A quantitative, non-interferometric X-ray phase contrast imaging technique

A quantitative, non-interferometricX-ray phase contrast imaging technique

Peter R.T. Munro,1,2,∗ Luigi Rigon,3 Konstantin Ignatyev,4

Frances C.M. Lopez,3,5 Diego Dreossi,6 Robert D. Speller,4 andAlessandro Olivo4

1Optical + Biomedical Engineering Laboratory, School of Electrical, Electronic andComputer Engineering, The University of Western Australia, 35 Stirling Highway, Crawley,

Western Australia 6009, Australia2Centre for Microscopy, Characterisation and Analysis, The University of Western Australia,

35 Stirling Highway, Crawley, Western Australia, 6009, Australia3INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy

4Department of Medical Physics and Bioengineering, University College London, MaletPlace, Gower Street, London WC1E 6BT, UK

5Physics Department, University of Trieste, Via Valerio 2, 34127 Trieste, Italy6Sincrotrone Trieste SCpA, S.S. 14 km 163.5, 34012 Basovizza (TS), Italy

∗[email protected]

Abstract: We present a quantitative, non-interferometric, X-ray differ-ential phase contrast imaging technique based on the edge illuminationprinciple. We derive a novel phase retrieval algorithm which requires onlytwo images to be acquired and verify the technique experimentally usingsynchrotron radiation. The technique is useful for planar imaging but isexpected to be important for quantitative phase tomography also. Theproperties and limitations of the technique are studied in detail.

© 2013 Optical Society of America

OCIS codes: (110.7440) X-ray imaging;(120.5050) Phase measurement; (340.6720) Syn-chrotron Radiation.

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1. Introduction

Conventional radiographs record the projection, along the direction of X-ray propagation, of anobject’s absorption properties. Image contrast thus arises due to spatial variation of an object’sthickness and constituent material absorption coefficients. Since the pioneering work of Bonseand Hart [1], a variety of techniques have been developed to acquire X-ray images possessingcontrast due to sample induced phase shifts [2–7]. Such images are referred to as phase contrastimages. Since then, techniques have been developed which are capable of quantifying the phaseshift. Such techniques, whilst being useful for planar imaging, are also essential for performingquantitative phase tomography.

Existing quantitative X-ray phase contrast imaging (XPCI) techniques may be divided intothree categories. The first category may be denoted analyser based imaging (ABI) and developscontrast using the rocking curve of an analyser crystal [2]. Using this technique, contributions toimage contrast due to the phase and absorption are able to separated. Furthermore, researchershave developed algorithms for quantifying the phase shift using various simplifying approxima-tions [8–11]. This method requires a beam of high spectral purity and small angular divergencethus limiting it to synchrotron sources or monochromated laboratory sources [2,12].

The second method for performing quantitative XPCI is based upon a technique known asin-line holography [4, 13]. This technique requires a source of high spatial coherence and somust be performed using synchrotron radiation or a microfocal source. Algorithms for phaseextraction have largely employed transport of intensity formalism and generally require one ormore assumptions to be made. For example, Nugentet al. [14] introduced a phase retrieval al-gorithm for the special case of uniform irradiance. Algorithms were introduced for performingquantitative phase imaging of phase objects [15] and weakly absorbing objects [16]. Paganinetal. [17] introduced a formalism for phase extraction based upon the assumption of a homoge-neous sample. Gureyev later combined the transport of intensity formalism with the first Bornformalism to obtain new phase retrieval algorithms [18, 19]. These techniques have all beenextended and we refer the reader to a recent review [20] for a thorough account.

The final method, known as grating interferometry, employs two or three gratings and theprinciple of Talbot self-imaging. A sample’s phase and absorption information is thus encodedin the relative shift and amplitude, respectively, of fringes formed through self-imaging. De-tectors with pixel sizes small enough to resolve the fringes are, however, unsatisfactory forimaging large fields of view with clinically compatible exposure times. Phase stepping [21] andthe moire [22] configuration, developed for use in optical interferometry, have both been em-ployed to overcome this problem. The generation of fringes requires a spatially coherent sourceor an array or such sources in order to achieve Talbot self-imaging. This method has thus been


employed using synchrotron radiation [23,24] and apertured laboratory sources [25–27].Theprinciple employed in this paper is called edge illumination XPCI (EIXPCI) [28] and is

depicted in Fig. 1(a). EIXPCI works by projecting an X-ray beam onto the edge of a sensitiveregion of a detector. When the arrangement in Fig. 1(a) is tiled as shown in Fig. 1(b), the codedaperture XPCI (CAXPCI) system results. Image formation is equivalent in both systems withthe exception that the sample must be scanned through the beam in the EIXPCI case. Theanalysis presented in this paper thus applies equally to both cases. For the remainder of thispaper we refer principally to EIXPCI assuming that the results also apply to CAXPCI.

Superficially, EIXPCI shares similarities with systems from two categories: ABI and grat-ing interferometry. On first inspection, EIXPCI seems to be in the family of grating basedinterferometers due to the presence of two sets of apertures. The technique is, however, non-interferometric since it does not employ Talbot’s self-imaging phenomenon [29]. In this regardthe edge illumination technique is related to that reported recently by Huanget. al [30] who,contrary to EIXPCI, employ phase stepping and do not make use of edge illumination. On theother hand, EIXPCI was first developed in analogy with ABI [28] due to the manner in whichthe pixel edge creates sensitivity to the angle of photon refraction. It is no surprise then, that thequantitative adaptation of EIXPCI shares more in common with ABI than grating interferome-try since an image of the object is taken for two orientations of the detector apertures relativeto the object. Furthermore, phase stepping is not employed.

In this paper we begin by deriving an algorithm for extracting the gradient of the phase shiftdue to a sample imaged using the EIXPCI system. We then derive the limitations and importantproperties of the technique before showing experimental results obtained using synchrotronradiation. Although the theoretical and experimental results of this paper imply the use of highlycoherent synchrotron sources, the technique is in fact amenable to use with laboratory sourcesusing the coded aperture technique [31–33].

2. Derivation of quantitative method

2.1. Outline of the problem

In the X-ray regime, the refractive index is normally expressed asn = 1−δ + iβ whereδ andβ are the refractive index decrements.δ can be as large as 10−6 whilst β can be as much asthree orders of magnitude less thanδ . A wavefront which propagates through a thin object willbe perturbed by the object’s complex transmission functionT(x,y) = exp(−iφ(x,y)−µ(x,y))whereφ andµ , both real valued, are defined as:

φ(x,y) = k∫

O

δ (x,y,z)dz µ(x,y) = k∫

O

β (x,y,z)dz (1)

whereO is the extent of the object andk is the wave number.µ results in attenuation of thewavefront amplitude, which is directly detectable. Please note that our definition ofµ differsfrom the attenuation coefficient as traditionally used in X-ray imaging which is represented bythe same symbol. For the sake of reference, the attenuation coefficient is given by 2kβ (x,y,z).Generally only the gradient or Laplacian ofφ can be measured using one of the previouslymentioned quantitative XPCI techniques.

As mentioned previously, the proposed quantitative imaging technique is based upon theEIXPCI method [28]. The system is designed to perform XPCI with laboratory sources, how-ever in the first instance we develop the theory of quantitative imaging assuming an idealisedmonochromatic point source. In practice, a synchrotron X-ray source can be assumed to closelyapproximate this condition. EIXPCI is generally performed using a system illustrated schemat-ically in Fig. 1(b), which employs two sets of apertures. The set of apertures labeledA1 in Fig.1(b) is called the pre-sample aperture as it is placed immediately upstream of the sample. The


set of apertures labeledA2 is called the detector aperture as it is positioned as close to the flatpanel detector as possible. The projected pitches ofA1 andA2 match that of the flat panel pixelwidth.

The key to EIXPCI is the principle of pixel edge illumination [28, 34]. This condition isachieved by offsetting aperturesA1 andA2 from each other as demonstrated in Fig. 1(b). Thepartial illumination condition is usually quantified by the illuminated pixel fraction (IPF) [29]which is defined as the total integrated X-ray intensity normalised by its maximum. The max-imum pixel intensity is usually observed when the two apertures are in alignment. Althoughan EIXPCI system sensitive to phase gradients in two dimensions has been developed [35,36],the system in Fig. 1(b) is sensitive to phase gradients in one direction only. As a result it ishenceforth assumed, for the mathematical derivations only, that the imaging system and objectsextend uniformly out of the page in Fig. 1(b). In practice, however, the stationary phase ap-proximation [37, Pgs. 29–34] shows that to within a good approximation, a region of a samplegenerates contrast only in the row of pixels that it projects onto. This allows the assumption ofa sample uniform in one direction to be made whilst at the same time forming an image of thesample.

Fla

t Pa

ne

l De

tec

tor

ΔP

ΔP

P

A2A1

zodzso

Source

X-ray beam

zodzso

Detector

a) b)

Fig. 1.(a) Demonstrationof the edge illumination principle whereby pre-sample apertures,located a distance zso from the source, shape a beam which is incident upon detector aper-tures and detector, placed a distance zod from the pre-sample apertures. The centre of thedetector apertures is offset from the centre of the beam by a distance∆P. (b) A CAXPCIsystem made by tiling the setup used in(a). A1 and A2 are apertures with the same pro-jected pitch which also matches the detector pixel width P. All parameters have the samemeaning as in(a). A1 creates X-ray beams which are incident upon A2 which is used toachieve the partial pixel illumination condition.The imaging system continues uniformlyout of the page.

2.2. Derivation for object with linear phase and absorption profile

For simplicity and without loss of generality, while deriving the equations for quantitative EIX-PCI we consider a single pair of apertures as demonstrated in Fig. 2. Quantitative EIXPCI isperformed by imaging the same object using two complementary positions ofA2. Both con-figurations are set to an IPF of 0.5, yet will result in inverted contrasts. For a non-absorbing,prism-like, object depicted in Fig. 2, the configuration denoted byI− will result in a detectedsignal which is lower than the flat field whilst theI+ configuration will result in a detected sig-nal which is greater than the flat field. If an absorbing object were employed,I− andI+ wouldalso be affected by absorption. We will derive the equations of the quantitative method by as-suming a monochromatic point source. Within the paraxial (i.e., small angle) approximation, a


point source results in an X-ray intensity which is uniform in they directionof Fig. 2. Then, byassuming that the imaging system and object are also uniform in they direction we need onlyconsider variations of intensity in thex direction. If the complex amplitude of X-rays incidentuponA2 is given byU(x) then, assuming a pixel of heightP, I− andI+ are given by:

I− =∫ P

0

∫ 0

−MW|U(x)|2dxdy I+ =

∫ P

0

∫ MW

0|U(x)|2dxdy (2)

whereM is the system magnification given by(zso+zod)/zso andzso andzod are defined in Fig.2.

SourceSource

Fig. 2. The two complementary aperture configurations used to perform quantitative EIX-PCI.W is the width of the pre-sample apertures,A1, M = (zso+zod)/zso is the magnifica-tion, ξ is the spatial coordinate within the open region of the pre-sample aperture andI−andI+ represent the signals measured by the detector pixels in the left and right configura-tions respectively.A2 represents the detector aperture.

We now turn to the task of calculating the complex amplitudeU(x). We begin by dropping they dependence of the complex transmission function thus re-writing it asT(x). Furthermore, inthe object space we use the symbolξ instead ofx to distinguish between the object and detectorspaces. We also assume that the phase and absorption functions may be described locally by alinear function giving the complex transmission function as:

T(ξ +ξs;ξi) = exp

(

−iφ(ξs)− i∂φ∂ξ

∣

∣

∣

∣

ξs

(ξ −ξi)−µ(ξs)−∂ µ∂ξ

∣

∣

∣

∣

ξs

(ξ −ξi)

)

(3)

where the parameterξs is used to represent a shift of the sample, the Taylor series expansionis taken aboutξ = 0 and the parameterξi is used to shift the Taylor series representation ofthe transmission function. The series expansion is centered onξ = 0 as this coincides withthe centre of the pre-sample aperture. Note that Eq. (3) means that we assume that the samplehas constant phase and absorption gradients within the transmitting region of the pre-sampleaperture. Then, by applying the paraxial approximation to the Fresnel-Kirchhoff diffraction


integral the complex amplitude incident uponA2 maybe found according to [38]:

U(x) = C∫ W/2

−W/2T(ξ −ξs;ξi)exp

(

ikξ 2 zso+zod

2zsozod

)

exp

(

−ikξx

zod

)

dξ

= Cexp

(

i

(

∂φ∂ξ

∣

∣

∣

∣

ξs

ξi −φ(ξs)

))

exp

(

∂ µ∂ξ

∣

∣

∣

∣

ξs

ξi −µ(ξs)

)

·

·∫ W/2

−W/2exp

(

−∂ µ∂ξ

∣

∣

∣

∣

ξs

ξ

)

exp

(

ikξ 2 zso+zod

2zsozod− ikξ

(

xzod

+1k

∂φ∂ξ

∣

∣

∣

∣

ξs))

dξ (4)

where, since we are ultimately interested in|U(x)|, complex phase terms dependent uponyhave been omitted and

C =U0

√

iλzsozod(zso+zod)exp(ik(zso+zod))exp

(

ik

(

x2

2zod

))

. (5)

WhereU0 represents the amplitude of the incident wave emitted by the source. As in Ref. [38]we employ the stationary phase approximation (SPA) to yield the first order asymptotic solutionto Eq. (4) as

U0 ∼ Cexp

(

i

(

∂φ∂ξ

∣

∣

∣

∣

ξs

ξi −φ(ξs)

))

exp

(

∂ µ∂ξ

∣

∣

∣

∣

ξs

ξi −µ(ξs)

)

√

iλzsozod

zso+zod·

·exp

[

−∂ µ∂ξ

∣

∣

∣

∣

ξs zsozod

zso+zod

(

xzod

+1k

∂φ∂ξ

∣

∣

∣

∣

ξs)]

exp

−ik2

zsozod

zso+zod

(

xzod

+1k

∂φ∂ξ

∣

∣

∣

∣

ξs)2

(6)

which is valid forx∈[

−MW/2−1/k(∂φ/∂ξ )|ξszod,MW/2−1/k(∂φ/∂ξ )|ξszod

]

. Outside of

this range the first order asymptotic solution is zero. Returning now to the ultimate objective ofevaluatingI− andI+, substituting Eq. (6) into Eqs. (2) yields:

I− = A

[

exp

(

∂ µ∂ξ

∣

∣

∣

∣

ξs

W

)

−B

]

I+ = A

[

−exp

(

−∂ µ∂ξ

∣

∣

∣

∣

ξs

W

)

+B

]

(7)

where A = |U0|2Pexp(2(∂ µ/∂ξ ) |ξsξi − 2µ(ξs))/

(

(zso+zod)2zso(∂ µ/∂ξ ) |ξs

)

and B =

exp(−2(∂ µ/∂ξ ) |ξszsozod(1/k) (∂φ/∂ξ ) |ξs/(zso+zod)). Any X-ray transmitted byA1 and notabsorbed by the sample should be detected in either theI− or I+ configurations, so long as it isnot scattered beyond the sensitive region of the pixel of both configurations. Further, any redis-tribution of X-rays between theI− andI+ configurations not due to absorption may be assumedto be due to a sample phase gradient. These assumptions, and the mathematical form of Eqs.(7), lead us to form the sum and difference of these two quantities as:

I+ + I− = 2A sinh

(

∂ µ∂ξ

∣

∣

∣

∣

ξs

W

)

(8)

I+ − I− = 2A

[

B−cosh

(

∂ µ∂ξ

∣

∣

∣

∣

ξs

W

)]

(9)

Note that Eq. (8) is equivalent to a conventional absorption image and an identical result couldbe obtained by integrating the X-ray intensity over the equivalent region of the sample aperture.


This also follows from conservation of energy within the paraxial limit. In the case that there isnosample or the sample is, not absorbing, Eq. (8) simplifies to:

I0 =|U0|

2MWP

(zso+zod)2 (10)

which we denote byI0 as it is equivalent to the flat field in a traditional absorption image. Inorder to perform quantitative XPCI we must extract the quantity 1/k(∂φ/∂ξ ) |ξs. We proceedby noting that an image, as opposed to a profile, may be obtained using one of two methods,either independently or in combination. One method makes use of periodic apertures and a flatpanel detector as demonstrated in Fig. 1(b). The other method employs a technique known asdithering whereby the object is scanned relative to the imaging system. These approaches areequivalent and, in keeping with the current derivation, we choose to scan the object througha single aperture pair. We thus assume that we have access toI− and I+ for object positionsξi = −∆ξ ,0,∆ξ . This allows us to determine(∂ µ/∂ξ ) |ξs as:

∂ µ∂ξ

∣

∣

∣

∣

ξs

=1

4∆ξlog

(

(I+ + I−)|ξi=∆ξ

(I+ + I−)|ξi=−∆ξ

)

(11)

If total acquisition time is limited we may have access to only two dithering stepsξi = −∆ξ ,0in which case(∂ µ/∂ξ ) |ξs may be found slightly less accurately as:

∂ µ∂ξ

∣

∣

∣

∣

ξs

=1

2∆ξlog

(

(I+ + I−)|ξi=0

(I+ + I−)|ξi=−∆ξ

)

(12)

It is then easy to obtain the total absorption function atξ = 0 as:

µ(ξs) =12

log

I0(I+ + I−)|ξi=0

sinh

(

∂ µ∂ξ

∣

∣

∣

ξsW

)

∂ µ∂ξ

∣

∣

∣

ξsW

(13)

≈12

log

[

I0(I+ + I−)|ξi=0

]

(14)

Having evaluated(∂ µ/∂ξ ) |ξs and sinceW is known,(1/k) (∂φ/∂ξ ) |ξs may be found accord-ing to:

(1/k)∂φ∂ξ

∣

∣

∣

∣

ξs

= − log

[

I+ − I−I+ + I−

∣

∣

∣

∣

ξi=0

sinh

(

∂ µ∂ξ

∣

∣

∣

∣

ξs

W

)

+cosh

(

∂ µ∂ξ

∣

∣

∣

∣

ξs

W

)]

(zso+zod)

2 ∂ µ∂ξ

∣

∣

∣

ξszsozod

(15)By making the approximations sinh(x) ≈ x, cosh(x) ≈ 1+x2/2 and log(1+x) ≈ x for x < 0.5,

Eq. (15) can be found in a simpler form, valid so long as∂ µ∂ξ

∣

∣

∣

ξsW < 1/2, as:

(1/k)∂φ∂ξ

∣

∣

∣

∣

ξs

= −

[

I+ − I−I+ + I−

∣

∣

∣

∣

ξi=0

+12

∂ µ∂ξ

∣

∣

∣

∣

ξs

W

]

W(zso+zod)

2zsozod(16)

Finally, we note that we have hitherto tacitly assumed thatβ is non-zero. Whenβ is zero, theentire derivation is considerably simpler resulting in:

(1/k)∂φ∂ξ

∣

∣

∣

∣

ξs

= −I+ − I−I+ + I−

∣

∣

∣

∣

ξi=0 W(zso+zod)

2zsozod(17)


meaning that the phase gradient can be extracted without performing the central differenceschemein Eq. (11). Thus the main results of this paper have been derived, in particular, Eqs.(11), (13) and (15) give the gradient of the absorption function, the total absorption functionand the gradient of the phase function respectively.

3. Examples and analysis

3.1. Limitations of the technique

The first restriction of the technique is that the object’s phase and absorption functions (i.e.,φ and µ) should be well approximated locally by a linear function. Furthermore, this linearapproximation should be reasonable over the range of the sample aperture opening (W). Thisrequirement is, however, common to all grating techniques since a projected fringe samplesthe average phase and absorption properties over a region of a sample. Secondly, Eqs. (11)and (15) were derived under the assumption that−MW/2−1/k(∂φ/∂ξ )|ξszod ≥ −MW andMW/2−1/k(∂φ/∂ξ )|ξszod ≤ MW. This assumption is employed in the evaluation of Eqs. (7)but it also follows from energy conservation. In particular, if photons deviate outside of theunion of the two exposed pixel regions in Fig. 2 then the quantityI+ + I− will not be equivalentto the conventional absorption image of the object. These restrictions require that:

(1/k)

∣

∣

∣

∣

∂φ∂ξ

∣

∣

∣

∣

ξs

≤MW2zod

. (18)

For the experimental system described in Sec. (3.2) this would require an object to have a gra-dient not exceeding 1.9×10−5/δ m−1. Given thatδ is seldom greater than 10−6 this restrictionwill be satisfied in most practical applications. Furthermore, if a typical laboratory based sys-tem were used withzso= 1.6m andzod = 0.4m [36], the object slope should not exceed the lessrestrictive value of 4×10−5/δ , assuming the same value ofW.

The next restriction concerns the gradient of the absorption functionµ(ξ ). For an objectwith properties described precisely by Eq. (3), the measured values ofI+ and I− will differslightly from those predicted by Eqs. (7) due to the SPA employed in the evaluation of Eq.(6). I+ and I− are, however, very well approximated by the expressions in Eqs. (7) and wedemonstrate this by an example, employing a wedge-like object with thickness described byh(ξ ) = 5ξ +1.5×10−4 where all quantities are in m.δ was fixed at 10−6 whilst β was varied.The system parameters werezso = 1.6m,zod = 0.4m,W = 40µm, ∆ξ = 10µm and the sourcehad a photon energy of 20keV. In this analysis, the phase extraction algorithm was applied todata found by direct numerical evaluation of Eq. (4), rather than applying the SPA. We thenseek to quantify the error in(∂φ/∂ξ ) |ξs which results from employing the SPA in deriving thephase and absorption extraction formulae.

For simplicity, we introduce the quantityρ to denote the ratio(I+− I−)/(I+ + I−), evaluatedusing the SPA. We then introduce the quantityε to represent the perturbation toρ , which be-comesρ(1+ε), whenI+ andI− are calculated by evaluating Eq. (4) directly. Figure 3 containsa plot ofε for a range of values ofβ and shows that the SPA indeed leads to a low value ofεacross the range of values ofβ considered.ε decreases asβ increases to 10−7. This is becausethe oscillations present in the actual X-ray intensity incident upon the detector aperture, but notrepresented by the SPA, become smaller due to the increased absorption.ε begins to increaseagain when the absorption becomes so high that the oscillation in the X-ray intensity achievesa magnitude on the same order as the average intensity predicted using the SPA.

In the next example, we show that the value of(∂φ/∂ξ ) |ξs, extracted from field data cal-culated by direct evaluation of Eq. (4), departs steadily from the true value asβ increases, asshown in Fig. 4, even thoughε decreases. In the absence of noise, the technique calculates


10−9

10−8

10−70.005

0.01

0.015

ε

β

Fig. 3. Plot of the error,ε, introduced into the ratio(I+ − I−)/(I+ + I−) by the SPA asβvaries for a wedge-like object.

(∂ µ/∂ξ ) |ξs very accurately. Thus, the error in the predicted value of(∂φ/∂ξ ) |ξs is due tothe absolute perturbation toρ which occurs when field data calculated using direct evalua-tion of Eq. (4) is used. The absolute error in(∂φ/∂ξ ) |ξs is thus the difference between thevalues found whenρ andρ(1+ ε) are used as input to Eq. (15). By assuming that, for largex,cosh(x)≈ sinh(x) and for smallx, log(1+x)≈ x, it can be shown that, to a good approximation,the absolute error in(∂φ/∂ξ ) |ξs is given by:

∂φ∂ξ

∣

∣

∣

∣

ξs

−

˜∂φ∂ξ

∣

∣

∣

∣

ξs

≈ ε(zso+zod)

4zsozod

(

1− I−I+

)

(1/k) ∂ µ∂ξ

∣

∣

∣

ξs(19)

where the tilde is used to represent the phase gradient extracted from values ofI− and I+

10−9

10−8

10−7

3

4

5x 10−6

1 k∂φ

∂ξ

∣ ∣ ∣

ξ 0

β

Fig. 4. Plot of the value of(1/k) (∂φ/∂ξ ) |ξs calculated for the example wedge-like objectfor a range of values ofβ . The value is calculated by simulating the values ofI+ and I−by evaluating the diffraction integrals of Eq. (4). The actual value of(1/k) (∂φ/∂ξ ) |ξs is5×10−6.

10−9

10−8

10−7−2

−1

0

(1−I −

/I +

)

(1/k)d

µ/dξ

β

Fig. 5. Plot of(1− I−/I+)/((1/k) (∂ µ/∂ξ ) |ξs) for the wedge-like object for a range ofvalues ofβ .


found by evaluating Eq. (4) directly. Examining this equation it is evident that the error isintroduceddue to the large disparity betweenI+ and I− which arises for large values of(∂ µ/∂ξ ) |ξs. This is demonstrated for this example by the plot of Fig. 5 which shows how(1− I−/I+)/((1/k) (∂ µ/∂ξ ) |ξs) varies withβ for the wedge-like example considered. Theacceptable value for(1− I−/I+)/((1/k) (∂ µ/∂ξ ) |ξs) will vary with the accuracy required forthe extracted phase. For design purposes, if an upper bound is placed upon(∂ µ/∂ξ ) |ξs, themaximum allowable value ofW can be found easily by assumingδ = 0 and substituting Eqs.(7) into the expression(1− I−/I+)/((1/k) (∂ µ/∂ξ ) |ξs).

As a final restriction we note that even in the limit of zero absorption, the estimated value for(∂φ/∂ξ ) |ξs in Fig. 4 is slightly below the real value of 5×10−6. This comparatively small erroris again due to the SPA which does not model diffraction by the pre-sample aperture. As a result,the estimated value of(∂φ/∂ξ ) |ξs is seen to have a small dependence on wavelength. Figure6 contains a plot of(∂φ/∂ξ ) |ξs against photon energy for the wedge-like object consideredpreviously withβ = 10−8 and all other parameters as specified previously. In practice, thiserror will be less significant than in the example as the effects of diffraction are less significantwhen sources with finite spectral width and spatial extent are employed.

20 30 40 50 604.9

4.95

5

5.05x 10

−6

photon energy (keV)

1 k(∂φ/∂

ξ)|ξ

0

Fig. 6. Plot of how(∂φ/∂ξ ) |ξs varies with the source photon energy for the wedge-likeobject example considered in this section withβ = 10−8.

3.2. Experimental setup

Experimental verification of the phase extraction technique was performed on the SYRMEPbending magnet of the Elettra synchrotron radiation facility in operation in Trieste, Italy. Thebeamline is described in more detail elsewhere [39, 40], we give here only the details relevantto the experiment. The sample stage was located, in the experimental room, approximately22m from the apparent X-ray source which has full width at half maximum dimensions of0.280(horizontal)×0.080(vertical) mm2. This results in a usable beam of dimension approx-imately 120×4 mm2. A channel-cut Si (1,1,1) crystal monochromates the beam to nominalphoton energy of 20 keV with a fractional bandwidth of 0.2%. A photon counting, linear arraysilicon microstrip detector known as PICASSO was employed [41]. The detector works in theso-called “edge-on” configuration and provides an array of 2368 pixels 50µm wide and 300µmhigh. An important property of the PICASSO detector is that global and channel specific thresh-olds can be adjusted in order to have negligible pixel cross talk [42,43].

The experimental system is shown schematically in Fig. 7. The diagram shows the narrowdimension of the beam and thus the beam extends 120mm into the page. The system in Fig. 7 issensitive to the phase gradients in the vertical direction and this is also the direction in which thesample was scanned. A 20µm slit was used as the pre-sample aperture, thus setting the valueof W to 20µm. A single edge was sufficient to act as the detector aperture and this was simplytranslated vertically to switch between theI+ andI− configurations. A single edge is able to be


Fig. 7. Schematic, side-on view of the experimental system employed to verify the phaseextraction algorithm. The diagram is not to scale.

used since a single X-ray beam is used, in contrast to the plurality of beams in the system shownin Fig. 1(b). If the system in Fig. 1(b) were employed, a larger value ofW would need to beemployed to ensure that nearly all photons which propagate through the object are counted ineitherI+ or I−. With the edge in place against the detector, each pixel recorded approximately4.5×104 counts per second.

Table 1. Properties of wires making up the sample. PEEK stands for polyetheretherketone.†The boron is supplied on a 5µm diameter tungsten core. The filaments were sourced fromGoodfellow (Goodfellow Cambridge Ltd., Huntingdon, UK) and the nominal diameterswere obtained from data sheets provided by Goodfellow.

Wire MaterialNominal Estimated δ/10−7 β/10−10

diameter (µm) diameter (µm) (20keV, [44]) (20keV, [44])A Titanium 250± 10% 260± 10 21.9 346B Sapphire 250± 20% 250± 10 20.3 39.9C Aluminium 250± 10% 260± 10 13.5 42.2D PEEK 450± 20% 480± 10 7.15 2.74E PEEK 200± 20% 210± 10 7.15 2.74F Boron† 200± 20% 220± 10 11.2 2.84

Figure 8 shows a photograph of the sample which was used in the experiment. It was com-posedof six filaments of varying diameter and material properties as outlined in Table 1 inwhich each filament is labeled with a letter from A to F which is used throughout the remain-der of this paper. The diameters were estimated from the experimental results shown in Fig. 9.The sample holder was positioned in the experimental setup of Fig. 7 such that the wires wereapproximately normal to the page and were thus roughly parallel with the beam and apertureedges. 770 vertical scan positions were taken for both theI+ andI− configurations. All 770 scanpositions were acquired for theI+ configuration before changing into theI− configuration andreturning the sample to its initial position. A scan step of 10µm was employed however severalscan steps were passed, without acquiring a detector signal, when the wires were not within thebeam. An exposure time of 1s was employed and each ofI+ andI− took approximately 35 min-utes in total to acquire. A motorised sample stage (Newport IMS300V) was employed whichhas a unidirectional reputability of 0.5µm and a guaranteed on-axis accuracy of 10±5µm. Thereputability of the sample position is important to ensure that theI+ andI− images are correctlyregistered.

3.3. Experimental results

Figure 9 shows images representing the raw data acquired during the experiment with the fil-aments arranged in the same order as in Fig. 8. Each row of pixels represents an acquisition


Fig. 8. Photograph of the sample used in the experiment.

from the detector for a particular vertical position of the sample. TheI+ and I− images werenormalised independently by a row of pixel values found by averaging over a sample-free re-gion, denoted by boxes in each image. The images exhibit a transition from predominantlyphase contrast, in the case of boron (lower filament), through to strong absorption contrast inthe case of titanium (upper filament). Note that the 5µm diameter tungsten core is clearly visi-ble in the image of the boron filament. Figure 10 shows line plots of the intensity along a singlecolumn of both images in Fig. 9 and a plot of their sum. Note that the left side of the line plotscorrespond to the upper part of the images in Fig. 9.

I+ I−

0.5

1

1.5

Fig. 9. Images of the sample taken in theI+ andI− configurations.Note that some verticalscan positions, in between filaments, were omitted to reduce the total number of acqui-sitions. The image shows 16mm of the sample in the horizontal direction. Each row ofpixels was normalised by the mean of the rows within the boxed region in each image. Thefilaments are arranged in the same order as in Fig. 8.

Figure 11 contains images of bothµ and∂φ/∂ξ which have been found by using the datafrom Fig. 9 as input to Eqs. (14) and (15) respectively. Note that all filaments are easily re-solved in the phase image, whilst only the three most absorbing filaments are resolvable in theabsorption image. The absorption profiles for filaments D, E and F (i.e., the right most three) inFig. 10 exhibit anomalies in the form of erroneous peaks. It is likely that these are due to smallmisalignments arising due to theI+ andI− images being taken during separate scan sequences.It should, however, be noted that this anomaly impacts minimally upon extracted phase in Fig.12. This demonstrates the robustness of the phase retrieval algorithm.

Fig. 12 compares the extracted phase gradient with the analytic value for(∂φ/∂ξ ) |ξs foundby differentiatingφ in Eq. (1) directly, assuming a cylindrical sample. The analytic value for theboron filament was calculated assuming a 5µm diameter tungsten core. Each of the analyticallycalculated phase gradients reach infinity at the filament edges. The extracted phase gradient


0.20.40.60.8

1(I

−+ I+)/2

0.5

1

1.5 I+

1 2 3 4 5 6 7

0.5

1

1.5

scan position (mm)

norm

alis

edph

oton

cou

nt

I−

Fig. 10. Profile plots for a single pixel column for the absorption image ((I− + I+)/2), I+andI−.

0 0.2 0.4 0.6 0.8−10

−510

−50

Fig. 11. Images ofµ (left) and(1/k)∂φ/∂ξ (right) derived from Fig. 9 according to Eqs.(14) and (15) respectively.

does not, however, reach infinity. There are a few reasons for this, firstly, a degree of smoothingis inevitable since the X-ray beam used in the imaging technique has a finite width and thussenses a segment of the filament. Secondly, the extracted value of(∂φ/∂ξ ) |ξs relies upon theextracted value of(∂ µ/∂ξ ) |ξs which is calculated by a central difference calculation. Thecentral difference calculation is accurate only for smooth functions and breaks down when asharp transition occurs, such as at the the filament edge. Despite these limitations, the extractedphase gradient is very close to the analytic phase gradient. Plot F of Fig. 12 contains two sharppeaks aroundξ = 0 due to the 5µm tungsten core on which the boron is constructed. The coreis below the spatial resolution of the imaging system and thus why it appears as a spike.


−200 0 200

−1

−0.5

0

0.5

1

x 10−5

ξ (µm)

(1/k

)∂φ/∂

ξ

A

−200 0 200

−1

−0.5

0

0.5

1

x 10−5

B

−200 0 200−10

−5

0

5

x 10−6

C

−200 0 200

−8

−6

−4

−2

0

2

4

x 10−6

D

−100 0 100

−4

−2

0

2

4

x 10−6

E

−100 0 100

−5

0

5

x 10−6

F

Fig. 12. Plots comparing the extracted value (broken line) for(1/k)∂φ/∂ξ for each wireand the theoretical value (solid line). The dots superimposed on the broken line representthe experimentally determined values of(1/k)∂φ/∂ξ . The wires are labeled with a letterwhich corresponds to a row in Table 1.

4. Conclusions

We have presented the derivation, analysis and experimental verification of a quantitative X-rayphase contrast imaging technique. The experiment was performed using synchrotron radiation,yet the technique has been adapted to be used with laboratory sources using the coded aperturetechnique [33]. The synchrotron experiments were important in establishing the absolute ac-curacy of the method by eliminating spectral averaging. The experimentally determined phasegradients matched the theoretically expected results very closely. The technique also works wellin close proximity to the infinite phase gradients near the filament edges. This work providesthe theoretical foundation for a number of experiments and theoretical developments, in bothplanar and three dimensional imaging, currently in progress within our laboratory.

Acknowledgments

This work was funded by the UK Engineering and Physical Sciences Research Coun-cil (EP/G004250/1 and EP/I021884/1). K.I. was supported by the Wellcome Trust(085856/Z/08/Z) and P.M. is supported by a Discovery Early Career Research Award fromthe Australian Research Council (DE120101331). F.L. is supported by the Prof. Giulio BrauttiPhD Memorial Fellowship. We would like to thank the personnel from the Elettra synchrotronand the University of Trieste working on the SYRMEP beamline for assistance with obtainingthe experimental results.


A quantitative, non-interferometric X-ray phase contrast imaging technique

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