# Transport-of-Intensity Model for Single-Mask X-ray Differential Phase
Contrast Imaging

Jingcheng Yuan1 and Mini Das1,2  1Department of Physics, University of
Houston, 3507 Cullen Blvd, Houston, TX 77204, USA
2Department of Electrical and Computer Engineering, Department of Biomedical
Engineering, University of Houston, 3517 Cullen Blvd, Houston, TX 77204, USA
<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

X-ray phase contrast imaging has emerged as a promising technique for
enhancing contrast and visibility of light-element materials, including soft
tissues and tumors. In this paper, we propose a novel model for a single-mask
phase imaging system based on the transport-of-intensity equation. Our model
offers an intuitive understanding of signal and contrast formation in single-
mask phase imaging systems. We also demonstrate efficient retrieval of
attenuation and differential phase contrast with just one intensity image
without requiring spectral information or mask/detector movement. The model
validity as well as the proposed retrieval method is demonstrated via both
experimental results on a system developed in-house as well as with Monte
Carlo simulations. Our proposed model overcomes the limitations of existing
models by providing an intuitive visualization of the image formation process.
It also allows optimizing differential phase imaging geometries for practical
applications, further enhancing broader applicability. Furthermore, the
general methodology described herein offers insight on deriving transport-of-
intensity models for novel X-ray imaging systems with periodic structures in
the beam path.

##  1 Introduction

Conventional X-ray imaging relies on the variations of X-ray attenuation
properties among different tissue types. However, it has limited contrast for
low atomic number materials such as organs, tumors, and other soft
tissue[1][2][3][4]. In recent years, X-ray phase contrast imaging (PCI) has
gained much attention for its potential to enhance this soft tissue contrast
by utilizing relative phase changes with X-ray propagation through the object.
Among the various techniques available, single-mask differential phase
contrast imaging method stands out as a simple and effective approach yielding
higher contrast than optic-free methods like propagation based phase imaging.
Propagation-based (PB) phase contrast imaging, which is one of the simplest
PCI techniques, does not require any additional optics in the beam path, but
only an increase in the object-to-detector distance and a partially coherent
source [5][6]. At a longer propagation distance, the wavefront distortions
caused by the object are recorded as intensity variations on the detector
plane. These variations can be modeled by the approximated form of the
transport-of-intensity equation (TIE)[7]:

$I(z,\vec{r})=I(0,\vec{r})-\frac{z}{k}(\nabla_{\perp}I(0,\vec{r})\cdot\nabla_{\perp}\phi(\vec{r})+I(0,\vec{r})\nabla_{\perp}^{2}\phi(\vec{r}))$
(1)

Here $I(z,\vec{r})$ and $I(0,\vec{r})$ are the x-ray intensity at the object
plane and detector plane respectively, $\phi(\vec{r})$ is the beam’s phase
shift caused by the object, $z$ is the object-to-detector distance, $k$ is the
wave number, and $\vec{r}$ is the coordinate in x-y plane. In most
applications of interest with predominantly soft materials in the beam path,
we can assume the intensity variation is slow in the x and y direction, so the
second term $\nabla_{\perp}I(0,\vec{r})\cdot\nabla_{\perp}\phi(\vec{r})$ can
be neglected[7][8]. Hence the equation becomes:

$I(z,\vec{r})=I(0,\vec{r})-\frac{z}{k}I(0,\vec{r})\nabla_{\perp}^{2}\phi(\vec{r})$
(2)

Thus in addition to the attenuation signal ($I(0,\vec{r})$), the intensity at
each detector pixel is predominantly influenced by the Laplacian of X-ray
phase shift caused by the object. This Laplacian phase signal manifests as
bright and dark borders along the edges, leading to edge enhancement.

(a)

(b)

Figure 1: Schematic of the single mask phase imaging method. (a) Top view of
the set up. The X-ray beam propagates in $z$ direction, and the detector
pixels are placed in the x-y plane. (b) Diagram of mask alignment with
detector pixels. The mask strips are along y direction.

A single-mask phase imaging technique [9] is similar to PB phase imaging but
with an added periodic X-ray absorption mask positioned between the source and
the object, in close proximity to the object (Fig.1(a)). The mask creates
X-ray beamlets by periodically blocking X-rays with thin strips of heavy-
element materials like gold. The mask is aligned with respect to the detector
such that the center of each thin and long strip of beamlet is aligned to
every other pixel boundary [9][10]. Hence, with a proper alignment, in the
absence of an object in the beam path, the signal intensity on each detector
pixel column is uniform, showing no discernible patterns (Fig.1(b))). When the
object is introduced, the heterogeneities within the object induce refraction
effects that alter the original directions of the beamlets. Thus, intensity
differences appear between neighboring pixels, resulting in the appearance of
bright and dark fringes on the detector. Fig.1b shows the schematic and 2(b)
shows experimental results to be described in detail later. In terms of wave
optics, this can also be explained as the modification of the Fresnel
diffraction pattern of the periodic mask with the introduction of the object.
These relative intensity variations can allow disentangling differential phase
information from attenuation-related intensity variations on the detector
plane when the appropriate light-transport model is known.
We note that the single-mask PCI method is a significantly simplified version
of the double-mask edge-illumination (EI) method developed earlier [11][12]
and also avoids ’wasting’ large number of photons that has already transmitted
through the object.

(a)

(b)

Figure 2: PCI intensity images of a PMMA rod and their cross-section profile
(blue curve) with (a) propagation-based method and (b) single-mask method.

The formulation of single-mask PCI has been previously attempted using both
refraction[9] and wave-optics[13] models. However, these existing models have
limitations in terms of providing intuitive visualizations of signal and
contrast formation in the images. In this paper, we present a new model based
on TIE and show how this model can be used for efficient retrieval of
absorption and differential phase. Furthermore, we show a single-shot (only
one acquisition with no movement of object or optical components), low-dose
phase-imaging that yields multiple image features and contrast types. While
our prior work has shown efficient phase retrieval methods with spectral data
(using photon counting detectors) [14][15][16][17][18], the retrieval shown
here does not require spectral data.

##  2 Methods

###  2.1 Formulation

Our formulation for single-mask PCI starts with the TIE, (Eqn.(1)). Unlike the
propagation-based method, here we have a high-contrast periodic absorption
mask, so the term $\nabla_{\perp}I(0,\vec{r})\cdot\nabla_{\perp}\phi(\vec{r})$
can no longer be neglected. Here, the transmitted intensity at the object
plane is $I(0,\vec{r})=T(\vec{r})\cdot M(x)$, where $T(\vec{r})$ and $M(x)$ is
the transmission function of the object and the mask, respectively. Therefore:

$\nabla_{\perp}I(0,\vec{r})=T\nabla_{\perp}M+M\nabla_{\perp}T\approx
T\partial_{x}M$ (3)

Here we applied the approximation that $\nabla_{\perp}I(0,\vec{r})$ is mainly
contributed by the mask so $M\nabla_{\perp}T$ can be neglected. After
substituting Eqn.(3) into Eqn.(1), the x-ray intensity measured by each
detector pixel can be calculated by integrating Eqn.(1) over the range of the
corresponding pixel:

$\displaystyle I_{n}$ $\displaystyle=\int_{x_{n}}^{x_{n+1}}T\cdot
M\,\mathrm{d}x-\frac{z}{k}\int_{x_{n}}^{x_{n+1}}T\cdot\partial_{x}M\cdot\partial_{x}\phi\,\mathrm{d}x-\frac{z}{k}\int_{x_{n}}^{x_{n+1}}T\cdot
M\cdot\nabla_{\perp}^{2}\phi\,\mathrm{d}x$ (4)

where $n$ is the pixel index in the horizontal direction when the masks with
slits in the vertical direction is used, $x_{n}$ and $x_{n+1}$ is the
coordinate of the left and right boundary of the corresponding pixel
respectively.
Here we assume that the attenuation, phase, and differential phase of the
sample vary slowly within the range of a pixel. Then
Eqn.(LABEL:Eqn_SM_Integrate1) becomes:

$\displaystyle I_{n}$
$\displaystyle=T_{n}\int_{x_{n}}^{x_{n+1}}M(x)\,\mathrm{d}x-\frac{z}{k}T_{n}\partial_{x}\phi_{n}\int_{x_{n}}^{x_{n+1}}\partial_{x}M(x)\,\mathrm{d}x-\frac{z}{k}T_{n}\nabla_{\perp}^{2}\phi_{n}\int_{x_{n}}^{x_{n+1}}M(x)\,\mathrm{d}x$
(5)
$\displaystyle=T_{n}(1-L_{n})\int_{np}^{(n+1)p}M(x)\,\mathrm{d}x-T_{n}D_{n}\int_{np}^{(n+1)p}\partial_{x}M(x)\,\mathrm{d}x$

Here $T_{n}=T(x_{n})$, which represents the object attenuation function
averaged within each pixel; $L_{n}=\frac{z}{k}\nabla^{2}_{\perp}\phi(x_{n})$,
which is the Laplacian of phase shift caused by the object;
$D_{n}=\frac{z}{k}\partial_{x}\phi(x_{n})$, which is the gradient of phase
shift, and is proportional to the x-ray refraction angle.
In the case of a perfect mask, as we have demonstrated in our previous paper
[16][19], the mask transmission function $M(x)$ can be expressed as a square
wave. Considering the imperfection of the mask, a more general form of its
transmission function can be expressed as a Fourier series:

$\displaystyle M(x)=\sum_{m}C_{m}\cos{\frac{2\pi mx}{(2p)}}$ (6)

where $p$ is the detector pixel size, which means the period of the mask is
two times of detector pixel size (See Fig.1(a) for reference on mask vs
detector period). Then we have:

$\displaystyle\int_{np}^{(n+1)p}M(x)\,\mathrm{d}x=\int_{np}^{(n+1)p}\sum_{m}C_{m}\cos{\frac{\pi
mx}{p}}\,\mathrm{d}x=C_{0}p$ (7a)
$\displaystyle\int_{np}^{(n+1)p}\partial_{x}M(x)\,\mathrm{d}x=M((n+1)p)-M(np)=-2\sum_{m}C_{2m+1}\cdot(-1)^{n}$
(7b)

The results of Eqn.(7) are related to the mask transmission function and are
not related to the object property. Thus, Eqn.(5) becomes:

$I_{n}=w_{e}T_{n}(1-L_{n})-\alpha(-1)^{n}T_{n}D_{n}$ (8)

One can see from this equation that the signal is a combination of two
distinct effects. The first term in Eqn.(8), which we refer to as the
propagation-based (PB) part, shares the same form as the propagation-based PCI
(Eqn.(2)).
The second term, referred to as the differential phase contrast (DPC) term,
gives rise to the characteristic bright and dark fringes within the image, as
demonstrated in the example depicted in Fig.2(b). This is because it contains
the factor $(-1)^{n}$, where $n$ denotes the pixel column index. The magnitude
of these fringes is directly proportional to the DPC signal $D_{n}$.
Also, the two parts of the signal are multiplied by two mask-related
coefficients $w_{e}$ and $\alpha$ respectively. The coefficients’ values are
determined by combining Eqn.(5) and Eqn.(7).

$\displaystyle w_{e}$ $\displaystyle=C_{0}p$ (9a) $\displaystyle\alpha$
$\displaystyle=2\sum_{m}C_{2m+1}$ (9b)

According to Eqn.(7a), the coefficient $w_{e}$ represents the integration of
the mask-transmission function within a pixel, which corresponds to the
average transmission of the mask. Thus, it can be interpreted as the effective
transparent width or aperture size. It is also similar to the $w_{e}$ in our
previous model for double mask method[19]. On the other hand, $\alpha$ is a
unit-less coefficient that depends on the odd Fourier coefficients of the
mask’s transmission function. It can be understood as the contrast of the mask
pattern.
The two coefficients can be interpreted as separate filters for the PB part
and the DPC part independently. In comparison with the PB method, the
intensity of the PB signal in the single-mask method is reduced by the
coefficient $w_{e}$. This implies that the mask selectively reduces the X-ray
intensity that contributes to the PB part, thereby allowing for a reduction in
X-ray radiation dose to the sample without affecting the signal intensity of
the DPC part. The second coefficient, $\alpha$, which represents the contrast
of the mask, determines the efficiency of obtaining the DPC signal.

###  2.2 Retrieval Method

From the last section we could see the attenuation, Laplacian phase and
differential phase have contributions to the measured intensity. Among them,
the DPC part is shown as high-frequency fringes in Fig.2(b). A retrieval
process is needed to separate the PB part and the DPC part.
In an experimental realization, a single image is taken with the object and
the mask in the beam path. This image (represented as $I_{n(M+S)}$) can be
compared with the image with mask only (flat field $I_{n(M)}$). The formula
for the mask-and-sample image $I_{n(M+S)}$ is shown in Eqn.(8); for the mask-
only (or flat-field) image, $I_{n(M)}=w_{e}$. After doing flat-field
correction, we obtain:

$\bar{I}_{n}=\frac{I_{n(M+S)}}{I_{n(M)}}=T_{n}(1-L_{n})-(-1)^{n}\frac{\alpha}{w_{e}}T_{n}D_{n}$
(10)

Thus we can write the corrected intensity for $n^{th}$ and $(n+1)^{th}$ pixels
in a same row:

$\displaystyle\bar{I}_{n}$
$\displaystyle=T_{n}(1-L_{n})-(-1)^{n}\frac{\alpha}{w_{e}}T_{n}D_{n}$ (11)
$\displaystyle\bar{I}_{n+1}$
$\displaystyle=T_{n+1}(1-L_{n+1})+(-1)^{n}\frac{\alpha}{w_{e}}T_{n+1}D_{n+1}$

We can separate PB and DPC signals by adding and subtracting the intensity
values on $n^{th}$ and $(n+1)^{th}$ pixels:

$\displaystyle\bar{I}_{n}+\bar{I}_{n+1}$ $\displaystyle\approx
2T_{n}(1-L_{n})$ (12a) $\displaystyle\bar{I}_{n}-\bar{I}_{n+1}$
$\displaystyle\approx 2(-1)^{n}\frac{\alpha}{w_{e}}T_{n}D_{n}$ (12b)

From Eqn.(12a), we can easily have the retrieval of the PB part (Eqn.(13a)).
In order to retrieve $D_{n}$, if we consider the intensity of the Laplacian of
phase to be relatively weak compared with 1, we can apply the approximation of
$1-L_{n}\approx 1$ when calculating differential phase $D_{n}$. Then we can
arrive at the retrieval formula for propagation based PCI and differential
phase:

$\displaystyle T_{n}(1-L_{n})\approx\frac{\bar{I}_{n}+\bar{I}_{n+1}}{2}$ (13a)
$\displaystyle
D_{n}\approx(-1)^{n}\frac{w_{e}}{\alpha}\frac{\bar{I}_{n}-\bar{I}_{n+1}}{\bar{I}_{n}+\bar{I}_{n+1}}$
(13b)

where (13a) is the retrieved PB image and (13b) is the retrieved DPC image.

###  2.3 Experiment

We used a polychromatic micro-focus x-ray tube (Hama-matsu L8121-03) operating
with a focal spot of $7\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and the tube
voltage of $40\text{\,}\mathrm{kV}$. The source-to-object and object-to-
detector distance were both around $60\text{\,}\mathrm{cm}$. The sample in
consideration is a PMMA rod with a diameter of $3\text{\,}\mathrm{mm}$. We
used a mask with gold strips, approximately
$52\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in periodicity, fabricated on a
silicon substrate. The data was collected using a Silicon photon-counting
detector with the pixel size of $55\text{\,}\mathrm{\SIUnitSymbolMicro
m}$[20], which was carefully calibrated and corrected[21][22]. While spectral
data is available with this detector, the methods presented here do not use
this spectral information and treats it as an energy integrating detector. The
raw image is shown in Fig.2(b) and the retrieved PB and DPC images are shown
in Fig.3. The results also include images of a dried wasp specimen shown in
Fig.4.

##  3 Results and Discussion

The raw image (Fig.2(b)) obtained from the single-mask method reveals distinct
signal components, including attenuation, the Laplacian of phase, and
differential phase. The presence of attenuation results in darker regions in
the middle of the cylinder. The Laplacian phase manifests as bright and dark
borders along the edges. Additionally, the differential phase appears as
bright and dark fringes specifically in regions with non-zero phase gradient.
As the differential phase signal varies across the sample, it gives rise to
variations in the intensity of the fringes. These observed signal components
align well with the outcomes predicted by our newly proposed model (Eqn.(8)),
validating its reliability in capturing and explaining the underlying physics
of the single-mask phase imaging method.

(a)

(b)

Figure 3: Retrieved (a) PB image and (b) DPC image of a PMMA rod taken with
single mask method in experiment, together with their average cross-section
profiles (light blue curve).

Additionally, The retrieved PB image (Fig.3(a)) obtained from our model
closely resembles the image captured using the propagation-based method
(Fig.2(a)). The minor difference between the attenuation levels can be
attributed to the shift of the spectrum induced by the mask’s silicon
substrate. Furthermore, the retrieved differential phase contrast (DPC) image
(Fig.3(b)) exhibits excellent contrast and visibility. These results indicate
that our proposed retrieval method, based on our formulated model, effectively
separates and provides visualization of different signal components all from a
single image.

(a)

(b)

Figure 4: Retrieved images of a wasp specimen taken with single mask method in
experiment. (a) Retrieved PB image; (b) Retrieved DPC image.

The retrieved PB and DPC images of the dried wasp using our model are shown in
Fig.4. Both these images, retrieved from a single-shot of single-mask phase
contrast intensity image, show fine details of the specimen. Both of these
images exhibit a high sensitivity to tissue boundaries within the sample, due
to the visualization of Laplacian and gradient of phase respectively. Note
that the retrieved PB image is a combination of the Laplacian of phase and the
attenuation.
Also, discernible differences exist between the two images. The retrieved PB
image captures edge information in every orientation in the 2D plane.
Conversely, the retrieved DPC image accentuates features that align
perpendicular to the mask strips.
In addition, it is interesting to note that the DPC images have higher
sensitivity to features with slower variations, such as the bubbles within the
adhesive used to affix the specimen shown at the lower region of the image.
For further verification of our TIE model, we compared the results obtained
from the TIE model and the Monte-Carlo simulation[23]. For the TIE model
(Eqn.(10)), the calculation is based on known $w_{e}$ and $\alpha$ without
using the mask transmission function. In contrast, the Monte-Carlo simulation
employs the mask’s transmission function. Both methods model the same
experimental geometry and sample. The results are shown in Fig.5. From the
figure, we observe that, the results obtained via our TIE model calculation
align consistently with the Monte-Carlo simulation outcomes with different
mask selections. It shows our new model demonstrates overall accuracy in
comparison with Monte-Carlo simulations.

Mask 1

$w_{e}=$4.06\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, $\alpha=1.0$

Mask 2

$w_{e}=$13.75\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, $\alpha=1.0$

(a)

(b)

(c)

Figure 5: Comparison of TIE model calculation and Monte-Carlo simulation with
different Mask the transmission functions. (a) Plot of transmission functions,
where Mask 1 represents a perfect square wave, while Mask 2 is defined as
$M(x)=0.5+0.5\cos(\frac{\pi x}{p})$; (b)-(c) Comparison of the flat-field
corrected raw image between TIE model calculation and Monte-Carlo simulation
of Mask 1 and Mask 2;

According to our model, the final signal depends not on the specific mask
transmission function but rather on two mask parameters: effective aperture
size ($w_{e}$) and mask contrast ($\alpha$). The flat-field corrected raw
image in Fig.5 reveals components corresponding to Eqn.(10), including
attenuation, DPC, and Laplacian phase. Notably, smaller $w_{e}$ values yield
higher DPC signal contrast, due to increased filtration of photons
contributing to the PB term. This enhances DPC signal proportion relative to
the PB signal, thus improving X-ray dose efficiency. However, it is important
to note that smaller $w_{e}$ values may present challenges in mask
manufacturing and potentially require longer exposure times to maintain image
quality. Careful consideration of trade-offs between dose efficiency, mask
fabrication feasibility, and exposure time is crucial in practical single-mask
method applications.

##  4 Conclusion

We have presented a novel light-transport model for single-mask (SM) X-ray
phase contrast imaging which yields strong differential phase signatures from
a simple system design. The measured X-ray intensity with the SM method
combines attenuation, Laplacian phase, and differential phase effects. Our
proposed model provides intuitive understanding of the relative contributions
of these effects to the detector pixel intensities. Our model also shows how
these effects depend on the design parameters of the imaging system. Aided by
our new model, we show an effective retrieval method yielding PB image
(combining attenuation with Laplacian phase) and a differential phase contrast
image in a single acquisition, thus yielding images with two types of edge
enhancement and shape-based contrast.
Our TIE model suggests that the mask transmission function can be
characterized by two parameters that have a significant influence on the final
signal. By considering these two parameters, one finds a flexibility and
adaptability in mask design and performance optimization in practical
applications. Our single-shot retrieval method combined with the simple system
design yields multiple ontrast. This offers a pathway for practical
translatability of PCI for a broad range of applications.

## References

  * [1] R. A. Lewis, “Medical phase contrast x-ray imaging: Current status and future prospects,” Phys. Med. Biol. 49, 3573 (2004).
  * [2] R. Fitzgerald, “Phase-Sensitive X-Ray Imaging,” Physics Today 53, 23–26 (2000).
  * [3] S. Tao, C. He, X. Hao, C. Kuang, and X. Liu, “Principles of Different X-ray Phase-Contrast Imaging: A Review,” Applied Sciences 11, 2971 (2021).
  * [4] S. D. Auweter, J. Herzen, M. Willner, S. Grandl, K. Scherer, F. Bamberg, M. F. Reiser, F. Pfeiffer, and K. Hellerhoff, “X-ray phase-contrast imaging of the breast—advances towards clinical implementation,” The British Journal of Radiology 87, 20130606 (2014).
  * [5] S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384, 335–338 (1996).
  * [6] Y. Suzuki, N. Yagi, and K. Uesugi, “X-ray refraction-enhanced imaging and a method for phase retrieval for a simple object,” Journal of Synchrotron Radiation 9, 160–165 (2002).
  * [7] C. Zuo, J. Li, J. Sun, Y. Fan, J. Zhang, L. Lu, R. Zhang, B. Wang, L. Huang, and Q. Chen, “Transport of intensity equation: A tutorial,” Optics and Lasers in Engineering 135, 106187 (2020).
  * [8] 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).
  * [9] F. Krejci, J. Jakubek, and M. Kroupa, “Hard x-ray phase contrast imaging using single absorption grating and hybrid semiconductor pixel detector,” Review of Scientific Instruments 81, 113702 (2010).
  * [10] G. K. Kallon, P. C. Diemoz, F. A. Vittoria, D. Basta, M. Endrizzi, and A. Olivo, “Comparing signal intensity and refraction sensitivity of double and single mask edge illumination lab-based x-ray phase contrast imaging set-ups,” Appl. Phys. p. 14 (2017).
  * [11] A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Applied Physics Letters 91, 074106 (2007).
  * [12] A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Noninterferometric phase-contrast images obtained with incoherent x-ray sources,” Applied Optics 50, 1765–1769 (2011).
  * [13] F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt., AO 52, 6940–6947 (2013).
  * [14] D. Gürsoy and M. Das, “Single-step absorption and phase retrieval with polychromatic x rays using a spectral detector,” Optics Letters 38, 1461 (2013).
  * [15] M. Das and D. Gursoy, “Single step X-ray phase imaging,” US Patent 9,237,876 (2016).
  * [16] M. Das and Z. Liang, “Spectral x-ray phase contrast imaging for single-shot retrieval of absorption, phase, and differential-phase imagery,” Optics Letters 39, 6343 (2014).
  * [17] M. Das, “Single step differential phase contrast x-ray imaging,” US Patent 9,445,775 (2016).
  * [18] I. Vazquez, I. E. Harmon, J. C. R. Luna, and M. Das, “Quantitative phase retrieval with low photon counts using an energy resolving quantum detector,” Journal of the Optical Society of America A 38, 71 (2021).
  * [19] M. Das and Z. Liang, “Approximated transport-of-intensity equation for coded-aperture x-ray phase-contrast imaging,” Optics Letters 39, 5395–5398 (2014).
  * [20] R. Ballabriga, J. Alozy, G. Blaj, M. Campbell, M. Fiederle, E. Frojdh, E. H. M. Heijne, X. Llopart, M. Pichotka, S. Procz, L. Tlustos, and W. Wong, “The Medipix3RX: A high resolution, zero dead-time pixel detector readout chip allowing spectroscopic imaging,” J. Inst. 8, C02016 (2013).
  * [21] M. Das, B. Kandel, C. S. Park, and Z. Liang, “Energy calibration of photon counting detectors using x-ray tube potential as a reference for material decomposition applications,” in _Medical Imaging 2015: Physics of Medical Imaging,_ vol. 9412 (SPIE, 2015), pp. 263–268.
  * [22] S. Vespucci, C. S. Park, R. Torrico, and M. Das, “Robust Energy Calibration Technique for Photon Counting Spectral Detectors,” IEEE Transactions on Medical Imaging 38, 968–978 (2019).
  * [23] J. Giersch and J. Durst, “Monte Carlo simulations in X-ray imaging,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 591, 300–305 (2008).