# Single-exposure x-ray dark-field imaging: quantifying sample microstructure
using a single-grid setup

Ying Ying How  School of Physics and Astronomy, Monash University, Clayton,
VIC 3800, Australia<EMAIL_ADDRESS>David M. Paganin  School of Physics
and Astronomy, Monash University, Clayton, VIC 3800, Australia  Kaye S. Morgan
School of Physics and Astronomy, Monash University, Clayton, VIC 3800,
Australia

###### Abstract

The size of the smallest detectable sample feature in an x-ray imaging system
is usually restricted by the spatial resolution of the system. This limitation
can now be overcome using the diffusive dark-field signal, which is generated
by unresolved phase effects or the ultra-small-angle x-ray scattering from
unresolved sample microstructures. A quantitative measure of this dark-field
signal can be useful in revealing the microstructure size or material for
medical diagnosis, security screening and materials science. Recently, we
derived a new method to quantify the diffusive dark-field signal in terms of a
scattering angle using a single-exposure grid-based approach. In this
manuscript, we look at the problem of quantifying the sample microstructure
size from this single-exposure dark-field signal. We do this by quantifying
the diffusive dark-field signal produced by 5 different sizes of polystyrene
microspheres, ranging from 1.0 µm to 10.8 µm, to investigate how the strength
of the dark-field signal changes with the sample microstructure size, $S$. We
also explore the feasibility of performing single-exposure dark-field imaging
with a simple equation for the optimal propagation distance given
microstructure with a specific size and thickness, and successfully verify
this equation with experimental data. Our theoretical model predicts that the
dark-field scattering angle is inversely proportional to $\sqrt{S}$, which is
consistent with our experimental data.

## Introduction

With advancements in x-ray generators and detectors, as well as the
introduction of computed tomography [1, 2], x-ray imaging has become a widely
used technique that non-invasively reveals the internal structure of a sample.
Conventional x-ray imaging manifests the difference in the attenuating ability
of different materials in the sample, to create image contrast. It is now one
of the standard imaging techniques used in clinical practice, materials
science and security screening [3]. However, the image contrast is
significantly degraded for samples made of weakly attenuating materials, such
as soft biological tissues. In recent decades, advanced x-ray imaging
techniques known as phase contrast x-ray imaging (PCXI) have been developed,
to enhance the image contrast for samples made up of materials that have
similar attenuating properties. PCXI covers a set of techniques that convert
the phase shift experienced by the x-ray wavefield while passing through the
sample, into an intensity modulation that can be measured on a detector.
Examples include propagation-based imaging (PBI) [4, 5, 6, 7], analyser-based
imaging (ABI) [8, 9, 10, 11], grating-interferometry (GI) [12, 13, 14, 15],
edge-illumination (EI) [16, 17, 18, 19], single-grid imaging [20, 21], and
speckle-based imaging [22, 23].

The size of the smallest detectable sample feature, in an x-ray imaging
system, is typically restricted by the spatial resolution of the system.
Diffusive dark-field imaging (henceforth termed ‘dark-field imaging’ for
simplicity) is a way around this limit. Such dark-field imaging looks at
diffuse scattering—e.g. small-angle x-ray scattering (SAXS) or ultra-small-
angle x-ray scattering (USAXS)—from sample microstructures, in order to detect
their presence. The dark-field signal is useful since it can reveal the
presence of spatially random sample microstructure, which is otherwise
invisible when using the full-field conventional or phase-contrast x-ray
imaging techniques. This imaging modality is also more dose-efficient, since
detectors with larger pixel sizes can be used than if the features were to be
resolved directly.

The dark-field signal has been measured qualitatively using most of the PCXI
techniques, either via an approach where the dark-field information is
extracted from contrast seen across a neighbourhood of pixels (such as PBI
[24, 25] and single-grid imaging [20, 26, 27]), or via an approach where the
dark-field is extracted on a pixel-by-pixel basis from multiple exposures
(such as ABI [28], GI [29], and EI [30, 31, 32]). One interesting case is
multiple-exposure speckle-tracking, where dark-field is extracted from local
neighbourhoods of pixels across multiple exposures. It is worth noting that
the dark-field signal has been retrieved from speckle-based set-ups using both
explicit [23, 33, 34] and implicit [35, 36, 37] approaches to analyse how the
speckles change. In explicit speckle-tracking approaches, the changes/motions
in the speckles are tracked in each local neighbourhood of pixels, whereas in
implicit speckle-tracking approaches, the changes in the speckles are tracked
by looking at how the intensity translates and diffuses across the whole
image, using one whole-image analysis step. The potential of dark-field
imaging can be further explored by quantifying the dark-field signal, and then
relating the signal strength to the properties of the sample microstructure,
such as the size, material or the arrangement of the individual
microstructures.

There are various fields that can potentially benefit from the quantification
of dark-field signal, including medical diagnosis, security screening and
materials science. Some possible biomedical applications include imaging lungs
with emphysema to measure airway size [38, 39], imaging breast tissues with
microcalcifications for early detection of cancer [40, 41], and imaging kidney
stones of different compositions and microscopic morphology for classification
[42]. Multiple animal studies have shown that lung diseases such as lung
cancer [43], emphysema [44] and fibrosis [45] can result in a weaker dark-
field signal from the lungs, due to the change in size or structure of the
alveoli. Recently, the diagnostic capability of a quantitative dark-field
signal has also been demonstrated on healthy individuals [46] and chronic
obstructive pulmonary disease (COPD) patients with emphysema [47], where the
dark-field signal was correlated to the lung volume and the diffusion capacity
of carbon monoxide, respectively. Other possible safety or industrial
applications include imaging and/or detecting goods that come in powder form,
such as drugs or explosives [48], and imaging industrial parts made from
carbon fibres [49].

A quantitative x-ray dark-field signal has been successfully extracted and
related to the sample properties, using ABI [39], GI [50, 51, 52, 53, 54] and
EI [55] techniques. The dark-field signal extracted from these techniques has
been related to different sample microstructure properties, such as (i) the
number of scattering interfaces, which can be related to the number of
microstructures [39], (ii) the kurtosis, which is a statistical quantity of
the scattering distribution that can be related to the microstructure size
[55], (iii) the correlation length, which is the length at which the
correlation between the microstructures is probed by GI [50, 54], and (iv) the
linear diffusion coefficient (or dark-field extinction coefficient), which is
analogous to the linear attenuation coefficient, and relates to the second
statistical moment or width of the scattering probability distribution
function of the sample [52, 51, 53]. However, the x-ray dark-field signal has
not yet been quantitatively related to sample properties using the single-grid
imaging technique.

The single-grid imaging technique [20, 21] is a grating-based PCXI technique,
which is similar to GI, but with a relatively simple setup compared to other
dark-field imaging techniques, such as ABI, GI, and EI. See Fig. 2. As its
name implies, the single-grid imaging technique only requires one optical
element (a grid), and neither calibration nor alignment is required prior to
the data acquisition. The detector needs to have a pixel size smaller than the
grid period, so that the intensity pattern formed by the grid can be fully
resolved. Both absorption grids [20, 56, 21] and phase grids [26, 57, 58] can
be used in this technique and the grid can be placed immediately upstream or
downstream of the sample. The grid can be replaced by any object that provides
an intensity pattern with high visibility, for example, a piece of sandpaper,
in which case the technique is known as speckle-based imaging [22, 23]. A
grating can also be used in this technique, but it is less favourable than a
grid, since the system will only be sensitive to the differential phase in one
direction (perpendicular to the grating lines) [59]. The two-dimensional
sensitivity provided by a grid is essential in the reconstruction of artefact-
free projected phase images [60], which can be useful in quantifying
microstructure when used in conjunction with dark-field images [39].

The data acquisition process of single-grid imaging involves only a single
sample exposure, where the grid patterns the illumination. The sample-and-grid
image is then compared to the reference image taken without the sample, where
only the grid is present. This permits the simultaneous extraction of
attenuation, phase-shift and dark-field signals produced by the sample. These
three quantities result in a decrease in mean, a shifting, and a broadening of
the intensity pattern, respectively. The short data acquisition time can
minimise motion blurring and x-ray radiation dose, which makes this technique
feasible for dynamic imaging.

Recently, a new retrieval algorithm by How & Morgan [61] was derived to
quantify the x-ray dark-field signal in single-grid imaging and relate the
signal to the number of microstructures, $N$. The algorithm was applied to a
sample with unresolved microstructures, made up of 1.0 µm polystyrene
microspheres. Below, we apply the same algorithm to samples made up of 5
different sizes of polystyrene microspheres, all smaller than the resolution
of the imaging system. This allows us to investigate how the strength of the
dark-field signal changes with the sample microstructure size, and determine
the feasibility of performing single-exposure quantitative dark-field imaging
using the single-grid imaging technique.

We first provide a mathematical model relating how the effective scattering
angle extracted is related to the sample microstructure size. This is achieved
by relating the number of microstructures in the beam path of an x-ray passing
through the sample to the total number of microstructures in the sample, which
is then related to the total volume of the microstructures and subsequently
the total thickness of the microstructures. We show that the experimental data
are consistent with this model, by plotting the scattering angle as a function
of sample thickness rather than sample microstructure size. We then derive an
expression for the propagation distance at which to perform single-exposure
quantitative dark-field imaging with maximum sensitivity. This is achieved by
analytically solving for the propagation distance at which the change in dark-
field signal with respect to the change in scattering angle is maximised. Via
this expression, we compute a range of suitable sample-to-detector distances
for each sample microstructure size, and we verify this by comparing (i) the
effective scattering angle extracted using a single exposure, to (ii) the
effective scattering angle extracted from multiple distances. Finally, we
explore the effects of propagation-based phase contrast effects overlying the
modelled grid and dark-field effects, discuss the properties of this
technique, and look at potential applications as well as future research
directions.

## Mathematical modelling

Here, we apply the algorithm developed by How & Morgan, to quantify the dark-
field signal in single-grid imaging [61]. In this approach, the dark-field
signal is extracted using an explicit cross-correlation approach between the
grid-only image and the grid-and-sample image. A summary of the algorithm is
given below, with full details available in How & Morgan [61].

### Extraction of the quantitative dark-field signal using an explicit cross-
correlation approach

The x-ray intensities seen at the detector in the presence of the grid,
$I_{g}(x)$, and the grid and sample, $I_{sg}(x)$, are defined as sine
functions in one-dimension, which are given by

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

and

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

respectively. Here, $x$ is the position across the sample, $a$ is the
amplitude, $b$ is the mean, and $p$ is the period of the intensity
oscillations due to the grid, $A$ is the change in amplitude of the grid
intensity oscillations that is introduced by the sample, and $t$ is the
transmission of the x-ray wavefield passing through the sample. The dark-field
signal, $DF$, which 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), $V^{r}$ for a grating-based method [29], is

$\begin{split}DF&=\frac{V^{s}}{V^{r}}=\frac{\frac{Aa}{tb}}{\frac{a}{b}}=\frac{A}{t}.\end{split}$
(3)

Here, the values of $A$ and $t$ are determined by fitting the local cross-
correlation results of the grid image $I_{g}(x)$ both with itself, and with
the sample and grid image $I_{sg}(x)$. The dark-field visibility signal has a
value between $0$ and $1$, where $1$ represents no dark-field
signal/scattering and $0$ represents maximum dark-field signal, where the grid
is invisible and the cross-correlation (or stepping curve) is ‘flat’.

By modelling the blurring kernel applied to the grid pattern in the presence
of the sample as a normalised zero-centred Gaussian function [62, Figure
5][63, 64], and $I_{sg}(x)$ as the convolution between $I_{g}(x)$ and the
Gaussian function, the dark-field signal is

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

Above, we used the scattering width, $d=z\theta$, where $z$ is the sample-to-
detector propagation distance and $\theta$ is the effective scattering angle.
Equation (4) describes how the dark-field signal changes with propagation
distance, which can be fitted to dark-field signals measured at one or many
different propagation distances, to accurately extract the effective
scattering angle.

The effective scattering angle can be related to the number of
microstructures, $N$, in the paraxial ray path, a number which is proportional
to the total sample thickness, $T$, if we assume these microstructures—which
in our experiment are microspheres—to have the same size. It has been
observed, using both crystal-analyser-based imaging and single-grid imaging,
that the x-ray scattering angle is proportional to $N$ to a power that is
greater than $\tfrac{1}{2}$ [39, 61]. This deviates from the random-walk model
proposed by von Nardroff [65], in which the scattering angle is proportional
to $\sqrt{N}$. The relationship between the scattering angle and the number of
microstructures can be written as

$\theta=kN^{\frac{1}{2}+\alpha},$ (5)

where $k$ is a positive coefficient and $\alpha$ is an anomalous diffusion
constant[66] that can be greater than or smaller than 0.

### Relating the scattering angle to the sample microstructure size

The algorithm proposed in How & Morgan [61] has only been applied to a sample
with microstructure size of 1.0 µm. Thus it is interesting to apply the
algorithm to samples that have different microstructure sizes, to investigate
how the strength of the dark-field signal changes with microstructure size.

We start by deriving the relationship between $N$ and the microsphere size.
First, assume a rectangular cuboid with x-rays normally incident on an area,
$A_{c}$, of one cuboid face. The x-rays pass through the thickness $T_{c}$,
which is randomly filled with microspheres of diameter $S$. The average number
of microspheres, $N$, along the ray path, is the ratio between (i) the total
projected line-of-sight area of the spheres in the cuboid, and (ii) the area
of the entrance face of the cuboid. Hence

$N=\frac{n_{T}\pi(S/2)^{2}}{A_{c}}=\frac{V_{T}}{\frac{4}{3}\pi(S/2)^{3}}\frac{\pi(S/2)^{2}}{A_{c}}=\frac{A_{c}T}{\frac{4}{3}\pi(S/2)^{3}}\frac{\pi(S/2)^{2}}{A_{c}}=\frac{3T}{2S},$
(6)

where $n_{T}$ is the total number of microspheres in the cuboid and $V_{T}$ is
the total volume of microspheres. Note that, after the third equals sign, we
replaced $V_{T}$ with $A_{c}T$. We can imagine this volume as melting all
microspheres into a rectangular cuboid with area $A_{c}$ and width $T$, which
is essentially the thickness of microspheres in the beam path of an x-ray
wavefield passing through the cuboid (as would be measured in an attenuation
signal).

Since we do not expect the anomalous diffusion coefficient $\alpha$ to change
for microspheres of different sizes (as shown by Kitchen et al. [39]), we
assume $\alpha$ to be $0$, which is consistent with the model of von
Nardroff[65], to compare the strength of the dark-field signal from
microstructures of different sizes via the $k$ value in Eqn. (5). Using Eqn.
(6), the $\alpha=0$ case of Eqn. (5) gives

$\theta=k\sqrt{N}=k\sqrt{\frac{3T}{2S}}=K\sqrt{T},$ (7)

where $K=k\sqrt{3/(2S)}$. We see that $\theta$ is proportional to
$1/\sqrt{S}$.

### Optimal distance for single-exposure dark-field imaging

The optimal distance for single-exposure dark-field imaging is the distance at
which the sensitivity of the imaging system towards the dark-field signal is
maximised. This means that a change in the scattering angle produced by the
sample (e.g. due to a different microstructure size) results in the biggest
possible change in the measured dark-field visibility signal (see the yellow
cross in Fig. 1). This can be determined by analytically solving for the
propagation distance, $z_{opt}$, for which
$\frac{\partial^{2}(DF)}{\partial\theta^{2}}=0$, where $DF$ is defined in Eqn.
(4). Hence

$z_{opt}=\frac{p}{\pi\theta}=\frac{p}{\pi
K\sqrt{T}}=\frac{p}{\pi\left(\frac{1}{\sqrt{mS}}\right)\sqrt{T}}=\frac{p}{\pi}\sqrt{\frac{mS}{T}}.$
(8)

Note, after the second equals sign, we used the expression for $\theta$ in
Eqn. (7). Furthermore, after the third equals sign, we substituted in
$K=1/\sqrt{mS}$, since by the definition of $K$ just below Eqn. (7),

$\frac{1}{K^{2}}=\left(\frac{2}{3k^{2}}\right)S=mS,$ (9)

where $m=2/(3k^{2})$ is the gradient of a plot of $1/K^{2}$ against
microstructure size $S$ (see Fig. 7 (b)).

Figure 1: A typical example of the dark-field signal measured as a loss in
visibility (at pixel (960, 1001), from the 8.0 µm sample discussed later) from
24 propagation distances. The dark-field signal starts to saturate at a
propagation distance of around 3.1 m (labelled with the red cross) and thus
the dark-field signals measured at 3.1 m and beyond are not included in the
fitting to extract the effective scattering angle using Eqn. (4). The yellow
cross labels the maximum-gradient point where the change in visibility with
respect to the change in scattering angle/propagation distance is maximised.
This specifies where the sensitivity of the dark-field imaging system is
maximised, providing the optimal distance to perform single-exposure
quantitative dark-field imaging. This optimal distance for sample
microstructure of different sizes is given by Eqn. (8). The orange line
indicates the ‘visibility floor’, at which the decrease in visibility is
dominated by the source-size blurring effect on the reference grid pattern,
and the signal retrieved is not representative of the sample anymore.

### Saturation of dark-field signal

Saturation of the dark-field signal can be observed at overly large
propagation distances. This introduces a limit to where the equations
described above are physically useful. Beyond this limit the grid intensity
pattern becomes too blurred, with the propagation distance being so large that
the algorithm can no longer recognise the reference pattern and thus fails to
properly fit the cross-correlation curves. In grating interferometry, the
sensitivity of the imaging system towards the dark-field signal is typically
tuned accordingly, to make sure that the dark-field visibility signal is
maintained above a given value, e.g. 0.1. In other words, the reduction in
visibility of the stepping curve is kept below 90$\%$ [67], to avoid obtaining
a saturated dark-field signal. Similarly, in the single-grid imaging
technique, we can tune the sensitivity to make sure the dark-field signal is
maintained above 0.3 or 0.4 (as suggested by our data). We use a higher
visibility threshold because the signal is extracted from a single exposure,
which is more susceptible to noise compared to the signal extracted from the
stepping curve obtained in GI using multiple exposures. Also, our technique
requires the visibility of the grid to be stronger than the visibility of
surrounding sample features, which is not the case in GI. This tuning can be
achieved by taking the images at a suitable propagation distance. In this
manuscript, when fitting across multiple distances we have excluded the dark-
field signal measured beyond a certain threshold distance—where the dark-field
signals begin to saturate (as shown in Fig. 1)—during the extraction of the
effective scattering angle. This threshold distance is determined by looking
at how the dark-field signal from samples of different thicknesses changes
with propagation distance. Note, the threshold distance for each sample is
different (as shown in the Numerical analysis section) since the number of
interfaces encountered by the x-ray beam is different for samples of the same
projected thickness but different microstructure sizes.

The dark-field signal may also become saturated due to additional visibility
contributed by the speckle pattern formed by the microspheres or other
surrounding sample features. It has been demonstrated, via experiment and
simulation, that a speckle pattern can be formed by randomly-packed glass
microspheres due to multiple-beam refraction and free-space propagation [62].
The intensity variations that make up the speckle pattern may locally enhance
the visibility of the observed reference intensity pattern and thus result in
an ‘increase’ of the dark-field visibility signal. A detailed simulation [62]
is required to determine the contribution of the speckle pattern formed by the
microspheres to the dark-field signal, which is outside of the scope of this
manuscript.

## Methods

We captured an experimental dataset of the sample shown in Fig. 2, which
includes different microstructure sizes and a range of projected thicknesses.
We imaged over a range of propagation distances, to investigate (i) how the
dark-field signal changes with the size of microstructure, and (ii) at which
distance quantitative single-exposure imaging is optimum.

### Experimental setup

Our experimental setup is shown in Fig. 2. This is a typical single-grid
imaging setup, with the addition that the propagation distance is allowed to
change. The experiment was performed at the Australian Synchrotron Imaging and
Medical Beamline (IMBL). An attenuating grid (a geological stainless steel
sieve, as used in How & Morgan [61]) was placed 43 cm upstream of the sample
(i.e. as close as possible). The sample and grid were placed on two different
stages which could be moved in a horizontal direction automatically, so that
they could be moved out of the field of view to capture grid-only and flat-
field images. A 25 µm thick Gadox phosphor was coupled to a pco.edge 5.5 sCMOS
detector to collect images and it was placed on a separate table which could
be moved to adjust the propagation distance $z$ between the sample and the
detector. The energy of the x-rays was 34 keV and the effective pixel size of
the setup was 9.8 µm.

### Sample preparation

The polystyrene microspheres were purchased suspended in 10 ml of water
(Microspheres-Nanospheres, Corpuscular Inc., Cold Spring New York, USA), with
the spheres making up 2.5% of the volume. Sample tubes were placed into a
centrifuge to separate the microspheres and water. The water was removed using
a pipette and the cap was left off to allow any remaining water to evaporate.
The sample tubes were placed in a water bath sonicator to break up clumps
formed during the liquid evaporation process. Nevertheless, there were obvious
clumps remaining, as seen in the inset of Fig. 2. The microspheres were then
transferred into the sample holder.

A sample holder with five ‘cells’ was custom-made with a piece of Kapton
sheet, which provides minimal attenuation to x-rays, with each ‘cell’
separated by rubber wedges, glued to the Kapton, to provide a range of sample
thicknesses as shown in Fig. 2. The five ‘cells’ held microspheres of diameter
1.0 µm, 4.1 µm, 6.2 µm, 8.0 µm and 10.8 µm.

Figure 2: Experimental setup for single-grid imaging with the detector placed
at multiple propagation distances, $z_{n}$. Images are taken with and without
the sample. Polystyrene microspheres of diameter 1.0 µm, 4.1 µm, 6.2 µm, 8.0
µm and 10.8 µm (from left to right, viewed from the source) were placed in a
custom-made sample holder made up of a Kapton sheet and rubber wedges. The
inset shows the zoomed-in view of each sample ‘cell’ through the orange
Kapton. The blur width increases as the propagation distance increases,
resulting in the grid intensity pattern being smeared out more significantly
and producing a stronger dark-field signal. Note that the scattering angle
$\theta$ has been exaggerated for visualisation purposes.

### Data acquisition

Flat-field (without grid or sample), grid-only and sample-grid images were
taken at sample-to-detector propagation distances of 0.28 m, then 0.4 m to 7 m
at steps of 0.3 m. One set of dark-current images was taken at the end of the
experiment. The exposure time was chosen to be 160 ms to fill the dynamic
range of the detector and 30 exposures were taken for each set of images,
which were averaged together prior to analysis to reduce the noise level.
Because the sample was wider than the field of view, two neighbouring sample-
grid images were taken at each distance (with the 6.2 µm sample appearing in
both the left and right images). The images shown in this manuscript place the
two sample images immediately next to each other.

### Numerical analysis

The raw images were first flat and dark-corrected, demagnified to account for
the subtle magnification effect seen especially at larger propagation
distances, and cropped to reduce the processing time. The images taken at 24
propagation distances were analysed using the algorithm described in How &
Morgan [61], with a cross-correlation window size of 14 pixels to match the
grid period. The dark-field signals measured at 24 distances were then fitted
to Eqn. (4) to extract the effective scattering angle, $\theta$. The dark-
field images taken at different propagation distances were realigned using the
linear alignment function in ImageJ software that applies the Scale Invariant
Feature Transform (SIFT). The image transformation matrices used to realign
the dark-field images were obtained from the alignment of the transmission
images. The dark-field images were also smoothed by a median square kernel of
size 14 pixels (i.e. the grid period) before the extraction of the scattering
angle, to reduce the noise level. As mentioned in the Saturation of dark-field
signal section, a threshold distance was determined for each sample to avoid
saturation of the dark-field signal, which was 1.6 m for the 1.0 µm
microspheres, 2.5 m for the 4.1 µm microspheres and 3.1 m for the remaining
samples. The dark-field signals measured at the threshold distance and beyond
are not included in the fitting to extract the effective scattering angle, as
shown in Fig. 1.

## Results

### Quantifying microstructure size from the x-ray dark-field signal

Figure 3 shows the results obtained from the images taken at propagation
distances of 0.4 m, 2.5 m and 4.6 m, including the change in amplitude ($A$),
transmission ($t$), dark-field signal ($DF$) and the effective scattering
angle ($\theta$) extracted using the dark-field signals captured from single
and multiple distances. The sample with the smallest microstructure, 1.0 µm
(first ‘cell’ from the left), produces the strongest dark-field signal and
thus the largest effective scattering angle. The strength of the dark-field
signal decreases with the sample microstructure size, which is consistent with
the inverse square root relationship between the $\theta$ and microstructure
size ($S$) in Eqn. (7). This is because the x-rays are being scattered by more
interfaces as they pass through a sample with a smaller microstructure size,
compared to the sample of the same thickness but with a larger microstructure
size. It is worth noting that the dark-field signals produced by the 8.0 µm
and 10.8 µm samples are stronger compared to the 6.2 µm sample. This is due to
the more effective packing of the microspheres in those two ‘cells’, seen by
fewer ‘clumps’, and resulting in fewer air gaps, more microstructures and
greater sample thickness than the 6.2 µm sample.

Figure 3: Dark-field imaging results obtained from single-grid images of
polystyrene microspheres of 5 different sizes (with the 6.2 µm sample
appearing in both the left and right images of the image pair), separated by
rubber wedges, shown here for three different propagation distances – 0.4 m,
2.5 m and 4.6 m. The diameter of the microspheres in each panel (from left to
right, similarly for the rest of the figures in this paper unless specified
otherwise) is 1.0 µm, 4.1 µm, 6.2 µm, 6.2 µm, 8.0 µm and 10.8 µm. The change
in amplitude of intensity oscillations, $A$ ((a), (b) & (c)), is divided by
the corresponding transmission of the x-ray wavefield, $t$ ((d), (e) & (f)) to
obtain the dark-field signal, $DF$ ((g), (h) & (i)). The effective scattering
angles, $\theta$, shown in panels (j), (k) and (l) are extracted from the $DF$
signal in panels (g), (h) and (i) respectively, using Eqn. (4), while $\theta$
in panel (m) is extracted from the $DF$ images taken at 24 distances, using
Eqn. (4). The microspheres with a larger size produce a weaker dark-field
signal and thus a smaller scattering angle, which agrees with Eqn. (7). It is
difficult to see this when comparing the 8.0 µm and 10.8 µm samples to the
others, since these microspheres are more densely packed. The $\theta$ image
extracted from the larger propagation distance has less noise than the smaller
propagation distance, where the blurring effect is weak. The scattering angle
extracted from a shorter propagation distance ((j)) is larger compared to
those extracted from a larger or multiple distances. The scattering angle
extracted from a propagation distance of 2.5 m was consistent with the
scattering angle extracted from multiple distances (except the 1.0 µm sample),
indicating that this is a suitable propagation distance for single-exposure
dark-field imaging of sample microstructure of these sizes.

To study the dependence of the dark-field signal on microstructure size, we
want an accurate measure of sample thickness, based on all collected data. The
sample thickness (shown in Fig. 4) is obtained using a Transport of Intensity
Equation (TIE)-based single-material phase retrieval algorithm [7] on the
transmission image (e.g. Fig. 3 (d)-(f)), and averaging the thickness
retrieved from images collected at 1.9 m, 2.2 m, 2.5 m and 2.8 m. Microsphere
clumps are clearly seen in both the photographs of the sample tubes in Fig. 2,
and the x-ray thickness image in Fig. 4. Since the rubber dividers were
significantly more attenuating than the microspheres, they were excluded and
those parts of the image were set to the average attenuation value, prior to
the thickness retrieval process. This was done to obtain an accurate measure
of the sample thickness from the single-material TIE algorithm, and avoid
smoothing the contrast from the rubber into the region of the image containing
microspheres, seen in Fig. 5, where the same cropping is not used. The 1.0 µm
sample reaches a larger thickness compared to the other samples since the air-
gaps between small microspheres are typically smaller. In Fig. 4 (b), the
contrast across all parts of the sample is adjusted by normalising the
thickness of each ‘cell’ between $0$ and $1$, to better visualise the details
of the sample.

Figure 4: Sample thickness, averaged from the thicknesses retrieved using the
transmission images obtained at 1.9 m, 2.2 m, 2.5 m and 2.8 m, via the TIE-
based single-material phase retrieval algorithm [7]. (a) Sample thickness
image with the same greyscale for the whole image. (b) Sample thickness image
with grey level of each sample ‘cell’ normalised between $0$ and $1$ for
visualisation purposes. Clumps with greater thicknesses shown in the figure
were consistent with the clusters of microspheres observed by visual
inspection of the sample (see Fig. 2). The shape of the rubber wedges provides
a wide range of projected sample thickness that increases gradually from the
bottom to the top of the image. The 1.0 µm and 8.0 µm samples reach greater
thicknesses compared to the other samples, due to the more effective packing
of the microspheres.

Figure 5 shows the complementarity between the attenuation signal, shown in
red, and the dark-field signal, shown in blue. The microspheres produce a
stronger dark-field signal relative to the attenuation signal when compared to
the rubber wedges, and thus the microspheres are shown in a stronger blue hue
than the rubber wedges. It is worth noting that certain regions in the 1.0 µm
and 10.8 µm microsphere ‘cells’ appear to have a slightly stronger red hue
compared to the surrounding regions, as a result of the microspheres being
packed more tightly together, potentially with some liquid remaining that
reduces the relative strength of the dark-field signal compared to the
attenuation signal. This agrees with what we observed in Fig. 4. In Fig. 5
(b), vertical red stripes (with no blue contribution) were observed near the
inner edges of all rubber wedges surrounding the microspheres, suggesting
maximum attenuation but minimum dark-field signal in these regions. We believe
this is an artefact from the phase retrieval algorithm[7] as a result of our
assumption that the whole sample is made up of a single material, polystyrene.
Since the rubber wedges are more attenuating than the polystyrene, the phase
retrieval algorithm ended up oversmoothing the phase near these edges and thus
resulting in ‘extra’ attenuation, interpreted as thickness, along the inner
edge of each ‘cell’.

Figure 5: Colour image of the samples that demonstrates the complementarity of
the attenuation and dark-field signals. (a) Colour image with the same colour
scale in the thickness and scattering angle image for all samples. (b) Colour
image with the colour scale of thickness and scattering angle images
normalised between 0 and 1 separately for each sample ‘cell’. The red and blue
channels of the image correspond to the sample phase-retrieved thickness and
scattering angle respectively. The rubber wedges have a stronger red hue but a
weaker blue hue compared to the microspheres since they are more attenuating
to x-rays, and contain fewer dark-field-producing unresolved microstructures.

We have now retrieved a quantitative measure of the scattering angle, which
depends on both the sample and the x-ray illumination, so the next step is to
relate the angle to the sample microstructure properties, such as the number
of microstructures. The effective scattering angle extracted from each sample
is then related to the sample thickness, which is proportional to the number
of microstructures since we have microspheres of the same size. This is
achieved by performing a least-squares fitting on the effective scattering
angle as a function of sample thickness. We noticed that the least-squares
fitting was greatly affected by the spread of data points. For example, the
few data points describing smaller sample thicknesses would have less
influence on the fit than the many data points at larger thicknesses since we
have significantly fewer pixels measuring the smaller thicknesses. We overcame
this by binning the data points based on their thickness and plotting the mean
scattering angle value in each bin instead of plotting the values extracted
from every pixel individually. The resulting plots are shown in Fig. 6. The
uncertainty of each data point is set to be the same, which is the median of
the standard deviation of the angles obtained from all the bins in the ‘cell’.

The Fig. 6 visibility plots are each fitted with both
$\theta=K^{\prime}\sqrt[p]{T-b^{\prime}}$ (red curves), which is similar to
Eqn. (7) but with the exponent of $T$ allowed to vary and $\theta=K\sqrt{T-b}$
(i.e. Eqn. (7)) (orange curves). The $x$-intercepts, $b^{\prime}$ and $b$,
which represent the thickness of the Kapton sheet are allowed to vary in both
fits since it was challenging to retrieve the precise Kapton thickness due to
the wide point spread function (PSF) of the detector. Note that the data
points in the 6.2 µm samples with thickness larger than 2.4 mm were excluded
during the fitting, since they only have a small number of pixels in each bin
compared to the rest of the sample, which makes those data points less
reliable. From Fig. 6, we can see that the red curves, which assume the
anomalous coefficient, $\alpha$, to be non-zero, provide a better fit to the
data points than the orange curves. This suggests the potential presence of
anomalous diffusion in the sample. We also observed that the exponent of $T$
is different for each sample, ranging from $\frac{1}{1.4}$ to $\frac{1}{1.8}$.
However, the relationship between the exponent or $\alpha$ and the sample
microstructure size $S$ is difficult to test with this many free variables in
the fit. We therefore focus on the orange curves which assume $\alpha=0$ and
relate the fitted coefficient, $K$, to the size of the microspheres as shown
in Fig. 7, which is consistent with a model where the effective scattering
angle, $\theta$, is inversely proportional to $\sqrt{S}$ as described in Eqn.
(7). From Fig. 7 (a), we observe an unexpectedly low value for $K$ associated
with a larger uncertainty value for the 6.2 µm sample, but also greater
uncertainty in that measure, due to the presence of air gaps, as we explain in
the Speckle pattern in the dark-field images section of the discussion.

Figure 6: Scattering angle, $\theta$, extracted from microspheres of different
diameters, $S$, as a function of sample thickness, $T$, which is proportional
to the number of microstructures, $N$, along a paraxial ray. Each plot is
fitted with both $\theta=K^{\prime}\sqrt[p]{T-b^{\prime}}$, where $K^{\prime}$
is a constant and the exponent is allowed to vary (red curves), and with
$\theta=K\sqrt{T-b}$, where $K$ is a constant, as suggested by von Nardroff
[65] (orange curves). The $x$-intercept, $b^{\prime}$ and $b$, which represent
the thickness of the Kapton sheet, is another variable in the fitting that can
change. The uncertainty (shown in blue) is constant for each data point to
make sure all data points have the same weighting during the fitting. The red
curves provide a better fit to the data, suggesting the presence of anomalous
diffusion. However, we focus on the orange curves since we are more interested
in how the strength of the dark-field signal changes with sample
microstructure size. The coefficient $K$ is then related to the sample
microstructure size as shown in Fig. 7. Figure 7: The relationship between the
fitted coefficient, $K$ (obtained from Fig. 6), and the sample microstructure
size, $S$, (a) before and (b) after linearisation. The results are consistent
with a model that says $K$ is inversely proportional to $\sqrt{S}$, which
agrees with Eqn. (7). Note that the data for 6.2 µm shown here is extracted
from Fig. 6 (d) and the uncertainty in each coefficient value is obtained by
fitting two new functions to the data $\pm$ uncertainties (green curves) in
Fig. 6. The gradient of the line of best fit in (b) was then used to solve for
the optimal distance to perform single-exposure dark-field imaging, as
explained in the Optimal distance for single-exposure dark-field imaging
section of the discussion.

### Single-exposure quantitative dark-field imaging

The technique described here can extract quantitative measurements from a
single sample exposure, provided that the sample-to-detector propagation
distance is (i) not so short as to provide insufficient signal, and (ii) not
so long as to saturate the dark-field signal. In light of this tradeoff, this
section examines the optimum distance for single-exposure imaging.

Figure 8 shows the scattering angle extracted from the single-exposure dark-
field signal obtained at 4 propagation distances. The scattering angle image
retrieved from a shorter propagation distance (Fig. 8 (a)) has significantly
more noise compared to the image retrieved from a larger propagation distance
(Fig. 8 (d)) due to the weaker dark-field sensitivity at a shorter propagation
distance. The scattering angle extracted from a shorter propagation distance
(Fig. 8 (a)) also has a greater magnitude compared to the scattering angle
extracted from a larger propagation distance (Fig. 8 (d)). The same trends and
observations are echoed in Fig. 3. This suggests that the effective scattering
angle may be overestimated at a propagation distance shorter than the optimal
distance, and underestimated at a propagation distance larger than the optimal
distance.

It is worth noting that the scattering angle from the 1.0 µm sample in Fig. 8
(a) is consistent with the angle retrieved from multiple distances (Fig. 3
(m)), indicating that 1.0 m is a suitable propagation distance to extract
quantitative dark-field signal for this sample. Similarly, the scattering
angle extracted from the other samples at 2.5 m (Fig. 3 (k)), 1.9 m and 3.1 m
(Fig. 8 (b) and (c)) are also consistent with Fig. 3 (m), which implies the
robustness of our technique towards the distance at which we perform single-
exposure dark-field imaging.

While there is some range of propagation distances that give qualitatively
similar images, and a substantial range within that set that produce images
that agree quantitatively, the choice of distance is important. In addition,
it should be considered how the optimal single-exposure propagation distance
changes for microstructures of different sizes.

Figure 8: Dark field scattering angle obtained at (a) 1.0 m, (b) 1.9 m, (c)
3.1 m and (d) 4.0 m, using a single exposure. At 1.0 m, only the scattering
angle retrieved for the 1.0 µm sample was consistent with the angle retrieved
from multiple distances (Fig. 3 (m)), whereas the scattering angle retrieved
for the 4.1 µm sample in (b) and the other samples in (b) and (c) are
consistent with Fig. 3 (m) and the scattering angle retrieved in (d) is
smaller compared to Fig. 3 (m). At larger propagation distances ((c) & (d)),
the retrieved dark-field scattering angle image has a lower noise level
compared to the images retrieved at shorter propagation distances ((a) & (b)).
Although the magnitude of the scattering angle from the 1.0 µm sample
decreases as the propagation distance increases, the measured scattering angle
remains relatively consistent for the other samples between distances, despite
having different noise levels. This implies our algorithm is robust in
extracting quantitative dark-field signals, even when the propagation distance
is not optimal.

We can use this experimental dataset to test the optimal distance formula
derived earlier in this paper. To do this, we substituted the median thickness
and median $\pm$ 2 standard deviations of the thickness of each sample into
Eqn. (8), to calculate an optimal distance and a range of suitable distances
for single-exposure quantitative dark-field imaging for each sample, shown in
green and blue respectively in Fig. 9, together with the optimal distance
determined from the experimental data, shown in red. The experimental optimal
distance is determined to be the distance at which the scattering angle,
retrieved at such a distance using a single exposure, matches best with the
scattering angle retrieved from multiple distances: see Fig. 10. The optimal
propagation distance obtained from the experimental data lies within the range
of the theoretical optimal propagation distance calculated using Eqn. (8),
showing that our result agrees with the theory. Figure 9 shows that to extract
a quantitative dark-field signal, a sample with a larger microstructure size
or smaller thickness needs to be imaged at a larger propagation distance,
while a sample with a smaller microstructure size or larger thickness needs to
be imaged at a shorter propagation distance, as described in Eqn. (8).
However, this is not evident for the 8.0 µm and 10.8 µm experimental datasets,
where both were found to match the multiple-distance data best at the same
experimental optimal distance, 2.5 m, which is smaller than the optimal
distance for the 6.2 µm sample. This may be due to the reference grid pattern
being blurred out significantly at larger propagation distances due to source-
size blurring, leaving a lower visibility reference pattern to image with.
This effect, not incorporated in the optimal-distance equation, renders the
cross-correlation less robust to noise and background intensity variations. A
lower-visibility reference pattern is more likely to result in a saturated
dark-field signal at a shorter distance, raising the ‘visibility floor’
indicated in Fig. 1.

Figure 9: Theoretical and measured optimal propagation distances to perform
single-exposure quantitative dark-field imaging on samples with various
microstructure sizes. The theoretical optimal propagation distance is
calculated using Eqn. (8) with $T=$ median thickness, while the range of
predicted viable propagation distances is obtained with $T=$ median thickness
$\pm$ 2 standard deviations in thickness. The experimental data (see Fig. 10
for the plot of each sample) is in agreement with the theoretical prediction.
Figure 10: Plots showing the dark-field scattering angle measured at the
optimal propagation distance (red) using a single exposure, compared to the
scattering angle extracted from multiple distances (blue) for each sample. In
all panels, the recovered scattering angle is plotted against the retrieved
sample thickness. Note that the optimal propagation distances chosen here are
the distances at which the two curves match the best. The two curves in all
samples were in agreement with each other. Moreover, the optimal propagation
distance for each sample falls within the range of theoretical prediction (see
Fig. 9).

## Discussion

We have extracted quantitative dark-field signals from polystyrene
microspheres of 5 different sizes and related the effective scattering angle
to the number of microstructures. The scattering angle and the number of
microstructures of all samples follows a relationship that deviates either
slightly or negligibly from the theoretical square-root relationship proposed
by von Nardroff[65]. As expected, it was observed that the sample with smaller
microstructures produces a stronger dark-field signal and thus a larger
effective scattering angle. Our theoretical model predicts that the scattering
angle is inversely proportional to the microstructure size, and our data is
consistent with this model. We have also determined a formula for the optimal
propagation distance to perform quantitative single-exposure dark-field
imaging. This was achieved by analytically solving for the distance at which
the change in the dark-field visibility signal with respect to the change in
scattering angle is maximised, and was confirmed with experimental data.

The saturation of the dark-field signal at a larger propagation distance was
observed for all samples, and depending on the sample microstructure size, the
dark-field signal began saturating at a different propagation distance.
Samples with smaller microstructures produced a saturated dark-field from a
shorter propagation distance than the samples with larger microstructures, for
equivalent thickness.

As noted for the case of the lower-visibility reference pattern seen at large
grid-to-detector distances, the visibility of the reference pattern relative
to noise and background variations also affects the distance at which
saturation is first observed. Therefore, within this discussion, we first
examine the origin of the speckle pattern observed from the sample and how
that affects the extraction of the quantitative dark-field signal. We then
explore the properties of this single-exposure quantitative dark-field imaging
technique, future research directions and potential applications.

### Speckle pattern in the dark-field images

From Fig. 3, we observed that the 6.2 µm sample produced stronger local
variations in the change in grid visibility, in the mean (transmission)
intensity, and hence the dark-field signal and scattering angle images,
compared to the rest of the sample. The speckle pattern observed in the dark-
field image of the 6.2 µm sample may be attributed in part to the propagation-
based phase contrast edge effects, contrast which is evident from the
transmission images (see Fig. 3 (d), (e) & (f)), increasing with propagation
distance. If this were the case, then by removing the propagation-based phase
effect, we should retrieve a dark-field signal image with no speckle pattern.
This correction can be achieved by dividing the raw data by the simulated
intensity of the x-ray wavefield propagated through free space after exiting
from the sample (as demonstrated in Groenendijk et al. [68]), before
performing our method of analysis. Figure 11 (b) shows the resulting intensity
seen after propagating an x-ray wavefield with uniform intensity through the
sample (with thickness shown in Fig. 11 (a)) and then an extra distance of 2.8
m, with the free-space propagation performed via the angular-spectrum approach
[69] with an assumption that the sample is not attenuating, using
$\delta=2.0463\times 10^{-7}$ (polystyrene at 34keV) and $\beta=0$, where the
complex refractive index is $n=1-\delta+i\beta$. Figure 11 (d) shows the
result obtained after dividing the raw sample-grid image (Fig. 11 (c)) by the
propagated grid-free intensity determined by simulation (Fig. 11 (b)). The
majority of the fringes formed within the sample itself and near the edges of
the glue down the bottom and near the rubber wedges can no longer be seen in
Fig. 11 (d), indicating that the propagation-based phase effects have been
removed.

The dark-field signals extracted from the images taken at a propagation
distance of 2.8 m before and after this correction are shown in Fig. 11 (e)
and (f) respectively. We observed that Fig. 11 (f) agrees with Fig. 11 (e)
except that (f) appears to be very slightly smoother. This not only suggests
that the speckle pattern formed in the 6.2 µm sample tube does not originate
from the propagation-based phase effects, but it also indicates that our
algorithm can still work effectively even in the presence of propagation-based
fringes. (Images of other samples after correction, and related relevant
results, can be found in Fig. S1 from the Supplementary Information.)

The presence of clusters and air gaps in the 6.2 µm tube, as described earlier
in the paper, can be further justified by examining Fig. 11 (c) & (d). Here,
we see that some regions in the grid intensity pattern are more blurred out,
compared to the rest of the pattern. As expected, this corresponds to the
regions with greater magnitude in the thickness image (Fig. 11 (a)) and also
to the regions with stronger dark-field signal in Fig. 11 (e) & (f). It is
also worth noting that the clusters formed in the 6.2 µm sample also act as
additional microstructure in the sample, with a length scale different from
the individual microspheres. The discontinuity observed from the dark-field
signal produced by the 6.2 µm sample across different propagation distances
(see Fig. S2 in the Supplementary Information) suggests that the blurring from
the extra sample microstructure may be of the same length scale as the grid
period, thus causing the grid intensity pattern to distort further, until the
cross-correlation analysis could not work properly and potentially also
leading to the saturation of the dark-field signal.

A small number of pixels in the 6.2 µm sample tube have shown an increase in
visibility, as demonstrated by the dark-field signals detected from these
pixels being greater than 1. This may be due to the focusing of x-rays by the
air gaps. It may also be due to clumps within the sample that are barely
resolved – i.e., not sufficiently smaller relative to the pixel size to
produce a dark-field signal, but also not sufficiently well-resolved to
produce a visible phase shift in the x-ray wavefield. Another contributing
factor may be grid-period-sized phase-induced distortions coming from the
barely-resolved sample clumps mentioned above, which may locally change the
visibility of the reference intensity pattern.

Figure 11: Dark-field retrieval while removing propagation-based phase
contrast edge effects. The diameter of the microspheres in each panel (from
left to right) is 6.2 µm, 8.0 µm and 10.8 µm. (a) Thickness image retrieved
via the TIE-based phase retrieval algorithm [7] using the transmission signal
extracted from images taken at a propagation distance of 2.8 m. (b) Simulated
intensity of an x-ray wavefield seen after propagating a uniform-intensity
x-ray wavefield by 2.8 m, via the angular spectrum approach[69], using the
phase information from (a) and assuming the sample is not attenuating. (c) The
raw sample-grid image at 2.8 m, and (d) the same, after dividing out the
propagation-based phase contrast edge effects (panel (b)). (e) & (f) The dark-
field signal extracted from panels (c) and (d), respectively. Some artefacts
can be observed near the edges of the rubber wedges in (b). The dark-field
signals in (e) and (f) are similar, suggesting that the speckle pattern
observed in the dark-field-signal image does not originate from the
propagation-based phase contrast fringes, but truly from local variations in
dark-field. This result also suggests that our algorithm can still extract the
dark-field signal accurately, even in the presence of propagation-based phase
contrast edge effects.

### Properties of the technique

One main advantage of this technique is that it only requires a single sample
exposure to extract the attenuation, phase-shift and dark-field signal. The
short data acquisition time can minimise the radiation dose delivered to the
sample and the potential motion blurring in the image, which makes this
technique more feasible for dynamic imaging or time-sequence imaging, compared
to other PCXI techniques that need multiple sample exposures to extract dark-
field images. This technique also has a relatively simple setup, compared to
other dark-field imaging techniques. It does not need any additional optical
elements other than an attenuating or phase grid. Furthermore, no calibration
or alignment is required for this technique, prior to the data acquisition
process. The sensitivity of the imaging system to the dark-field signal can
also be tuned, by changing the pixel size and the propagation distance. We can
increase the sensitivity by using a smaller pixel size or a larger propagation
distance.

The single-grid imaging technique can also be performed in a laboratory-based
setup using polychromatic x-ray sources with non-negligible source size [20,
70], since it only requires a certain degree of spatial coherence of the
source [70], which could be improved by placing a grid immediately downstream
of the source. Although the single-grid imaging technique only has a weak
requirement on the temporal coherence of the source, the polychromaticity of
the source could be an issue for quantitative dark-field imaging, due to the
beam hardening effect, which can contribute to a ‘pseudo’ dark-field signal
[71].

Our technique is robust with respect to propagation distance, with the
effective scattering angle extracted from a single exposure consistent within
a range of propagation distances, centred on the optimal distance. The optimal
distance to perform single-exposure dark-field imaging for a sample with
estimated thickness and microstructure size can be determined using Eqn. (8),
where the value of $m$ can be obtained from a calibration curve of samples
made up of the same material, with known thickness and size. A simple general
rule—in choosing a suitable sample-to-detector distance experimentally—is to
image the sample at a propagation distance that is large enough such that the
blurring from the sample can be observed clearly in the intensity pattern
formed on the detector, but not so large that the intensity pattern is
completely blurred out.

Another advantage of this technique is that the spatial-mapping/explicit-
tracking method [21] we applied here can provide a higher spatial resolution
than a Fourier-analysis approach [72], since we are comparing the windows
pixel-by-pixel. Such Fourier analysis can fail when the grid frequency
overlaps with the sample feature frequency (as shown in Fig. 4 from Morgan et
al. [26]). Although the results from Morgan et al. [26] focused on phase
imaging, we believe the same applies for dark-field imaging. This spatial-
mapping approach also allows the algorithm to be successfully applied on
images taken using speckle-based imaging, as shown in Section 6.2 of How &
Morgan [61], where the grid is replaced with a piece of sandpaper [23, 22].

However, the single-grid imaging technique has some limitations. One major
limitation is that this technique is primarily suitable for millimetre-to-
centimetre-sized samples. This is due to the fact that it requires a
relatively small pixel size to capture the changes to the grid pattern
directly. To obtain high-quality dark-field images, the pixel size usually has
to be smaller than the blurring width introduced by the sample. Although
technically this technique can be used to image a large sample by using a
detector with a larger area but a sufficiently small pixel size, this can
introduce a higher radiation dose on the sample and may not be feasible due to
the cost and/or lack of availability of such a detector.

### Potential applications and future directions

As explained in the previous section, this technique can be useful for imaging
dynamic or irreversible processes, such as in vivo biological response to a
treatment [73], chemical reactions [74, 75] and the production of metal foams
[76]. This technique can also be extended into three dimensions to acquire
fast tomography, which can be useful for biomedical applications, such as lung
imaging [38, 39], breast tissue imaging for early detection of cancer [40],
and kidney stone imaging for classification [42]. In particular, the dark-
field signal can provide information about the size of alveoli [38, 39], and
thus this technique has the potential to be used as a diagnostic tool for lung
diseases such as emphysema [44], fibrosis [45], chronic obstructive pulmonary
disease (COPD) [47], and lung cancer [43], which can produce changes in the
size and structure of the alveoli. This technique can also be used to study
chemical reactions that involve the forming or decomposition of substances in
an aqueous solution, which act as unresolved microstructures that provide the
dark-field signals [74, 75]. Moreover, by quantifying the dark-field signal
and relating the signal to the sample microstructure size and material, this
technique can also be used to identify powder-like goods, which can be useful
for airport security to detect explosive substances or powdered drugs [48].

In this manuscript, we have extracted the effective scattering angle from the
dark-field signal using multiple distances, and related this angle to the
sample microstructure size. However, this does not fully explain the
relationship between the dark-field signal and the scattering from sample
microstructures. It would be interesting to further understand how the dark-
field signal relates quantitatively to the small-angle or ultra-small-angle
x-ray scattering (SAXS or USAXS). This model can be further improved by
investigating how the dark-field signal is related to the statistical moments
that describe the shape of the SAXS or USAXS distribution [77, 55]. It would
also be interesting to determine the fraction of the propagating x-ray
wavefield that contributes to the dark-field signal [78].

In applying the method to a range of sample sizes, it may be important to
separate the edge-related phase effects from the dark-field effects. In some
regimes, if a phase fringe has a length scale comparable to the reference grid
period, the fringe could be misinterpreted as a dark-field signal, although we
have not seen that here (Fig. 11). Future work could investigate how the
strength of the dark-field signal is determined by the sample material and
microstructure shape, and how this relates to the optimal propagation distance
to perform quantitative single-exposure dark-field imaging.

Another direction is to investigate quantities associated with directional
dark-field signal[79, 80], which comes from elongated microstructures that are
oriented in a certain direction. By modelling the blurring function as a two-
dimensional Gaussian distribution[80], Croughan et al. [27] have successfully
applied this technique to extract the directional dark-field signal, including
both the angle at which the microstructures are oriented and the eccentricity
of the blurring from the microstructures. This quantitative directional dark-
field imaging technique can be used to determine the diameter of fibres, which
have a length many times their width, and hence a scattering angle much larger
in one direction.

## Conclusion

This manuscript has derived and tested guidelines for performing single-
exposure dark-field x-ray imaging to quantify the size of dark-field-
generating microstructure. To provide experimental evidence, we have extracted
the diffusive dark-field signal from polystyrene microspheres of diameter 1.0
µm, 4.1 µm, 6.2 µm, 8.0 µm and 10.8 µm, using multiple propagation distances,
employing the single-grid algorithm developed by How & Morgan [61]. We
observed that a sample with smaller microstructures produces a stronger dark-
field signal, compared to a sample with larger microstructures of the same
projected thickness. The retrieved dark-field scattering angle was consistent
with our theoretical model stating the angle is inversely proportional to the
square root of sample microstructure size.

We determined an expression for the optimal sample-to-detector distance range
for single-exposure dark-field imaging, by analytically solving for the
distance at which the change in dark-field visibility signal with respect to
the change in scattering angle is maximised. This also avoids both the
insufficient signal seen at short sample-to-detector distances and the dark-
field saturation seen at large distances. According to the analytical
solution, the optimal distance for single-exposure dark-field imaging depends
on the grid period, sample microstructure size and sample thickness. We have
verified the theoretical model by comparing the effective scattering angle
extracted from a single exposure at a single distance, to the angle extracted
from multiple distances. Here, the experimental optimal distance falls within
the theoretical optimal distance range, for all samples.

Single-grid dark-field imaging can characterise sample microstructure, once
calibrated with known microstructures of comparable material and size to those
we wish to investigate. Imaging can then be performed with a single sample
exposure, allowing time-sequence or low-dose imaging.

## Data availability

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

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## Acknowledgements

We acknowledge ARC Future Fellowship funding (FT180100374). The images in this
paper were captured at the Imaging and Medical Beamline at the Australian
Synchrotron, part of ANSTO, under application 16818a. We also thank Michelle
Croughan for providing constructive feedback on the manuscript.

## Author contributions

Y.Y.H and K.S.M. designed the technique. Y.Y.H and K.S.M. performed the
experiment and wrote the analysis code. The paper was mainly written by
Y.Y.H., with input from K.S.M. and D.M.P., who also provided guidance and
supervision of the project.

## Competing interests

The author(s) declare no competing interests.