x-ray phase contrast image...

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Nuclear Instruments and Methods in Physics Research B 254 (2007) 307–318

NIMBBeam Interactions

with Materials & Atoms

X-ray phase contrast image simulation

A. Peterzol a,*, J. Berthier a, P. Duvauchelle a, C. Ferrero b, D. Babot a

a Institut National des Sciences Appliquees de Lyon, Laboratoire de Controle Non Destructif par Rayonnements Ionisants (CNDRI),

20 Av. A. Einstein, 69621 Villeurbanne, Franceb European Synchrotron Radiation Facility, BP220 rue Horowitz, 38043 Grenoble, France

Received 3 July 2006; received in revised form 16 October 2006Available online 21 December 2006

Abstract

A deterministic algorithm is proposed to simulate phase contrast (PC) X-ray images for complex three-dimensional (3D) objects. Thisalgorithm has been implemented in a simulation code named VXI (virtual X-ray imaging). The physical model chosen to account for PCtechnique 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 transmissionfunction. 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 sim-ulate the polychromatic case, the intensity image is integrated over the considered X-ray energy spectrum. Then, in order to account forthe system spatial resolution properties, the high spatial resolution image (mono or polychromatic) is convolved with the total pointspread 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 almost10 years a very active field of X-ray science since the tech-nique offers greatly enhanced image quality over conven-tional radiology [27]. Phase contrast (PC) arises becauseboth the amplitude and phase of X-rays are modified asan X-ray beam propagates through an object. A detailedunderstanding of the underlying physics requires that theradiation be treated as a wave field rather than by meansof simple geometrical optics. In that context, a big effort

0168-583X/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.nimb.2006.11.042

* Corresponding author.E-mail address: [email protected] (A. Peterzol).

has been dedicated to develop a comprehensive theoryfor PC imaging.

In order to develop clinical and industrial applicationsof this promising technique, it is of primary importanceto well characterize the physical background in order toquantitatively analyze the PC image performance.

Moreover, to develop and optimize a new imagingsystem, and to recognize the influence of the various adjust-able parameters, simulation can be a helpful tool.

In previous papers [28–30], it was reported on a com-puter code developed to simulate the operation of radio-graphic, radioscopic or tomographic systems. This code,named VXI (virtual X-ray imaging), is based on ray tracingtechniques and is completely deterministic. It enables to

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Fig. 1. Schematic display of the wave-optical approach to the PCformation mechanism for a circular cross-section object being irradiatedby a spherical and monochromatic X-ray wave generated by a pointsource located at a distance r0 from the object plane. The image is detectedat a distance r1 from the latter.

308 A. Peterzol et al. / Nucl. Instr. and Meth. in Phys. Res. B 254 (2007) 307–318

simulate direct images [28] and first-order scattering [29,30]in complex configurations (intricate three-dimensional(3D) objects, polychromatic spectra, focal spots causinggeometric unsharpness, etc.).

In this work we report on the implementation of analgorithm designed to simulate X-ray PC images of com-plex 3D objects in the VXI code.

The new code version (VXI PC) consists mainly of twoparts. The first one exploits the VXI ray-tracing approachto compute the object transmission function. The new partsimulates the PC image due to the wave front distortionintroduced by the sample.

In the first part, the use of computer-aided design(CAD) models enables simulations to be carried out withcomplex 3D objects. Differently from the VXI original ver-sion, which makes use of an object description via triangu-lar facets, the new code requires a more ‘‘sophisticated’’object representation based on non-uniform rational B-spline (NURBS), which offer one common mathematicalground for both standard analytical shapes and free formshapes.

In this work the most significant aspects of the mathe-matical derivation of the PC formulas for the imageintensity are reviewed and discussed. Furthermore, ourapproach to simulate PC images including algorithmsand equations is presented. Finally, the results obtainedby the new code are examined with the help of some repre-sentative examples.

2. Theory of phase contrast imaging

Most of the theoretical frameworks found in the litera-ture, [1,2,4,6–10,13,14,17,18,20,25,26] being some exam-ples, and describing the so-called in-line PC imaging arebased on the Fresnel–Kirchhoff diffraction theory [31–33].In particular Wu and Liu [18] provided a general theoreti-cal formalism covering both near field diffraction andholography. Differently from previous works, the formulasthey proposed did not assume a low overall phase pertur-bation [9] or limiting conditions on the partial derivativesof the phase and attenuation terms [10]. In this way theyprovided the tools to handle a broader ensemble of cases.

In addition, also simpler models based on ray-opticalapproximations have been proposed [12,16,21]. Thesemodels are based on the refraction of X-rays within anobject.

It is possible to show that the ray-optical results can alsobe obtained by including in the diffraction formalism somerestrictions on the spatial frequencies present in the finalimage, but without limitations on the maximum phaseshift. A detailed description of the simplified ray-opticalapproach to describe PC images can be found in [24].

In writing VXI PC we followed the more general Fres-nel–Kirchhoff diffraction formalism.

For sake of completeness, it should be mentioned herethat a third approach based on the Wigner distribution for-malism was recently developed [23]. This approach is more

general than the Fresnel–Kirchhoff diffraction theory, sinceit takes into account also the partial coherence effects of theincident X-ray beam.

The Fresnel–Kirchhoff diffraction formalism assumes aspatially coherent X-ray source (either a point X-ray sourceor a plane wave), however the spatial incoherence effects ofthe finite source size are introduced in a second step bymeans of a convolution with the source response function.This approach is still valid when considering incoherentX-ray sources such as X-ray tubes.

For the more general case of partially coherent illumina-tion, the readers are referred to a recent work [26] where amodel corresponding to a simple generalization of theShell-model source [33] is presented.

In the following the Fresnel–Kirchhoff theory is brieflyreviewed and the fundamental equations which VXI PCis based on are recalled. More details can be found in[25], where the general principles and results of the optimi-zation of the in-line PC imaging system performances forspatially incoherent sources are investigated.

In order to apply the Kirchhoff’s integral [31] to PCimaging, let us first consider a monochromatic sphericalwave of wavelength k, generated from a point sourcelocated at the point P0 of coordinates (x0,y0,�r0) (seeFig. 1), and propagating through a sample virtually lyingon the so-called object plane, which is positioned just afterthe sample and perpendicular to the z-axis. The distancesource-to-object plane is r0. In order to write the diffractedwave field in P(x1,y1, r1) at a distance r1 from the objectplane, we assume that the wave field at the object plane

A. Peterzol et al. / Nucl. Instr. and Meth. in Phys. Res. B 254 (2007) 307–318 309

U(x,y,z = 0) can be expressed as the product of an objecttransmission function t(x,y) with U0(x,y,z = 0), i.e. thespherical wave that would have been observed at the objectplane in the absence of the sample.

We assume also that there exists a finite surface A on theobject plane outside which t(x,y) = 1. Under these circum-stances, the wave perturbation in P can be written as

Uðx1; y1; r1Þ ¼ U 0ðx1; y1; r1Þð1þ cðx1; y1; r1ÞÞ; ð1Þ

where U0(x1,y1, r1) represents the spherical wave thatwould have been detected at the image plane to which P

belongs, in the absence of the sample (free space propaga-tion), and the term c(x1,y1, r1) is the function related to theformation of the PC image

cðx1; y1Þ ¼ �i

2kse�i2ps

k

Z ZA

dxdyðtðx; yÞ � 1Þ

� ei2pðs0þs1Þ

k

s0s1

r1

s1

þ r0

s0

� �; ð2Þ

where for the sake of simplicity, the z coordinate has beenomitted, and s denotes the distance between P0 and the im-age plane point (x1,y1), while s0 and s1 denote the distancesfrom P0 and P to the object plane point (x,y,z = 0), respec-tively. From Eq. (1) it is straightforward to write the nor-malized intensity in(x1,y1) as

inðx1; y1Þ ¼ j1þ cðx1; y1Þj2; ð3Þ

where we chose to normalize the intensity I to I0, i.e. theX-ray beam intensity that would have been detected onthe image plane at the P position in the absence of thesample (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 numer-ically. This procedure can be facilitated if the so-calledsmall angle or paraxial approximation [31] is introduced.Under this assumption, this integral can be rewritten as

cðx1; y1Þ ¼M

ikr1

e�ipk

ðx1�x0Þ2þðy1�y0Þ2ðr0þr1Þ

h i Z þ1

�1

Z þ1

�1dxdyðtðx; yÞ � 1Þ

� eipkðx�x0Þ

2þðy�y0Þ2

r0þðx�x1Þ2þðy�y1Þ2

r1

h i; ð4Þ

where M = (r0 + r1)/r0 represents the image magnification,and the term (t(x,y) � 1) is non-zero only over the objectplane area A. Expression (4) has two important properties:(i) in the case of a one-dimensional (1D) sample, for whicht(x,y) will be a function of only one variable, the integralcan be evaluated separately along the x- and y-directions,(ii) it contains the convolution [34] of (t(x,y) � 1) with anexponential term called propagator. This second propertyis very important since it allows computing the term c alsoin the Fourier space. Let us define T(u,v) as the Fouriertransform of the term (t(x,y) � 1). From (4) the Fouriertransform of c, C(u,v), is given by

Cðu; vÞ ¼M2T ðMu;MvÞ

� exp �pikr1Mðu2þ v2Þ� �

exp 2pir1

r0

ðx0uþ y0vÞ� �

:

ð5ÞThe term exp(�ipkr1M(u2 + v2)) in (5) represents the opti-cal 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 simulatePC signals by inverse Fourier transforming C(u,v).

The transmission function t(x,y) represents the phaseshift and the attenuation effect due to the sample. In liter-ature, [7,9,10,18–20] being same examples, t(x,y) is usuallywritten as

tðx; yÞ ¼ expðiUðx; yÞ � Bðx; yÞÞ; ð6Þwhere U(x,y) and B(x,y) correspond to the object phaseand linear attenuation term, respectively. The definitionsof U(x,y) and B(x,y) are

/ðx; yÞ ¼ �2pk

Zdrdðx; y; zÞ;

Bðx; yÞ ¼ 2pk

Zdrbðx; y; zÞ;

ð7Þ

where the integration is performed along the direction link-ing 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 realand imaginary part, respectively, of the X-ray refractive in-dex n decrement

n ¼ 1� dþ ib: ð8ÞU(x,y) and B(x,y) represent the projections of the object’sd(x,y,z) and b(x,y,z), respectively, along the X-ray travel-ling direction. In order to be able to model an object via (6)and (7), the object is supposed to be ‘‘thin’’ for X-rays sothat the projection approximation holds true. If d is theobject thickness, the object can be deemed thin [19] as longas the size of the finest feature to image is larger than(kd)0.5.

It is important to underline that, to our knowledge, theformulas describing the X-ray beam intensity impingingupon image planes reported in literature – and as a conse-quence the corresponding signal simulations – are based ontwo important assumptions: (i) small angle approximation,which justifies the use of (4) and (5), and (ii) the projectionapproximation, which allows to describe the object trans-mission function by (6) and (7).

Up to now the mathematical PC framework we pre-sented assumes a monochromatic point source and an idealdetector. In practice, the source has a finite size and theimage detector has a limit on the maximum detectablespatial frequency.

The intensity distribution in the image obtained with atotally incoherent, finite size source and a finite resolutiondetector is the convolution of the intensity distribution inthe image corresponding to the point source and the sys-tem point spread function (PSF) [25]. The latter is the


convolution of two functions, the geometric PSF, whichrepresents the effect of the geometric blurring caused bythe size and shape of the focal spot and the magnificationfactor used, and the PSF of the acquisition device.

Another aspect which has to be taken into account is thetemporal coherence of the source. Differently from syn-chrotron X-ray facilities, which can provide monochro-matic beams (DE/E � 10�4), X-ray tubes are actuallypolychromatic sources. In such a case it is necessary to inte-grate the intensity formula over the emitted spectrum [31].

3. The VXI PC code

As previously mentioned, VXI is a computer code devel-oped to simulate the operation of radiographic, radioscopicor tomographic devices [28–30]. The completely determin-istic simulation is based on ray-tracing techniques and onthe X-ray attenuation law. In order to be able to carryout image simulations with a large variety of samples, thecode was designed to accept CAD files in standard formatto describe the sample geometry. Many software packagesenable complex 3D objects to be drawn and CAD files tobe 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 tothe size of the meshes, can be adjusted. The object mayconsist of different parts, possibly made of different materi-als, assumed to be homogeneous. The CAD model of eachpart can be processed independently.

Once the object and the position of the point source aredefined, a set of rays is emitted from the source towardsevery pixel centre of the detector (see Fig. 2). Each raymay intersect a certain number of meshes at the sample sur-face or at the interfaces between different parts of theobject. The path length in every part of the object is calcu-

Fig. 2. Principle of the VXI simulation code. The ray SK intersects twomeshes in the points A and B. Geometrical calculations enable determin-ing the attenuation path length AB. Ray (1): transmitted photons. Rays(2) and (3): scattered photons.

lated by determining the coordinates of all the intersectionpoints.

This procedure, which is used in the VXI code to com-pute the number of photons N(E) which emerge from thesample 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) projec-tions along the X-ray travelling direction, i.e. U(x,y) andB(x,y). In this way it is possible to evaluate the 2D mapof the transmission function t(x,y) on the object plane.For this purpose a virtual detector plane is positioned justafter the sample, and for every pixel coordinates (xn,ym) atthe detector plane the total path length di through eachmaterial i of the object is calculated. Subsequently,U(xn,ym) and B(xn,ym) are computed as

/ðxn; ym;EÞ ¼�2pk

Xi

diðEÞdi;

Bðxn; ym;EÞ ¼2pk

Xi

biðEÞdi;

ð9Þ

where di(E) and bi(E) designate the real and imaginarypart, respectively, of the refraction index decrement associ-ated with the material i at the energy E.

Therefore, with a simple modification of the existingcode it is possible to compute the 2D map of the objecttransmission function t(x,y). Given the latter, the PC signalcan be simulated by computing the parameter c troughnumerical integration (i.e. Eq. (4)) or by discrete Fouriertransformation (i.e. Eq. (5)). In the new PC imaging pro-gram, the parameter c is computed exploiting a discrete fastFourier transform algorithm, which reduces the computa-tion time in comparison with the numerical integration.The latter approach is employed only when the small angleapproximation 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). Thisis a spatial high resolution image, which assumes a mono-chromatic point source. In order to reproduce the case of apolychromatic X-ray source, the intensity image is com-puted as a weighted sum of all monoenergetic imagesobtained for each energy Ei belonging to the consideredspectrum. The latter is treated as input information suchas for example the source-to-object distance r0.

The last step consists in convolving the spatial high res-olution polychromatic image with the PSF of the imagingsystem being considered.

4. The effects of the CAD model on PC image simulation

With our method it is possible to simulate the PC signalsof a wide range of 3D complex objects. By now, mainly theintensity patterns of edges [4,13,17], cylindrical phantoms[2,11,16,19,21,24], and spheres [14,22], for which the ana-lytical calculation of phase and attenuation projections isstraightforward, have been reported. For these simpleobjects, the experimental data agree well with the expected

Fig. 3. The CAD facetted model of a 150 lm radius PMMA sphere.


ones [2,4,11,13,19,21] indicating that the Fresnel–Kirchhoffdiffraction theory is well suited to describe the PC imageformation.

For this reason the software was firstly tested by simu-lating the PC images of cylinders and spheres. In particu-lar, the 2D map of t(x,y) was computed (1) analyticallyand (2) using VXI, and the corresponding image intensitieswere compared. By that it was possible to point out the‘‘weak points’’ of the 3D-object CAD representation. Asan example, the CAD facetted model of a sphere(radius = 150 lm) is reported in Fig. 3.

While this object ‘‘discretisation’’ does not pose anyproblem (the object ‘‘sampling step’’ is supposed to be suf-ficiently smaller than the detector pixel size) in simple

Fig. 4. 1D profile of a wire (PMMA as a material, radius = 150 lm, X-ray enerthe facetted CAD model (VXI). Due to the object symmetry, only half of the sigpolygon-like shape of the VXI phase profile.

attenuation imaging, it introduces not negligible artefactsin the PC case. To better understand this aspect, the follow-ing considerations are necessary.

In order to simulate the PC signal, t(x,y) has to beknown/sampled with a precision of one micrometer or evenless. Actually, if we look at (4), the propagator in the con-volution integral oscillates at very high frequencies (if weconsider the case of a 20 keV monochromatic and parallelbeam and an object-to-detector distance of 1 m, the peak-to-peak distance of the propagator oscillations starts witha value of 12 lm and ends up with values smaller than1 lm). A fine sampling of the propagator and, as a conse-quence, of t(x,y) is therefore needed to correctly computethe convolution integral. On the other hand, the Fouriertransform of the propagator oscillates relatively slowly(Eq. (5)), but still a fine sampling of t(x,y) is necessary tocorrectly reproduce T(u,v) in a wide spatial frequencyrange.

Now, the triangular facets’ smallest dimension is about10–20 lm. As a consequence, the 1 lm sampled phase pro-jection profile of a curved object presents a ‘‘polygon’’-likeshape.

In Fig. 4(a) is reported as an example the 1D profile ofthe phase function U(x) for a wire calculated using VXIalong with the analytical expression: U(x) = �4pd(R2 �x2)0.5/k, where R is the wire radius. In the reported exam-ple, a 150 lm radius wire of polymethyl methacrylate(PMMA) irradiated with a parallel and monochromaticbeam of 15 keV is considered. For the sake of simplicity,only half of the phase profile is reported owing to its sym-metry with respect to the wire centre.

In order to highlight the polygon-like shape of the VXIphase profile, a detail of Fig. 4(a) is reported in Fig. 4(b).Unlike the analytical phase, whose derivative changes con-tinuously, the derivative of the VXI phase profile is con-stant over a polygon side and changes when moving from

gy = 15 keV) phase, computed using both the analytical formula (ana) andnal is reported in (a), while (b) shows a detail of (a) in order to highlight the

Fig. 5. Simulated signal for a 150 lm radius PMMA wire, assuming aparallel and monochromatic beam of 15 keV and an ideal detector (i.e.with an infinite spatial resolution). They were computed using both theanalytical (ana) and the facetted (VXI) CAD model for the transmissionfunction evaluation.


one side to the next. This property is the source of the arte-facts we encountered when calculating t(x,y) by means ofVXI to simulate the PC signal.

In Fig. 5 are reported the relative intensity profiles com-puted using the analytical and VXI transmission functionsplotted in Fig. 4, and sampled with a step of 0.1 lm. Thesignals are calculated for a sample-to-detector distance of1 m. The agreement between the two profiles would begood, if not the presence of low frequency oscillations,indicated by an arrow in Fig. 5 (the frequency is low com-pared with the oscillations of the wire interference pattern)in the VXI signal. These oscillations are due to the polyg-onal shape of the VXI transmission function and have nophysical meaning: they are artefacts coming from therelated CAD model.

Note that the peak-to-peak distance of these oscillationsis linked to the polygon side length. In fact, the facettedobject contains a Fourier component due to the quasi-peri-odic object description: the frequency of the triangular fac-ets repetition comes up in the PC signal.

In Fig. 6(a) and (b) are reported as a 2D example theintensity images at 1 m sample-to-detector distance for aPMMA sample sphere of 150 lm radius irradiated with aparallel monochromatic beam of 15 keV, calculated usingthe analytical and the VXI transmission function, respec-tively. In both cases t(x,y) was mapped with 0.2 lm preci-sion. Also in this case, the VXI image presents the artefactsdue to the facetted CAD model (Fig. 3).

It has to be pointed out that in general the PC signaldepends only on the derivative terms (of different orders)of the phase: for example, the intensity distribution in theimage of a pure phase object is proportional (in the lowspatial resolution approximation) to the Laplacian of thephase distribution in the object wave. Hence, the image

reveals the object edges, which are typically associated witha dark-white fringe structure.

Differently from conventional radiography, the PC tech-nique is very sensitive to derivative discontinuities of t(x,y).Returning to Fig. 6, the artefacts introduced by the facettedmodel can persist even after the spatial resolution proper-ties 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 convo-lution with the PSF:

PSFðx; yÞ ¼ erf axðx� bxÞ½ � � erf axðx� cxÞ½ �ð Þ erf ayðy � byÞ� �

� erf ayðy � cyÞ� �

; ð10Þ

where erf is the error function, and ax,y, bx,y and cx,y areparameters defined as follows: the quantity (cx,y � bx,y)corresponds to the FWHMs along the x-, y-directions ofthe PSF, which have been set equal to 30 lm for both thex-, y-axes, since this value represents the smallest pixel sizefor digital mammography systems (SenoScan DigitalMammography system, Fischer Imaging Corporation,Denver, USA). In practice, (10) represents the 2D exten-sion of Eq. (1) in [35]. The parameters ax,y indicate thespreading of the PSF along the x-, y-directions and wereset equal to 0.2 lm�1 for both axes. This spreading is asso-ciated with the PSF (shown in Fig. 7) which was used toproduce the images of Fig. 6(c) and (d). Here, the high fre-quency signal contribution to I/I0, which was present in thedata prior to convolution, has been removed and the imagecontrast has been significantly reduced. Nevertheless,Fig. 6(d) still presents the faceting artefacts even if the dif-ference between the two images (generated with the twodifferent t(x,y)) is smaller than the original unconvolvedcase. In Fig. 6(e), which shows the intensity profiles (pass-ing through the sphere centre) extracted from Fig. 6(c) and(d), one can clearly see the low frequency oscillations insidethe sphere originating from the CAD model.

From these preliminary examples, it clearly emerges thata PC simulation tool needs an exact object modelling. Forthis reason a new CAD object description has been envis-aged and the related code has been developed. The 3Dobjects are described following a parametric approach.

In particular, the present CAD model makes use of non-uniform rational B-splines (NURBS) [36], which offer onecommon mathematical basis to represent both standardanalytical shapes (e.g. conic sections and quadric surfaces)and free form shapes while maintaining mathematicalexactness and resolution independence. The NURBS arewidely used in industrial design to represent complex geo-metrical surfaces (for more details the readers are referredto [36]). The majority of the CAD tools allow to exportIGES or STEP type files, which are standard formats mak-ing provision for the NURBS geometric entity description.

One of the key features of NURBS curves (NURBS sur-faces are the straightforward 2D generalization of NURBScurves) is that their shape is determined by (among otherthings) the positions of a set of points called control points.

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 werecomputed 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 centralprofiles extracted from the sphere images (c) and (d) are reported in (e).


Each control point influences the part of the curve nearestto it, but has little or no effect on parts of the curve that arefarther away. The basic idea is to represent a curve C as aweighted average of all the control points

CðuÞ ¼Xn

i¼0

N i;pðuÞBi; a 6 u 6 b; ð11Þ

where Bi represents the ith control point (each Bi is gener-ally identified by three coordinates {xi,yi,zi}), Ni,p(u) arethe pth-degree B-spline basis functions and u is the indepen-dent variable used in the parametric method (where a curveis represented as: C(u) = [x(u),y(u), z(u)]). The B-splinebasis function of degree p is defined using the recurrenceformula, well suited to a computerised implementation:

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 asfollows: (cx,y � bx,y) = 30 lm and ax,y = 0.2 lm�1.

Fig. 8. Constructing a circle with the NURBS approach using sevencontrol points, the coordinates and weights of which are reported in thefigure. The associated knot vector contains the following elements:U = {0,0,0,1/3,1/3,2/3,2/3,1,1,1}.


N i;0ðuÞ ¼1 if ui 6 u < uiþ1;

0 otherwise;

N i;pðuÞ ¼u� ui

uiþp � uiN i;p�1ðuÞ þ

uiþpþ1 � uuiþpþ1 � uiþ1

Niþ1;p�1ðuÞ;

ð12Þwhere the sequence U = u0, . . . ,um (which is a non-decreas-ing 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. Thelatter demarcate the intervals along u associated to eachcontrol point. The relative length of each interval is notconstant in order to allow some control points to affect alarger portion of the curve and others a smaller portion.This property justifies the NU in NURBS, as it standsfor non-uniform (knot vector).

In Eq. (12) a ‘‘0/0’’ ratio can occur; in this case the ratiois defined to be zero. The knot vector of a B-spline curve isa non-periodic and non-uniform knot vector of the form

U ¼ f a; . . . ; a|fflfflfflffl{zfflfflfflffl}pþ1

; upþ1; . . . ; um�pþ1; b; . . . ; b|fflfflfflffl{zfflfflfflffl}pþ1

g: ð13Þ

As a general rule the curve starts at the (p + 1)th knot fromthe beginning of U and stops at the (p + 1)th knot from itsend. A curve of order p + 1 (or degree p) is defined only ifp + 1 basis functions are non-zero.

Since only rational functions can represent conics, onecould generalize the B-spline curve, defined by Eq. (11),to a rational expression. This generalization is the actualNURBS and is defined as

CðuÞ ¼Pn

i¼0Ni;pðuÞwiBiPni¼0N i;pðuÞwi

; ð14Þ

where wi are the weights associated to each control pointBi. Increasing the weight of an individual control pointhas the effect of ‘‘pulling’’ the curve toward that point.

As a practical example, Fig. 8 shows how to represent acircle (degree 2) by a NURBS curve using seven controlpoints, the coordinates and weights of which arereported in the same figure. The associated knot vectorcontains 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 twoindependent parameters, u and v, as follows:

Sðu; vÞ ¼Pn

i¼0

Pmj¼0Ni;pðuÞN j;qðvÞwi;jBi;jPn

i¼0

Pmj¼0N i;pðuÞNj;qðvÞwi;j

: ð15Þ

In particular the spheres and cylinders of our examples fea-ture straightforward NURBS representations. The controlpoints Bi,j(xi,j,yi,j,zi,j) of a sphere of radius r are defined in aspherical coordinates parameterization:

xi;j ¼ r cosðhiÞ cosð/jÞ;yi;j ¼ r cosðhiÞ sinð/jÞ;zi;j ¼ r sinðhiÞ;

ð16Þ

where / is the azimuthal and h the polar angle. The weightswi,j are given by: wi,j = cos(hi)cos(/j). For a cylinder of ra-dius r in a cylindrical coordinates parameterization, thecoordinates of the control points are expressed as

xi;j ¼ r cosðhiÞ;yi;j ¼ r sinðhiÞ;zi;j ¼ tj;

ð17Þ

where h is the rotation angle and t is the elevation in thecylinder. The related weights are: cos(hi).

As a first example of object representation by NURBSin VXI, the intensity image of the usual PMMA sphere is

Fig. 9. 3D objects of different shapes ‘‘virtually’’ contained in a cube of1.5 mm side. For the purposes of the simulation, the objects weresupposed to be PMMA made and irradiated with a polychromatic X-raybeam (see Fig. 10) impinging on the side indicated by the arrow.


illustrated in Fig. 6(f). In this case all the artefacts presentin Fig. 6(b) vanished. In addition, there is an excellentagreement (differences less than 5%) between the imagesobtained analytically and those computed using theNURBS model; a virtually perfect agreement was obtainedalso for the cylinder (differences less than 1%).

The CAD NURBS model was therefore adopted as abasis for the new VXI version, by means of which it is

Fig. 10. A typical mammography spectrum as from a

possible to compute the PC images of all 3D complexobjects describable in the IGES or STEP file format.

As another example, let us consider the four 3D objectsof different shapes contained in a bounding box of 1.5 mmside (see Fig. 9). The objects are supposed to be made ofPMMA and irradiated with a polychromatic X-ray beam,the spectrum of which is reported in Fig. 10, impingingon the box side indicated by the arrow in Fig. 9; thet(x,y) sampling step is 0.5 lm and r1 = 1 m. The X-rayspectrum is a typical mammography spectrum obtainablewith a molybdenum (Mo) anode target and with a0.03 mm Mo filter. This spectrum presents two characteris-tic lines at 17.4 and 19.6 keV, and allows reducing the PCsignal degradation due to the deployment of a polychro-matic beam as compared to a broader spectrum withoutlines. Actually, the PC image in Fig. 11(a), which corre-sponds to the four objects in Fig. 9 displays a good contrast(the peak-to-peak difference at the object edges is about1.8 in a relative intensity scale).

For comparison purposes the image with the objects ofFig. 9 obtained using the facetted model for the evaluationof t(x,y) is also shown in Fig. 11(b). In this case, the imageexhibits the low frequency periodic artefacts.

A detailed comparison between the two images can beseen in Fig. 11(c), which shows the absolute value of thedifference between Fig. 11(a) and (b). It should be pointedout that the large differences at the object edges are alsodue to a misalignment of the PC signal oscillations, asshown in Fig. 12, where the horizontal profiles extractedfrom Fig. 11(a) and (b) and passing trough the sphere cen-tre are displayed.

In particular, Fig. 12(a) shows the two profiles corre-sponding to the sphere, while Fig. 12(b) shows the profilescorresponding to the object located on the right of thesphere looking at the image (see Fig. 9). Also in this case,the high spatial resolution (0.5 lm) images have been

Mo anode target and using a 0.03 mm Mo filter.

Fig. 11. Simulated images corresponding to the four objects of Fig. 9, as obtained with the polychromatic beam of Fig. 10 for mammography applicationsand an ideal detector. The images were obtained using the NURBS CAD (a), and the facetted CAD (b) models, respectively for the transmission functionevaluation. 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 detailcomparison 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.


Fig. 12. The horizontal profiles extracted from Fig. 10(a) (NURBS model) and (b) (facetted model). Both horizontal profiles pass through the spherecentre. In particular, (a) displays the profiles corresponding to the sphere, while (b) shows the profiles corresponding to the object placed to the right of thesphere.


convolved with the PSF defined in (10) and shown inFig. 7. The convolution effects are clearly noticeable inFig. 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 thedifference between the two images.

After convolution, the difference between the imagesobtained using the two different CAD models becomessmaller, since the convolution reduces the artefacts withouteliminating them completely.

5. Conclusions and future perspectives

In this work a computer code able to simulate PCimages has been presented. The code is based on the Fres-nel–Kirchhoff diffraction theory and computes the objecttransmission function t(x,y) according to the classical pro-jection approximation.

The proposed code is an upgrade of the existing VXIprogram platform exploiting in particular a ray-tracingtechnique for computing the t(x,y) 2D map. This code isdesigned to process CAD object files containing a facettedsample description and also objects described by NURBS.This second approach has been followed, after observingthat the polyhedral object representation introduces notnegligible artefacts in the PC images, since the PC techniqueis 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, sincemost geometric objects can be described by means of aclass of parametric curves and surfaces. The advantage ofthe NURBS description of surfaces is that of providinggeometrically smooth objects, thus not causing artefacts.

The results supplied by the NURBS code version are ofdoubtlessly better quality. There is a perfect agreementbetween the images obtained by means of the analyticaland of the NURBS-based description of t(x,y).

It has to be said however, that using the NURBS objectrepresentation entails computation times larger than withthe original VXI version. As an example, the time neededto compute the t(x,y) referring to Fig. 11(a) (3200 · 3200pixels) without optimized ray-tracer is 3h:46 0 with theNURBS and a few minutes with the facetted objectdescription on a standard desktop computer.

The facetted sample description is nevertheless useful fordescribing biological samples for which it is not possible touse CAD tools [28]. In these cases, the artefacts involved bythe polyhedral description are less detectable when consid-ering a low spatial resolution system as it is the case of con-ventional medical imaging.

With the new VXI version the PC images of complex 3Dobjects can be easily produced taking into account also theproperties of the source (size, energy spectrum) and thedetector (PSF).

PC VXI is a useful tool for estimating the achievableimage quality by means of the PC imaging technique. Itcan be employed at synchrotron or lab facilities dealingwith PC experiments or [37–40] developing systems forPC imaging.

Prospectively we envisage to improve the model for thetransmission function t(x,y). The new t(x,y) evaluation willtake into account the refraction of X-rays as they passthrough the sample. The aim of this approach is to providea simulation tool capable to account for PC imaging ofthick samples (several cm) under spatial high resolutionconditions (1 lm or less).

Acknowledgments

The authors gratefully acknowledge A. Bravin for themany constructive discussions they had with him. P. Ble-uet, M. Sanchez del Rio are also acknowledged for theiruseful suggestions and enthusiastic participation in the ini-tial phase of the present investigation. The authors thank


the French Ministry of National Education, AdvancedEducation, and Research for supporting this work andare very grateful to the IN2P3’s Computing Centre for pro-viding them with computing resources and assistance.

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