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ReviewCite this article:Wilkins SW, Nesterets YI,Gureyev TE, Mayo SC, Pogany A, Stevenson AW.2014 On the evolution and relative merits ofhard X-ray phase-contrast imaging methods.Phil. Trans. R. Soc. A 372: 20130021.http://dx.doi.org/10.1098/rsta.2013.0021

One contribution of 16 to a Discussion MeetingIssue ‘Taking X-ray phase contrast imaginginto mainstream applications’ and its satelliteworkshop ‘Real and reciprocal space X-rayimaging’.

Subject Areas:optics

Keywords:phase contrast, imaging, X-ray imaging

Author for correspondence:A. W. Stevensone-mail: [email protected]

†Prof. Stephen (‘Steve’) Wilkins passed awayvery suddenly on 25 March 2013. His manyfriends and colleagues from around the worldwere greatly shocked at this sad news. Stevewas the driving force in the preparation of this,his final manuscript.‡Present address: 8 Victor Rd, Glen Iris, VIC3146, Australia.

On the evolution and relativemerits of hard X-rayphase-contrast imagingmethodsS. W. Wilkins†, Ya. I. Nesterets, T. E. Gureyev,

S. C. Mayo, A. Pogany‡ and A. W. Stevenson

CSIRO Materials Science and Engineering, PB33, Clayton South,Victoria 3169, Australia

This review provides a brief overview, albeit froma somewhat personal perspective, of the evolutionand key features of various hard X-ray phase-contrast imaging (PCI) methods of current interestin connection with translation to a wide rangeof imaging applications. Although such methodshave already found wide-ranging applications usingsynchrotron sources, application to dynamic studiesin a laboratory/clinical context, for example for in vivoimaging, has been slow due to the current limitationsin the brilliance of compact laboratory sources andthe availability of suitable high-performance X-raydetectors. On the theoretical side, promising newPCI methods are evolving which can record bothcomponents of the phase gradient in a single exposureand which can accept a relatively large spectralbandpass. In order to help to identify the mostpromising paths forward, we make some suggestionsas to how the various PCI methods might becompared for performance with a particular viewto identifying those which are the most efficient,given the fact that source performance is currentlya key limiting factor on the improved performanceand applicability of PCI systems, especially in thecontext of dynamic sample studies. The rapid ongoingdevelopment of both suitable improved sources anddetectors gives strong encouragement to the view thathard X-ray PCI methods are poised for improvedperformance and an even wider range of applicationsin the near future.

2014 The Author(s) Published by the Royal Society. All rights reserved.

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rsta.royalsocietypublishing.orgPhil.Trans.R.Soc.A372:20130021

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1. Introduction and overviewThe observation of new physical phenomena and their understanding on the basis of acceptedphysical laws are the essence of progress in science. However, there can be significant time delaysbetween the first and second ingredients being realized even though the underlying physicsrequired may be well known. This review seeks to give a brief overview of the evolution of X-rayphase-contrast imaging (PCI) in various forms and to reflect en passant upon some of the phasesin the development of the field, albeit from a somewhat personal perspective. In this process, itseeks to trace some of the key ideas and works involved in the evolution of X-ray PCI with a viewto looking at the comparative advantages and limitations of the main hard X-ray PCI techniquescurrently in practice, particularly in the context of applications using conventional laboratorysources [1]. There have been a number of valuable recent reviews of hard X-ray PCI, in general[2], and some especially in the context of using synchrotron sources, for example [3]. This reviewtries to focus on methods that are of current interest in connection with hard X-ray applicationsusing compact laboratory sources and the relevant issues with regard to performance.

A key quantity involved in the interaction of electromagnetic radiation with matter is thecomplex refractive index, namely

n(r, E) = 1 − δ(r, E) − iβ(r, E), (1.1)

here expressed in the usual form for X-rays, where δ, β 1. The expressions for δ and β awayfrom an absorption edge may be found in [4]. For present purposes, we note that in the regimedominated by photoelectric absorption

δ(r, E) = roh2c2

2πE2 ρ(r) ∼ O(

1E2

), (1.2a)

whereas

β(r, E) = hc4πE

μo(r, E) ∼ O(

1E4

), (1.2b)

where E is the photon energy, ρ is the electron density, μo is the photoelectric component of thelinear attenuation coefficient, ro is the classical electron radius. As is well known, the β term leadsto normal absorption while the δ term leads to purely refractive contributions.

In considering imaging with electromagnetic radiation, we term images that contain featuresarising from the purely refractive term, δ, in the complex refractive index above (and are relatedto interference effects that are recorded via intensity distributions), phase-contrast images.

In this review, we first trace some early observations of PCI in the visible light context as alead in to considering some of the manifestations in which X-ray PCI has been developed inrecent times.

For a wave initially propagating along the optic axis assumed in the z-direction andundergoing a phase change φ(r⊥) in the r⊥ = (x, y)-plane (at right angles to z-direction) on passingthrough a thin object (such that the projection approximation is valid), this phase change isgiven by

φ(r⊥; z, k) = −k∫ z

−∞δ(r⊥, z′; k)dz′ = −2πro

k

∫ z

−∞ρ(r⊥, z′)dz′ = O

(1E

), (1.3)

where k = 2π/λ = 2πE/hc, so that the x-component of angle of deviation from the propagationdirection, αx, for small phase gradients owing to δ is given for X-rays by [1]

αx ≈ −1k

∂φ(x, y; k)∂x

≈ O(

1E2

), (1.4)

while we might note by comparison that the scattering angle in SAXS/USAXS in the kinematicalapproximation varies as 1/E.



3


.........................................................

2. Historical background

(a) Optical imaging(i) ‘Magic mirrors’ (ancient China and Japan)

Perhaps the earliest example of PCI being put into practice involves the art and demonstrationof so-called ‘magic mirrors’ [5–7] that date back to at least the fifth century CE in China andapparently even earlier to the Han Dynasty (206 BCE–24 CE) [7,8], and, the performance of whichremained steeped in mystery until the twentieth century. The basic feature of these magic mirrorslay in the fact that a cast or embossed design on the back surface of these bronze mirrors becamevisible when the highly polished and smooth convex front surface of the mirror was illuminatedby direct sunlight and the reflected beam from the front surface of the mirror projected on to awall. This magnified image of the back surface of the mirror did not involve any apparent focusingeffect and was equally sharp at almost any distance. It appeared to have the mystical qualityof imaging the back surface of the mirror by visible light penetrating the thick bronze mirror(typically of order, say, 5 mm thick) with no apparent features existing on the polished reflectingsurface. Such mirrors were also known in Japan and termed ‘Makyoh’ for ‘wonder mirror’.

Although the properties of magic mirrors were known in the West from 1832, a properexplanation for their imaging properties baffled scientists for around 100 years until 1933 whenW. H. Bragg [5,6,9] provided a satisfactory explanation, in general, physical terms for theirproperties. The underlying explanation was due to height distortions in the front surface of themirror, believed caused by strains induced when polishing the front surface of the mirror as aresult of the structure on the back surface. These minute height distortions led to visible lightimages of the back-surface structure being produced in the reflected beam via Fresnel diffractionin the near-field regime [6].

According to the analysis provided by Berry [6], the relationship between image intensity andheight deviations from an assumed ideal spherical form for the convex mirror is of Laplacianform, namely

I(r⊥, z) = 1 + z∇2⊥h(r⊥), (2.1)

where h(r) is the height deviation (surface relief) of the mirror and r⊥ = (x, y) denotes position inthe mirror reference plane and z is a reduced distance that asymptotes to Ro, the mirror radius,as distance from source to mirror and mirror to observation plane become large. The form ofequation (2.1) clearly indicates that the form of the image is invariant with respect to change inthe reduced distance, and the treatment lies within a geometrical optics approximation.

A key feature predicted by Berry’s treatment is the presence of black–white fringes at the edgesof sharp height distortions. Such fringes are observed to be bright–dark when the step on themirror surface goes from low to high and vice versa for high to low steps.

(ii) Contrast by defocus in early optical microscopes (Antoni van Leeuwenhoek)

Although not the inventor of the optical microscope, in his pioneering experiments with his earlyoptical microscopes (1670s), van Leeuwenhoek was able to observe bacteria for the first time in1683. His microscopes were quite simple and involved a single tiny spherical lens, essentially amagnifying glass, but of such small focal length that he could achieve magnifications of up toapproximately 300 times [10].

His method of lens-making involved manufacture of miniature glass spheres by fusing ofglass and was a key and well-kept secret that gave him a competitive advantage. He carriedout numerous microscopic studies of biological organisms and other samples that were recordedin hand-drawn sketches, and in 1673 (at the age of 40) published his first paper (in PhilosophicalTransactions of the Royal Society, vol. 8, pp. 6037–6038, 6116–6118).

In his microscopic studies of bacteria, it is inevitable that structures he studied would havebecome more readily visible if they were observed slightly out of focus. This very early work



4


.........................................................

was followed around 250 years later by Zernike’s [11] development in 1932 (patented 1934) of theoptical phase-contrast microscope, involving a phase plate introduced in the back-focal plane tohelp convert phase shifts produced by the sample into intensity variations in the imaging plane,that opened up imaging of unstained biological cells.

(b) Developments in X-ray phase-contrast imaging(i) X-ray interferometry using single crystals (IB-PCI)

The earliest example of X-ray PCI lies in the work of Bonse & Hart [12,13] published in 1965.Using a Laue-Laue-Laue monolithic X-ray interferometer and a laboratory tube source in line-focus mode, they recorded thickness fringes in an X-ray interferometer for a thin plastic plano-concave epoxy plate that essentially acted as a pure phase object [13]. Subsequent, early work byAndo & Hosoya [14] further helped to demonstrate the potential of this technique. In this case,the image recorded phase changes induced in the wavefront by the sample, albeit modulo 2π . Forsimple objects, the phase unwrapping problem can readily be handled to yield the phase changeinduced by the sample.

Impressive applications of this technique to biological samples demonstrating very highsensitivity to minute electron density differences in biological samples have been achieved byMomose and co-workers [15,16].

(ii) Double-crystal (or analyser)-based methods (AB-PCI)

Here, the first crystal acts as a monochromator/collimator, whereas the second crystal acts asan angular or spatial frequency filter. The wider the rocking curve of the analyser crystal, thegreater the angular acceptance (spatial frequency bandpass) of the information recorded in theimage. Wavefront distortions produced by the sample are modulated and filtered by dynamicalscattering in the analyser crystal and free-space propagation. In most cases, the detector is closeto the sample, and propagation effects after the sample are typically ignored.

This approach has a long history with various groups having made significant contributions.The earliest would appear to be those of Goetz et al. in 1979 [17] and Forster et al. in 1980 [18], bothworks involved the use of a double-crystal diffractometer to image a weakly absorbing sample,namely a fusion pellet. Impressive application of the double-crystal technique to biologicalsamples was demonstrated by Ingal & Beliaevskaya [19]. Davis et al. [1] used a monolithic double-crystal arrangement to demonstrate key properties of the AB-PCI approach, for example contrastreversal with change in sign of the rocking angle for a phase object. Ingal & Beliaevskaya [19,20]described methods for combining images, whereas Chapman et al. [21] developed methods forseparating absorption and refraction information from AB-PCI images. This line of approachwas termed diffraction-enhanced imaging (DEI) [21]. Regimes of AB-PCI have been discussedby Gureyev & Wilkins [22].

By the nature of the method, each AB-based image is sensitive only to the phase-gradient component lying in the plane of the diffraction apart from propagation-based effects.Determination of the other phase-gradient component requires a separate measurement or theinvocation of appropriate boundary conditions (but this is often numerically unstable and tendsto lead to artefacts), see [23]. An approach for recording two phase-contrast images (but withthe same diffraction plane) simultaneously using a Laue-case analyser and extracting phaseinformation has been described by Kitchen et al. [24,25].

Although the AB-PCI method was first developed using conventional laboratory sources,AB-PCI techniques have been widely implemented at synchrotron sources, because such sourcesreadily provide high-intensity beams that are both quasi-monochromatic and quasi-planewavein character, and so are well matched to the acceptance properties of the monochromator. Ifconventional laboratory sources are used, the high degree of curvature of the wavefront is notwell matched to the use of flat crystals. Some improvement in angular acceptance of the system



5


.........................................................

can be achieved by the use of an asymmetric collimator–monochromator or potentially by the useof curved crystals.

Analyser-based imaging can yield very sensitive images, for example of soft tissue [26] andwith low background owing to high angular selectivity of scattering from the sample. However,such analyser-based imaging suffers from the following disadvantages and limitations in termsof widespread application:

— need for high mechanical stability of the crystals;— significant loss of intensity owing to filtering of energy spectrum by the monochromator.

This will become worse at higher E. This poses a significant limitation on the use ofconventional X-ray sources for AB-PCI;

— limited field of view (FoV) which is determined by the maximum practical size of crystalsand Bragg diffraction condition;

— spatial frequency bandpass is proportional to the rocking curve width of the analysercrystal [27] and decreases as E becomes large; and

— for moderate E, signal-to-noise ratio (SNR) and figure of merit (FoM) decrease only as1/E [28,29]. As E becomes large, the resolution of the system might be thought to degradeowing to increase in extinction, however this effect would appear to be compensated forowing to the change in projection on the image plane (see equation (4) in [23]; figure 1).

Application of AB-PCI methods for clinical medical diagnostics will need to overcome the variouslimitations indicated above. Some examples of significant progress in this area in the case ofapplication to mammography can be found in [26].

(c) Pursuit of methods of phase-contrast imaging that do not require monochromaticX-rays

In our own work, after exploring aspects of AB-PCI [1,30] and conscious of the limitations ofexploitation of AB-PCI with conventional sources, we sought other methods of PCI that did notnecessitate using essentially monochromatic X-rays. Among them was an aperture-based method[31] that involved measuring the deflection of X-ray beamlets on introduction of a sample intothe beam. Measurement of the distributions of beamlets (figure 2) recorded using a high spatialresolution detector offers the possibility of simultaneously measuring refractive, absorptive andscattering (SAXS) contributions made by the sample with essentially no strict requirement onmonochromaticity or involving a loss of photons after passing through the sample. Such animaging system has a requirement for highly collimated beamlets that might be produced usinga microfocus source. The approach could readily be incorporated in a conventional radiographic-type imaging system with the possibility of rapid switching from conventional to aperture-basedimaging mode. Requirements on energy bandpass are not very strict. We note that the refractionangle for the purely refractive contribution of the sample varies as 1/E2 (see equation (1.4))which would affect the peak profile but one might expect not to affect the peak displacementto first-order (see also refs [31–33]). Deconvolution of the beamlet profiles for the known energydistribution of the beamlets may be beneficial if the source is not essentially monochromatic.

Various groups have developed and demonstrated aperture-based methods of X-ray PCI,including some involving only a single exposure of the sample, and that show promise forpractical application in biomedical studies. These are discussed in more detail in the relevantsection below.

(d) Propagation-based hard X-ray phase-contrast imaging (PB-PCI)In the case of looking for early encounters with phase effects in (moderately hard) X-ray imaging,one might go back to the work of Cosslett & Nixon [34] on the projection X-ray microscope.Using a magnetically focused electron beam incident on a thin foil target as a microfocus X-ray



6


.........................................................

(a)

(c)

(b)

(d)

beam expander andcollimator crystal

sample

dual-faceanalyser

X-ray film

source

slit

distortedwavefront

Figure 1. (a) Diagram showing one variant of analyser-based imaging also showing (b) monolithic monochromator-analysersystem used, see [1]. (c) Comparison of a conventional-type radiograph and (d) an AB-PCI radiograph of a eucalyptus leafrecorded using the system with fixed analyser crystal shown in figure 1 [1]. Note that (d) is, in fact, a dark-field radiographowing to tilt of analyser away from the exact Bragg condition, because vein structure is bright on a dark background.

P

absorption contrast

shift

area

phase contrastinformation

phase contrastand homogeneityinformation

position

position

position

width

B'

D

0

A

H

BS

(a)

(b)

(c)

Figure 2. Schematic outline of an aperture-based method of phase-contrast imaging, showing possible types of informationthat could be measured (adapted from [31]).



7


.........................................................

softwood

Kevex microfocus source

hardwood

(a)

(c)

(b)

Figure 3. Some images recorded by A. Stevenson in 1993 of thin slices (of the order 100µm thick) of (a) softwood and(b) hardwood samples imaged using (c) a nominal 10µm spot size microfocus X-ray source (Kevex) with Cu target.

source, they observed Fresnel diffraction effects from samples, but were operating in the contextof shadow microscopy and such effects were basically regarded as a form of unwanted blurringin the image and essentially to be avoided.

In our case, the quest for methods of PCI that did not involve the need for highlymonochromatic radiation led us to more carefully analyse some experiments that we had carriedout on imaging thin wood samples (around 100 µm thick) using a low-power laboratory-basedmicrofocus source (Kevex, with nominal 10 µm spot size from a Cu target). We observed that theimages changed as a function of the distance, R2, between the sample and detector (X-ray film).Some examples of such images are illustrated in figure 3.

Although Fresnel diffraction seemed a possible explanation, the fact that we had an essentiallypolychromatic source seemed at first to point away from this explanation. However, furtherexperiments with a variety of samples and varying the source–sample and sample–detectordistance combined with theoretical considerations confirmed the basic explanation in terms ofFresnel diffraction [35,36], especially when imaging weakly absorbing objects.

Key features of the Fresnel-diffraction-based approach to propagation-based PCI can be seenfrom the expression below for the intensity in the detection plane for diffraction from a pure phaseobject, namely

I(Mx, My; R1 + R2; k) = Io

M2

1 + R2

kM∇2

⊥φ(x, y; R1, k)

= Io

M2

1 − 2π

ro

k2R2

M∇2

⊥∫R1

−∞ρ(x, y, z)dz

, (2.2)



8


.........................................................

0.5

partially coherentCTF

Pt source CTF

resolution(source size)

0

cont

rast

tran

sfer

fun

ctio

n (C

TF)

–0.5

–1.0

0.5

1.0resolution

(source size) partially coherentCTF phase

contrastCTFfor perfectcoherence

1.0 1.5÷(lz)u

Figure 4. Contrast transfer function for the contribution owing to δ (purely refractive) in PB-PCI for perfect coherence (redcurve) and for finite source size (blue curve). The green curve shows loss of features owing to finite source size. Based on [37].

where R1 is the source-object distance and M = (R1 + R2)/R1 is the geometrical magnification.Expression (2.2) can be derived from Fresnel diffraction in the paraxial approximation under aweak-object assumption (see [37]) or from the transport of intensity equation (TIE)

−k∂

∂zI(r⊥, z) = ∇⊥ · (I(r⊥, z)∇⊥φ(r⊥, z))

≡ ∇⊥I(r⊥, z) · ∇⊥φ(r⊥) + I(r⊥, z)∇2⊥φ(r⊥, z), (2.3)

see, for example, references [2,38].From expression (2.2), one can readily see that

— essentially pure phase objects will show no contrast in contact radiography mode but canshow significant phase contrast, especially at sharp boundaries, in projection mode; and

— image structure in the near-field regime relevant to (2.2) is essentially independent ofenergy bandpass of the incident beam. This helpful and somewhat surprising findingmakes PCI readily achievable, e.g. even with polychromatic laboratory microfocussources (say E/E ≤ 30%), in a certain regime [37].

Importantly, the magnitude of phase-contrast is strongly determined by the lateral spatialcoherence length of the incident wave in the object plane, namely

lx,y = λR1

2.355σx,y(src). (2.4)

For significant phase contrast to occur, lx,y needs to be comparable to or larger than the inversespatial frequency (1/ufeature) of the feature size of interest, see [37] and also figure 4 (CTF).

Related work for propagation-based imaging was published independently in the contextof the synchrotron case by Snigirev et al. [39], using monochromatic radiation and a highlycollimated parallel beam.



9


.........................................................

(e) Some key features of PB-PCIThe resolution for a propagation-based PCI system at the object plane is given by [40,41]

σ 2sys ≈ (1 − 1/M)2σ 2

src + M−2σ 2det + σ 2

min, (2.5)

where σ 2sys is the variance of the system point-spread function (assumed Gaussian) and σ 2

src andσ 2

det are the variances of the source distribution and the detector spatial resolution, respectively.The diffraction term, with σmin = 1/2(λR2/M)1/2, becomes significant beyond the first Fresnelzone but less significant for a given geometry as E becomes larger (cf. AB-PCI).

For a pure phase object, the contrast transfer function is illustrated in figure 4 and shows theloss of phase-contrast information both for large and small ξ = u

√(λz). For small ξ , it is due to

the intrinsic nature of propagation-based phase contrast and for large ξ it is due to geometricalblurring. The quantity 1/u is the feature size of interest and ξ < 2−1/2 corresponds to the near-fieldregime. Equation (2.5) also forms the basis for geometrical resolution in other methods discussedbelow (see [28]).

(f) Wavefront sensing and coded aperture methodsWavefront sensing in visible light optics may be traced back to the work of Hartmann at the turnof the twentieth century using regularly spaced apertures in an opaque material and improvedupon by Shack using lenslet arrays and an imaging detector [42].

Various groups have recently developed and demonstrated coded aperture-based methodsfor hard X-ray PCI that show considerable promise for practical application in biomedical studiesand potentially in clinical medical studies if suitable laboratory-based sources become available.

In particular, the work of the following groups should be mentioned in this connection:

(1) Olivo and co-workers [43–45] using two matched coded apertures one just before thesample and one just in front of the detector (or a high spatial resolution detector withsharp pixel boundaries). For each component of the phase gradient, two images arerequired.

(2) Morgan et al. [46–48], who have used a single two-dimensional coded aperture in frontof the sample combined with a high spatial resolution detector and correlation analysisof shifts in intensity in small regions of images taken with and without the sample in thebeam, in order to simultaneously determine two components of the phase gradient. Seealso Wen et al. [49] for treatments of this general approach.

(3) Krejci et al. [32,33,42,50], who have used subpixel resolution analysis with a micrometre-scale source and a high spatial resolution detector to determine very small displacementsin two-dimensions of essentially rectangular profile X-ray beamlets to obtain bothcomponents of the phase gradient simultaneously from a single image. This has includedstudies at 70 kVp [32] (figure 5).

(4) Mayo & Sexton [51] have used lenslet arrays to carry out Shack–Hartman-type wavefrontanalysis for less than 10 keV X-rays and this would also appear to be of interest for hardX-rays if suitable lenslet arrays can be produced.

All the coded aperture methods mentioned above have the potential to operate with fairlybroadband radiation, which can be a considerable advantage for application with conventionalsources. Methods (2) and (3), that involve only a single-coded aperture, appear to have stricterrequirements on source size than method (1). In a general sense, single-coded aperture methodsmight be viewed as an extension of PB-PCI in that the combination of sample and coded apertureplaced immediately in front of or behind the object may be effectively regarded as a single object.

However, one should note that for highly absorbing coded apertures, the further assumptionoften made that the intensity at the exit from the combined aperture-object is slowly varying islikely not to be valid. This means that simple theoretical treatments of wave propagation and



10


.........................................................

I1 I2

I3

I4

Figure 5. Schematic of beamlet displacementmeasurement using pixel-array detector as outlined in [32,50]with the capabilityfor subpixel resolution. Light-shaded area is beamlet distribution before insertion of sample and solid-shaded area is centralbeamlet distribution after insertion of sample.

phase retrieval for these cases may not be valid, e.g. the first term on the RHS of the expressionfor the TIE given in equation (2.3) cannot be ignored and may, in fact, be helpful in leading toamplification of phase contrast owing to such sharp features [52].

3. Grating-interferometer-based phase-contrast imagingGrating-interferometer-based (GIB) methods of imaging contain elements of single-crystalinterferometry, AB-PCI and PB-PCI. Early discussion of possible implementation of GIB methodsfor X-rays can be found in the work of Clauser [53,54], whereas practical implementationmay be traced to the work of David et al. [55], who used a two-grating system plus analysercrystal (to filter out unwanted diffraction orders) and also to Momose et al. [56], who initiatedimplementation of a Talbot-grating-based (TGB) interferometer with two gratings alone andalso demonstrated measurement of the phase gradient for a simple object. A detailed review ofdevelopments in TGB methods for X-rays can be found in a recent article by Pfeiffer [57].

A detailed consideration of the theoretical performance of grating-based imaging systemsis given in [29,58]. There are significant requirements on spatial coherence relative to the firstgrating and lesser requirements on beam monochromaticity. Introduction of a linear aperturearray in close proximity to the source (Talbot–Lau case) can increase the intensity in the imagein proportion to the number of apertures but is at the expense of spatial resolution, becausespatial resolution is determined by the width of the array, not the individual effective sizeof the sourcelets. Spatial resolution is also determined by the period of the gratings and/ordetector resolution.

In principle, if a high spatial resolution detector (better than the Talbot fringe spacing) isavailable, only one grating, namely the phase grating, is required. This approach to grating-interferometer-based phase-contrast imaging (GIB-PCI) has been demonstrated by Takedaet al. [59].

Extension of the Talbot-grating-interferometer approach to the simultaneous determination ofthe two components of phase gradient using two-dimensional gratings has been described byItoh and colleagues [60,61] and by Wu et al. [52].

4. Summary of key features of the five methods of phase-contrast imagingIn table 1, we try to summarize some of the key system performance requirements and limitationsfor the five different classes of PCI methods considered here. In preparing this, we have



11


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Table1.Com

parisonofbasicsystemrequirementsfordifferenthardX-rayphase-contrastimaging

methods.

phase-contrast

chromaticcoherence

spatialcoherenceofsource

commentsandkeyissuesforapplication

imaging

requirementsand

requirementsandlimitson

quantity

withconventionalsources.

technique

optics

performanceathighE

resolution

sensed

alsosom

ekeyreferences

1.interferom

etryusing

single

crystals(IB-PCI)

monolithicsingle

crystal

first-crystalwaferacts

asa

monochrom

ator/beam

splitter.Involves

single-crystal-type

bandpass

resolution

islimitedbysource

size/beam

collim

ation

and

extinctionlength.

φ‘phaseunwrapping’problemneedstobetreatedasmeasured

phaseism

odulo

2πsingle-crystalsplitter/m

onochrom

atorhaslow

photonthroughput

andFoVlimitedbyprojectedsize

ofcrystalin

theplaneof

diffraction[12–15]

.............................................................................................................................................................................................................................................................................................................................................

2.analyser-based

(AB-PCI/DEI)

single-crystal

monochrom

ator

andanalyser

firstcrystalacts

asa

monochrom

ator/colli

mator.Involves

single-crystal-type

bandpass

resolution

islimitedbysame

factorsasPB-PCIandalso

inplaneofdiff

ractionby

extinctionlengthinanalyser

crystal,see[27–29]

∇ xφ

onlythecom

ponentofthephasegradientintheplaneof

diffractionisdirectlyderivedfromasinglemeasurement

(heredenoted‘x’)

notwellmatchedtoa‘pointsource’=

>smallFoV(inplaneof

diffraction)andimagedistortioninout-of-diffraction-plane

direction

requiresvariousadditionalassumptionsonformofrocking

curve

andnatureofwavefrontforphaseretrieval[1,19–21,27,30]

.............................................................................................................................................................................................................................................................................................................................................

3.propagation-based

(PB-PCI)

noopticsneeded

essentially,nobandpass

restriction

innear-field

regim

e

highspatialcoherencerequired.

Resolution

limitedbysource

size(M

1)anddetector

resolution

(M≈1).Alsoby

diffraction,unlessinthe

near-field.See[40]

∇2φ

canyieldtwo-dimensionalphasemapdirectlyfrommeasured

imageorim

ages.Essentiallyenergybandpassindependentin

near-field

regim

e.No

needforoptics(therefore,low

on

aberrations).CanprovidelargeFoV

sensitivetosharp/smallfeaturesbutnotverysensitive

tobroad

features[29]

resolution

limitedbysource

sizeordetectorresolutionandby

diffractionatveryhighspatialresolution

[31,35,37,39–41,62]

.............................................................................................................................................................................................................................................................................................................................................

(Continued.)



12


.........................................................

Table1.(Continued.)

phase-contrast

chromaticcoherence



imaging

requirementsand


quantity


technique

optics

performanceathighE

resolution

sensed

alsosom

ekeyreferences

4Shack–Hartm

ann/coded

aperturemethodsin

non-interferom

etric

mode

canoperatewithbroad

bandpass

resolution

limitedbysize

and

spacing

ofaperturesintheCA

array

requirehighspatialresolution

detector

.............................................................................................................................................................................................................................................................................................................................................

spatialresolution

isultimately

limitedbydetectorresolution(or

sourcesize),i.e.expression(2.5)

(i)codedaperturemethods

involving

e.g.two

two-dim

ensional

gratings

twom

atchedcoded

apertures

smallsourcesizerequiredtogive

well-definedbeamlets

∇ xφ

onlymeasureonecom

ponentofphasegradientatatimeand

requirestwo

imagesofobjectforeachcomponent[43–45,63]

.............................................................................................................................................................................................................................................................................................................................................

(ii)singleabsorbergrating

justpriortotheobject

anduseofsingle

image

ofobject

single-codedaperture

orsuitable

opticto

modulatethe

wavefront

highspatialcoherencerequired

(microfocussource)

∇ x,yφ

Morganetal.[46–48]involving

methodofanalysisvia

autocorrelation

functionevaluationforintensitym

easuredw

ith

andw

ithoutsample,typicallyforone‘cell’oftheimage.Can

userandomphaseCA(e.g.paper).Spatialresolution

limitedby

areausedinautocorrelation

analysis

Seealso

Wenetal.foraFouriersatelliteapproachtoanalysis[49]

.............................................................................................................................................................................................................................................................................................................................................

(iii)subpixelresolution

(Krejciandcolleagues)

single-codedaperture

array(withsharp

boundaries)

matchedtodetector

pixelspacings

micro-/nanofocussourceneededto

givew

ell-definedbeamlets.

Subpixelresolution

possible

frommultipleimages

∇ x,yφ

involvesuseoflow-noisehighspatialresolution

pixel-array

detectortogive

subpixelresolution

viasamplingofbeamlet

acrossmultiplepixels

Beamletsneedtohavesharplydefinedprofiles

Yields

2com

ponentsofphasegradientsimultaneouslyfroma

single

imageoftheobject[31–33,50,64]

.............................................................................................................................................................................................................................................................................................................................................

(iv)usinglensletarrays

lensletarray

highspatialresolution

required

∇ x,yφ

canimproveSNR

butrequiressuitable

X-raylenses.E.g.Mayo&

Sexton[51]

.............................................................................................................................................................................................................................................................................................................................................

(Continued.)



13


.........................................................

Table1.(Continued.)

phase-contrast

chromaticcoherence



imaging

requirementsand


quantity


technique

optics

performanceathighE

resolution

sensed

alsosom

ekeyreferences

5.grating-based

Interferometery

(GIB-PCI)

phasegratingadjacent

toobjectand

possibly

oneor

moreothergratings

depending

on

sourcesizeand

detectorresolution

basicmethodisnotvery

sensitiveto

E/Ebut

willinvolvedispersion

andlossofperformance

owing

todispersionby

sample

requiresahighdegreeofspatial

coherence.Forbroadsource,

canbeobtainedbyuseof

source-grating,butthis

reduces

spatialresolution.Spatial

resolution

limitedbysame

expressionasforPB-PCIand/or

secondgrating

period

∇ xφ

usuallytreatedingeom

etricalopticsapproxim

ation.Systemitself

isnotverysensitive

toE/E,exceptfordispersioneffects

inducedbythesample.Also

bright-anddark-fieldmodesof

imaging

possible

givenmultipleimagesforstepped2nd

grating.See[56,65,66]

.............................................................................................................................................................................................................................................................................................................................................

Resolution

effectively

limitedbyexpression(2.5),wheresourcesize

istotalsource

sizeintheTalbot–Laucase

useofoneormoregratings

plushigh

spatial

resolution

detectorto

resolve

Talbotimages

∇ xφ

single

grating

methods,seeTakedaetal.[59].

Thisareaoverlapsw

ithsingle

opticCAmethods[32,46,48,50]

.............................................................................................................................................................................................................................................................................................................................................

extension

to

two-dim

ensional

gratings

∇ x,yφ

requireshigh

spatialresolution

detector[60,61]

.............................................................................................................................................................................................................................................................................................................................................



14


.........................................................

particularly drawn on references [28,29,58], in addition to specific references mentioned elsewherein this review.

Some general considerations relating to the performance of the various PCI methods areas follows:

— while some methods may be a form of incoherent imaging, a key parameter is angularsource size viewed from the sample. When comparing different systems operating at thesame energy, this quantity corresponds to a measure of lateral spatial coherence length;

— determination of a single component of the phase gradient does not unambiguouslylead to phase retrieval in principle, unless the object is isolated in the directionperpendicular to the phase-gradient component measured. Even then, retrieval of thephase distribution, φ, may be impractical due to artefacts introduced in the data analysisvia noise;

— if broadband polychromatic radiation is used, then dispersion by the sample may affectthe ability to determine the phase gradient;

— single-image methods are highly desirable, if the sample is not stationary in time or if thesource of illumination is not stationary;

— avoidance of apertures after the sample, where possible, is desirable in order to minimizedose and maximize information gathered. New types of two-dimensional pixel-arraydetectors offer exciting possibilities for practical application of coded aperture methodsthat can provide subpixel resolution information with little crosstalk between pixels andeven possibly energy analysis; and

— coded aperture methods can potentially simultaneously provide phase gradient,absorption and scattering (dark field) images from a single image.

5. Optimization of performance and comparative studiesA useful theoretical measure of the relative performance of different PCI methods at a givenphoton energy (or energy spectrum) is SNR for a given sample, feature and dose to the sample aswell as given characteristics for source size and detector. A useful FoM in this regard is [28,29]

FoM = SNRD1/2 , (5.1)

where the SNR is based on the number of photons detected, D is the dose at the entry surfaceof the sample. This essentially quantifies the statistically significant information derived about agiven feature per unit dose and allows useful comparison of quality of the information providedby different X-ray imaging methods.

While contrast afforded by a method is important, it might be noted that contrast is only oneindication of performance, e.g. observing one photon in an experiment constitutes huge contrast,but provides essentially no information.

A theoretical comparison of performance parameters for AB-PCI, PB-PCI and GIB-PCImethods for a range of instrumental parameters and model samples has recently been publishedby Diemoz et al. [28] and also together with some related experimental studies [29], providingexpressions and some experimental results for FoMs for each of the above methods. Of particularinterest are the results for the variation of the FoM of the various methods with photon energy,E, which were presented by these workers, namely, PB-PCI (1/E2), AB-PCI (1/E) and GIB-PCI(1/E2). These results all assume a certain fluence (photons/area), Io, incident on the sample suchthat dose, D= KdoseIo, where Kdose depends on sample properties and E.

In considering the relative merits of different PCI methods for a particular application andfeature type, as well as a particular X-ray source, the following cases might typically arise:

(1) the selected imaging method and desired SRN is dose limited, or(2) it is exposure-time (i.e. X-ray source power) limited.



15


.........................................................

If case (1) applies, then expression (5.1) is a useful practical measure of performance of a method.However, if case (2) applies, then the relative efficiency of the different methods and their specificconfigurations in using a given source becomes relevant.

To help to give practical guidance in regard to the relative merits of different methods andconfigurations for case (2), i.e. when they are exposure-time limited, we here introduce anadditional figure-of-merit-type quantity, namely the efficiency

Eff(method; source) = SNR

time1/2 = FoM × dose rate1/2, (5.2)

which may be viewed as the rate of information collection for a given PCI method andexperimental implementation. For present purposes, a crude approach to describing the factorsaffecting the dose rate might be expressed by

dose rate(method; source) ∝ EE

× B × Sarea used × f × 1n

, (5.3)

where B is the brightness of the source (photons mm−2 s−1 per 0.1% bandpass), Sarea used isthe area of the source that contributes to the dose at the entry plane to the sample, E is theenergy bandpass of the imaging system for the given source, f is the fraction of photons that aretransmitted to the entry plane of the object (e.g. after losses owing to apertures and possible gainsby focusing effects), whereas n is the number of images required to obtain the final image.

On this basis, we note, for example, that while AB-PCI would appear to have the best trendfor an FoM given by equation (5.1) as E becomes large, E/E for AB-PCI will vary roughly as 1/Eowing to reduced energy bandpass of the monochromator for AB-PCI, but not vary much with Efor the other methods considered here.

It should also be remarked that any comparison of relative performance of different methodsinevitably involves the selection of a model sample, and so any comparison of methods wouldbe, to a greater or lesser extent, sample-specific. For detailed comparison of specific methods,configurations, model samples and operating conditions, computer simulation can provide a veryconvenient, efficient and flexible approach, and is the one that we and others have used widely.

6. Phase retrieval and omnimicroscopyThe various methods of PCI mentioned above become unified when one includes quantitativephase retrieval as an outcome of the investigations. This has now become a major focus ofactivity in the X-ray imaging area and extends to phase-contrast computerized X-ray tomography.Basically, it involves inverting the diffraction problem to determine the exit wave function forthe object and thereby both the absorption and projected electron density, via (1.3). In the caseof propagation-based imaging for X-rays, the earliest demonstration lies in the work of Nugentet al. [67] and was based on solution of the TIE (see equation (2.3)). There are many variants ofthe approach to phase retrieval for PB-PCI including using images at two or more distances, e.g.holotomography (see Cloetens et al. [68]) and TIE-based [69], multi-energy phase retrieval [70]and the single-homogeneous material case [71] that is widely used.

For methods of PCI that measure only a single gradient of the phase, phase retrieval, whiletheoretically possible for objects isolated in at least one direction, can involve instabilities andartefacts as mentioned earlier. Given that methods are becoming available that can determineboth gradients of the phase simultaneously at each point in the image, these would seem to bepreferable.

Detailed discussion of phase retrieval methods lies outside the scope of this review. However, itis perhaps useful to note the unifying role that phase/amplitude determination plays with regardto the different PCI methods which is outlined in the article by Paganin et al. on ‘Omnimicroscopy’[72], where the determination of the phase and amplitude modulation at the exit surface of theobject by one method in principle allows the generation of synthetic images for any of the otherPCI methods.



16


.........................................................

7. ConclusionThe field of hard X-ray PCI is rapidly developing on many fronts. This review has tried to givea brief overview of the development and key features of the main classes of methods, includingsome, which are at an emerging stage and which appear to have some desirable features. Amongdesirable features for widespread practical application in general are ability to be readily andbe efficiently implemented on conventional sources, ability to record complete information in asingle image/exposure, mechanical simplicity and stability as well as moderate cost.

Practical application of PCI to static objects using conventional sources is already welldeveloped. However, a major limitation at the present time in studies of dynamic systems andespecially in vivo studies is the low brilliance of conventional/laboratory sources compared withsynchrotrons. Using synchrotron sources, PCI methods have found widespread application to invivo biomedical and also some clinical medical areas, but transition to in vivo laboratory-/clinic-based applications has been slow. Notwithstanding, some promising areas for early applicationof laboratory-based PCI in the biomedical and medical context would appear to be where doseand/or exposure time is not important, e.g. biopsies and where a small FoV is of interest, e.g.region of interest diagnostics and where suitable high spatial resolution detectors may already beavailable. Application to a wider range of clinical and biomedical studies awaits the developmentof higher performance laboratory X-ray sources and large-area high-performance detectors suitedto the implementation of hard X-ray PCI methods mentioned here. These are technologies thatare undergoing rapid advances at the present time, which bodes well for widespread practicalapplication of PCI methods in the near future.

In conclusion, we might recall the words of Fritz Zernike in the context of phase-contrastoptical microscopy, namely that “I went in 1932 to the Zeiss Works in Jena to demonstrate. It wasnot received with such enthusiasm as I had expected. Worst of all was one of the oldest scientificassociates, who said ‘If this had any practical value, we would ourselves have invented it longago’ ” with the hope that this barrier is not encountered in applications of X-ray PCI methods.

Acknowledgements. We acknowledge a wide range of collaborators with whom we have worked. In many cases,their names are included in the common author lists of the references cited in this review. We also thank MrsJulie McInerney for her help with compiling the bibliography.

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http://dx.doi.org/doi:10.1364/OL.36.003551



http://dx.doi.org/doi:10.1088/0031-9155/54/15/n02

http://dx.doi.org/doi:10.1088/1748-0221/6/01/c01073

http://dx.doi.org/doi:10.1038/nphys265

http://dx.doi.org/doi:10.1364/opex.13.006296

http://dx.doi.org/doi:10.1364/opex.13.006296

http://dx.doi.org/doi:10.1103/PhysRevLett.77.2961


http://dx.doi.org/doi:10.1364/josaa.15.000579

http://dx.doi.org/doi:10.1016/S0030-4018(97)00637-8

http://dx.doi.org/doi:10.1046/j.1365-2818.2002.01010.x

http://dx.doi.org/doi:10.1111/j.0022-2720.2004.01315.x


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