively. The main advantage of
single-grid imaging is that only a single sample exposure is required during
the data acquisition process. The short data acquisition time can help in
minimising any motion blur and thus makes this technique feasible for dynamic
imaging and time sequence imaging. This technique is closely related to
single-exposure speckle-based imaging [33, 34, 35], where the reference grid
is simply replaced by a piece of sandpaper to create reference speckle. In
this manuscript, we demonstrate the quantification of the x-ray dark-field
signal in single-grid imaging and relate the signal to the number of sample
microstructures.

We first describe the mathematical modelling of the dark-field signal in
single-grid imaging and derive a method for extracting this signal using an
explicit cross-correlation approach. This is achieved by modelling the
intensity pattern seen behind a grid with a sinusoidal function and looking at
how this is reflected in the local cross-correlation between a section of the
intensity pattern seen with and without the sample present. We then apply the
numerical analysis to synchrotron images, converting visibility loss into an
effective scattering angle and relating the angle to the number of
microstructures in the sample. Finally, we discuss conditions for quantitative
dark-field extraction with a single sample exposure, apply the technique to
edge-scattering and to speckle-based images, and look at future directions.

##  2 Mathematical model for dark-field effects in single-grid imaging

This method of analysis is based on a ‘single-grid’ experimental set-up, as
shown in Fig. 1, where an absorption grid is placed in the illuminating beam,
some distance upstream of a detector, and a reference image captured. The
sample is then introduced close to the absorption grid, such that with
propagation it distorts the grid pattern according to the phase properties of
the sample and blurs the grid pattern according to the dark-field properties
of the sample [36, 32, 8, 37, 9].

###  2.1 Modelling the intensity behind a grid

We start by modelling the x-ray intensity seen behind the grid with and
without the sample, in one-dimension. Although for an ideal system, we might
model the intensity behind a grid as a square wave, we modelled the intensity
of the grid as a sinusoidal function here as we observed that the intensity
behind the grid is better modelled by a sinusoid in the experimental data. It
is worth noting that any regular intensity oscillations, for example, a
smoothed square or triangular wave, captured in an experimental setting
(finite point spread function and source size) are likely to be well-modelled
by a sinusoid, when sampled at around 7-15 pixels per period (typical for this
imaging method to balance spatial resolution and range of sensitivity). This
is also an appropriate model for the periodic intensity produced by a phase
grid [9, 38, 39].

We define the intensity seen in the presence of the grid as a function of
position $x$ across the image, $I_{g}(x)$, as a sine function,

$I_{g}(x)=a\sin\left(\frac{2\pi x}{p}\right)+b,$ (1)

where $a$ represents the amplitude, $b$ represents the mean and $p$ represents
the period of the grid. We will refer to $I_{g}(x)$ as the ‘grid intensity’.
When the sample is introduced, the intensity captured by the detector is
shifted transversely and more importantly, has a lower mean gray value and
smaller amplitude oscillations due to the presence of sample. This suggests
that we can model the ‘sample-grid intensity’, $I_{sg}(x)$, which refers to
the intensity seen in the presence of the grid and sample, with

$I_{sg}(x)=Aa\sin\left(\frac{2\pi x}{p}\right)+tb,$ (2)

where $A$ represents the change in the amplitude of the grid intensity
oscillations that is introduced by the sample and $t$ represents the
transmission of x-ray wavefield passing through the sample, which can have a
value ranging from 0 to 1, where 0 means no x-ray wavefield passes through the
sample and 1 means the x-ray wavefield is not attenuated. The phase shift of
the x-ray wavefield is not taken into consideration here since the sample-grid
intensity will effectively be shifted to align with the grid intensity during
cross-correlation analysis, as explained later, before we perform the dark-
field retrieval. A cross-correlation can be used in single-exposure
quantitative single-grid or speckle-based phase retrieval [34, 9], to compare
the local ‘grid intensity’ with the ‘sample-grid intensity’, and we use the
same approach here for dark-field retrieval, for reasons described in the
discussion.

The next step is to perform a cross-correlation between $I_{sg}(x)$ and
$I_{g}(x)$, and then also $I_{g}(x)$ with itself to allow normalisation of
variations in the visibility and local mean intensity of the reference grid
image. These two cross-correlations result in $k(i)$,

$\begin{split}k(i)=(I_{sg}\star
I_{g})(i)&=\int^{p}_{0}\int^{p}_{0}\overline{I_{sg}(x-i)}I_{g}(x)\,dx\,dy\\\
k(i)&=\frac{1}{2}Aa^{2}p^{2}\cos\left(\frac{2\pi
i}{p}\right)+b^{2}p^{2}t,\end{split}$ (3)

and $h(i)$,

$\begin{split}h(i)=(I_{g}\star
I_{g})(i)&=\int^{p}_{0}\int^{p}_{0}\overline{I_{g}(x-i)}I_{g}(x)\,dx\,dy\\\
h(i)&=\frac{1}{2}a^{2}p^{2}\cos\left(\frac{2\pi
i}{p}\right)+b^{2}p^{2},\end{split}$ (4)

respectively. The cross-correlation is performed over one period of grid-
induced intensity oscillations instead from $-\infty$ to $\infty$ since the
sinusoidal function does not converge at infinity, and the cross-correlation
is typically performed across one period of the grid within an experimental
image.

By comparing the amplitude of $k(i)$ to $h(i)$, we can obtain

$\frac{\text{Amplitude(k)}}{\text{Amplitude(h)}}=\frac{\frac{1}{2}Aa^{2}p^{2}}{\frac{1}{2}a^{2}p^{2}}=A,$
(5)

and by comparing the mean of $k(i)$ to $h(i)$, we can obtain

$\frac{\text{Mean(k)}}{\text{Mean(h)}}=\frac{b^{2}p^{2}t}{b^{2}p^{2}}=t.$ (6)

###  2.2 Modelling how the dark-field is reflected in the cross-correlation
analysis

The dark-field signal, $DF$, extracted from a grating-based technique is
defined as the relative change in visibility between the sample-grid intensity
(or stepping curve), $V^{s}$, and the grid-intensity (or stepping curve),
$V^{r}$ [5],

$\begin{split}DF&=\frac{V^{s}}{V^{r}}=\frac{\frac{I^{s}_{max}-I^{s}_{min}}{I^{s}_{max}+I^{s}_{min}}}{\frac{I^{r}_{max}-I^{r}_{min}}{I^{r}_{max}+I^{r}_{min}}}=\frac{\frac{Aa}{tb}}{\frac{a}{b}}=\frac{A}{t},\end{split}$
(7)

where we have used Michelson’s definition of visibility,
$V=(I_{max}-I_{min})/(I_{max}+I_{min})$, where $I_{max}$ is the local maximum
intensity and $I_{min}$ is the local minimum intensity [40].

We now have a way to measure $A$ and $t$ from the grid-only and sample-grid
images (Section 2.1), and to use $A$ and $t$ to quantify changes in visibility
of the reference pattern (Eqn. 7). This would be sufficient to collect a
visibility-reduction image to describe the dark-field, as typically used in
grating interferometry. However we would like to relate the dark-field to
quantities that describe the sample, so the next step is to link visibility to
a less set-up-specific quantity. We do this by understanding how the set-up
parameters such as the propagation distance and grid period relate to the
collected visibility-reduction dark-field signal.

###  2.3 Modelling how the dark-field signal depends on propagation distance
and blur width

The dark-field effect can be modelled as a local blurring of the collected
intensity image [41, 42], and we can define the blur width geometrically as
shown in Fig. 1. Since we assume the effective scattering angle, $\theta$, to
be very small, it can be written as,

$\begin{split}\tan\left(\frac{\theta}{2}\right)\approx\frac{\theta}{2}&=\frac{d/2}{z}\\\
\theta&=\frac{d}{z},\end{split}$ (8)

which can be rearranged into $d=z\theta$, where $d$ is the blur width and $z$
is the distance between the sample and detector (propagation distance).

We can model the blurring kernel applied to the grid pattern in the presence
of sample as a normalised Gaussian function centred at zero, $G(x)$ [43,
Figure 5][44, 45],

$G(x)=\frac{1}{\sqrt{2\pi}(d/2)}\exp{\left(-\frac{x^{2}}{2(d/2)^{2}}\right)},$
(9)

where we have set mean $\mu=0$ and we model the blur width, $d$ as twice the
root-mean-squared width of the Gaussian (equivalently the standard deviation),
$\sigma$. The sample-grid intensity can then be modelled by the convolution
between the grid intensity and $G(x)$,

$\begin{split}I_{sg}(x)=I_{g}(x)*G(x)=a\exp\left(-\frac{2\pi^{2}(d/2)^{2}}{p^{2}}\right)\sin\left(\frac{2\pi
i}{p}\right)+b,\end{split}$ (10)

where $i$ is the coordinate in the convolution space. The dark-field signal
(defined in Eqn (7)) is given by,

$DF=\frac{V^{s}}{V^{r}}=\frac{a\exp\left(-\frac{2\pi^{2}(d/2)^{2}}{p^{2}}\right)}{b}\frac{b}{a}=\exp\left(-\frac{2\pi^{2}(d/2)^{2}}{p^{2}}\right)=\exp\left(-\frac{2\pi^{2}z^{2}\theta^{2}}{4p^{2}}\right),$
(11)

where we substituted the definition of scattering width, $d=z\theta$. Note
that a sin or a cos function can be used interchangeably in Eqns. 2 and 10,
since the cross-correlation sinusoid is centred during the analysis process.

###  2.4 Modelling how the dark-field signal relates to the sample properties

We now have a measure of the dark-field, $\theta$, which is independent of the
set-up used to take that measurement. The next step is to relate that
measurement to properties of the sample microstructure. It is possible to
model the microstructures as many spheres, in which case the average total
deviation angle of an x-ray passing through a number of spheres, $\alpha_{0}$
is described by [31]

$\alpha_{0}=\left(4\delta^{2}N(\log{\frac{2}{\delta}+1})\right)^{1/2},$ (12)

where $\delta$ is the complex refractive index decrement of the sphere
material and $N$ is the number of microstructures an x-ray beamlet encounters
while propagating along the optical axis. By assuming the sample
microstructures, which in our experiment are microspheres, to have the same
diameter, we can say that the total thickness of the microspheres, $T$ is
directly proportional to the number of microstructures, $N$. This implies that
the effective scattering angle $\theta$ is directly proportional to $\sqrt{T}$
in the case where all the scatterers are of the same size and material.

##  3 Experimental details

We captured an experimental dataset to test this approach to dark-field
retrieval and the quantitative nature of the extracted dark-field measure,
over a range of propagation distances and hence set-up sensitivities.

###  3.1 Experimental setup

The setup of our experiment is the same as the typical single-grid imaging
technique, with the addition that we allowed the propagation distance to
change as shown in Fig. 1. The experiment was performed at the Australian
Synchrotron Imaging and Medical Beamline (IMBL). An attenuating grid (a
geological stainless steel sieve with holes of 90 $\mu$m) was placed 150cm
upstream of the sample (as close as the set-up permitted). The sample was
allowed to move along the $x$ axis so that it could be moved out of the field
of view to capture a grid-only (reference pattern) image. Both the grid and
sample were placed together on a table which could be moved to adjust the
distance $z$ between the sample and the detector, which was on a separate
table. The energy of the x-ray beam used was 30 keV and the exposure time for
each image was 650 ms (chosen to fill the dynamic range of the detector). The
effective pixel size for the setup was 11.5 $\mu$m, using a 25 $\mu$m thick
Gadox phosphor coupled to a pco.edge 5.5 sCMOS detector.

Figure 1: Set-up for single-grid imaging with detector being placed at
multiple propagation distances, $z$. The sample can be placed immediately
upstream or downstream of the grid. The linear relationship between blur
width, $d$ and $z$ is described by Eqn. 8. The sample introduces attenuation
and significant blurring on the grid pattern formed on the detector especially
at larger propagation distance. Note that the scattering of the x-ray has be
exaggerated in this figure for visualisation purpose.

###  3.2 Sample preparation

Our sample began as a PMMA sample tube containing polystyrene microspheres
suspended in 10ml of water, with the spheres making up 2.5% of the volume.
Prior to imaging, the samples were spun in a centrifuge for 3 minutes or
longer until most of the microspheres precipitated at the bottom of the tubes
and formed a solid and an aqueous layer. We then removed the solution using a
pipette and left the cap off to allow the liquid to evaporate. Two tubes were
imaged, one containing microspheres of 1.0 $\mu$m diameter and the other
containing microspheres of 8.0 $\mu$m diameter. Images were captured at
propagation distances ranging from 0.2m to 2.7m with an increment of 0.1m and
also at 0.25m to improve sampling at short distance. At each propagation
distance, we took 30 exposures of the ‘grid intensity’ and 30 of the ‘sample-
grid intensity’ and averaged each to improve the signal-to-noise ratio.

##  4 Numerical analysis/results

Before analysing the images, we performed a few pre-processing steps including
adjusting for temporal changes in flux, adjusting for grid movement and
calculating the grid period in pixels. We then performed cross-correlation
analyses using the spatial mapping method proposed by Morgan et al. [37],
which determines the distortion of reference pattern directly in real space on
an area-by-area basis. We then fitted Eqns. 3 and 4 to the cross-correlation
profiles in $x$\- and $y$-direction to extract the amplitude and mean of each
profile, while fixing $p$ as the period of the grid and setting $i=0$ as the
coordinate of the maximum of the cross-correlation, with sub-pixel accuracy,
to account for phase-induced shifts. Examples of this fitting process
performed at pixels with and without the presence of sample, along with a
histogram to show the confidence of the fitting process are shown in Figs. S2,
S3 and S1 respectively in the Supplementary Materials.

###  4.1 Extracting the sample-induced visibility loss

The next step following the fitting is to extract the change in visibility
induced by the sample from the cross-correlation result. It is worth
mentioning that this method is designed for isotropic scattering which gives
rise to dark-field signal that is not directional. This means that the dark-
field signal in each direction is expected to be the same.

Since our samples are microspheres, we made a reasonable assumption that the
blurring of the grid pattern imposed by the sample is symmetric. Following
this assumption, we can simplify the 2D cross-correlation result into 1D by
focusing on the central row or column of the phase-adjusted cross-correlation
when we were trying to fit for the amplitude and mean values. In addition,
performing a 2D fit would be more time-consuming and may also be more
sensitive to noise. Given that most microstructure is not both elongated and
oriented all in the same direction, this approach will work for most samples.

To calculate how the amplitude of the intensity oscillations changes, $A$, we
calculate the RMS amplitude in each of the two directions (ie.
$\textrm{Amplitude}(k)=\sqrt{\textrm{Amplitude}(k_{x})^{2}+\textrm{Amplitude}(k_{y})^{2}}$),
and then follow Eqn. 5. If there is indeed a directional dark-field signal,
the RMS measure accurately extracts the typical reduction in visibility. To
obtain the transmission value, $t$, we followed Eqn. 6, and then simply
averaged the transmission from each of the $x$\- and $y$-direction fits, since
no asymmetry should be present in a scalar value. The transmission $t$ was
corrected in this dataset by an addition of 0.003 to account for the effect of
the wide point spread function of the detector, which resulted in transmission
value lower than the unity at areas where the sample is not present. This
effect should ideally be corrected by performing a deconvolution between the
measured point spread function and the raw images, if the measured point
spread function is available. The dark-field signal is then calculated by
dividing the fitted $A$ by $t$, as shown in Eqn 7.

###  4.2 Converting visibility measurements to effective scattering angle

The images taken at different distances are analysed numerically through the
same procedure mentioned in the previous sections to give us a total of 27
dark-field images that each show the change in the visibility of the grid
pattern induced by the sample. We then extracted the effective scattering
angle by plotting how the dark-field visibility loss at each pixel changes
with propagation distance, and fitting to Eqn. 11, as shown in Fig. 2. While
we retrieved the dark-field signal pixel-by-pixel, since the dark-field signal
measured across the neighbouring pixels within the same period of the grid
were expected to be similar, we chose to smooth the dark-field signal images
at each distance with an averaging kernel of the same size as the grid period
(13 pixels $\times$ 13 pixels) to reduce the noise in the retrieved dark-field
images. The effect of averaging the dark-field image can be seen in Fig. 2,
where the dark-field signals at different distances gave a closer fit after
averaging (Fig. 2 (b) & 2 (d)) compared to the more noisy signals without any
averaging (Fig. 2 (a) & 2 (c)).

The dark-field visibility signal remained constant at 1.0 when no sample is
present (ie. no loss in visibility, see Fig. 2 (c) & (d)), but decreases as
the propagation distance increases with the presence of sample (Fig. 2 (a) &
(b)). This is due to the fact that the blurring width is directly proportional
to the propagation distance.

Figure 2: A typical example of the dark-field signal measured as a loss in
visibility at one pixel from 27 propagation distances (a) & (c) before and (b)
& (d) after smoothing with a mean square kernel of size 13 pixels. Stronger
dark-field signals are measured from pixels in the shadow of the sample ((a) &
(b)) compared to pixels outside the shadow of the sample ((c) & (d)). The
effective scattering angle extracted by fitting Eqn. 11 to (a) and (b) were
$36.94\mu$rad and $35.25\mu$rad, and $0.0006437\mu$rad and $0.2716\mu$rad on
(c) and (d).

###  4.3 Relating the effective scattering angle to sample properties

The effective scattering angle is then related to the thickness of the sample,
$T$, which is directly proportional to the number of microstructures, $N$,
since the microstructures in our experiment are all the same size. The
thickness of the sample (Fig. 3 (b)) is recovered from the transmission image
obtained from images taken at a propagation distance of 2.7m (Fig. 3 (a)) by
using the Transport of Intensity Equation (TIE)-based phase retrieval
algorithm [46], using $\beta=9.2028\times 10^{-11}$ and $\delta=2.6286\times
10^{-7}$(polystyrene at 30 keV). This provides a way to test if our results
follow the trend described by Eqn. 12, since $N$, the number of
microstructures in projection, is proportional to $T$.

Figure 3: (a) Transmission image extracted from single-grid images of two
tubes of polystyrene microspheres taken at a propagation distance of 2.7m,
after correction for the wide point spread function. (b) Projected thickness
recovered from (a) using TIE-based single-image phase retrieval [47].

##  5 Results

We performed the curve fitting analysis on the images taken at 27 different
propagation distances to extract the change in amplitude, $A$ and
transmission, $t$. Examples of $A$ and $t$ extracted from images taken at two
different propagation distances - $0.2$m and $1.4$m are shown in Fig. 4
(a),(b) and (c),(d). The dark-field signal, $DF$ is then calculated by
dividing $A$ by $t$ as suggested in Eqn. 7 and the result is shown in Fig. 4
(e) and (f).

Figure 4: Dark-field imaging results obtained from single-grid images of two
tubes of polystyrene microspheres, shown here for two different propagation
distances - $0.2$m and $1.4$m. The change in amplitude of intensity
oscillations, $A$ ((a) & (b)), is divided by the corresponding transmission of
x-ray wavefield, $t$ ((c) & (d)) to obtain the dark-field signal, $DF$ ((e) &
(f)). The scattering angles (from the sample tube on the left), $\theta$ shown
in panel (g) and (h) are extracted from the $DF$ signal in panel (e) and (f)
respectively using Eqn. 11 while $\theta$ in panel (i) is extracted from the
$DF$ images measured at all 27 distances shown in Fig. 2. The $\theta$ image
extracted from the larger propagation distance has less noise than the smaller
propagation distance, where the blurring effect is weak. The $\theta$ image
retrieved using dark-field signal measured from 27 distances has less noise
compared to $\theta$ image retrieved from single distance.

###  5.1 Effective scattering angle from 27 exposures

To examine the quantitativeness of the extracted dark-field scattering angle,
we look at how well this data fits the dark-field scattering model proposed by
von Nardroff. Figure 5 plots the scattering angle extracted from the dark-
field signal measured at 27 distances (as shown in Fig. 4 (i)) as a function
of the sample thickness (see Section 4.3), corrected to remove the simulated
sample tube thickness. We fitted the data with $\theta=m\times{T}^{1/p}$,
where $m$ and $p$ were allowed to vary. From Fig. 5, we can see that the data
was best fitted with $\theta=m\times^{2.19}\sqrt{T}$ (see Fig. S5 for how the
coefficient of determination ($R^{2}$) value changes with different p values)
instead of $\theta=m\sqrt{T}$, where $m$ is a constant, as proposed by von
Nardroff [31]. This suggested that the effective scattering angle measured
here, $\theta$, is proportional to ${}^{2.19}\sqrt{N}$.

Figure 5: The scattering angle, $\theta$ as a function the sample thickness,
$T$, which is proportional to the number of microstructures $N$. The plot is
fitted with $\theta=m^{p}\sqrt{T}$, where the exponent of $T$ is allowed to
vary, and with $\theta=m\sqrt{T}$, where $m$ is a constant, as suggested by
von Nardroff [31].

In Fig. 6, we aim to show how well each pixel in the image fits the model. The
data points that are located within the range of 4 $\mu$rad from the line of
best fit are labelled in cyan and are distributed evenly across the image.

Figure 6: Colour-coded mapping of the pixel values within the image of the
left tube, showing how well each pixel in (a) the $\theta$ image retrieved
from $DF$ measured at 27 distances (i.e Fig. 4(i)) are described by our model
$\theta=m^{2.19}\sqrt{T}$. The majority of pixels in (a) is coloured in cyan,
indicating that they are well described by the model and sit within the
positive and negative $4\mu$rad offset from the model. Note that there are a
small amount of data points coloured in pink located at the bottom right of
the plot in (b). The thickness and scattering angle pixel values are shown for
the entirety of the experimental image in Fig. S4.

###  5.2 Effective scattering angle from a single sample exposure

Given the motivation to minimise imaging time and dose and the single-exposure
nature of the single-grid technique, it is interesting to examine how
quantitative the results are at a single distance instead of using images from
multiple distances. The scattering angle, $\theta$ was extracted from images
obtained at propagation distance of 0.2m, 0.7m, 1.4m, 2.1m and 2.7m. Figure 7
shows how the scattering angle extracted at each single distance relates to
the sample thickness (i.e number of sample microstructures). Examples of the
scattering angle extracted at 0.2m and 1.4m are shown in Fig. 4(g) and 4(h)
respectively.

Figure 7: The scattering angle, $\theta$ as a function of the sample
thickness, $T$ (which is proportional to $N$), extracted from dark-field
signals measured at images taken at a single distance of (a) 0.2m (b) 0.7m (c)
1.4m (d) 2.1m and (e) 2.7m . The line of best fit is
$\theta\propto^{p}\sqrt{T}$, where $p$ corresponds to $2.456$, $2.204$,
$2.244$, $2.539$ and $2.627$ respectively. The shorter distances show
significant noise and the larger distances show saturation for large
thicknesses, suggesting that quantitative dark-field imaging with single
exposure needs to be performed with a reasonable (not too short or too long)
propagation distance to obtain consistent quantitative results.

A significantly wider spread in scattering angle is extracted at shorter
distances, such as 0.2m and 0.7m (Fig. 7(a) and (b)). A saturation in the
dark-field signal/scattering angle is observed from images taken at larger
distances such as 2.1m and 2.7m (Fig. 7(d) and (e)).

The magnitude of the spread in scattering angle at a given thickness is
related to the position of the data points in the plot of $DF$ against $z$
(see Fig. 2 (b)). The fit is more robust to noise for data points at larger
propagation distances where the blurring effect is strong and easily measured,
which results in less uncertainty in the $\theta$ values extracted. However,
if the propagation distance is too large and the grid pattern is so smoothed
that it reaches comparable visibility to sample features, then the dark-field
signal begins to saturate and hence does not correctly predict the scattering
angle for those large thicknesses where the signal is strongest. This effect
can be seen in the flattening of the curve in Fig. 7 (e).

##  6 Further applications of the approach

While the key aim of this paper was to design an analytical approach to
quantitatively retrieve x-ray dark-field from a single-grid set-up, we also
want to show how well this approach works in a range of scenarios.

###  6.1 Dark-field from edges

Dark-field signal is primarily seen from microstructure, but can also be seen
from sample edges. To test the ability of our technique to detect dark-field
from edges, we examined some high-resolution single-grid images. Figure 8 (a)
shows a raw image taken using a phase grid to study a gold EM grid and PMMA
spheres, also used in [48], using 25 keV x-rays and 0.722 $\mu$m effective
pixel size. From the magnified image in Fig. 8 (b), we can see that the grid
pattern has lower visibility near the edges of the PMMA spheres due to the
edge scattering effect. Figure 8 (c) shows the dark-field visibility signal,
$DF$, extracted from Fig. 8 (a) using the algorithm presented in this paper.
The blurring of the grid pattern near the edges of the PMMA spheres had been
successfully detected as indicated by the stronger signals near the edges of
the spheres as shown in Fig. 8 (c). It is worth mentioning that dark-field
from edges was not observed in the dark-field images extracted from the
microsphere sample that this manuscript focuses on. We believe this is due to
the fact that the dark-field signals produced by our microspheres were much
stronger than the edge effect near the edges of the sample tube and thus the
edge effect was not visible in our dark-field images. In addition, the pixel
size used in the experiment of Fig. 8 was 0.722 $\mu$m [48], which was only 6%
of the pixel size used to image the microsphere sample, and thus provided a
higher sensitivity to the dark-field signal. Our algorithm is therefore able
to detect the blurring of grid pattern due to the scattering of the x-ray
wavefield from the edges of an object.

Figure 8: (a) Raw image of a phase grid illuminating a gold EM grating and
PMMA spheres[48] (see main text for experimental details). (b) Magnified
region from the yellow rectangular area in (a), showing reduced visibility of
the grid pattern due to edge scattering. (c) The dark-field signal $DF$
extracted from (a) using the algorithm presented here.

###  6.2 Application to speckle-based imaging

Theoretically, the approach outlined in this paper could be applied to images
captured using any reference pattern. This is because the dark-field signal is
extracted from the cross-correlation analyses, which works equally well with
any type of reference pattern [37, 34] and does not require a periodic
reference in the way a Fourier technique does. Furthermore, the sine function
we use to fit the local intensity is an appropriate model for any local
oscillation in intensity. Therefore, we tested our algorithm on a speckle-
based dataset [34, 33, 35], acquired at the European Synchrotron Radiation
Facility (ESRF), shown in both Berujon et al., 2016 [49] and Pavlov et al.,
2020 [50], where sandpaper with grit size P800 was used in the system as a
reference pattern instead of an attenuating grid. The sample used in this
experiment was a red currant, imaged with 17keV x-rays, and the effective
pixel size of the setup was 5.8 $\mu$m. The sample was placed 0.5m downstream
of the sandpaper and 1m upstream of the detector. Images were taken with the
sandpaper at 6 different transverse locations with and without the sample.

Figure 9 (a) and 9 (c) shows the raw speckle-only image and sample-speckle
image (along with the magnified image of the yellow-rectangular area in the
corresponding image) at one position of the sandpaper. In Fig. 9 (d), the
sample has locally reduced the visibility of the speckle pattern. We performed
the analysis described in this manuscript using the speckle-only image and the
sample-speckle image with the sandpaper located in the same position. Here, to
reduce the noise in the images, we show the median value of each pixel over
the 6 images for both the transmission (Fig. 9 (e)) and dark-field (Fig. 9
(f)) images. We observed dark-field signal near the edges of the sample,
inside the currant where the seed/stem sits, and also from the mounting under
the currant. The dark-field image also agrees qualitatively with that seen in
Pavlov et al., 2020 [50], where an intrinsic approach is used to capture the
dark-field, indicating that our approach is also applicable to speckle-based
imaging.

Figure 9: The raw (a) speckle-only image and (b) magnified region from the
yellow rectangular area in (a) [49]. The raw (c) sample-speckle image and (d)
magnified region from the yellow rectangular area in (c), where the sample is
a red currant (see main text for experimental details). Images of (e)
transmission, $t$ and (f) the dark-field signal, $DF$ are extracted using the
approach described in this manuscript, and taking the median value over 6
images.

##  7 Discussion

In this paper, we have designed and demonstrated a new approach to extract and
quantify the dark-field signal using a single-grid imaging set-up. This has
been achieved by constructing a mathematical model (Section 2.1) and designing
a numerical algorithm (Section 4) to extract quantitative dark-field images.
We see qualitative agreement between the results reported in Section 5 and
dark-field images seen at other set-ups, with signal generated by unresolved
microstructure. In addition, we have shown quantitative dark-field signal
retrieval from a single-grid PCXI set-up. The recovered dark-field signal is
consistent with the model proposed by von Nardroff in that the scattering
angle is roughly proportional to the square root of the number of
microstructures through which the beam passes. We have successfully applied
our algorithm on data obtained using a single-grid imaging setup and on data
obtained using speckle-based imaging. Nevertheless, some unexpected behaviour
was observed in the dark-field signals we extracted. We have split the
remainder of the discussion into a number of sections focused on interesting
observations and future directions.

###  7.1 Accounting for propagation-based edge effects

Some abnormal dark-field values were observed, especially at larger
propagation distances, where a small number of pixels near the edges of the
sample tube had visibility greater than 1, which was outside the range of the
expected values for the dark-field signal ($0-1$). This may be due to the fact
that the propagation-based phase contrast effect is generally not taken into
consideration in the grating-based imaging techniques, including single-grid
imaging. Our model/algorithm has considered the bright fringes near the edges
of the sample tubes that comes from the focusing of the x-ray beam as
“negative” attenuation due to the increase in mean value or amplitude. One
possible way to tackle this problem is by removing the propagation-based phase
contrast effect from the raw images prior to the dark-field analysis, using an
approach similar to that performed by Groenendijk et al. [48]. In the approach
taken by Groenendijk et al., the propagation-based phase effect was simulated
by propagating a wavefield with uniform intensity numerically through the
angular spectrum approach [51] using phase information retrieved from the
images. The simulated propagation-based phase effect was then divided out of
the attenuation images in order to perform material decomposition. Another way
to solve this issue would be by incorporating the propagation-based phase
contrast effects into the retrieval process using the X-ray Fokker-Planck
equation [41] as suggested by Morgan & Paganin [42].

###  7.2 A quantitative dark-field model

A second interesting point to discuss is the extracted quantitative measures
for the scattering angle as a function of the number ($N$)/thickness($T\propto
N$) of microstructures of typical thickness $a$. In Section 5, we have shown
that the scattering angle measured by our data was best-fitted as proportional
to ${}^{2.19}\sqrt{T}\propto^{2.19}\sqrt{N}$ instead of $\sqrt{N}$ as
suggested by von Nardroff [31]. This is not inconsistent with the result
obtained by Kitchen et al. [14], who used an analyser-crytal approach to
measure the scattering angle from PMMA microspheres and found the angle to be
directly proportional to ${}^{2.2627}\sqrt{N}$. The scattering angles in both
cases were proportional to a root with a factor greater than 2 of the number
of microstructures in the beam path. We suspected that the deviation of our
result from the model proposed by von Nardroff may be due to anomalous
diffusion of the scattered x-ray wavefield. To further explore the
relationship between the sample microstructures and the scattering angle, we
could repeat the analysis on the images obtained with other diameter
microstructures.

###  7.3 Optimal single-exposure imaging

In this experiment, we acquired images at 27 different propagation distances
to accurately measure how the scattering angle relates to the number of
microstructures in the sample. However, in practice, it is often useful to use
a single sample exposure only, to minimise motion blurring due to a moving
sample and radiation dose delivered to the sample. A single-exposure approach
also allows time-sequence imaging to capture a dynamic process like a
biological response to a treatment [52] (noting that repeated processes can
use multiple images [53]). Therefore, the question then turns to whether the
propagation distance between the sample and the grid can affect the accuracy
of the results obtained. Generally, it is not recommended to image the sample
with the detector being placed immediately downstream of the sample since this
will result in a weak blurring effect and it is also not recommended to image
the sample with a large propagation distance since this can result in the grid
pattern being blurred out too significantly and the cross-correlation analysis
may fail. In the second case, we have a saturated dark-field signal where the
visibility of the grid falls below the visibility of other features in the
image, and sensitivity to a greater dark-field signal is lost.

In Section 5 and 5.2, we showed the images for the scattering angles extracted
using dark-field signals measured using one distance only, for 0.2m and 1.4m
(see Fig. 4 (g) and (h)) and the plots showing how these dark-field effective
scattering angles measured at a single distance were related to the thickness
of the microspheres (see Fig. 7) respectively. At smaller propagation
distances, we obtained a wider range of scattering angles for a given
thickness compared to larger propagation distances, showing the low
sensitivity at these short distances makes measurements less reliable.
However, the scattering angle obtained with dark-field signals extracted at
shorter distance still varied with sample thickness with a fit that is closely
matched with the fit obtained in Fig. 5, despite having a wider range for each
sample thickness. For the larger propagation distances, it seems the dark-
field is saturated in the sense that we can only just see the grid in the raw
images, and the visible intensity variations are largely PBI speckle from the
microspheres. In Fig. 7 (e), the plot is flattened out for large sample
thickness when it should be increasing in dark-field, which may indicate the
data is not fitting the model anymore. Figure 2 (b) also shows that the
visibility signal does not decrease as expected after around 2m propagation,
suggesting a saturated dark-field signal. While Fig. 7 (d) and 7 (e) show the
scattering angle measured is consistent up to 5mm thickness, the scattering
angle then ‘saturates’ for larger thicknesses. Although Fig. 4 (g) and (h) had
different ranges of values, they are qualitatively similar. In summary,
shorter propagation distance can result in weaker dark-field signal with more
variability, while larger propagation distance typically results in a stronger
dark-field signal with a smaller variability but measurements will not follow
the model laid out here if the visibility signal is saturated. Essentially, we
can tune the sensitivity of the system to the dark-field signal by changing
the propagation distance. Custom tests would be required to determine the
optimum propagation distance for each sample, but we would expect that a
suitable propagation distance would show a noticable drop in grid visibility
across the sample (e.g. 0.5, see Fig. 2).

###  7.4 Directional dark-field imaging

Our approach to quantifying the dark-field signal could be extended into two
dimensions, which would be particularly useful for samples that produce a
directional dark-field signal [30]. The grid-only intensity, $I_{g}(x,y)$ can
be modelled as a two-dimensional sine function and the sample-grid intensity,
$I_{sg}(x,y)$ can be modelled as the convolution between grid-only intensity
with a zero-centred asymmetric Gaussian blurring function. A two-dimensional
fitting can be performed on the cross-correlation results to extract the
magnitude and direction of the blurring contributed by the sample and thus
opening up a new area of study on directional dark-field signal.

###  7.5 Advantages and disadvantages of this approach

The primary advantage of single-grid imaging is that only a single sample
exposure is required for the data acquisition process. This makes the approach
more feasible for dynamic imaging and time sequence imaging, compared to the
other phase contrast x-ray imaging techniques such as analyser-based and
grating-based imaging which require multiple sample exposures to extract the
phase and dark-field signals. Another advantage of single-grid imaging is that
it has a relatively simple setup compared to analyser-based and grating-based
imaging techniques since it only requires one optical element - a grid - and
no calibration or alignment is required prior to the data acquisition process.
In addition, we are able to tune the sensitivity of the system to the dark-
field signal, decreasing the pixel size and/or increasing the propagation
distance to increase sensitivity.

We have chosen an approach that uses spatial-mapping/explicit-tracking by
cross-correlation, as used for phase retrieval by Morgan et al. [37], instead
of a Fourier approach [54], since the Fourier analysis method can fail when
the grid frequency overlaps with the sample feature frequency [37]. The
spatial mapping method can also provide better spatial resolution than the
Fourier analysis (see [9, Figure 4]) since we compare the windows pixel-by-
pixel instead of only focusing on a small area in the Fourier space. The
spatial mapping method can also be applied on images taken with other
reference patterns, such as the sandpaper as shown in Section 6.2.

However, the single-grid imaging technique does have a few disadvantages. One
of the major disadvantages is that this technique requires a relatively small
pixel size to capture both phase and dark-field signals. Generally, the pixel
size has to be smaller than the blurring width to obtain high quality dark-
field images. Thus, this technique is best-suited to small samples (e.g.
millimetre-to-centimetre scale) due to the requirement for a small pixel size.
Although it is technically possible to image large samples, which could be
achieved by using a detector with a large area but small pixel size, it may
not be feasible due to the cost, availability or radiation dose.

###  7.6 Future directions

In addition to the directions mentioned above, there are a few other possible
directions for future investigation. One subsequent investigation would be to
explore how the dark-field signal and the scattering angle change with the
microstructure size. This could be done by imaging additional samples over a
range of microstructure sizes. We could also investigate the dependence of the
dark-field signal on the microstructure material and/or the type of medium
surrounding the microstructures.

Other possible future directions include exploring the relationship between
the dark-field signal with x-ray energy by performing the experiment with an
energy-sensitive detector [55, 56]. Finally, an investigation could look at
quantifying the edge scattering effects.

##  8 Conclusion

The x-ray dark-field signal that originates from the SAXS or USAXS of
unresolved sample features can provide complementary information on sample
properties that is inaccessible from the absorption- and phase-contrast
images. A quantitative dark-field signal can be useful in various fields
including medical diagnosis [14], material science [57] and airport security
[18].

In this manuscript, we proposed a new method to extract and quantify the dark-
field signal from a single-grid imaging set-up, utilising a mathematical model
and a numerical analysis algorithm. The dark-field signal was quantified by
curve-fitting and comparing the cross-correlation results between the grid-
only image with itself and between the grid-only image and sample-grid image.
The quantitative dark-field signal was then converted into the effective
scattering angle and then related to the number of microstructures, $N$, that
the beam passes through. The scattering angle was found to be directly
proportional to ${}^{2.19}\sqrt{N}$, which was roughly consistent with the
model proposed by von Nardroff which suggests that the scattering angle is
proportional to $\sqrt{N}$, and matched closely with analyser-crystal-based
measurements obtained by Kitchen et al. [14]. We showed that the algorithm is
able to detect edge scattering and is also compatible with speckle-based
imaging [34, 33, 35].

The feasibility of quantifying the dark-field signal using a single sample
exposure was investigated in this manuscript. It was observed that the
propagation distance has played an important role in this. We hope that this
easily-implemented single-grid dark-field imaging approach can be of use in
quantitatively imaging mm-cm-sized samples, particularly in the case of time-
sequence imaging or dose-sensitive applications.

The analysis could be improved by incorporating the propagation-based phase
contrast effects into the model through the X-ray Fokker-Planck equation.
Further studies can also be done to find optimal propagation distance for
single sample exposure imaging of a given sample, to extend into two-
dimensions to extract a directional dark-field signal and to investigate the
dark-field signal dependence on sample microstructure size, material and x-ray
energy.

## Acknowledgements

We acknowledge useful discussions with David Paganin and recommended
references from Marcus Kitchen. We thank Celebrity Groenendijk for sharing the
data she had previously published[48], which was analysed in Fig. 8 here and
captured at BL20XU at SPring-8 under application 2019A1151. We also thank
Sebastien Berujon, Eric Ziegler and Emmanuel Brun for sharing the speckle-
based x-ray dataset analysed in Fig. 9, originally published in [49] and also
in [58]. The key images in this paper were captured at the Imaging and Medical
Beamline at the Australian Synchrotron, part of ANSTO.

## Funding

Dr Morgan was supported by FT180100374 from the Australian Research Council.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available
at this time but may be obtained from the authors upon reasonable request.

##

See Supplement 1 for supporting content.

## References

  * [1] W. C. Röntgen, “Uber eine neue art von strahlen,” Sitzungsber Phys Med Ges Wurtzburg 9, 132–141 (1895).
  * [2] D. Jordan, “Handbook of x-ray imaging: Physics and technology,” Medical Physics 47, 2577–2578 (2020).
  * [3] S. Wilkins, Y. I. Nesterets, T. Gureyev, S. Mayo, A. Pogany, and A. Stevenson, “On the evolution and relative merits of hard x-ray phase-contrast imaging methods,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 372, 20130021 (2014).
  * [4] L. Rigon, F. Arfelli, and R.-H. Menk, “Generalized diffraction enhanced imaging to retrieve absorption, refraction and scattering effects,” Journal of Physics D: Applied Physics 40, 3077 (2007).
  * [5] F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nature materials 7, 134–137 (2008).
  * [6] M. Endrizzi, P. C. Diemoz, T. P. Millard, J. Louise Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard x-ray dark-field imaging with incoherent sample illumination,” Applied Physics Letters 104, 024106 (2014).
  * [7] M. Endrizzi, F. Vittoria, L. Rigon, D. Dreossi, F. Iacoviello, P. Shearing, and A. Olivo, “X-ray phase-contrast radiography and tomography with a multiaperture analyzer,” Physical review letters 118, 243902 (2017).
  * [8] H. H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission gratings,” Optics letters 35, 1932–1934 (2010).
  * [9] K. S. Morgan, P. Modregger, S. C. Irvine, S. Rutishauser, V. A. Guzenko, M. Stampanoni, and C. David, “A sensitive x-ray phase contrast technique for rapid imaging using a single phase grid analyzer,” Optics letters 38, 4605–4608 (2013).
  * [10] I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x-ray phase-contrast and dark-field imaging with a laboratory source,” Physical review letters 112, 253903 (2014).
  * [11] T. Gureyev, D. Paganin, S. Taba, S. Lewis, P. Brennan, and H. Quiney, “Dark-field signal extraction in propagation-based phase-contrast imaging,” arXiv preprint arXiv:2003.12248 (2020).
  * [12] N. Matsunaga, K. Yano, M. Endrizzi, and A. Olivo, “Detection of individual sub-pixel features in edge-illumination x-ray phase contrast imaging by means of the dark-field channel,” Journal of Physics D: Applied Physics 53, 095401 (2019).
  * [13] S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg _et al._ , “Emphysema diagnosis using x-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proceedings of the National Academy of Sciences 109, 17880–17885 (2012).
  * [14] M. J. Kitchen, G. A. Buckley, L. T. Kerr, K. L. Lee, K. Uesugi, N. Yagi, and S. B. Hooper, “Emphysema quantified: mapping regional airway dimensions using 2d phase contrast x-ray imaging,” Biomedical Optics Express 11, 4176–4190 (2020).
  * [15] K. Willer, A. Fingerle, W. Noichl, F. De Marco, M. Frank, T. Urban, R. Schick, A. Gustschin, B. Gleich, J. Herzen, T. Koehler, A. Yaroshenko, T. Pralow, G. Zimmermann, B. Renger, A. Sauter, D. Pfeiffer, M. Makowski, E. Rummeny, P. Grenier, and F. Pfeiffer, “X-ray dark-field chest imaging can detect and quantify emphy-sema in copd patients,” medRxiv (2021).
  * [16] F. Yang, F. Prade, M. Griffa, I. Jerjen, C. Di Bella, J. Herzen, A. Sarapata, F. Pfeiffer, and P. Lura, “Dark-field x-ray imaging of unsaturated water transport in porous materials,” Applied Physics Letters 105, 154105 (2014).
  * [17] J. Glinz, J. Šleichrt, D. Kytỳř, S. Ayalur-Karunakaran, S. Zabler, J. Kastner, and S. Senck, “Phase-contrast and dark-field imaging for the inspection of resin-rich areas and fiber orientation in non-crimp vacuum infusion carbon-fiber-reinforced polymers,” Journal of Materials Science 56, 9712–9727 (2021).
  * [18] E. A. Miller, T. A. White, B. S. McDonald, and A. Seifert, “Phase contrast x-ray imaging signatures for security applications,” IEEE Transactions on Nuclear Science 60, 416–422 (2013).
  * [19] M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Physics in Medicine & Biology 48, 3875 (2003).
  * [20] G. Khelashvili, J. G. Brankov, D. Chapman, M. A. Anastasio, Y. Yang, Z. Zhong, and M. N. Wernick, “A physical model of multiple-image radiography,” Physics in Medicine & Biology 51, 221 (2005).
  * [21] M. J. Kitchen, D. M. Paganin, K. Uesugi, B. J. Allison, R. A. Lewis, S. B. Hooper, and K. M. Pavlov, “X-ray phase, absorption and scatter retrieval using two or more phase contrast images,” Optics express 18, 19994–20012 (2010).
  * [22] A. Astolfo, M. Endrizzi, G. Kallon, T. P. Millard, F. A. Vittoria, and A. Olivo, “A first investigation of accuracy, precision and sensitivity of phase-based x-ray dark-field imaging,” Journal of Physics D: Applied Physics 49, 485501 (2016).
  * [23] Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Applied Physics Letters 95, 094105 (2009).
  * [24] M. Bech, O. Bunk, T. Donath, R. Feidenhans, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Physics in Medicine & Biology 55, 5529 (2010).
  * [25] S. K. Lynch, V. Pai, J. Auxier, A. F. Stein, E. E. Bennett, C. K. Kemble, X. Xiao, W.-K. Lee, N. Y. Morgan, and H. H. Wen, “Interpretation of dark-field contrast and particle-size selectivity in grating interferometers,” Applied optics 50, 4310–4319 (2011).
  * [26] F. Prade, A. Yaroshenko, J. Herzen, and F. Pfeiffer, “Short-range order in mesoscale systems probed by x-ray grating interferometry,” EPL (Europhysics Letters) 112, 68002 (2016).
  * [27] R. P. Harti, M. Strobl, B. Betz, K. Jefimovs, M. Kagias, and C. Grünzweig, “Sub-pixel correlation length neutron imaging: Spatially resolved scattering information of microstructures on a macroscopic scale,” Scientific Reports 7, 44588 (2017).
  * [28] S. Gkoumas, P. Villanueva-Perez, Z. Wang, L. Romano, M. Abis, and M. Stampanoni, “A generalized quantitative interpretation of dark-field contrast for highly concentrated microsphere suspensions,” Scientific reports 6, 35259 (2016).
  * [29] P. Modregger, M. Kagias, S. C. Irvine, R. Brönnimann, K. Jefimovs, M. Endrizzi, and A. Olivo, “Interpretation and utility of the moments of small-angle x-ray scattering distributions,” Physical review letters 118, 265501 (2017).
  * [30] T. H. Jensen, M. Bech, O. Bunk, T. Donath, C. David, R. Feidenhans, and F. Pfeiffer, “Directional x-ray dark-field imaging,” Physics in Medicine & Biology 55, 3317 (2010).
  * [31] R. von Nardroff, “Refraction of x-rays by small particles,” Physical Review 28, 240 (1926).
  * [32] E. E. Bennett, R. Kopace, A. F. Stein, and H. Wen, “A grating-based single-shot x-ray phase contrast and diffraction method for in vivo imaging,” Medical physics 37, 6047–6054 (2010).
  * [33] S. Berujon, H. Wang, and K. Sawhney, “X-ray multimodal imaging using a random-phase object,” Physical Review A 86, 063813 (2012).
  * [34] K. S. Morgan, D. M. Paganin, and K. K. Siu, “X-ray phase imaging with a paper analyzer,” Applied Physics Letters 100, 124102 (2012).
  * [35] M.-C. Zdora, “State of the art of x-ray speckle-based phase-contrast and dark-field imaging,” Journal of Imaging 4, 60 (2018).
  * [36] Y. Takeda, W. Yashiro, Y. Suzuki, S. Aoki, T. Hattori, and A. Momose, “X-ray phase imaging with single phase grating,” Japanese journal of applied physics 46, L89 (2007).
  * [37] K. S. Morgan, D. M. Paganin, and K. K. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Optics express 19, 19781–19789 (2011).
  * [38] J. Rizzi, P. Mercere, M. Idir, P. Da Silva, G. Vincent, and J. Primot, “X-ray phase contrast imaging and noise evaluation using a single phase grating interferometer,” Optics Express 21, 17340–17351 (2013).
  * [39] A. Gustschin, M. Riedel, K. Taphorn, C. Petrich, W. Gottwald, W. Noichl, M. Busse, S. E. Francis, F. Beckmann, J. U. Hammel _et al._ , “High resolution and sensitivity bi-directional x-ray phase contrast imaging using 2d talbot array illuminators,” arXiv preprint arXiv:2105.07347 (2021).
  * [40] A. A. Michelson, _Studies in optics_ (Courier Corporation, 1995).
  * [41] D. M. Paganin and K. S. Morgan, “X-ray fokker–planck equation for paraxial imaging,” Scientific reports 9, 1–18 (2019).
  * [42] K. S. Morgan and D. M. Paganin, “Applying the fokker–planck equation to grating-based x-ray phase and dark-field imaging,” Scientific reports 9, 1–14 (2019).
  * [43] M. J. Kitchen, D. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and S. T. Mudie, “On the origin of speckle in x-ray phase contrast images of lung tissue,” Physics in medicine & biology 49, 4335 (2004).
  * [44] A. Khromova, F. Arfelli, H. Besch, H. Plothow-Besch, R. Menk, and L. Rigon, “Monte carlo simulation of x-ray multiple refractive scattering from fine structure objects imaged with the dei technique,” in _IEEE Symposium Conference Record Nuclear Science 2004._ , vol. 6 (IEEE, 2004), pp. 4014–4018.
  * [45] G. Khelashvili, J. G. Brankov, D. Chapman, M. A. Anastasio, Y. Yang, Z. Zhong, and M. N. Wernick, “A physical model of multiple-image radiography,” Physics in Medicine & Biology 51, 221 (2005).
  * [46] D. Paganin, _Coherent X-ray optics_ , 6 (Oxford University Press on Demand, 2006).
  * [47] D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” Journal of microscopy 206, 33–40 (2002).
  * [48] C. F. Groenendijk, F. Schaff, L. C. Croton, M. J. Kitchen, and K. S. Morgan, “Material decomposition from a single x-ray projection via single-grid phase contrast imaging,” Optics Letters 45, 4076–4079 (2020).
  * [49] S. Berujon and E. Ziegler, “X-ray multimodal tomography using speckle-vector tracking,” Physical Review Applied 5, 044014 (2016).
  * [50] K. M. Pavlov, D. M. Paganin, H. T. Li, S. Berujon, H. Rougé-Labriet, and E. Brun, “X-ray multi-modal intrinsic-speckle-tracking,” Journal of Optics 22, 125604 (2020).
  * [51] D. McMorrow and J. Als-Nielsen, _Elements of Modern X-ray physics_ (Wiley Hoboken, 2011).
  * [52] K. S. Morgan, M. Donnelley, N. Farrow, A. Fouras, N. Yagi, Y. Suzuki, A. Takeuchi, K. Uesugi, R. C. Boucher, K. K. Siu _et al._ , “In vivo x-ray imaging reveals improved airway surface hydration after a therapy designed for cystic fibrosis,” American journal of respiratory and critical care medicine 190, 469–472 (2014).
  * [53] R. Gradl, K. Morgan, M. Dierolf, C. Jud, L. Hehn, B. Günther, W. Möller, D. Kutschke, L. Yang, T. Stoeger _et al._ , “Dynamic in vivo chest x-ray dark-field imaging in mice,” IEEE transactions on medical imaging 38, 649–656 (2018).
  * [54] M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” JosA 72, 156–160 (1982).
  * [55] K. Taphorn, F. De Marco, J. Andrejewski, T. Sellerer, F. Pfeiffer, and J. Herzen, “Grating-based spectral x-ray dark-field imaging for correlation with structural size measures,” Scientific Reports 10, 1–9 (2020).
  * [56] T. Sellerer, K. Mechlem, R. Tang, K. A. Taphorn, F. Pfeiffer, and J. Herzen, “Dual-energy x-ray dark-field material decomposition,” IEEE Transactions on Medical Imaging 40, 974–985 (2020).
  * [57] J. Valsecchi, M. Strobl, R. P. Harti, C. Carminati, P. Trtik, A. Kaestner, C. Grünzweig, Z. Wang, K. Jefimovs, and M. Kagias, “Characterization of oriented microstructures through anisotropic small-angle scattering by 2d neutron dark-field imaging,” Communications Physics 3, 1–8 (2020).
  * [58] K. M. Pavlov, D. M. Paganin, K. S. Morgan, H. T. Li, S. Berujon, L. Quénot, and E. Brun, “Directional dark-field implicit x-ray speckle tracking using an anisotropic-diffusion fokker-planck equation,” Physical Review A 104, 053505 (2021).