PPhase-sensitive x-ray ghost imaging
Margie P. Olbinado ∗ The European Synchrotron – ESRF, CS40220, 38043 Grenoble, France
David M. Paganin
School of Physics and Astronomy, Monash University, Victoria 3800, Australia
Yin Cheng and Alexander Rack
The European Synchrotron – ESRF, CS40220, 38043 Grenoble, France (Dated: March 26, 2019)Imaging with hard x-rays is an invaluable tool in medicine, biology, materials science, and culturalheritage. Propagation-based x-ray phase-contrast imaging [1–3] and tomography have been mostlyused to resolve micrometer-scale structures inside weakly absorbing objects as well as inside densespecimens. Indirect x-ray detection has been the key technology to achieve up to sub-micrometerspatial resolutions [4], albeit inefficiently and hence at the expense of increased radiation doseto the specimen. A promising approach to low-dose imaging and high spatial resolution even athigh x-ray energies is ghost imaging [5–11], which could use single-pixel, yet efficient direct x-raydetectors made of high-density materials. However, phase contrast has not yet been realised withx-ray ghost imaging. We present an approach which exploits both the advantages of x-ray ghostimaging and the high sensitivity of phase-contrast imaging. In comparison with existing techniques,our method is efficient and achieves high-fidelity x-ray ghost images with phase contrast, accuratedensity resolution and dramatically higher spatial resolution. The method is scalable to practicaltomography with large fields of view, micrometer spatial resolution, and with high-energy x-raysabove 100 keV. It is also applicable to other phase-sensitive imaging techniques [12–15] and withother probes such as neutrons, alpha rays, and muons, for which high spatial resolution detectorsare limited or even not available.X-ray ghost imaging [5–11] is a newly developed imagingtechnique, derived from visible light optics [16–19], which hasthe potential to achieve ultra-low radiation dose imaging andhigh spatial resolution. It utilises optical correlations betweenspatially resolved photons that never pass through an ob-ject of interest and non-spatially resolved photons that dopass through the object. Similarly to classical ghost imaging[18, 19], experimental x-ray ghost imaging [5–10] has beenrealised with speckled illumination. This has been done byusing intrinsic noise of a synchrotron x-ray source [5], andby phase-contrast-generated [6–9] or attenuation-contrast-generated [10] speckle patterns. A ghost image is retrievedfrom intensity correlations between a series of speckle fieldsthat illuminate an object and the total intensities transmit-ted by the object. Remarkably, the photons passing throughthe object are detected using only a so-called single-pixel‘bucket’ detector. Since single-pixel, high-Z (atomic number),direct x-ray detectors are significantly more efficient than two-dimensional (2D) indirect x-ray detectors commonly used inx-ray phase-contrast imaging, the potential reduction of theradiation dose to the specimens is highly appealing [5, 10].The efficiency of indirect detectors is worse at high x-ray en-ergies. Consequently, resolutions of only several micrometershave been achieved above 100 keV. In contrast, since the spa-tial resolution of a ghost image is determined by the specklesize and not by the detector [8, 20], ghost imaging may achievethe much-coveted high spatial resolution with high x-ray en-ergies. For example, large, single-pixel detectors made withthe high-Z material CdTe have a quantum efficiency close to100% up to or even above 100 keV x-ray energy. ∗ [email protected]; Corresponding author In essence, classical ghost imaging expresses the spatial dis-tribution of an object’s transmission function as a linear com-bination of speckle fields [18, 19]. Each speckle field is consid-ered as a linearly independent basis vector. This is concep-tually similar to building functions as a sum of sinusoids in aFourier series. A classical ghost imaging setup is composed ofa beam path (or reference arm) in which the speckle fields aredetected by a 2D detector, and a beam path (or object arm)in which the object’s transmitted intensities are detected bya bucket detector. In computational x-ray ghost imaging, thereference images are pre-determined or pre-characterised.Figure 1a shows a schematic diagram of a computationalx-ray ghost imaging setup using speckle-generating masks,which are either x-ray absorbing or phase-shifting. A seriesof speckled illuminating fields is obtained by raster scanningthe mask in the transverse plane.Let I in be a uniform, incident illumination intensity andˆ T M ,j ( r ⊥ , z = R M ) be the transmission function of the j thspeckle mask (M) at a mask-to-sample propagation distance R M . Here, r ⊥ = ( x, y ) denotes coordinates in the planesperpendicular to the optical axis z . The j th illumination in-tensity onto the sample is I j ( r ⊥ ) = ˆ T M ,j ( r ⊥ , z = R M ) I in . (1)Letting ˆ T S ( r ⊥ , z = R S ) be the transmission function of thesample (S) at the sample-to-bucket detector distance R S , thesignal collected by the bucket detector may be written as: b j = (cid:90) (cid:90) Ω ˆ T S ( r ⊥ , z = R S ) I j ( r ⊥ ) d r ⊥ , (2)where Ω is the surface over which the beam intensity isrecorded and within which the sample is entirely contained. a r X i v : . [ ee ss . I V ] M a r FIG. 1.
Computational x-ray ghost imaging (XGI). a , A conventional XGI set-up with a structured illumination approachusing speckles. Intensity correlations between a series of known illuminating speckle fields (reference images) and the corresponding totalintensity transmitted by the sample (bucket signals) collected by a bucket detector are utilised to synthesise an attenuation-contrast x-rayghost image. b , Our phase-contrast XGI experimental set-up with a structured detection approach using periodic structures instead ofspeckles. By interchanging the mask and the sample in the sequence, the bucket signal is sensitive to x-ray phase shifts from the sampleand a phase-contrast x-ray ghost image can be synthesised. 1D gratings are used as masks in combination with a 1D bucket detector,which is a collection of ‘mailbox’ detectors. The ghost image reconstruction formula is applied for each mailbox detector ( c ). d , Referencex-ray images of N = 10 grating patterns with 1 to 10 lines per mm. e , Calculated point-spread-function PSF( x, x (cid:48) ) of the phase-contrastx-ray ghost image without the sample for one of the mailbox detectors ( l = 1 mm, N = 10). The spatial resolution (FWHM of the PSF)is 10 × the mailbox detector length. The hat ( ˆ ) indicates that the transmission function is anoperator and that the order of operation is crucial.Using N speckle masks, an x-ray ghost image may be syn-thesised using [18, 19]: G ( r ⊥ ) = 1 N N (cid:88) j =1 ( b j − ¯ b ) I j ( r ⊥ ) . (3)The bucket signal b j subtracted by the mean ¯ b acts as aweighting coefficient for the corresponding reference specklefield I j in the superposition. Interestingly, even though nei-ther of the detectors used in the measurement yields an imageof the object, and though no photons passing through the ob-ject are ever registered by a position-sensitive detector, animage can be obtained by harnessing intensity correlations .This scheme, named classical ghost imaging with x-rays[5, 7–10] only retrieves the attenuation-contrast component ofthe object’s x-ray transmission image independent of R S , thuslosing the much-enhanced contrast or sensitivity to density variations in the sample that could be obtained by exploitingpropagation-induced x-ray phase contrast. This inability toexploit phase contrast is the case with the usual mask–samplesequence. Here we show that the key to achieve phase contrastin x-ray ghost imaging is to reverse the sequence of sampleand mask: a structured detection approach [21–23] instead ofa structured illumination approach.The origin of contrast in an x-ray image is related to theobject’s complex refractive index: n ( r ⊥ , z, λ ) = 1 − δ ( r ⊥ , z, λ ) + iβ ( r ⊥ , z, λ ) . (4)For simplicity, consider a homogeneous object of projectedthickness t ( r ⊥ ), and quasi-monochromatic, plane-wave x-rayradiation with wavelength λ and initial intensity I in . Underthe projection approximation, the spatial variations of theoptical density, D ( r ⊥ ) = 4 πλ βt ( r ⊥ ) (5) FIG. 2.
X-ray transmission images of interconnected aluminium lamellae cut from a sponge sample. a , Directx-ray phase-contrast image ( I S /I flat field ): calculated from radiographs directly recorded using a 2D imaging detector. b , Ghostx-ray phase-contrast image ( G S /G flat field ): calculated from synthesised ghost images both with the sample ( G S ) and without( G flat field ). Scale bars in a and b represent 1 mm. c , Horizontal line profiles. d , Vertical line profiles. e, f Magnified views ofthe insets in a and b showing phase-contrast enhancement at the edges of representative thick and thin region of the lamellae.Scale bars in e and f represent 250 µ m.and the x-ray phase shift, φ ( r ⊥ ) = − πλ δt ( r ⊥ ) (6)generate the attenuation contrast and phase contrast in the x-ray image, respectively [24]. In the hard x-ray regime, phase-contrast imaging is up to three orders of magnitude moresensitive than attenuation-contrast imaging. This is becauseabsorption decreases with the fourth power of photon energywhile phase contrast decreases with the square of the energy.A phase-contrast image is readily achieved through free-space propagation or Fresnel diffraction. At an object-to-detector propagation distance R ( R ≤ d λ , where d is thecharacteristic length scale over which the object appreciably changes), a near-field phase-contrast image or Fresnel imagemay be derived from the so-called transport-of-intensity equa-tion [25] and is given by [24]:ˆ T ( r ⊥ , z = R ) I in = (cid:18) − Rδµ ∇ ⊥ + 1 (cid:19) e − µt ( r ⊥ ) I in , (7)where the linear attenuation coefficient µ = 4 πβ/λ and ∇ ⊥ is the gradient operator in the x - y plane. The transmis-sion function contains both the the attenuation contrast andpropagation-induced phase-contrast components. By conser-vation of energy, the average intensity of the Fresnel imageover the entire area A of the surface Ω( r ⊥ ), at any distance R is equal to that of the contact image ( R = 0).By inserting Eqn. 7 in Eqn. 2 and invoking the conservation of energy, the resulting bucket signal is given by: b j = (cid:90) (cid:90) Ω (cid:18) − R S δ S µ S ∇ ⊥ + 1 (cid:19) e − µ S t S ( r ⊥ ) I j ( r ⊥ ) d r ⊥ = (cid:90) (cid:90) Ω e − µ S t S ( r ⊥ ) I j ( r ⊥ ) d r ⊥ . (8)Therefore, with the usual setup, where the mask is upstreamof the sample, the bucket signal (Eqn. 8) is not sensitive to thepropagation-induced phase-contrast component of ˆ T S . Onlythe attenuation-contrast component exp[ − µ S t S ( r ⊥ )] is recov-ered in the ghost image.The key to achieving phase contrast in x-ray ghost imagingis to interchange the sequence of the sample and the mask inthe beam. The mask is placed at the desired phase-contrastimage plane, a distance R S downstream from the sample. Thebucket signals, which measure the weighting coefficients of thesuperposition in the ghost image reconstruction (Eqn. 3), donot register a constant signal that is insensitive to phase con-trast. We emphasise the crucial order of the sample and themask by the order of operation of the transmission functionin the bucket signal equation, which we write as: b j = (cid:90) (cid:90) Ω ˆ T M ( r ⊥ , z = R M ) ˆ T S ( r ⊥ , z = R S ) I in ( r ⊥ ) d r ⊥ . (9)In this way, the bucket signal is sensitive to the phase-contrast component of the sample’s transmission image,ˆ T S ( r ⊥ , z = R S ) I in ( r ⊥ ). Notice also that depending on R S ,a phase-contrast x-ray ghost image in the Fresnel (near-field)or Fraunhofer (far-field) regime can be synthesised. Further-more, the ghost image synthesis can even be extended to anyx-ray phase-contrast imaging approaches such as Talbot in-terferometry [12, 13], and near-field speckle-tracking [14, 15].This approach is also compatible with ghost imaging com-bined with x-ray diffraction topography and crystallography.Simulations have indeed shown that our detection approach isa means to achieve analyser-based x-ray phase-contrast ghostimaging [26]. Equation 9 also clearly indicates that only theattenuation-contrast component of ˆ T M matters, hence ampli-tude masks should be used.Our setup for x-ray phase-contrast ghost imaging is de-picted in Fig. 1b. Instead of using speckle patterns, whichform a non-orthogonal basis and require N (cid:29) p measure-ments in order to synthesise a ghost image consisting of p pixels, we employ a linear combination of periodic fields withvarying frequencies, which form a nearly-orthogonal set of ba-sis patterns. This eliminates the inherent redundancy of thespeckle-based approach. We implemented this using trans-mission gratings, which is practical since the fabrication ofsuch gratings for hard x-rays is well-established. For exam-ple, high-aspect-ratio gratings with 2 µ m pitch and 160 µ mthickness used for up to 180 keV x-ray energy have been re-ported [27]. Deterministic orthogonal basis patterns such asthe Hadamard and Fourier basis patterns [21–23] have beenused instead of non-orthogonal random patterns or speck-les. Compressive sensing concepts [18], orthogonalisation ofspeckle fields [8] and iterative refinement [9] may also be em-ployed to reduce N .We opted not to use 2D gratings. Instead, we consideredthat a combination of 1D gratings with a 1D bucket detectoris equivalent to using 2D gratings with a single-pixel (0D)detector. A 1D bucket detector constitutes a set of what wecall mailbox detectors. Recently, high-Z, 2D direct detectorssuch as the EIGER2 CdTe (Dectris Ltd., Switzerland) with pixel size of 75 µ m have become commercially available. Linearrays or even pixels of such 2D detectors may be used asmailbox detectors. The ghost image reconstruction formulais applied for each mailbox detector (Fig. 1c). The advantagesare two-fold: (1) N is reduced by a factor equal to the numberof mailbox detectors used. (2) The spatial resolution may betuned independently in two directions: one with the smallestgrating line width w , and the other with the mailbox detectorheight h .The position-dependent point-spread function (PSF) of anx-ray ghost image without sample was calculated for a givenmailbox detector in order to check whether a set of gratings(Fig. 1d) constitute a complete set of basis elements. Thecorresponding completeness relation [8, 20] that needs to besatisfied is given by:PSF( x − x (cid:48) ) = 1 N N (cid:88) j =1 [ I j ( x (cid:48) ) − ¯ I ][ I j ( x ) − ¯ I ] , (10)where ¯ I is the average intensity of the j th illumination in-stead of an average over all illuminations and x runs overpixels equal to the mailbox detector length l . By way of ex-ample we show the calculated PSF( x − x (cid:48) ) for l = 1 mm and N = 10 (Fig. 1e) with a near-diagonal matrix, proving that theset of gratings is nearly orthogonal up to a resolution givenby the width of the diagonal. The measured average full-width-at-half-maximum of the PSF( x − x (cid:48) ), which representsthe spatial resolution of the system, was 100 µ m. This wasexpected and is equal to the smallest grating width w N =10 .With N = p , where p = l/w , the resulting periodic illumi-nating fields indeed constitute a nearly-orthogonal set. Notethat neither orthogonalisation nor compressive sensing meth-ods were applied prior to the calculation of the PSF.Figure 2 shows a comparison of x-ray transmission imagesof interconnected aluminium lamellae cut from a sponge sam-ple. The sample’s transmission image ( I S /I flat field ) in Fig. 2awas calculated from images directly recorded by a 2D imag-ing detector. The sample’s transmission image in Fig. 2bwas calculated from synthesised ghost images both with thesample ( G S ) and without ( G flat field ). The phase-contrast-enhanced edges characteristic of a near-field Fresnel image[1–3] are clearly visible in the ghost image just as in the directimage. The ghost image shows quantitative accuracy compa-rable to the direct image as illustrated in the line profilesshown in Figs. 2c-d. The transmission is unity at the pores,greater than unity at the material edges (phase contrast) anddecreases at regions with increasing material density (attenu-ation contrast). Strong phase contrast, which is expected at athick edge where a large x-ray phase gradient occurs, can beresolved along the horizontal (Fig. 2e). Due to the better spa-tial resolution along the vertical, a fringe pattern at the thinedge can be resolved (Fig. 2f). The white-black-white contrastis the familiar Fresnel diffraction phenomenon. The centralminimum is due to destructive interference of waves symmet-rical with the edge and the oscillations are Fresnel fringes [2].We emphasise that with attenuation contrast alone, this edgewould have been invisible as the x-ray transmissions by theair and near the thin edge are essentially unity (see line profilein Fig. 2f). More importantly, this quantitative accuracy wasachieved despite the fact that the resolution along the hori-zontal is an order of magnitude less than along the vertical ( w = 100 µ m and h = 6.5 µ m). This ensures that the same highfidelity and quantitative accuracy can be achieved with large-pixel direct detectors when combined with micrometer-pitch gratings. For example, a combination of w = 6.5 µ m and h = 100 µ m should achieve a similar ghost image. The quan-titative accuracy is a consequence of using a near-complete,near-orthogonal set of basis patterns versus a non-orthogonalset. Finally, we note that, similar to standard propagation-based phase-contrast imaging, phase-sensitive ghost imagingonly requires sufficient spatial coherence via a small sourcesize but essentially no temporal coherence.On the basis of these results we conclude that the presentedmethod is a significant step forward in x-ray ghost imaging.The specific merit of our structured detection approach isthe achievement of phase-contrast images with a resolutionthat goes beyond the spatial resolution of a bucket or mail-box detector. Furthermore, the use of gratings instead ofspeckles produces x-ray ghost images with high fidelity andquantitative accuracy. As the fabrication of large-area, high-aspect ratio transmission gratings is already well-established,the technique is easily scalable to large fields of view andmicrometer spatial resolutions with high energy x-rays. Ourmethod of x-ray phase-contrast ghost imaging can be imple-mented, and combined with phase retrieval and computed to-mography. Our computational ghost imaging approach usinggratings is also applicable with other probes such as neutrons,alpha rays, and muons, for which high spatial resolution de-tectors are limited or do not even exist. Single-pixel detectorswould need to be combined with 2D gratings, or could be usedwith slits to form mailbox detectors that can be used with 1Dgratings. Methods
The experiment was carried out at beamline ID19 of TheEuropean Synchrotron - ESRF (Grenoble, France). A U-17type undulator was used, with the gap tuned to generate 19keV pink x-ray beam. The vertical and horizontal x-ray sourcesizes (full-width-at-half-maximum) were 25 µ m and 150 µ m,respectively. The sample was located 140 m from the source,while the mask and the detector were located 13 m from thesample. The sample was a metal foam (Mayser GmbH &Co. KG, Germany) made of 99.7% aluminium and with av-erage pore size of 2.5 mm. An off-the-shelf x-ray test pat-tern (Type 23, H¨uttner, Germany) composed of 0.5 mm thicklead patterns on 1 mm plexiglass was used for the gratings.The x-ray transmissions through the plexiglass and the leadwere 96% and 2%, respectively. The 10 grating patterns usedhave 1 to 10 lines per mm. The set of 10 reference images(Fig. 1d) was obtained by scanning the test pattern in stepsof 65 µ m parallel with the middle grating line. This scanningwas repeated for the measurements with the sample. An in-direct x-ray image detector composed of an sCMOS camera(pco.edge; pixel size: 6.5 µ m, PCO AG, Germany) coupledwith a 100 µ m-thick LuAG:Ce scintillator using a tandem oflenses (Hasselblad, Sweden) with 100 mm focal lengths wasused. The mailbox detectors used were line arrays of 1000 µ m length and 6.5 µ m width (of the same indirect x-ray de-tector). A total of 1700 line arrays was used in the presentedghost image (Fig. 2b). The 11-mm horizontal field of viewwas achieved by stitching 21 images with 500 µ m width each,that were cropped from 1 mm width ghost images. Both thescanning and the stitching would not have been necessary ifwe had a large rectangular grating.[1] A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, andI. Schelokov, On the possibilities of x-ray phase contrastmicroimaging by coherent high-energy synchrotron radi-ation, Rev. Sci. Instrum. , 5486 (1995).[2] P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, andM. Schlenker, Phase objects in synchrotron radiationhard x-ray imaging, J. Phys. D: Appl. Phys. , 133(1996).[3] S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, andA. W. Stevenson, Phase-contrast imaging using polychro-matic hard X-rays, Nature , 335 (1996).[4] A. Koch, C. Raven, P. Spanne, and A. Snigirev, X-rayimaging with submicrometer resolution employing trans-parent luminescent screens,
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