X-ray phase contrast image simulation - X

NIM B
Beam Interactions
with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 254 (2007) 307–318
www.elsevier.com/locate/nimb
X-ray phase contrast image simulation
A. Peterzol
a
a,*
, J. Berthier a, P. Duvauchelle a, C. Ferrero b, D. Babot
a
Institut National des Sciences Appliquées de Lyon, Laboratoire de Contrôle Non Destructif par Rayonnements Ionisants (CNDRI),
20 Av. A. Einstein, 69621 Villeurbanne, France
b
European Synchrotron Radiation Facility, BP220 rue Horowitz, 38043 Grenoble, France
Received 3 July 2006; received in revised form 16 October 2006
Available online 21 December 2006
Abstract
A deterministic algorithm is proposed to simulate phase contrast (PC) X-ray images for complex three-dimensional (3D) objects. This
algorithm has been implemented in a simulation code named VXI (virtual X-ray imaging). The physical model chosen to account for PC
technique is based on the Fresnel–Kirchhoff diffraction theory.
The algorithm consists mainly of two parts. The first one exploits the VXI ray-tracing approach to compute the object transmission
function. The second part simulates the PC image due to the wave front distortion introduced by the sample.
In the first part, the use of computer-aided drawing (CAD) models enables simulations to be carried out with complex 3D objects.
Differently from the VXI original version, which makes use of an object description via triangular facets, the new code requires a more
‘‘sophisticated’’ object representation based on non-uniform rational B-splines (NURBS).
As a first step we produce a spatial high resolution image by using a point and monochromatic source and an ideal detector. To simulate the polychromatic case, the intensity image is integrated over the considered X-ray energy spectrum. Then, in order to account for
the system spatial resolution properties, the high spatial resolution image (mono or polychromatic) is convolved with the total point
spread function of the imaging system under consideration.
The results supplied by the proposed algorithm are examined with the help of some relevant examples.
2006 Elsevier B.V. All rights reserved.
PACS: 81.70.q; 87.59.Bh; 42.15.Dp; 42.50.Dr
Keywords: Phase contrast; X-ray imaging; Deterministic simulation; Ray-tracing
1. Introduction
Phase contrast X-ray imaging [1–26] has been for almost
10 years a very active field of X-ray science since the technique offers greatly enhanced image quality over conventional radiology [27]. Phase contrast (PC) arises because
both the amplitude and phase of X-rays are modified as
an X-ray beam propagates through an object. A detailed
understanding of the underlying physics requires that the
radiation be treated as a wave field rather than by means
of simple geometrical optics. In that context, a big effort
*
Corresponding author.
E-mail address: angela.peterzol@insa-lyon.fr (A. Peterzol).
0168-583X/$ - see front matter 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.nimb.2006.11.042
has been dedicated to develop a comprehensive theory
for PC imaging.
In order to develop clinical and industrial applications
of this promising technique, it is of primary importance
to well characterize the physical background in order to
quantitatively analyze the PC image performance.
Moreover, to develop and optimize a new imaging
system, and to recognize the influence of the various adjustable parameters, simulation can be a helpful tool.
In previous papers [28–30], it was reported on a computer code developed to simulate the operation of radiographic, radioscopic or tomographic systems. This code,
named VXI (virtual X-ray imaging), is based on ray tracing
techniques and is completely deterministic. It enables to
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A. Peterzol et al. / Nucl. Instr. and Meth. in Phys. Res. B 254 (2007) 307–318
2. Theory of phase contrast imaging
general than the Fresnel–Kirchhoff diffraction theory, since
it takes into account also the partial coherence effects of the
incident X-ray beam.
The Fresnel–Kirchhoff diffraction formalism assumes a
spatially coherent X-ray source (either a point X-ray source
or a plane wave), however the spatial incoherence effects of
the finite source size are introduced in a second step by
means of a convolution with the source response function.
This approach is still valid when considering incoherent
X-ray sources such as X-ray tubes.
For the more general case of partially coherent illumination, the readers are referred to a recent work [26] where a
model corresponding to a simple generalization of the
Shell-model source [33] is presented.
In the following the Fresnel–Kirchhoff theory is briefly
reviewed and the fundamental equations which VXI PC
is based on are recalled. More details can be found in
[25], where the general principles and results of the optimization of the in-line PC imaging system performances for
spatially incoherent sources are investigated.
In order to apply the Kirchhoff’s integral [31] to PC
imaging, let us first consider a monochromatic spherical
wave of wavelength k, generated from a point source
located at the point P0 of coordinates (x0, y0, r0) (see
Fig. 1), and propagating through a sample virtually lying
on the so-called object plane, which is positioned just after
the sample and perpendicular to the z-axis. The distance
source-to-object plane is r0. In order to write the diffracted
wave field in P(x1, y1, r1) at a distance r1 from the object
plane, we assume that the wave field at the object plane
Most of the theoretical frameworks found in the literature, [1,2,4,6–10,13,14,17,18,20,25,26] being some examples, and describing the so-called in-line PC imaging are
based on the Fresnel–Kirchhoff diffraction theory [31–33].
In particular Wu and Liu [18] provided a general theoretical formalism covering both near field diffraction and
holography. Differently from previous works, the formulas
they proposed did not assume a low overall phase perturbation [9] or limiting conditions on the partial derivatives
of the phase and attenuation terms [10]. In this way they
provided the tools to handle a broader ensemble of cases.
In addition, also simpler models based on ray-optical
approximations have been proposed [12,16,21]. These
models are based on the refraction of X-rays within an
object.
It is possible to show that the ray-optical results can also
be obtained by including in the diffraction formalism some
restrictions on the spatial frequencies present in the final
image, but without limitations on the maximum phase
shift. A detailed description of the simplified ray-optical
approach to describe PC images can be found in [24].
In writing VXI PC we followed the more general Fresnel–Kirchhoff diffraction formalism.
For sake of completeness, it should be mentioned here
that a third approach based on the Wigner distribution formalism was recently developed [23]. This approach is more
Fig. 1. Schematic display of the wave-optical approach to the PC
formation mechanism for a circular cross-section object being irradiated
by a spherical and monochromatic X-ray wave generated by a point
source located at a distance r0 from the object plane. The image is detected
at a distance r1 from the latter.
simulate direct images [28] and first-order scattering [29,30]
in complex configurations (intricate three-dimensional
(3D) objects, polychromatic spectra, focal spots causing
geometric unsharpness, etc.).
In this work we report on the implementation of an
algorithm designed to simulate X-ray PC images of complex 3D objects in the VXI code.
The new code version (VXI PC) consists mainly of two
parts. The first one exploits the VXI ray-tracing approach
to compute the object transmission function. The new part
simulates the PC image due to the wave front distortion
introduced by the sample.
In the first part, the use of computer-aided design
(CAD) models enables simulations to be carried out with
complex 3D objects. Differently from the VXI original version, which makes use of an object description via triangular facets, the new code requires a more ‘‘sophisticated’’
object representation based on non-uniform rational Bspline (NURBS), which offer one common mathematical
ground for both standard analytical shapes and free form
shapes.
In this work the most significant aspects of the mathematical derivation of the PC formulas for the image
intensity are reviewed and discussed. Furthermore, our
approach to simulate PC images including algorithms
and equations is presented. Finally, the results obtained
by the new code are examined with the help of some representative examples.
A. Peterzol et al. / Nucl. Instr. and Meth. in Phys. Res. B 254 (2007) 307–318
U(x, y, z = 0) can be expressed as the product of an object
transmission function t(x, y) with U0(x, y, z = 0), i.e. the
spherical wave that would have been observed at the object
plane in the absence of the sample.
We assume also that there exists a finite surface A on the
object plane outside which t(x, y) = 1. Under these circumstances, the wave perturbation in P can be written as
U ðx1 ; y 1 ; r1 Þ ¼ U 0 ðx1 ; y 1 ; r1 Þð1 þ cðx1 ; y 1 ; r1 ÞÞ;
ð1Þ
where U0(x1, y1, r1) represents the spherical wave that
would have been detected at the image plane to which P
belongs, in the absence of the sample (free space propagation), and the term c(x1, y1, r1) is the function related to the
formation of the PC image
Z Z
i i2ps
cðx1 ; y 1 Þ ¼ se k
dx dyðtðx; yÞ 1Þ
2k
A
i2pðs0 þs1 Þ e k
r1 r0
þ
;
ð2Þ
s0 s1 s1 s0
where for the sake of simplicity, the z coordinate has been
omitted, and s denotes the distance between P0 and the image plane point (x1, y1), while s0 and s1 denote the distances
from P0 and P to the object plane point (x, y, z = 0), respectively. From Eq. (1) it is straightforward to write the normalized intensity in(x1, y1) as
2
inðx1 ; y 1 Þ ¼ j1 þ cðx1 ; y 1 Þj ;
ð3Þ
where we chose to normalize the intensity I to I0, i.e. the
X-ray beam intensity that would have been detected on
the image plane at the P position in the absence of the
sample (I = jUj2 and I0 = jU0j2).
Hence, in order to simulate the beam intensity at the P
position, the integral in Eq. (2) has to be computed numerically. This procedure can be facilitated if the so-called
small angle or paraxial approximation [31] is introduced.
Under this assumption, this integral can be rewritten as
h
i
2
2 Z þ1 Z þ1
1 y 0 Þ
M ipk ðx1 x0ðrÞ0 þðy
þr1 Þ
cðx1 ; y 1 Þ ¼
e
dxdyðtðx; yÞ 1Þ
ikr1
1
1
h
i
ip
ek
ðxx0 Þ2 þðyy 0 Þ2 ðxx1 Þ2 þðyy 1 Þ2
þ
r1
r0
;
ð4Þ
where M = (r0 + r1)/r0 represents the image magnification,
and the term (t(x, y) 1) is non-zero only over the object
plane area A. Expression (4) has two important properties:
(i) in the case of a one-dimensional (1D) sample, for which
t(x, y) will be a function of only one variable, the integral
can be evaluated separately along the x- and y-directions,
(ii) it contains the convolution [34] of (t(x, y) 1) with an
exponential term called propagator. This second property
is very important since it allows computing the term c also
in the Fourier space. Let us define T(u, v) as the Fourier
transform of the term (t(x, y) 1). From (4) the Fourier
transform of c, C(u, v), is given by
309
Cðu; vÞ ¼ M 2 T ðMu; MvÞ
r1
2
2
exp pikr1 Mðu þ v Þ exp 2pi ðx0 u þ y 0 vÞ :
r0
ð5Þ
2
2
The term exp(ipkr1M(u + v )) in (5) represents the optical transfer function under Fresnel diffraction conditions,
and it behaves like a filter in the spatial frequency space
[31]. Eq. (5) can be used instead of (4), in order to simulate
PC signals by inverse Fourier transforming C(u, v).
The transmission function t(x, y) represents the phase
shift and the attenuation effect due to the sample. In literature, [7,9,10,18–20] being same examples, t(x, y) is usually
written as
tðx; yÞ ¼ expðiUðx; yÞ Bðx; yÞÞ;
ð6Þ
where U(x, y) and B(x, y) correspond to the object phase
and linear attenuation term, respectively. The definitions
of U(x, y) and B(x, y) are
Z
2p
/ðx; yÞ ¼
drdðx; y; zÞ;
k
Z
ð7Þ
2p
Bðx; yÞ ¼
drbðx; y; zÞ;
k
where the integration is performed along the direction linking the source point P0 and the object plane position (x, y).
d(x, y, z) and b(x, y, z) are the 3D distributions of the real
and imaginary part, respectively, of the X-ray refractive index n decrement
n ¼ 1 d þ ib:
ð8Þ
U(x, y) and B(x, y) represent the projections of the object’s
d(x, y, z) and b(x, y, z), respectively, along the X-ray travelling direction. In order to be able to model an object via (6)
and (7), the object is supposed to be ‘‘thin’’ for X-rays so
that the projection approximation holds true. If d is the
object thickness, the object can be deemed thin [19] as long
as the size of the finest feature to image is larger than
(kd)0.5.
It is important to underline that, to our knowledge, the
formulas describing the X-ray beam intensity impinging
upon image planes reported in literature – and as a consequence the corresponding signal simulations – are based on
two important assumptions: (i) small angle approximation,
which justifies the use of (4) and (5), and (ii) the projection
approximation, which allows to describe the object transmission function by (6) and (7).
Up to now the mathematical PC framework we presented assumes a monochromatic point source and an ideal
detector. In practice, the source has a finite size and the
image detector has a limit on the maximum detectable
spatial frequency.
The intensity distribution in the image obtained with a
totally incoherent, finite size source and a finite resolution
detector is the convolution of the intensity distribution in
the image corresponding to the point source and the system point spread function (PSF) [25]. The latter is the
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convolution of two functions, the geometric PSF, which
represents the effect of the geometric blurring caused by
the size and shape of the focal spot and the magnification
factor used, and the PSF of the acquisition device.
Another aspect which has to be taken into account is the
temporal coherence of the source. Differently from synchrotron X-ray facilities, which can provide monochromatic beams (DE/E 104), X-ray tubes are actually
polychromatic sources. In such a case it is necessary to integrate the intensity formula over the emitted spectrum [31].
3. The VXI PC code
As previously mentioned, VXI is a computer code developed to simulate the operation of radiographic, radioscopic
or tomographic devices [28–30]. The completely deterministic simulation is based on ray-tracing techniques and on
the X-ray attenuation law. In order to be able to carry
out image simulations with a large variety of samples, the
code was designed to accept CAD files in standard format
to describe the sample geometry. Many software packages
enable complex 3D objects to be drawn and CAD files to
be generated, for example in stereolithographic (STL)
format. These files contain a list of nodes and meshes
(triangular facets) that approximate the object surface.
The precision of the approximation, which is linked to
the size of the meshes, can be adjusted. The object may
consist of different parts, possibly made of different materials, assumed to be homogeneous. The CAD model of each
part can be processed independently.
Once the object and the position of the point source are
defined, a set of rays is emitted from the source towards
every pixel centre of the detector (see Fig. 2). Each ray
may intersect a certain number of meshes at the sample surface or at the interfaces between different parts of the
object. The path length in every part of the object is calcu-
lated by determining the coordinates of all the intersection
points.
This procedure, which is used in the VXI code to compute the number of photons N(E) which emerge from the
sample and reach a pixel of the detector (Eq. (1) in [28]),
can be applied to compute the d(x, y, z) and b(x, y, z) projections along the X-ray travelling direction, i.e. U(x, y) and
B(x, y). In this way it is possible to evaluate the 2D map
of the transmission function t(x, y) on the object plane.
For this purpose a virtual detector plane is positioned just
after the sample, and for every pixel coordinates (xn, ym) at
the detector plane the total path length di through each
material i of the object is calculated. Subsequently,
U(xn, ym) and B(xn, ym) are computed as
2p X
di ðEÞd i ;
/ðxn ; y m ; EÞ ¼
k
i
ð9Þ
2p X
Bðxn ; y m ; EÞ ¼
b ðEÞd i ;
k i i
where di(E) and bi(E) designate the real and imaginary
part, respectively, of the refraction index decrement associated with the material i at the energy E.
Therefore, with a simple modification of the existing
code it is possible to compute the 2D map of the object
transmission function t(x, y). Given the latter, the PC signal
can be simulated by computing the parameter c trough
numerical integration (i.e. Eq. (4)) or by discrete Fourier
transformation (i.e. Eq. (5)). In the new PC imaging program, the parameter c is computed exploiting a discrete fast
Fourier transform algorithm, which reduces the computation time in comparison with the numerical integration.
The latter approach is employed only when the small angle
approximation does not apply. In this case the parameter c
is being calculated by numerical integration of Eq. (2).
Once the parameter c has been worked out, the relative
intensity image is derived in accordance with Eq. (3). This
is a spatial high resolution image, which assumes a monochromatic point source. In order to reproduce the case of a
polychromatic X-ray source, the intensity image is computed as a weighted sum of all monoenergetic images
obtained for each energy Ei belonging to the considered
spectrum. The latter is treated as input information such
as for example the source-to-object distance r0.
The last step consists in convolving the spatial high resolution polychromatic image with the PSF of the imaging
system being considered.
4. The effects of the CAD model on PC image simulation
Fig. 2. Principle of the VXI simulation code. The ray SK intersects two
meshes in the points A and B. Geometrical calculations enable determining the attenuation path length AB. Ray (1): transmitted photons. Rays
(2) and (3): scattered photons.
With our method it is possible to simulate the PC signals
of a wide range of 3D complex objects. By now, mainly the
intensity patterns of edges [4,13,17], cylindrical phantoms
[2,11,16,19,21,24], and spheres [14,22], for which the analytical calculation of phase and attenuation projections is
straightforward, have been reported. For these simple
objects, the experimental data agree well with the expected
A. Peterzol et al. / Nucl. Instr. and Meth. in Phys. Res. B 254 (2007) 307–318
Fig. 3. The CAD facetted model of a 150 lm radius PMMA sphere.
ones [2,4,11,13,19,21] indicating that the Fresnel–Kirchhoff
diffraction theory is well suited to describe the PC image
formation.
For this reason the software was firstly tested by simulating the PC images of cylinders and spheres. In particular, the 2D map of t(x, y) was computed (1) analytically
and (2) using VXI, and the corresponding image intensities
were compared. By that it was possible to point out the
‘‘weak points’’ of the 3D-object CAD representation. As
an example, the CAD facetted model of a sphere
(radius = 150 lm) is reported in Fig. 3.
While this object ‘‘discretisation’’ does not pose any
problem (the object ‘‘sampling step’’ is supposed to be sufficiently smaller than the detector pixel size) in simple
311
attenuation imaging, it introduces not negligible artefacts
in the PC case. To better understand this aspect, the following considerations are necessary.
In order to simulate the PC signal, t(x, y) has to be
known/sampled with a precision of one micrometer or even
less. Actually, if we look at (4), the propagator in the convolution integral oscillates at very high frequencies (if we
consider the case of a 20 keV monochromatic and parallel
beam and an object-to-detector distance of 1 m, the peakto-peak distance of the propagator oscillations starts with
a value of 12 lm and ends up with values smaller than
1 lm). A fine sampling of the propagator and, as a consequence, of t(x, y) is therefore needed to correctly compute
the convolution integral. On the other hand, the Fourier
transform of the propagator oscillates relatively slowly
(Eq. (5)), but still a fine sampling of t(x, y) is necessary to
correctly reproduce T(u, v) in a wide spatial frequency
range.
Now, the triangular facets’ smallest dimension is about
10–20 lm. As a consequence, the 1 lm sampled phase projection profile of a curved object presents a ‘‘polygon’’-like
shape.
In Fig. 4(a) is reported as an example the 1D profile of
the phase function U(x) for a wire calculated using VXI
along with the analytical expression: U(x) = 4pd(R2 x2)0.5/k, where R is the wire radius. In the reported example, a 150 lm radius wire of polymethyl methacrylate
(PMMA) irradiated with a parallel and monochromatic
beam of 15 keV is considered. For the sake of simplicity,
only half of the phase profile is reported owing to its symmetry with respect to the wire centre.
In order to highlight the polygon-like shape of the VXI
phase profile, a detail of Fig. 4(a) is reported in Fig. 4(b).
Unlike the analytical phase, whose derivative changes continuously, the derivative of the VXI phase profile is constant over a polygon side and changes when moving from
Fig. 4. 1D profile of a wire (PMMA as a material, radius = 150 lm, X-ray energy = 15 keV) phase, computed using both the analytical formula (ana) and
the facetted CAD model (VXI). Due to the object symmetry, only half of the signal is reported in (a), while (b) shows a detail of (a) in order to highlight the
polygon-like shape of the VXI phase profile.
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A. Peterzol et al. / Nucl. Instr. and Meth. in Phys. Res. B 254 (2007) 307–318
reveals the object edges, which are typically associated with
a dark-white fringe structure.
Differently from conventional radiography, the PC technique is very sensitive to derivative discontinuities of t(x, y).
Returning to Fig. 6, the artefacts introduced by the facetted
model can persist even after the spatial resolution properties of the specific imaging system are taken into account.
As an example, the intensity images of Fig. 6(a) and (b),
are reported in Fig. 6(c) and (d), respectively, after convolution with the PSF:
PSFðx; yÞ ¼ ðerf ½ax ðx bx Þ erf ½ax ðx cx ÞÞ erf ay ðy by Þ
ð10Þ
erf ay ðy cy Þ ;
Fig. 5. Simulated signal for a 150 lm radius PMMA wire, assuming a
parallel and monochromatic beam of 15 keV and an ideal detector (i.e.
with an infinite spatial resolution). They were computed using both the
analytical (ana) and the facetted (VXI) CAD model for the transmission
function evaluation.
one side to the next. This property is the source of the artefacts we encountered when calculating t(x, y) by means of
VXI to simulate the PC signal.
In Fig. 5 are reported the relative intensity profiles computed using the analytical and VXI transmission functions
plotted in Fig. 4, and sampled with a step of 0.1 lm. The
signals are calculated for a sample-to-detector distance of
1 m. The agreement between the two profiles would be
good, if not the presence of low frequency oscillations,
indicated by an arrow in Fig. 5 (the frequency is low compared with the oscillations of the wire interference pattern)
in the VXI signal. These oscillations are due to the polygonal shape of the VXI transmission function and have no
physical meaning: they are artefacts coming from the
related CAD model.
Note that the peak-to-peak distance of these oscillations
is linked to the polygon side length. In fact, the facetted
object contains a Fourier component due to the quasi-periodic object description: the frequency of the triangular facets repetition comes up in the PC signal.
In Fig. 6(a) and (b) are reported as a 2D example the
intensity images at 1 m sample-to-detector distance for a
PMMA sample sphere of 150 lm radius irradiated with a
parallel monochromatic beam of 15 keV, calculated using
the analytical and the VXI transmission function, respectively. In both cases t(x, y) was mapped with 0.2 lm precision. Also in this case, the VXI image presents the artefacts
due to the facetted CAD model (Fig. 3).
It has to be pointed out that in general the PC signal
depends only on the derivative terms (of different orders)
of the phase: for example, the intensity distribution in the
image of a pure phase object is proportional (in the low
spatial resolution approximation) to the Laplacian of the
phase distribution in the object wave. Hence, the image
where erf is the error function, and ax,y, bx,y and cx,y are
parameters defined as follows: the quantity (cx,y bx,y)
corresponds to the FWHMs along the x-, y-directions of
the PSF, which have been set equal to 30 lm for both the
x-, y-axes, since this value represents the smallest pixel size
for digital mammography systems (SenoScan Digital
Mammography system, Fischer Imaging Corporation,
Denver, USA). In practice, (10) represents the 2D extension of Eq. (1) in [35]. The parameters ax,y indicate the
spreading of the PSF along the x-, y-directions and were
set equal to 0.2 lm1 for both axes. This spreading is associated with the PSF (shown in Fig. 7) which was used to
produce the images of Fig. 6(c) and (d). Here, the high frequency signal contribution to I/I0, which was present in the
data prior to convolution, has been removed and the image
contrast has been significantly reduced. Nevertheless,
Fig. 6(d) still presents the faceting artefacts even if the difference between the two images (generated with the two
different t(x, y)) is smaller than the original unconvolved
case. In Fig. 6(e), which shows the intensity profiles (passing through the sphere centre) extracted from Fig. 6(c) and
(d), one can clearly see the low frequency oscillations inside
the sphere originating from the CAD model.
From these preliminary examples, it clearly emerges that
a PC simulation tool needs an exact object modelling. For
this reason a new CAD object description has been envisaged and the related code has been developed. The 3D
objects are described following a parametric approach.
In particular, the present CAD model makes use of nonuniform rational B-splines (NURBS) [36], which offer one
common mathematical basis to represent both standard
analytical shapes (e.g. conic sections and quadric surfaces)
and free form shapes while maintaining mathematical
exactness and resolution independence. The NURBS are
widely used in industrial design to represent complex geometrical surfaces (for more details the readers are referred
to [36]). The majority of the CAD tools allow to export
IGES or STEP type files, which are standard formats making provision for the NURBS geometric entity description.
One of the key features of NURBS curves (NURBS surfaces are the straightforward 2D generalization of NURBS
curves) is that their shape is determined by (among other
things) the positions of a set of points called control points.
A. Peterzol et al. / Nucl. Instr. and Meth. in Phys. Res. B 254 (2007) 307–318
313
Fig. 6. Simulated images for a 150 lm radius PMMA sphere, assuming a parallel and monochromatic beam of 15 keV and an ideal detector. They were
computed using the analytical (a), the facetted CAD (b), and the new NURBS CAD (f) approach to the transmission function evaluation. The images (c)
and (d) are the results of the convolution of the images (a) and (b), respectively, with the 30 lm FWHM detector response function (see Fig. 7). The central
profiles extracted from the sphere images (c) and (d) are reported in (e).
Each control point influences the part of the curve nearest
to it, but has little or no effect on parts of the curve that are
farther away. The basic idea is to represent a curve C as a
weighted average of all the control points
CðuÞ ¼
n
X
i¼0
N i;p ðuÞBi ;
a 6 u 6 b;
ð11Þ
where Bi represents the ith control point (each Bi is generally identified by three coordinates {xi, yi, zi}), Ni,p(u) are
the pth-degree B-spline basis functions and u is the independent variable used in the parametric method (where a curve
is represented as: C(u) = [x(u), y(u), z(u)]). The B-spline
basis function of degree p is defined using the recurrence
formula, well suited to a computerised implementation:
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Fig. 7. The 2D PSF used to obtain the sphere images of Fig. 6(c) and (d).
The PSF is defined in (10) where the function parameters were set as
follows: (cx,y bx,y) = 30 lm and ax,y = 0.2 lm1.
Fig. 8. Constructing a circle with the NURBS approach using seven
control points, the coordinates and weights of which are reported in the
figure. The associated knot vector contains the following elements:
U = {0, 0, 0, 1/3, 1/3, 2/3, 2/3, 1, 1, 1}.
1 if ui 6 u < uiþ1 ;
0 otherwise;
u ui
uiþpþ1 u
N i;p1 ðuÞ þ
N iþ1;p1 ðuÞ;
N i;p ðuÞ ¼
uiþpþ1 uiþ1
uiþp ui
N i;0 ðuÞ ¼
ð12Þ
where the sequence U = u0, . . . , um (which is a non-decreasing sequence of real numbers, i.e. ui 6 ui+1 for i = 0, . . . , m)
is named the knot vector and the ui are called knots. The
latter demarcate the intervals along u associated to each
control point. The relative length of each interval is not
constant in order to allow some control points to affect a
larger portion of the curve and others a smaller portion.
This property justifies the NU in NURBS, as it stands
for non-uniform (knot vector).
In Eq. (12) a ‘‘0/0’’ ratio can occur; in this case the ratio
is defined to be zero. The knot vector of a B-spline curve is
a non-periodic and non-uniform knot vector of the form
U ¼ f a; . . . ; a; upþ1 ; . . . ; umpþ1 ; b; . . . ; b g:
|fflfflfflffl{zfflfflfflffl}
|fflfflfflffl{zfflfflfflffl}
pþ1
ð13Þ
pþ1
As a general rule the curve starts at the (p + 1)th knot from
the beginning of U and stops at the (p + 1)th knot from its
end. A curve of order p + 1 (or degree p) is defined only if
p + 1 basis functions are non-zero.
Since only rational functions can represent conics, one
could generalize the B-spline curve, defined by Eq. (11),
to a rational expression. This generalization is the actual
NURBS and is defined as
Pn
N i;p ðuÞwi Bi
CðuÞ ¼ Pi¼0
;
ð14Þ
n
i¼0 N i;p ðuÞwi
where wi are the weights associated to each control point
Bi. Increasing the weight of an individual control point
has the effect of ‘‘pulling’’ the curve toward that point.
As a practical example, Fig. 8 shows how to represent a
circle (degree 2) by a NURBS curve using seven control
points, the coordinates and weights of which are
reported in the same figure. The associated knot vector
contains the following elements: U = {0, 0, 0, 1/3, 1/3, 2/3,
2/3, 1, 1, 1}.
NURBS surfaces are represented as a function of two
independent parameters, u and v, as follows:
Pn Pm
i¼0
j¼0 N i;p ðuÞN j;q ðvÞwi;j Bi;j
:
ð15Þ
Sðu; vÞ ¼ Pn Pm
i¼0
j¼0 N i;p ðuÞN j;q ðvÞwi;j
In particular the spheres and cylinders of our examples feature straightforward NURBS representations. The control
points Bi,j(xi,j, yi,j, zi,j) of a sphere of radius r are defined in a
spherical coordinates parameterization:
xi;j ¼ r cosðhi Þ cosð/j Þ;
y i;j ¼ r cosðhi Þ sinð/j Þ;
ð16Þ
zi;j ¼ r sinðhi Þ;
where / is the azimuthal and h the polar angle. The weights
wi,j are given by: wi,j = cos(hi)cos(/j). For a cylinder of radius r in a cylindrical coordinates parameterization, the
coordinates of the control points are expressed as
xi;j ¼ r cosðhi Þ;
y i;j ¼ r sinðhi Þ;
ð17Þ
zi;j ¼ tj ;
where h is the rotation angle and t is the elevation in the
cylinder. The related weights are: cos(hi).
As a first example of object representation by NURBS
in VXI, the intensity image of the usual PMMA sphere is
A. Peterzol et al. / Nucl. Instr. and Meth. in Phys. Res. B 254 (2007) 307–318
Fig. 9. 3D objects of different shapes ‘‘virtually’’ contained in a cube of
1.5 mm side. For the purposes of the simulation, the objects were
supposed to be PMMA made and irradiated with a polychromatic X-ray
beam (see Fig. 10) impinging on the side indicated by the arrow.
illustrated in Fig. 6(f). In this case all the artefacts present
in Fig. 6(b) vanished. In addition, there is an excellent
agreement (differences less than 5%) between the images
obtained analytically and those computed using the
NURBS model; a virtually perfect agreement was obtained
also for the cylinder (differences less than 1%).
The CAD NURBS model was therefore adopted as a
basis for the new VXI version, by means of which it is
315
possible to compute the PC images of all 3D complex
objects describable in the IGES or STEP file format.
As another example, let us consider the four 3D objects
of different shapes contained in a bounding box of 1.5 mm
side (see Fig. 9). The objects are supposed to be made of
PMMA and irradiated with a polychromatic X-ray beam,
the spectrum of which is reported in Fig. 10, impinging
on the box side indicated by the arrow in Fig. 9; the
t(x, y) sampling step is 0.5 lm and r1 = 1 m. The X-ray
spectrum is a typical mammography spectrum obtainable
with a molybdenum (Mo) anode target and with a
0.03 mm Mo filter. This spectrum presents two characteristic lines at 17.4 and 19.6 keV, and allows reducing the PC
signal degradation due to the deployment of a polychromatic beam as compared to a broader spectrum without
lines. Actually, the PC image in Fig. 11(a), which corresponds to the four objects in Fig. 9 displays a good contrast
(the peak-to-peak difference at the object edges is about
1.8 in a relative intensity scale).
For comparison purposes the image with the objects of
Fig. 9 obtained using the facetted model for the evaluation
of t(x, y) is also shown in Fig. 11(b). In this case, the image
exhibits the low frequency periodic artefacts.
A detailed comparison between the two images can be
seen in Fig. 11(c), which shows the absolute value of the
difference between Fig. 11(a) and (b). It should be pointed
out that the large differences at the object edges are also
due to a misalignment of the PC signal oscillations, as
shown in Fig. 12, where the horizontal profiles extracted
from Fig. 11(a) and (b) and passing trough the sphere centre are displayed.
In particular, Fig. 12(a) shows the two profiles corresponding to the sphere, while Fig. 12(b) shows the profiles
corresponding to the object located on the right of the
sphere looking at the image (see Fig. 9). Also in this case,
the high spatial resolution (0.5 lm) images have been
Fig. 10. A typical mammography spectrum as from a Mo anode target and using a 0.03 mm Mo filter.
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A. Peterzol et al. / Nucl. Instr. and Meth. in Phys. Res. B 254 (2007) 307–318
Fig. 11. Simulated images corresponding to the four objects of Fig. 9, as obtained with the polychromatic beam of Fig. 10 for mammography applications
and an ideal detector. The images were obtained using the NURBS CAD (a), and the facetted CAD (b) models, respectively for the transmission function
evaluation. The images (d) and (e) are the results of the convolution of the images (a) and (b), respectively, with the detector PSF (of Fig. 7). A detail
comparison of the results supplied by the two different CAD models is given in (c) and (f), in terms of the absolute value of the difference between (a) and
(b), and (d) and (f), respectively.
A. Peterzol et al. / Nucl. Instr. and Meth. in Phys. Res. B 254 (2007) 307–318
317
Fig. 12. The horizontal profiles extracted from Fig. 10(a) (NURBS model) and (b) (facetted model). Both horizontal profiles pass through the sphere
centre. In particular, (a) displays the profiles corresponding to the sphere, while (b) shows the profiles corresponding to the object placed to the right of the
sphere.
convolved with the PSF defined in (10) and shown in
Fig. 7. The convolution effects are clearly noticeable in
Fig. 11(d) and (e) (compare with Fig. 11(a) and (b)).
A detailed comparison between the images in Fig. 11(d)
and (e) is given in Fig. 11(f) via the absolute value of the
difference between the two images.
After convolution, the difference between the images
obtained using the two different CAD models becomes
smaller, since the convolution reduces the artefacts without
eliminating them completely.
5. Conclusions and future perspectives
In this work a computer code able to simulate PC
images has been presented. The code is based on the Fresnel–Kirchhoff diffraction theory and computes the object
transmission function t(x, y) according to the classical projection approximation.
The proposed code is an upgrade of the existing VXI
program platform exploiting in particular a ray-tracing
technique for computing the t(x, y) 2D map. This code is
designed to process CAD object files containing a facetted
sample description and also objects described by NURBS.
This second approach has been followed, after observing
that the polyhedral object representation introduces not
negligible artefacts in the PC images, since the PC technique
is very sensitive to discontinuities of the t(x, y) derivatives.
As a consequence an exact object modelling is needed.
The NURBS approach meets this requirement, since
most geometric objects can be described by means of a
class of parametric curves and surfaces. The advantage of
the NURBS description of surfaces is that of providing
geometrically smooth objects, thus not causing artefacts.
The results supplied by the NURBS code version are of
doubtlessly better quality. There is a perfect agreement
between the images obtained by means of the analytical
and of the NURBS-based description of t(x, y).
It has to be said however, that using the NURBS object
representation entails computation times larger than with
the original VXI version. As an example, the time needed
to compute the t(x, y) referring to Fig. 11(a) (3200 · 3200
pixels) without optimized ray-tracer is 3h:46 0 with the
NURBS and a few minutes with the facetted object
description on a standard desktop computer.
The facetted sample description is nevertheless useful for
describing biological samples for which it is not possible to
use CAD tools [28]. In these cases, the artefacts involved by
the polyhedral description are less detectable when considering a low spatial resolution system as it is the case of conventional medical imaging.
With the new VXI version the PC images of complex 3D
objects can be easily produced taking into account also the
properties of the source (size, energy spectrum) and the
detector (PSF).
PC VXI is a useful tool for estimating the achievable
image quality by means of the PC imaging technique. It
can be employed at synchrotron or lab facilities dealing
with PC experiments or [37–40] developing systems for
PC imaging.
Prospectively we envisage to improve the model for the
transmission function t(x, y). The new t(x, y) evaluation will
take into account the refraction of X-rays as they pass
through the sample. The aim of this approach is to provide
a simulation tool capable to account for PC imaging of
thick samples (several cm) under spatial high resolution
conditions (1 lm or less).
Acknowledgments
The authors gratefully acknowledge A. Bravin for the
many constructive discussions they had with him. P. Bleuet, M. Sanchez del Rio are also acknowledged for their
useful suggestions and enthusiastic participation in the initial phase of the present investigation. The authors thank
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A. Peterzol et al. / Nucl. Instr. and Meth. in Phys. Res. B 254 (2007) 307–318
the French Ministry of National Education, Advanced
Education, and Research for supporting this work and
are very grateful to the IN2P3’s Computing Centre for providing them with computing resources and assistance.
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