Single grating method for low dose 1-D and 2-D phase contrast X-ray imaging

Abstract

X-ray phase contrast imaging (XPCI) using a single absorption grating and a hybrid semiconductor pixel detector is a newly introduced approach with great potential for application in medicine, biology and material research. In comparison with a conventional grating interferometer technique, which requires a multiple-exposure (phase-stepping) procedure, our method is greatly simplified, because both phase gradient and absorption images are obtained from just one exposure. Consequently, the approach can significantly reduce the time-consuming scanning and also possibly the unnecessary dose. Examples of application of the single-grating approach as an imaging tool for investigations in biology are presented. Particularly, we present the extension of our 1-D single grating method to a two-direction sensitive technique. The novel 2-D sensitive XPCI method is based on precise sub-pixel position determination of the X-ray pattern projected by the two-dimensional transmission grating directly from the pattern image. In a single exposure, phase gradient images in two perpendicular directions together with the conventional attenuation image are produced. Results of the proof-of-concept experiment are presented.

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Single grating method for low dose 1-D and 2-D phase contrast X-ray imaging
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2011 JINST 6 C01073
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2011 JINST 6 C01073
PUBLISHED BY IOP PUBLISHING FOR SISSA
RECEIVED:October 31, 2010
ACC EPTED :December 17, 2010
PUBLISHED:January 11, 2011
12th INTERNATIONAL WORKSHOP ON RAD IATI ON IMAGING DETECTORS,
JULY 11th –15th 2010,
ROBINSON COLLEGE, CAMBRIDGE U.K.
Single grating method for low dose 1-D and 2-D
phase contrast X-ray imaging
F. Krejci,a,b,1J. Jakubekaand M. Kroupaa
aInstitute of Experimental and Applied Physics, Czech Technical University in Prague,
Horska 3a/22, 128 00, Prague 2, Czech Republic
bFaculty of Biomedical Engineering, Czech Technical University in Prague,
Nam. Sitna 3105, 272 01 Kladno, Czech Republic
E-mail: frantisek.krejci@utef.cvut.cz
ABS TR ACT: X-ray phase contrast imaging (XPCI) using a single absorption grating and a hybrid
semiconductor pixel detector is a newly introduced approach with great potential for application in
medicine, biology and material research. In comparison with a conventional grating interferometer
technique, which requires a multiple-exposure (phase-stepping) procedure, our method is greatly
simplified, because both phase gradient and absorption images are obtained from just one expo-
sure. Consequently, the approach can significantly reduce the time-consuming scanning and also
possibly the unnecessary dose. Examples of application of the single-grating approach as an imag-
ing tool for investigations in biology are presented. Particularly, we present the extension of our
1-D single grating method to a two-direction sensitive technique. The novel 2-D sensitive XPCI
method is based on precise sub-pixel position determination of the X-ray pattern projected by the
two-dimensional transmission grating directly from the pattern image. In a single exposure, phase
gradient images in two perpendicular directions together with the conventional attenuation image
are produced. Results of the proof-of-concept experiment are presented.
KEYWORDS: Inspection with x-rays; X-ray radiography and digital radiography (DR); Image re-
construction in medical imaging; X-ray detectors
1Corresponding author.
c
2011 IOP Publishing Ltd and SISSA doi:10.1088/1748-0221/6/01/C01073

2011 JINST 6 C01073
Contents
1 Introduction 1
1.1 Conventional X-ray phase contrast imaging (XPCI) 1
1.1.1 Grating interferometer approach 2
1.1.2 Single absorption grating & high spatial resolution detector approach 2
1.1.3 Single absorption grating & hybrid semiconductor pixel detector approach 3
1.2 Two-dimensional (2-D) XPCI 4
2 Principles and methods 4
2.1 1-D setup geometry 5
2.2 2-D setup geometry 6
3 Experimental 7
3.1 One-dimensional single grating XPCI 7
3.2 Two-dimensional single grating XPCI 8
4 Conclusions & future work 8
1 Introduction
X-ray absorption imaging is a standard tool in medical diagnostics and is increasingly used in other
research areas such as biology and materials science. Despite the progress in detector technology,
for certain types of samples, such as biological tissue or polymers, the use of conventional X-ray
radiography is limited because, for a reasonable dose, these objects show poor absorption con-
trast. It has been demonstrated, that phase sensitive X-ray imaging, which uses phase shift rather
than beam attenuation as a contrast imaging signal, is a powerful imaging technique providing
high sensitivity even for weakly absorbing objects [1]–[10]. Furthermore, the measurement of the
phase shifts of the X-rays waveform traveling through the inspected object enables one to reach
substantially higher contrast with significantly lower dose [5].
1.1 Conventional X-ray phase contrast imaging (XPCI)
During the last four decades XPCI has been intensively studied and several approaches for X-
ray phase retrieval enabling to measure the phase variations induced by an investigated object
have been developed. They can be classified into free-space propagation techniques [2], interfer-
ometric methods [4], setups using an analyzer crystal [1], or grating interferometers [11]. The
crystal interferometer uses Bragg reflection as a beam splitter, and the recorded signal measures
the phase shift (Φ)directly. With the analyzer-based imaging method, the Bragg crystal enables
to measure the first spatial derivative of the phase (∇Φ). For propagation-based phase imaging,
where the measured quantity corresponds to the second derivative of the phase (∆Φ), the in-line
method is often used, where the effects of phase contrast become evident, as the sample-detector
distance is increased.
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However, because of the use of crystal optics, there are very stringent requirements on the
X-ray source parameters (very high intensity, high spatial and temporal coherence) and the ex-
perimental setup as a whole (large source-to-detector distance, extreme requirements on the setup
stability) restricting the methods mainly to synchrotron radiation facilities.
1.1.1 Grating interferometer approach
At present, the largest potential by the XPCI methods, enabling high-quality imaging in a table-top
setup, is the grating based approach [11]–[13]. This method provides all the benefits of contrast-
enhanced phase-sensitive imaging, but is also fully compatible with conventional absorption radio-
graphy. During the last years it has been demonstrated that phase contrast imaging with a grating
interferometer can be efficiently performed with a conventional, low-brilliance X-ray source [13].
Furthermore, the ability of the grating interferometer to measure concurrently images based on the
local scattering power of the sample (dark field images) was demonstrated [14].
For the phase-gradient retrieval, current grating-based methods need a phase or an absorption
grating to produce the intensity distribution as well as a separate absorption grating to analyze the
intensity distribution by a phase-stepping method [12] (see figure 1a). During phase-stepping, the
grating is scanned transversely to the incident beam while acquiring multiple projections. The sam-
ple is supposed to be static, and the resulting poor time resolution is one of the major drawbacks of
this method. Moreover, phase stepping necessarily implies multiple exposures and, in comparison
with current conventional absorption imaging techniques, the dose upon phase-stepping tends to be
too high. This limitation prevents the widespread use of the technique, e.g., into clinical practice.
Thus, the development of an approach enabling, in a single exposure to retrieve concurrently
the phase, absorption and possibly local scattering power information is desirable. This feature
opens possibilities for widespread applications (e.g., medical and imaging of dynamical processes,
direct extension to 2-D phase contrast sensitivity in a single exposure, low dose phase-contrast
tomography). Up to now, most of these issues remain unsolved or solved unsatisfactorily in relation
to real-world applications.
1.1.2 Single absorption grating & high spatial resolution detector approach
The method utilizing high resolution X-ray detector together with a single absorbing grating is
a single exposure approach enabling quantitative XPCI. The method was recently discussed by Z.
Wang et. al [15] and the application of the approach for dynamical XPCI studies was demonstrated
[16]. The approach has also its own two-direction sensitive variation, as demonstrated in ref. [17].
For the image retrieval, the intensity distribution downstream of the grating is recorded and ana-
lyzed directly from a single exposure data (i.e., there is no need of phase-stepping) using a spatial
phase demodulation method. Simplicity and minimal requirements on the setup alignment are the
main advantages of the approach. However, to retrieve the phase-gradient image, the pattern pro-
jected by the attenuation grating has to be sampled sufficiently by the X-ray detector used. This
requirement naturally results in an unpleasant reduction of the natural spatial resolution of the de-
tector appearing in the final XPCI image. A possible solution to these obstacles is the application
of an extremely high granularity detector (pixel pitch ∼1µm). However, there is naturally a poor
compromise between pixel pitch and the respective pixel signal-to-noise ratio, resulting again in
dose limitations.
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Figure 1. Current grating-based methods (a) require a phase or an absorption grating (1) to produce the
intensity distribution and another absorption grating (2) to analyze the intensity distribution by the phase-
stepping method. In the single-grating absorption XPCI technique (b) the detector grating (2) are removed
and the phase gradient signal is calculated directly from the detector signal. The investigated object can
be placed in front of or behind the grating. If the detector resolution is sufficiently high to sample each
projected pattern strip, the phase gradient image can be retrieved using the Fourier based demodulation
method. Another approach for single grating XPCI is the usage of a highly sensitive hybrid semiconductor
detector such as Medipix type with sub-pixel resolution data capability [19].
1.1.3 Single absorption grating & hybrid semiconductor pixel detector approach
In our previous work [18], we have demonstrated that the phase gradient information can be ob-
tained thanks to sub-pixel resolution, which can be directly obtained from the high-sensitivity
counting pixel detector in a setup with micro-focus X-ray tube and simple coded aperture. Based
on this sub-pixel resolution principle, we have introduced a new XPCI approach using the single
absorption grating and hybrid semiconductor pixel detector [19] (see figure 1b). The approach en-
ables achieving high quality phase gradient and absorption images (both images obtained from one
exposure) without time-consuming phase stepping and with possible better dose utilization. The
technique works with a fully polychromatic spectrum and gives ample variability in object magni-
fication. Consequently, the approach can open the way to further widespread application of phase
contrast imaging, e.g., into clinical practice.
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1.2 Two-dimensional (2-D) XPCI
The advantages of the two-dimensional sensitivity have been discussed since the XPCI technique
was pioneered. In the grating based XPCI, the quantity that serves as imaging information is not the
wave-front phase profile Φ(x,y), but its first derivative along the axis perpendicular to the grating
lines Φx=∂Φ(x,y)/∂y (see figure 1). Thus this technique is called differential phase-contrast
imaging (DPC).
Some quantitative analysis and signal processing can require retrieval of the wave front pro-
file Φ(x,y). In principle, the full phase profile, which is equivalent to the integral of the index of
refraction along to the beam path of the investigated object, can be done by a straightforward one-
dimensional integration. In practice, the integration very often results, however, in phase images
that suffer from unfavorable artifacts due to noise in the image (e.g., low number of detected pho-
tons) or unknown boundary conditions (the investigated object is larger than the field of view). The
natural solution to these obstacles is a measurement of the phase gradient in two independent direc-
tions enabling, using special integration algorithms, reconstruction of the phase profile of excellent
quality even in the case when the 1-D algorithm completely fails [20].
Together with the method allowing to measure 2-D DPC images under reasonable conditions
(with exception of the approach discussed in ref. [17], for real-world measurement unacceptable
multiple application of the 1-D approach is used) this clearly opens the way, e.g., for further dose
reduction. In this contribution, we focus on the extension of our 1-D single grating method [19]
to the 2-D sensitive approach. The novel 2-D sensitive XPCI method is based on precise sub-pixel
position determination of the X-ray pattern projected by the two-dimensional transmission grating
directly from the pattern image. In a single exposure, phase gradient images in two perpendicular
directions together with the conventional attenuation image are measured.
2 Principles and methods
The phase of spatially coherent X-ray waves passing through the object is shifted according to
the gradient of the effective index of refraction (the gradient is perpendicular to the beam path).
Consequently, X-rays are deflected from their original direction of propagation which results in a
projected pattern shift as indicated in figure 1. The local beam deviation α(or, equivalently, the
pattern shift on the detector) is linearly proportional to the local gradient of the phase waveform
Φ(y,z) and can be quantified as [21]
α(y,x) = λ
2π
∂Φ(x,y)
∂y=
+∞
Z
−∞
∂ δ (x,y,z)
∂ydz ,(2.1)
where λis the wavelength of the incident X-rays, yis the direction of the phase gradient calcula-
tion (perpendicular to the beam propagation), and δ(x,y,z) is the decrement of the real part of the
object’s refractive index. In the grating based XPCI, due to the shape of the gratings and, more im-
portantly, due to the phase stepping procedure, the pattern shift is calculated only in one direction
perpendicular to the grating lines. However, according to eq. (2.1), the phase gradient image is
given as a map of displacement of the pattern projected by the grating in the cases with and without
the object. Thus, when an appropriate 2-D attenuation grating is used, the pattern shift (equivalent
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Figure 2. Illustration of the setup alignment for the 1-D single grating approach: (a) position of the projected
X-ray pattern (pink stripes) on the detector in the case when the condition of eq. (2.2) is fulfilled (n = 2) and
in the situation when the grating strips are aligned parallel to the detector columns. Positions of the projected
X-ray beams irradiating two adjacent pixels without (b) and with (c) the measured object. The alignment
enables to measure the shift d and attenuation of the projected strips just using the respective intensities I1,
I2,I3,I4utilizing eq. (2.3) and eq. (2.4), respectively [19].
to the phase wave-front gradient) can be measured concurrently in two directions perpendicular to
each other and to the beam path. In principle, there is no need for multiple exposures (as well as in
the case of the 1-D XPCI imaging).
2.1 1-D setup geometry
In our one-dimensional single grating XPCI approach, an absorption grating is imaged first without
and then with the investigated object. The displacement calculations are performed using sub-pixel
resolution in a setup aligned according to the condition
T·M=n·s(2.2)
where Tis the periodicity of the grating , Mis the geometrical magnification of the grating, n
= 2,3,4. . . and sis the detector pitch. Fulfillment of eq. (2.2) together with the alignment of the
grating strips parallel to the detector columns ensure that every projected X-ray sub-beam formed
at a certain gap of the grating irradiates the detector in the same position in relation to the pixel
matrix (see figure 2). The pattern shift drepresenting the phase gradient information and the
corresponding absorption image Acan be then calculated simply from one-exposure data (intensity
labeled as I1, I2, I3, I4, see figure 2) [19]:
d=p2·I1
I2
−I3
I4·1+I3
I4−1
(2.3)
A=I3+I4
I1+I2
.(2.4)
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Figure 3. Setup for the 2-D single grating method. The phase of the spatially coherent X-ray waves passing
through the object is shifted according to the gradient of the effective index of refraction. Consequently,
X-rays are deflected from their original direction of propagation which results in a projected pattern shift.
In the proposed method, the pattern shift in two perpendicular directions is calculated directly from the
detector signals I1,I2,I3,I4using sub-pixel resolution made possible by the use of highly sensitive hybrid
pixel semiconductor detector of the Medipix type.
2.2 2-D setup geometry
A newly introduced two-dimensional XPCI method is the direct extension of the 1-D XPCI ap-
proach discussed above and, in more detail, in ref. [19]. This novel 2-D method works similarly
with a single absorbing grating and a hybrid semiconductor pixel detector of the Medipix type [22].
The pattern downstream is directly recorded by the detector and, using sub-pixel resolution data
handling, the phase gradient images in two perpendicular directions together with the conventional
attenuation image are calculated.
Instead of a 1-D transmission grating (parallel strips), a two-dimensional transmission grating
is used (see figure 3). In principle, the shape of the grating opening (projected pattern shape)
can be chosen independently from the shape of the detector pixel; nevertheless, the usage of a
square-shaped geometry is very desirable in relation to the pattern shift calculation. When the
squared geometry of the openings is used (see figure 4) and when eq. (2.2) remains valid for both
pixel matrix directions, the pattern shift calculation in both directions can be performed simply
according to the same relation which is used in the 1-D approach.
Labeling the intensity in the respective neighboring four pixels I’1, I’2, I’3and I’4in the case
without the object and I1, I2, I3and I4in the case with the object (see figure 4), then the horizontal
shift dH, vertical shift dVand attenuation image A can be calculated as:
dH=p1I1
I2
−I0
2
I0
1·1+I1
I2−1
dV=p3I1
I3
−I0
3
I0
1·1+I1
I3−1
,(2.5)
A=I1+I2+I3+I4
I0
1+I0
2+I0
3+I0
4
.(2.6)
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Figure 4. Position of the original (dashed line) and refracted beam (in pink). The non-symmetrical initial
beam alignment is more convenient for practical measurements. In this case the intensity signals I0
1,I0
2,I0
3,I0
4
(without the object) has to be measured.
Figure 5. Phase gradient images of a cylinder of Plexiglas of 2 mm diameter measured with the ideal grating
design (n = 2 in eq. (2.2)). In a single exposure, the natural spatial resolution of the detector 256 x 256 is
reduced by a factor of 2 in the direction of the phase contrast measurement. Image taken with a non-filtered
tungsten spectrum at 50 keV. The cross sectional view profile (one row data without any filtering) taken at
positions indicated by the red line shows perfect agreement with the theoretical prediction (differentiation of
the circle function), which proves that the method generates clear quantitative information with possibility
to use two different direction of the phase wave-front differentiation for its backward reconstruction.
3 Experimental
3.1 One-dimensional single grating XPCI
The single grating method not only generates a form of phase contrast, but also provides clear
quantitative phase information, as demonstrated in figure 5. This feature is crucial in relation to
final image quality, because (as discussed in section 1.2) using two different directions of the phase
gradient measurement one can perform advanced image analysis, e.g., backward phase wave-front
reconstruction.
The single grating method is fully compatible with conventional attenuation imaging. The
method provides all the benefits (non-destructive technique with possibility for tomographic recon-
struction, imaging of dynamical processes, etc.). Moreover, the phase gradient image is acquired
from the same single exposure (see figure 6).
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Figure 6. Example of application of the single grating object for non-destructive imaging in biology: (a)
Absorption and (b) phase gradient image of the bone dry head of a hornet, (c) absorption and (d) phase
gradient image of a mouse leg. All images were taken on a 35 keV tungsten spectrum.
3.2 Two-dimensional single grating XPCI
As verification of reliability of the proposed 2-D method we reproduced the image of a well defined
geometrical object with expected changes in the effective index of refraction in both the horizontal
and vertical directions (see figure 7).
The source-to-detector and source-to-grating distance were for this measurement 890 mm and
405 mm, respectively. The approach has been tested for various geometrical magnifications (there
is no principal restriction in placing the investigated object in front of or behind the 2-D grating)
as well as for various X-ray spectra (from quasi-monochromatic radiation of the K-alpha line of
a Cu-Be target spot to a broad polychromatic spectrum of a W-Be target spot at 70 keV). An
X-ray tube FXE 160.50 FeinFocus with spot of Gaussian shape (sigma ∼1µm) equipped with
a changeable anode was used. The measurement was performed with the Timepix detector [23]
operated in counting mode. Parameters of the grating: size of the square-shaped openings was 25
µm, periodicity of the grating was 50 µm (i.e., the width of the gold parts in between was 25 µm).
The thickness of the gold layer electroplated on a 100 µm thick glass base was 30 µm.
4 Conclusions & future work
In this contribution, we have demonstrated that the novel XPCI approach utilizing a single absorp-
tion grating and a hybrid semiconductor pixel detector is an imaging tool with great application
potential, e.g., in medicine, biology or material research, because both absorption and clear quan-
titative phase gradient information are acquired in a single exposure.
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Figure 7. Results of the proof-of-principle experiment performed for the proposed 2-D sensitive approach:
(a) absorption image (b) phase gradient image measured in the horizontal direction, (c) phase gradient image
measured in the vertical direction, (d) photograph of the testing object (2 cylinders of PMMA of 2 mm
diameter fixed together by a piece of double-sided tape). All images (a,b,c) were retrieved from a single
exposure data. Image taken with a non-filtered tungsten spectrum at 50 keV. The cross sectional views
profiles (e) and (f) indicated by the yellow lines in (b) and (c), respectively, show good agreement with the
theoretical prediction.
As a direct extension of this 1-D single grating approach, the novel 2-D sensitive XPCI method
has been introduced. In a single exposure, phase gradient images in two perpendicular directions
together with the conventional attenuation image are measured. The application of the 2-D phase
gradient information in relation to the phase wave-front integration is underway.
Acknowledgments
This work has been carried out in framework of the Medipix Collaboration. The authors acknowl-
edge the financial support of the Projects LC06041 and 6840770040 of the Ministry of Education,
Youth and Sports of the Czech Republic. The authors also gratefully appreciate V. Jurka and K.
Hruska for their work in the design and fabrication of the gratings.
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References
[1] T.J. Davis et al., Phase-contrast imaging of weakly absorbing materials using hard X-rays,Nature
373 (1995) 595.
[2] S.W. Wilkins et al., Phase-contrast imaging using polychromatic hard X-rays,Nature 384 (1996) 335.
[3] R. Fitzgerald, Phase-sensitive X-ray imaging,Phys. Today 53 (2000) 23.
[4] A. Momose, T. Takeda, Y. Itai and K. Hirano, Phase-contrast X-ray computed tomography for
observing biological soft tissues,Nat. Med. 2(1996) 473.
[5] R.A. Lewis, Medical phase contrast x-ray imaging: current status and future prospects ,Phys. Med.
Biol. 49 (2004) 3573.
[6] M. Bech et al., Soft-tissue phase-contrast tomography with an x-ray tube source,Phys. Med. Biol. 54
(2009) 2747.
[7] L.D. Chapman et al., Diffraction enhanced x-ray imaging,Phys. Med. Biol. 42 (1997) 2015.
[8] J. Jakubek et al., Phase contrast enhanced high resolution X-ray imaging and tomography of soft
tissue,Nucl. Instrum. Meth. A 571 (2007) 69.
[9] J. Jakubek et al., Spectrometric properties of Timepix pixel detector for X-ray color and phase
sensitive radiography,IEEE Nucl. Sci. Symp. Conf. Proc. 3(2007) 2323.
[10] J. Jakubek et al., Compact system for high resolution x-ray transmission radiography, in-line phase
enhanced imaging and micro CT of biological samples,IEEE Nucl. Sci. Symp. Conf. Proc. (2006)
1077.
[11] A. Mamose et al., Demonstration of x-ray Talbot interferometry,Jap. J. Appl. Phys. 42 (2003) L866.
[12] T. Weitkamp et al., X-ray phase imaging with a grating interferometer,Opt. Express 13 (2005) 6296.
[13] F. Pfeiffer, T. Weitkamp, O. Bunk and C. David, Phase retrieval and differential phase-contrast
imaging with low-brilliance X-ray sources,Nat. Phys. 2(2006) 258.
[14] F. Pfeiffer et al., Hard-X-ray dark-field imaging using a grating interferometer,Nat. Mater. 7(2008)
134.
[15] Z. Wang et al., Fast x-ray phase-contrast imaging using high resolution detector,IEEE Trans. Nucl.
Sci. 56 (2009) 1383.
[16] A. Momose, W. Yashiro, H. Maikusa and Y. Takeda, High-speed x-ray phase imaging and X-ray
phase tomography with Talbot interferometer and white synchrotron radiation,Opt. Express 17
(2009) 12540.
[17] H.H. Wen et al., Single-shot x-ray differential phase-contrast and diffraction imaging using
two-dimensional transmission gratings,Opt. Lett. 35 (2010) 1932.
[18] F. Krejci, J. Jakubek, M. Kroupa, X-ray phase contrast imaging using single absorption coded
aperture, article in press in Nucl. Instrum. Meth. A(corrected proof).
[19] F. Krejci, J. Jakubek and M. Kroupa, Hard x-ray phase contrast imaging using single absorption
grating and hybrid semiconductor pixel detector,Rev. Sci. Instrum. 81 (2010) 113702.
[20] C. Kottler, C. David, F. Pfeiffer and O. Bunk, A two-directional approach for grating based
differential phase contrast imaging using hard x-rays,Opt. Express 15 (2007) 1175.
[21] M. Born and E. Wolf, Principles of optics, Pergamon Press, Oxford U.K. (1980).
[22] X. Llopard et al., Medipix2, a 64k readout chip with 55µm square elements working in single photon
counting mode,IEEE Trans. Nucl. Sci. NS-49 (2001) 2279.
[23] X. Llopart et al., Timepix, a 65k programmable pixel readout chip for arrival time, energy and/or
photon counting measurements,Nucl. Instrum. Meth. A 581 (2007) 485.
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