Quantitative 3D refractive index decrement reconstruction using single-distance phase-contrast tomography data

Abstract

X-ray propagation-based phase-contrast imaging is an attractive phase-sensitive imaging technique that has found applications in many research fields. Here, we report the investigations of a method which can quantitatively reconstruct in 3D the refractive index decrement of a quasi-homogeneous object using single-distance phase-contrast tomography data. The method extends the Born-type approximation phase-retrieval algorithm, which is based on the phase-attenuation duality (ε = δ/β, with constant ε) and suitable for homogeneous objects, to tomography and we study its application to quasi-homogeneous objects. The noise performance and the phase-attenuation duality influences of the method are also investigated. In simulation, the method allows us to quantitatively reconstruct the 3D refractive index decrement for quasi-homogeneous and weakly absorbing samples and it performs well in the practical noise situation. Furthermore, it shows a substantial contrast increase and successfully distinguishes different materials in a quasi-homogeneous and weakly absorbing sample from experimental data, even with inappropriate ε value.

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Quantitative 3D refractive index decrement reconstruction using single-distance phase-
contrast tomography data
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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS
J. Phys. D: Appl. Phys. 44 (2011) 495401 (9pp) doi:10.1088/0022-3727/44/49/495401
Quantitative 3D refractive index
decrement reconstruction using
single-distance phase-contrast
tomography data
R C Chen1, L Rigon1and R Longo1,2
1INFN, Sezione di Trieste, Trieste 34012, Italy
2Department of Physics, University of Trieste, Trieste 34127, Italy
E-mail: rongchang.chen@ts.infn.it
Received 1 July 2011, in final form 21 October 2011
Published 22 November 2011
Online at stacks.iop.org/JPhysD/44/495401
Abstract
X-ray propagation-based phase-contrast imaging is an attractive phase-sensitive imaging
technique that has found applications in many research fields. Here, we report the
investigations of a method which can quantitatively reconstruct in 3D the refractive index
decrement of a quasi-homogeneous object using single-distance phase-contrast tomography
data. The method extends the Born-type approximation phase-retrieval algorithm, which is
based on the phase-attenuation duality (ε=δ/β, with constant ε) and suitable for
homogeneous objects, to tomography and we study its application to quasi-homogeneous
objects. The noise performance and the phase-attenuation duality influences of the method are
also investigated. In simulation, the method allows us to quantitatively reconstruct the 3D
refractive index decrement for quasi-homogeneous and weakly absorbing samples and it
performs well in the practical noise situation. Furthermore, it shows a substantial contrast
increase and successfully distinguishes different materials in a quasi-homogeneous and weakly
absorbing sample from experimental data, even with inappropriate εvalue.
(Some figures may appear in colour only in the online journal)
1. Introduction
For about one century, x-ray imaging has always exploited
the absorption differences of the samples. In contrast, x-ray
phase-sensitive imaging, which uses the phase shift rather than
the absorption as the imaging signal, substantially extends
the possibilities of x-ray absorption imaging, especially
when imaging low-Zsamples [1,2]. Several phase-sensitive
imaging techniques have been developed since the mid-
1990s, such as the interferometric method [3,4], the analyser
based method [5,6], the grating based method [7,8], the
coded-aperture based method [9] and the propagation-based
method [10,11]. These methods differ enormously in the
recorded signal, the experimental setup and the radiation
source requirements. Because of the use of interferometer or
crystal optics, the interferometric and analyser based methods
rely on a highly parallel and monochromatic x-ray beam, which
are typically available at synchrotron radiation facilities. On
the other hand, the latter three methods have been demonstrated
to work well with a micro-focus x-ray tube sources [11–13].
Among these three methods, the x-ray propagation-based
phase-contrast imaging (PPCI) has the simplest experimental
setup, which requires no additional optics in the imaging
geometry and is identical to the conventional radiographies
except for providing that the beam is sufficiently spatially
coherent and increasing the sample-to-detector distance (SDD)
[14]. PPCI has attracted increasing attention and its application
continues in many research fields [15–17].
X-ray computed tomography (CT) is a nondestructive
technique for visualizing interior features within samples,
and for obtaining quantitative information on their three-
dimensional (3D) geometries and properties [18,19]. PPCI
0022-3727/11/495401+09$33.00 1© 2011 IOP Publishing Ltd Printed in the UK & the USA

J. Phys. D: Appl. Phys. 44 (2011) 495401 R C Chen et al
has been successfully extended to CT mode, resulting in
x-ray propagation-based phase-contrast CT (PPCT) [20–22].
Qualitative PPCT can be performed by applying the standard
filtered back-projection CT reconstruction algorithm to the
PPCT data. The results are proportional to the Laplacian
of sample refractive index distribution providing an edge
enhancement, which allows us to visualize the boundaries of
regions with different refraction properties [14]. In addition,
PPCT radiographies fringes contain phase information which
could be extracted by means of phase retrieval [23]. Several
linear phase-retrieval algorithms have been proposed, such
as the transport of intensity equation (TIE) method [24],
the contrast transfer function (CTF) method [23], the mixed
approach between the CTF and TIE method [25,26], the Born-
type approximation method [27] and the Bronnikov algorithm
[28]. In order to reconstruct the 3D refractive index of the
object, all the above-mentioned methods require at least double
SDD intensity measurements that will sharply increase the
experiment time and deliver a higher radiation dose to the
samples, which could hinder their biomedical applications.
Definitely, phase retrieval employing only a single SDD PPCT
data will boost its applications, and make the procedure much
easier.
Several phase-retrieval algorithms using a single SDD
PPCT data has been proposed, such as the modified Bronnikov
algorithm method [29], which modifies based on the Bronnikov
algorithm by introducing an absorption correction factor and
it is further investigated by different groups on the basis of
the phase-attenuation duality property of low-Zsamples [30–
32]; the CTF based method [33], which explores the validity
conditions for the linear CTF expression and applies to a
substantially wider class of objects; the TIE based method
[34], which quantitatively reconstructs complex refractive
index distribution of a multi-material object, and it requires
a priori knowledge of sample; and the nonlinear phase-
retrieval method [35], which performs beyond the solution to
the linearized TIE equation. Other approaches can be found
in [20,27,36,37].
In this paper, we extend the Born-type approximation
phase-retrieval algorithm [27] to tomography and we study
its application to quasi-homogeneous objects. The noise
performance and the phase-attenuation duality influences of
the method are also investigated via simulation. Moreover, the
method is tested by experimental PPCT data of polymers and
biological samples.
2. Theory
As shown in figure 1, an object is illuminated by a
monochromatic plane x-ray beam, and the PPCT projection
images are recorded in the image plane. The object can be
described by its 3D complex refractive index distribution,
n(x,y,z) =1−δ(x,y,z) +iβ(x,y, z), where δand β
are the refractive index decrement and the absorption index,
respectively, and (x,y,z)are the spatial coordinates. Because
of the weak interaction of x-rays with matter, the beam
propagation path inside the sample can be assumed to be
straight and Ldenotes the linear path in the sample. The
wave–object interaction can then be represented as object
transmittance function [38]
Tθ(x, y ) =exp[−γθ(x , y) −iφθ(x , y)],(1)
where θrepresents the CT rotation angle; the sample
phase φθ(x, y ) function and absorption function γθ(x , y) are,
respectively,
φθ(x, y ) =kLθ
δ(x,y, z)dz,
γθ(x, y ) =kLθ
β(x,y, z)dz,
(2)
where k=2π/λ is wavenumber and λis the wavelength, and
Lθ denotes the line integral over the object along the beam
path Lat CT rotation angle θ.
Under the paraxial approximation, the propagation over
free space can be described by the Fresnel diffraction integral.
Thus the intensity distribution Iθ,z at SDD =zcan be
represented as [39]
Iθ,z(x, y) =|hz(x, y ) ∗∗Tθ(x, y )|2,(3)
where
hz(x, y ) =exp(ikz)
iλz exp iπ
λz (x2+y2)
is the Fresnel propagator, the ∗∗ denotes two-dimensional
convolution.
Let us now assume imaging an object of weak absorption,
γ(x,y) 1, and slowly varying phase shift, i.e. Guigay’s
condition |φ(x +λzξ , y +λzη) −φ(x,y)|1 is fulfilled
[40], with (ξ , η) corresponding to (x, y ) in the Fourier
space. According to the Born-type approximation PPCI theory,
the intensity distribution Izat SDD=zof PPCT can be
approximated by the following equation [27]:
F[(Iθ,z/Iθ,0−1)/2](ξ, η) =ˆγθcos χ+ˆ
φθsin χ, (4)
where χ=πλz(ξ2+η2),ˆ
φand ˆγdenote the Fourier transform
of phase and absorption function, φand γ, respectively.
The sin χ, phase optical transfer function, reaches its first
maximum at z≈1/2λ(ξ2+η2).
Suppose imaging a pure phase object, i.e. γθ(x, y ) =0
and Iθ,0=1, combining with equation (4) the following result
can be obtained [27]
φθ(x, y ) =F−1F[(Iz,θ −1)/2]
sin χ.(5)
Equation (5) allows retrieving the phase function of a pure
phase sample from only a single SDD PPCI radiograph. For
convenience we name it as pure phase object Born algorithm
(PO-BA). However, even though the absorption level for low-
Zsamples is very low, pure phase object condition is not
satisfied and the retrieved result will be severely corrupted by
the residual absorption artefact, as is shown in the following.
Fortunately, the residual absorption artefact of PO-BA can be
removed based on the phase-attenuation duality property of
low-Zsample.
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J. Phys. D: Appl. Phys. 44 (2011) 495401 R C Chen et al
Figure 1. Schematic of PPCT scanning geometry.
Figure 2. Simulation phantom: (a) 3D phantom, (b) a simulated PPCT projection at a given rotation angle, (c)–(e) phantom slices
according to the line positions in (a).
When a sample is quasi-homogeneous, the real and
imaginary parts of its complex refraction index are proportional
to each other (phase-attenuation duality property) [41]:
δ(x,y, z) =εβ(x,y,z), (6)
where εis a constant. With equations (2) and (6), and
the linearity property of Fournier transform, the following
equation can be gained:
ˆ
φθ(ξ, η) =εˆγθ(ξ , η). (7)
Equation (7) indicates that the sample can be quasi-
homogeneous if the all the materials have the same εvalue.
In the case of weakly absorbing sample, i.e. I0≈1, substitute
equation (7) into (4), and the following equating can be
obtained [27]:
φθ(x, y ) =F−1F[(Iz,θ −1)/2]
ε−1cos χ+ sin χ.(8)
Equation (8) retrieves the phase function of a phase-amplitude
sample using a single SDD PPCI radiograph. For convenience
and in contrast to PO-BA, we name as phase-attenuation
duality Born algorithm (PAD-BA). The εvalue can be obtained
from x-ray database [42] or estimated via different methods
[32,43], here we treat it as a priori knowledge of the sample.
After retrieving the phase function for the entire set of
projection images for the PPCT, the 3D refractive index can be
reconstructed by applying the standard filter back-projection
algorithm to φθ(x, y ), that is
δ(x,y, z) =k−1π
0
φθ(x, y ) ∗∗νdθ, (9)
where νis CT reconstruction filter [18]. Similar results of
equations (8) and (9) were previously obtained by Gureyev
[30], see equation (7) and (8) in [30].
3. Materials and methods
3.1. Simulations
The performance of the PAD-BA was first investigated via
simulation. The simulated phantom is shown in figure 2,in
which figure 2(a) is the 3D phantom, figure 2(b) is a simulated
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J. Phys. D: Appl. Phys. 44 (2011) 495401 R C Chen et al
Figure 3. Reconstructed phantom slices without implementing phase retrieval and after implementing the PO-BA or PAD-BA, the slices
positions are the same as in figures 2(c)–(e): (a)–(c) without implementing phase retrieval, (d)–(f) after implementing the PO-BA, (g)–(i)
after implementing the PAD-BA, (j) profiles of (e), (h) and figure 2(d) according to the line position in (h).
PPCT projection at a given rotation angle and figures 2(c)–(e)
are phantom slices according to the line positions in figure 2(a).
As figure 2shows, the 3D phantom is made up of an ellipsoid
and two spheres inside. Three different complex refractive
index values are assigned to the three phantom regions,
respectively, i.e. the (δ, β ) values of the phantom are (0.0, 0.0)
(black, background), (1.0×10−7,1.0×10−10), (2.0×10−7,
2.0×10−10),(3.0×10−7,3.0×10−10 )(white), which means
ε=1000 for the quasi-homogeneous phantom. The PPCT
data are created via CT projection theory and the Fresnel
diffraction theory with energy of 14 keV, SDD=0.6 m, and
a pixel size of 9 µm. The pixels of the 3D phantom are
128 ×128 ×128 and 220 PPCT projections were generated
within 180◦CT scan range. With the simulation parameters,
we obtain |φ(x +λzξ , y +λzη) −φ(x, y)|≈0.08 rad., which
fulfils Guigay’s condition.
At first, for comparison, the phantom phase function
was retrieved by implementing PO-BA (equation (5)) and
PAD-BA (equation (8) with ε=1000)to the simulated
PPCT projections separately, then the 3D refractive index
was reconstructed by implementing the standard filter back-
projection algorithm with Shepp–Logan filter.
Following, the phase-attenuation duality influences of
the PAD-BA were investigated. With the same simulated
PPCT data, the phantom phase functions were retrieved by
implementing PAD-BA with εvalue of 700, 800, 900, 1000,
1100, 1200 and 1300, respectively, and the corresponding
slices were reconstructed. The slices in the same position
as in figure 2(d) for all εvalues were employed for
investigation.
Next, photon counting statistics noise (PCSN) was added
to the PPCT data in order to investigate the noise performance
of the PAD-BA. We considered this type of noise because
it is unavoidable, which means that even if a sensor could
count each photon hitting it without added noise, there would
still be noise due to the statistics of counting photons. Such
statistics is regulated by the Poisson distribution. Thus, PCSN
was included in the simulated data on a pixel-by-pixel basis,
generating random deviates drawn from a Poisson distribution
with mean value Nequal to the expected number of photons
reaching each pixel in the noiseless image. Six different
levels of noise were added and the corresponding dataset was
generated, the average Nvalue being 10000, 5000, 1000, 500,
100 and 50 for the respective data set. The reconstructed slices,
obtained this time with the appropriate value of ε=1000, were
in the same position as in figure 2(d) for all noise levels and
figure 2(d) was employed for quantitative evaluation.
The relative root-mean-square (RMS) error metric is
introduced to quantitatively evaluate the difference between
phase retrieved and phantom slices. The RMS value is
calculated via the following formula:
RMS =
i,j |δpr(i, j ) −δp(i, j )|2/
i,j |δp(i, j )|2×100%,
(10)
where δp(i, j ) and δpr (i, j ) are the values of the phantom and
phase-retrieved images at pixel (i, j), respectively.
3.2. Experiments
PAD-BA was also tested by experimental PPCT data which
were collected at the SYRMEP beamline [44] at the ELETTRA
synchrotron facility, Italy. The SYRMEP beamline employs
a bending magnet source with a Si (1 1 1) double-crystal
monochromator, which provides photon energy ranging from
8.5 to 35 keV. The energy of 14 keV, SDD =0.6 m and a CCD
detector (Photonic Science, UK) with 2004 ×1336 pixels and
an effective pixel size of 9 µm were used to acquire all PPCT
data sets.
Three samples were investigated. The first one is a
polymer sample, which is a plastic tube (∅≈5 mm) filled with
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J. Phys. D: Appl. Phys. 44 (2011) 495401 R C Chen et al
Figure 4. Reconstructed phantom slices of the PAD-BA with different εvalues, the slice position are the same as in figure 2(d), (a)–(f)
reconstructed slices for εvalues of 700, 800, 900, 1100, 1200 and 1300, respectively, (g) profiles of all slices in figures 4and 2(d) along the
line position in figure 4(a).
Figure 5. Reconstruction slices with different levels of noise after implementing PAD-BA: (a)–(f) reconstructed slice with average photon
number per pixel (N) value of 10 000, 5000, 1000, 500, 100 and 50, respectively.
poly(methyl methacrylate) (PMMA) powder (mean particle
size 600 µm); the second one is a coffee bean wrapped
in parafilm and the third one is an ant sample, which for
convenience was put into a plastic tube (∅≈5 mm) during
PPCT data acquisition. We obtain |φ(x +λzξ, y +λzη) −
φ(x,y)|≈0.5 rad for the polymer sample; even though this
value does not strictly obey Guigay’s condition, the sample
is well reconstructed by the PAD-MBA as is shown in the
following. For all three samples, 900 PPCT projections were
collected within 180◦CT scan range.
4. Results and discussion
4.1. Simulations
Figure 3presents the reconstructed phantom slices without
implementing phase retrieval and after implementing the PO-
BA or PAD-BA. The positions of reconstructed slices are the
same as figures 2(c)–(e). More in detail, figures 3(a)–(c)
are reconstructed slices without implemented phase retrieval,
figures 3(d)–(f) are reconstructed slices with PO-BA, while
figures 3(g)–(i) are reconstructed slices with PAD-BA, and
figure 3(j) are profiles of figures 3(e), (h) and 2(d) according
to the line position in figure 3(h). It can be seen that,
without implementing phase retrieval (phase contrast), the
reconstruction slices have strong edge enhancement which
are shown with the dark-white fringe; after PO-BA, which
treat the phase-amplitude sample as pure phase sample, the
reconstruction slices are corrupted by the residual absorption
artefact and the reconstructed refractive index are totally
different with respect to the phantom values. In contrast, after
implementing PAD-BA, as the inset in figure 3(j) shows, the
residual absorption artefact was removed and the refractive
index matches the phantom value well except for some small
errors near the edges of the objects, which can be explained
by the influence of the interpolation in the back-projection
step.
Figure 4represents the reconstructed phantom slices of
the PAD-BA with different εvalues. Figures 4(a)–(f) are
reconstructed slices for εvalues of 700, 800, 900, 1100,
5

J. Phys. D: Appl. Phys. 44 (2011) 495401 R C Chen et al
Table 1. RMS values between phantom and the retrieved slice for different levels of noise with ε=1000.
Noise (PCSN)
Noise level Noiseless N10 000 5000 1000 500 100 50
1/√N×100% 1.0 1.4 3.2 4.5 10.0 14.1
RMS (%) 12.5 12.9 13.2 15.5 18.4 30.1 43.1
Figure 6. Experimental results of polymer sample without implementing phase retrieval: (a) reconstructed slice, (b) histogram of (a).
Figure 7. Experimental results of polymer sample with implementing the PO-BA: (a) reconstructed slice, (b) histogram of (a).
1200 and 1300, respectively, figure 4(g) shows the profiles of
figures 2(d) and 4(a)–(f) along the line position in figure 4(a).
From figure 4, especially figure 4(g), it is obvious that the
reconstructed slices with too large εvalues will blur the result
while the phase-retrieval effect will be diminished with too
small εvalues, which will cause some degree of residual edge
enhancement in the result image. It should be noted that, even
in the extreme situations (ε=700, 1300), the three different
regions can still be distinguished well although the quantitative
result is lost. This means it is possible to distinguish
different materials in a sample even with inappropriate ε
value, which is not easy to precisely estimate for an unknown
sample.
Figure 5shows the reconstruction slices with different
levels of noise after implementing PAD-BA: Figures 5(a)–(f)
are reconstructed slices with the appropriate value of ε=1000
and average photon number per pixel (N) of 10000, 5000,
1000, 500, 100 and 50, respectively. Obviously, the noise
level in the image depends on N, and can be quantified by
means of relative standard deviation 1/√N×100%. It is
clearly seen that the reconstructed slice is stable for N=
10 000, which is often easily achievable in practice. On
the other hand, with the decrease of counted photons, the
result is progressively corrupted by the noise and the visibility
decreases. These results were confirmed by the RMS values
between the phantom and the retrieved slices (table 1).
4.2. Experiments
The experimental results of polymer sample are shown from
figures 6–9, in which figure 6is a reconstructed slice without
implementing phase retrieval, figure 7is a reconstructed slice
with implementing the PO-BA, figure 8is a reconstructed
slice with implementing the PAD-BA with ε=1384 for
6

J. Phys. D: Appl. Phys. 44 (2011) 495401 R C Chen et al
Figure 8. Experimental results of polymer sample with implementing the PAD-BA (ε=1384): (a) reconstructed slice, (b) histogram of
(a), the labels indicate the peak for different material.
Figure 9. Experimental results of polymer sample with implementing the PAD-BA (ε=500): (a) reconstructed slice, (b) histogram of (a),
the labels indicate the peak for different materials.
the PMMA at 14 keV [42] and figure 9is a reconstructed
slice with implementing the PAD-BA with ε=500. In
all these figures, the (a) image depicts the reconstructed
slices, while the (b) image reports the histogram of (a). In
figure 6(a), the outline of the PMMA powder and plastic tube
is very clear since there is strong edge enhancement, but it
is impossible to distinguish between two different materials
and that is confirmed of its histogram in figure 6(b), in which
only one peak is obtained. After implementing the PO-
BA, as figure 7shows, the result is totally corroded by the
residual absorption artefact. In contrast, after implementing
the PAD-BA, the residual absorption artefact is removed as
both images of figure 8show; moreover the plastic tube and
PMMA can be well distinguished. The reconstructed δvalue
is 8.56 ×10−7, which is quite different from the ideal value
1.36 ×10−6, however, it is obvious that the each material has
its own separate histogram peak. In figure 9, even though
ε=500 is far from the ideal value 1384 of PMMA, both
images show that the plastic tube and PMMA can still be well
distinguished.
Figures 10 and 11 represent the 3D rendering images of
coffee bean and ant samples after applying the PAD-BA (ε=
1000 for both samples), as examples of possible application of
the technique. In both cases, the details of samples are well
reconstructed, such as the interstices of coffee bean and the
Figure 10. 3D rendering image of coffee bean sample after
applying the PAD-BA.
legs and antennas of the ant. It should be noted that promising
results are obtained for both samples even though we just
used the typically ε=1000 for low-Zsamples instead of
optimizing it. Since PAD-BA requires only one single SDD
PPCT data, this could be interesting for botany and biology
application researchers, whose samples can often be treated as
quasi-homogeneous and weakly absorbing.
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J. Phys. D: Appl. Phys. 44 (2011) 495401 R C Chen et al
Figure 11. Two different views of 3D rendering images of ant sample after applying the PAD-BA.
5. Conclusions
In this paper, we extended the phase-attenuation duality Born-
type approximation phase-retrieval algorithm (PAD-BA) to
tomography and we applied it to quasi-homogeneous object.
Combining with CT, PAD-BA simplifies the experimental
setup and imposes no additional dose compared with
absorption CT. The simulation results demonstrate that PAD-
BA allows us to quantitatively reconstruct the 3D refractive
index for quasi-homogeneous and weakly absorbing samples,
from single SDD PPCT data, and it performs well in the
practical noise situation. The experimental results show
a substantial contrast increase and successfully distinguish
among different materials in a quasi-homogeneous and weakly
absorbing sample, even with inappropriate εvalue. We believe
that PAD-BA will find application in many different fields, such
as botany, biomedical and material science.
Acknowledgments
The authors wish to thank Nicola Sodini for his invaluable
help during data acquisition. They are grateful to Edoardo
Castelli and Tiqiao Xiao for encouraging this study and for
helpful discussion. The authors also would like to thank the
reviewers for pointing out some weaknesses of the paper and
for providing helpful comments. RCC was partially supported
by ICTP TRIL programme.
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