Practical x-ray ghost imaging with synchrotron light
Daniele Pelliccia, Margie P. Olbinado, Alexander Rack, David M. Paganin
aa r X i v : . [ phy s i c s . i n s - d e t ] S e p Practical x-ray ghost imaging with synchrotron light
Daniele Pelliccia ∗ Instruments & Data Tools Pty Ltd, Victoria 3178, Australia andSchool of Science, RMIT University, Victoria 3001, Australia
Margie P. Olbinado and Alexander Rack
European Synchrotron Radiation Facility, 38043 Grenoble, France
David M. Paganin
School of Physics and Astronomy, Monash University, Victoria 3800, Australia
We present a practical experimental realization of transmission x-ray ghost imaging using syn-chrotron light. Hard x-rays from an undulator were split by a Si 200 crystal in Laue geometry toproduce two copies of a speckled incident beam. Both speckle beams were simultaneously measuredon a CCD camera. The sample was inserted in one of the two beams, and the corresponding imagewas integrated over the camera extent to synthesize a bucket signal. We show the successful x-rayghost image reconstruction of two samples and discuss different reconstruction strategies. We alsodemonstrate a method for measuring the point spread function of a ghost imaging system, whichcan be used to quantify the resolution of the ghost imaging reconstructions and quantitatively com-pare different reconstruction approaches. Our experimental results are discussed in view of futurepractical applications of x-ray ghost imaging, including a means for parallel ghost imaging.
INTRODUCTION
Ghost imaging [1–3] was first proposed [4, 5] and thenexperimentally achieved [6, 7] in visible-light quantumoptics. It utilizes intensity correlations between (i) spa-tially resolved photons which never pass through a sam-ple of interest and (ii) non-spatially resolved photons thatdo pass through the sample. It is remarkable that thisparallelized form of the Hanbury Brown–Twiss experi-ment [8–10] permits images of a sample to be obtained,despite the fact that photons passing through the ob-ject are detected using only a single “bucket” detector[11, 12]. Interestingly, the use of delocalized photons cre-ated by passage through a beam-splitter, in the standardquantum-optics setup for ghost imaging, negates the clas-sical dualism of a spatially confined photon either locallyinteracting with an object or locally not interacting withan object. Accordingly, there has been much debate as towhether ghost imaging is or is not intrinsically quantummechanical [1].Previous successful implementations of this intriguingimaging method using visible light [6, 7, 11, 12], atoms[13] and x-rays [14–16] open profound new pathways toimaging. Exploration of these new pathways remains inits infancy.Ghost imaging retains some level of mystery on ac-count of the aforementioned debate as to whether theprocess is inherently quantum mechanical (relying, forexample, on photon entanglement), or intrinsically clas-sical [1]. We concur with the currently-dominant opinionthat some ghost imaging scenarios may be understood inpurely classical terms, while other forms of ghost imag-ing rely intrinsically on quantum processes [1]. The classof ghost imaging experiment, to which the present paper belongs, may be understood in purely classical terms.Ghost imaging has several particularly attractive fea-tures: (i) In its computational-imaging variant [11, 17,18], ghost imaging permits images of an object to beformed without any position-sensitive detectors whatso-ever. Radiation is only detected using a large one-pixel“bucket detector”, with the illuminating intensity pat-terns being known and therefore not needing to be mea-sured [3, 11, 12, 18]. This may permit fuller utilisation ofall radiation scattered by a sample of interest, via mul-tiple buckets. This possibility may be particularly at-tractive for imaging using X-rays, atoms and neutrons.(ii) Turbulence robustness [19–22] is another attractivefeature of the method. Since it relies on intensity corre-lations , the ghost signal is unaffected by non-correlatedrandom perturbations that perturb the input pair of sig-nals. (iii) Forms of ghost imaging may exist which leadto reduced radiation dose on the sample [15, 23], a pos-sibility which is especially appealing when using ionizingradiation. One compelling proposal considers x-ray free-electron lasers, utilising parametrically down-convertedphotons passing through the object that are entangledwith x-ray-photons that never pass through the object[23]. More broadly, an attractive avenue for further re-search is to seek ghost-imaging protocols that offer re-duced radiation dose to a sample of interest.We restrict attention to x-ray ghost imaging for theremainder of this paper. The current literature on ex-perimental realisations of x-ray ghost imaging is sparse.Inspired by earlier proposals [24, 25], two papers werepublished in 2016, both reporting the experimental re-alisation of ghost imaging in one transverse dimensionusing x-ray synchrotron sources [14, 15]. Ghost imagingusing a laboratory x-ray source, again in one transversedimension, was reported in 2017 [16]. Very recently, andexperimental realization of 2D x-ray ghost imaging witha laboratory source was reported [26], providing the firstexperimental evidence of the possibility of dose reductionafforded by the ghost imaging protocol. Even taking intoaccount its evident roots in experimental x-ray studies[27–32] on the Hanbury Brown–Twiss (HBT) effect [8–10], the paucity of extant literature on experimental re-alisations of x-ray ghost imaging is an obvious indicatorof rich avenues for future work.With the notable exception of the extremely recentmanuscript by Zhang at al. [26], all published reconstruc-tions [14–16] are one-dimensional. Note, in this context,that the earlier x-ray HBT studies [27–31] may be viewed,at least in retrospect, as zero-dimensional ghost imaging.The key point is the dearth of experimental x-ray ghostreconstructions in higher than one dimension. Anotherlimitation of existing approaches [15] is in the method bywhich speckle bases are produced not being readily scal-able to higher-dimensional and higher-resolution ghostimaging. Of the currently-published approaches, that ofYu et al. [14] with a synchrotron x-ray source, Schoriand Schwartz [16] and Zhang et al. with a tabletop x-ray tube source utilise spatially random masks to gen-erate the ensemble of speckle patterns that form the setof linearly independent basis functions used in the ghostreconstruction. The use of masks appears particularlyamenable to scaling up to both higher-dimension andhigher-resolution x-ray ghost imaging. Another difficultyinherent to current approaches is the very precise tim-ing resolution needed in the approach of Pelliccia et al. [15], since this employs the pulsed nature of third genera-tion x-ray synchrotron emission from individual electronbunches to provide an ensemble of speckle patterns asinput into the ghost reconstruction. Note however thatuse of x-ray speckles produced by pulsed sources is likelyto be very promising for SASE free electron lasers, due totheir high spatial coherence (and therefore high specklecontrast).Here we present a two-dimensional experimental real-isation of x-ray ghost imaging, with a higher number ofpixels in each transverse dimension than all of the previ-ously cited experimental studies. We believe this methodto be practical in the sense of being scalable to higher di-mensional imaging (i.e. tomography) and finer spatialresolutions. The method is also amenable to highly par-allel geometries, in which ghost images of a large numberof objects may be acquired simultaneously. Moreover,our analysis explicitly benefits from the previously men-tioned turbulence robustness of the method, by being in-sensitive to uncorrelated fluctuations between the objectand reference arms of our ghost-imaging setup.We close this introduction with a brief outline ofthe remainder of the paper. Section 2 gives an out-line of the x-ray synchrotron-based experimental setupand measurement process for practical two-dimensional ghost imaging. Section 3 presents our reconstructed two-dimensional images, of both a perforated lead-sheet sten-cil and the tungsten filament of an incandescent lightglobe. Two different methods of ghost-image reconstruc-tion are considered, together with a brief study com-paring spatial resolution in the two different methods ofghost imaging. Section 4 comprises a discussion on di-verse topics such as a means for determining the pointspread function associated with a given ghost-imagingspeckle basis, the turbulence robustness of the method,and future possibilities including parallelised x-ray ghostimaging, computational x-ray ghost imaging, and x-rayghost tomography.
EXPERIMENTAL REALIZATION OF X-RAYGHOST IMAGINGExperimental setup
The experiment was carried out at beamline ID19 ofthe European Synchrotron ESRF in Grenoble (France).Undulator light with a mean energy of 26.3 keV was fo-cused by a stack of compound refractive lenses to a fo-cal spot of about 5.5 mm diameter at the sample. Toproduce variable (and controllable) speckles in the beamwe inserted a 1 cm thick perspex container filled withglass powder in the beam, 5.8 m upstream of the sample.The powder (Oberfl¨achentechnik Seelmann, Germany)was composed of grains, irregular in shape, with typi-cal size distributed in the range 200–1000 µ m. Propa-gation based phase contrast from the beads generated aspeckle pattern on the sample, that could be controlledby raster scanning the glass-beads slab in the transverseplane. A silicon crystal beam splitter, placed 20 cm up-stream of the sample, was used to produce two copiesof the beam by means of Laue diffraction (transmissiongeometry) from the (220) planes of the silicon. The pri-mary beam was mostly transmitted by the silicon wafer(see scheme in Fig. 1), thus creating two non-identicalcopies of the beam. Both beams were then recorded bythe same pixel array detector (scintillator lens-coupled toa FReLoN camera), placed immediately downstream ofthe sample. The effective pixel size of the camera was 30 µ m. The attenuated image of the primary beam, with FIG. 1. Schematic diagram of the experimental setup.
FIG. 2. (a) Image of the primary beam on the FReLoN cam-era acquired with 2 s exposure time. The beam was attenu-ated by stacking a 500 µ m thick Cu foil and a 500 µ m thickGaAs wafer to protect the camera. Scale bar = 2 mm. (b)Corresponding image of the diffracted beam. No attenuatorwas placed in the diffracted beam path. (c) Blurred version ofthe image in (a) to highlight the similarities in the speckle pat-tern distribution between the direct and the diffracted beam. the glass-beads slab in place, is shown in Fig. 2(a). Anattenuator composed of a 500 µ m thick Cu foil and a500 µ m thick GaAs wafer was inserted in the primarybeam to avoid saturation and protect the camera dur-ing the prolonged exposures. The corresponding imageof the diffracted beam (cropped from the same frame ofthe camera) is shown in Fig. 2(b). The image was ac-quired with 2 s exposure time. Notice the different in-tensity of the two beams, with the diffracted beam beingmuch weaker compared to the primary beam. The ra-tio of the average intensities in the center of the directand diffracted beam was estimated by successive mea-surements with and without attenuators to be 1 . × − .Two main features of the diffracted beam are worth re-marking. (i) Due to unavoidable vibration of the siliconcrystal, the image of the diffracted beam appears blurredwhen compared to the primary beam. To facilitate com-parison between the two, we show the blurred version ofthe primary beam image in Fig. 2(c). The blurring wasperformed using a Gaussian kernel with σ = 60 µ m (2pixels). The size of the speckles in the blurred imageis comparable with the corresponding size of the speck-les of the diffracted beam, so that the similarity betweenthe two is more evident. (ii) Likely due to some straingenerated in the crystal mount, the diffracted beam ap-pears to be slightly compressed, with more intensity be-ing diffracted around the top and bottom edge of thebeam. This led, as we will see in Sec. 3.2, to a slightdemagnification of the ghost images when compared tothe original images. Ghost imaging measurement procedure
The sample was inserted in the diffracted beam. Wedecided on this configuration to ensure that the sam-ple was being illuminated by the weaker beam. Givenour estimated intensity ratio, and assuming the measuredcounts to be proportional to the number of photons (we are in a nearly monochromatic case), we expect that thesample receives a fraction of 0.014% of the photons thatare in the reference beam. The variable speckle illumi-nation was obtained by raster scanning the glass-beadsslab in the transverse plane. The range of the raster scanwas 150 mm ×
90 mm (H × V), with a step size of 750 µ m × µ m (H × V). These parameters were chosen toensure the step size to be about 2 times larger than thetypical speckle size (compatible with the total size of theglass beads sample), so that images taken at neighbour-ing positions along the scan were nearly independent.A speckle image was acquired at each position of theglass-beads slab, with 2 s exposure time. The bucketsignal was then synthesized by summing the number ofcounts in an area of 200 ×
180 pixels comprising thediffracted beam. The reference image at each positionwas generated by cropping an area of 260 ×
230 pixelscentered around the primary beam from the raw (attenu-ated) camera image. We acquired a total of 5000 framesfor each ghost imaging reconstruction.Due to the periodic electron injections into the ESRFstorage ring, a low frequency time structure (with pe-riod of 1 h) was present in the intensity signal. To pre-vent this feature affecting the ghost imaging reconstruc-tion, the data (both bucket signal and reference images)were Fourier filtered to remove such low frequency com-ponents. The filter was a simple low-pass designed toremove all frequency below 5% of the Nyquist (tempo-ral) frequency, and therefore discard all slow variationsof the beam intensity due to the injections.
GHOST IMAGING RECONSTRUCTIONRESULTSGhost imaging reconstruction strategy
Conventional ghost imaging reconstruction can be ob-tained with the superposition formula [11, 12]: v i = 1 m m X j =1 (cid:0) b j − ¯ b (cid:1) A ij . (1)In Eq. (1), v i is the i -th pixel of the (rasterized) ghostimage v , written as the superposition of the correspond-ing pixels of the measured speckle images A ij ( i -th pixelof the j -th measurement). Each term of the superposi-tion is weighted by the corresponding bucket signal b j subtracted by its mean ¯ b . The total number of imagesused in the process is m . In our experiment m = 5000.The ghost image v consists of n = n × n pixels. De-noting by A the m × n matrix of reference images, thebucket signal is b = Av .Equation (1) can be written in the compact form v = h ( b − h b i ) A i , (2)where the symbol hi denotes ensemble average.The use of a random measurement matrix A however,is not optimal: to attain a good signal-to-noise ratio(SNR) a large number of measurements is generally re-quired ( m ≫ n ), which makes the basic protocol unsuit-able for applications demanding low dose. As noted in[12], the reason for that becomes apparent by interpretingEq. (1) as a conventional expansion using linearly inde-pendent basis functions. The rows of the matrix A repre-sent the basis vectors, and the process of ghost imagingis in fact a projection operation of the image on sucha basis. A better basis is made of orthonormal ratherthan merely linearly independent vectors, and thereforethe quality of the ghost image recovery depends on howwell the rows of the random measurement matrix ap-proximate an orthonormal basis—see Appendix for moredetail on this point. Further to this, it is also worth not-ing that genuinely random illumination can be hard toproduce in practice. In our case, we took care to scan theglass-beads slab by a transverse step size that was muchlarger than the transverse speckle size, however residualcorrelations are still present.To overcome such limitations causing a non optimalchoice of the measurement matrix, several approaches arecurrently used. Compressive ghost imaging speeds imagerecovery using ideas and techniques of compressive sens-ing [12]. Image recovery can be improved by compressivesensing by identifying an orthonormal basis in which theimage to be recovered is sparse [12, 33, 34].In an alternative approach, commonly used in singlepixel cameras, image recovery can be much improved ifone starts from an orthonormal measurement matrix (orsensing matrix) in the first place. A common choiceis the Hadamard matrix H implementing Hadamard–Walsh functions via a spatial light modulator (see forinstance [35]). In this case, redefining the bucket signalas b H = Hv , in the absence of noise, the expansion inEq. (1) is exact when m = n . While this is a convenientchoice in the visible part of the spectrum (where efficientspatial light modulators exist), this option is not easy toimplement for x-ray imaging.Our approach was to assume that the speckle illumi-nating function is approximately orthogonal, and per-form an effective orthogonalization using the QR decom-position of the matrix A : A = QR . (3)Here, Q is the orthogonal matrix we seek, and R is anupper triangular matrix.The only non-trivial step is to rearrange the bucketvector b accordingly. Specifically we seek a vector ˜b = Qv , that is the bucket signal that would be measured ifthe measurement matrix were the orthogonal matrix Q .Since b = Av , we can write b = QRv = QRQ − Qv = AQ − ˜b . (4)Therefore, by inverting the previous expression: ˜b = QA − b . (5)Equations (3) and (5) constitute the algorithm we useon our data to produce an approximately orthonormaldecomposition. The ghost imaging reconstruction canthen be obtained using Eq. (2) with b → ˜b and A → Q (note that h ˜b i = 0): v = h ˜b Q i . (6)Using Eq. (6) for the reconstruction has two main ad-vantages over the conventional superposition formula inEq. (2). First, it guarantees optimal use of the informa-tion, as the new measurement matrix is now composedof orthogonal rows. Second, since the QR decompositionscrambles the basis, the typical speckle size of the orthog-onal measurement matrix Q become effectively smallerthan the typical size of the physical speckles used in themeasurement. This means that the resolution of the re-constructed ghost image is no longer limited by the realspeckle size, at the price of an increased noise in the re-construction. FIG. 3. Measurement of the stencil sample. (a) Direct im-age of the sample when illuminated by one realization of thespeckle pattern. (b) Conventional ghost imaging reconstruc-tion using m = 5000 measurements. (c) Ghost imaging recon-struction after QR decomposition of the measurement matrix,obtained using the same measurement as the previous case.(d) Median image of 150 ghost images obtained by QR de-composition of the randomly permuted measurement matrix.The scale bar corresponds to 1 mm. Notably, one could also overcome the noise problem,by noting that for underconstrained problems ( m < n )one could perform a QR decomposition multiple timesby permuting the rows of A (and, correspondingly, of b )each time. In practice one obtains a different reconstruc-tion each time, but the difference in the reconstructionsis mostly in the noise background. By averaging multiplereconstructions (using always the same data, hence notincreasing the radiation dose), one could reduce the noiseand increase the resolution of the reconstruction at thesame time. Experimental results
The first sample we imaged was a stencil obtained bydrilling three holes in a lead sheet. The direct image ofthe sample in the diffracted beam (before synthesizingthe bucket signal) is shown in Fig. 3(a). The ghost im-age, obtained using the conventional reconstruction for-mula in Eq. (2) with m = 5000, is shown in Fig. 3(b).Note that the ghost image size is equal to the referenceimage size of 260 ×
230 = 59800 pixels. Therefore weacquired a little more than 8% of the Nyquist sampling.The ghost imaging reconstruction clearly reproduces thesample features albeit at reduced resolution, as dictatedby the speckle size. Significant background noise is also
FIG. 4. Measurement of the tungsten coil. (a) Direct imageof the sample when illuminated by a realization of the specklepattern. (b) Conventional ghost imaging reconstruction using m = 5000 measurements. (c) Ghost imaging reconstructionafter QR decomposition of the measurement matrix, obtainedusing the same measurement as the previous case. (d) Medianimage of 150 ghost images obtained by QR decomposition ofthe randomly permuted measurement matrix. The scale barcorresponds to 1 mm. present as a consequence of the limited number of mea-surements and the camera noise.As discussed in the previous section, resolution canbe improved via QR decomposition of the measurementmatrix. The result of this operation (see Eqs. (4) – (6))is shown in 3(c). The resolution of this image is muchimproved, to the detriment of the image noise which isincreased. Incidentally, this noise–resolution trade-off isconsistent with the noise–resolution uncertainty princi-ple, recently introduced by Gureyev et al. [36]. Con-versely, by repeating the QR decomposition 150 times(each time performing a random permutation of the rowsof the measurement matrix A and, correspondingly, ofthe bucket values b ) and taking the median of those im-ages, the map in Fig. 3(d) can be synthesized. This lastimage is comparable to the conventional reconstructionin terms of noise, and displays higher resolution.The same set of images for the second sample is shownin Fig. 4. In this case the sample was a tungsten coil.The map obtained by the median of 150 ghost imagesobtained after QR decomposition shows a marked im-provement over the conventional ghost image, which re-flects the advantage of the QR decomposition method inoptimising the use of the available information.When compared to the stencil reconstruction, theghost image of the coil looks noisier. Both images havebeen reconstructed using the same number of measure-ments. The difference is to be found in the sample ex-tent compared to the beam size. The stencil sample iseffectively composed of three holes only, whose size is rel-atively small compared to the beam. That means that,variations in the speckles position amount to relativelylarge excursions of the buckets signal. Conversely, thebucket signal after the coil sample will vary comparablymuch less, as it receive contribution from most of thebeam size. Therefore we expect that the ghost imagingprocedure is much more sensitive when reconstructingthe stencil, as opposed to the tungsten coil.Finally, as anticipated in Sec. 2.1, due to the beamcompression effected by the silicon beamsplitter, theghost images appeared demagnified when compared tothe original images. This is especially evident observingthe coil images in Fig. 4, while however being presentto the same extent in the ghost image reconstructions ofboth samples. DISCUSSIONCompleteness relation and Point Spread Function ofthe ghost imaging system
The standard ghost imaging formula in Eq. (1) consid-ers the ensemble of linearly independent random speckleimages as a basis from which to synthesize the reconstruc-tion. Indeed, as explored in more detail in the Appendix,the standard ghost imaging formula may be viewed as asuperposition of approximately orthogonal functions. Adirect consequence is that the rows of the measurementmatrix A should obey an “approximate completeness re-lation”, which can be written as:1 m m X j =1 (cid:0) A ij − ¯ A (cid:1) (cid:0) A kj − ¯ A (cid:1) ≈ δ jk (7)where δ jk is the Kronecker delta. Note that, subtractingthe average ¯ A from each coefficient is required to havezero-mean terms, as each of the A ij is non-negative onaccount of it being a measured intensity value. The pre-vious equation would be exact only in the ideal case inwhich the measurement matrix (after subtracting its av-erage) forms a complete orthonormal set.To make the previous argument more apparent, let usexplicitly rewrite the rows of the measurement matrixas the measured speckle images I j ( x, y ), where x, y arethe coordinates on the detector plane and index j runsover the number of measurements. With this new, moretransparent, notation the completeness relation in Eq. 7can be rewritten as:1 m m X j =1 (cid:2) I j ( x, y ) − ¯ I (cid:3) (cid:2) I j ( x ′ , y ′ ) − ¯ I (cid:3) ≈ δ ( x − x ′ , y − y ′ ) , (8)where δ ( x, y ) is the Dirac delta. As before, Eq. (8) isexact only when the speckle images subtracted by theiraverage form a complete orthonormal set. In all practicalcases the previous equation can be used instead to definean effective Point Spread Function (PSF), δ ( x − x ′ , y − y ′ ) → PSF( x − x ′ , y − y ′ ) centered around the point x ′ , y ′ (see Eq. (16) in the Appendix for a more detailed analysison this point).Such a PSF defines in fact the spatial resolution of theghost imaging system. In the light of this idea, we calcu-lated Eq. (8) for the conventional ghost imaging situation(where the I j ( x, y ) are the original speckle images) as wellas for the modified speckle images after the QR decom-position. The results are shown in Fig. 5. When usingthe original mask the FWHM of the PSF (fitted with aGaussian function) turns out to be about 125 µ m, whichreduces to about 80 µ m after the QR decomposition.The form of ghost imaging presented here – equippedwith the definition of the PSF as discussed above – maybe therefore viewed as a form of scanning probe imag-ing [37] using a completely delocalized probe and a largeintegrating bright-field detector. While the resolution ofscanning probe imaging is usually dictated by the size ofa localised scanning probe, for our delocalized scanningprobe the resolution is limited by the smallest character-istic length scale present in the intensity fluctuations ofthe ensemble of illuminating speckle fields (cf. Eq.(16) FIG. 5. (a) PSF of the ghost imaging system, calculated usingEq. (8), for the conventional ghost imaging situation. Thescale bar represents 500 µ m. (b) Corresponding PSF calcu-lated after QR decomposition. The PSF appears noticeablynarrower, reflecting the resolution improvement afforded bythe QR decomposition. (c) Line profile taken across the cen-tral horizontal line in the maps in (a) – black solid line – and(b) – red solid line. When fitted with a Gaussian function, thetwo peaks have a FWHM of 125 µ m and 80 µ m respectively. in the Appendix), which is in turn the width of the PSFcalculated using Eq. (8). Other remarks on future practical aspects of x-rayghost imaging
We have previously mentioned the turbulence robust-ness of ghost imaging [19–22]. This turbulence robust-ness arises from the invariance of the ensemble averagein Eq. (2), with respect to the addition of statistically un-correlated fluctuations (“turbulence”) in the object andreference arms of the ghost-imaging setup. Stated moreprecisely, the ensemble average in Eq. (2) is unchangedunder either or both of the replacements b → b + δ b and A → A + δ A , where δ b is a zero-average random fluc-tuation added to the bucket signal and δ A is a randomfluctuation added to the speckle images, provided that δ b and δ A are not correlated. This robustness enhancesthe practicality of the method.Another practical aspect, of x-ray ghost imaging in theexperimental setup used here, is its ability to be paral-lelised. Inspired by the parallel form of computationalghost imaging proposed by Yuan et al. [38], considerthe setup shown in Fig. 6. Here, an x-ray source σ il-luminates an ensemble of m random speckle-producingmasks { A j ( x, y ) } , where j = 1 , · · · , m labels each re-alisation of the mask and ( x, y ) are coordinates in theplane perpendicular to the optic axis. A series of beam-splitters B , B , · · · then illuminate a series of objects α, β, · · · , giving associated non-spatially-resolved signalsin the bucket detectors b , b , · · · Each bucket signal ineach detector may be correlated with the same ensembleof speckle images registered by the pixellated array detec-tor for each realisation of the mask, to yield independentparallelised ghost imaging. The objects in in Fig. 6 arestaggered so as to keep constant the source-to-object dis-tance, thereby ensuring that Fresnel diffraction and otherfree-space-propagation effects are accounted for, with theregistered speckle pattern measured over the pixellatedarray detector being equal (up to a multiplicative con-stant) to the speckle patterns illuminating each object.Note that each beamsplitter only needs to remove a neg-ligible fraction of the total energy from the beam whichultimately illuminates the pixellated array detector; theresulting attenuation of the speckle-basis images regis-tered by the array detector can be trivially taken intoaccount in the parallel ghost reconstructions. Note alsothat, while Fig. 6 indicates one object per beam-splitter,one could also have multiple objects per beam-splitter,using multiple Bragg or Laue reflections from a crystalbeam-splitter, or multiple Laue reflections from a poly-crystal beam-splitter.Irrespective of whether or not the ghost-imaging ge-ometry is parallelised, one can also consider x-ray ghostimaging scenarios which reduce or even eliminate the re-liance on a position sensitive detector. With this endin mind, suppose that the ensemble of spatially randomscreens in Figs 1 or 6 may be generated in a reproduciblefashion using a radiation-hard mask. Such masks mightbe generated by transversely displacing a highly struc-tured object (such as the layer of glass beads used in thepresent study), and also by rotation of a suitable highlystructured object (such as a thick metallic foam). Inthis scenario, one need measure the ensemble of referencespeckle fields only once —the pixellated array detector canthen be switched off, or even removed, and ghost imagessubsequently recorded using only the bucket detectors,by running the mask through the previously mentionedreproducible set of positions/orientations. Note, more-over, that the acquisition time in this geometry can bedriven down significantly, since there are no longer anyposition-sensitive x-ray detectors, together with their as-sociated readout times. Note also that the same set ofspeckle images could be used with each orientation of anobject, in a tomographic x-ray ghost imaging context.Interestingly, in the spirit of computational (ghost)imaging [11, 17, 18], one may even dispense altogetherwith the pixellated array detector. To do this, one wouldhave a highly structured mask whose three-dimensionalmicro-structure is so well characterised, and the illumi-nating beam so stable and well characterised, that onecan use a numerical implementation of the x-ray scatter-ing and diffraction to calculate the ensemble of referencespeckle fields that one would have measured had an ar-ray detector been used; these images therefore do notneed to be measured. One would then have a form of computational x-ray ghost imaging using only bucket de-tectors. In this context, we point out that the functionplayed by spatial light modulators in visible-light com-putational imaging is replaced with the known micro-structured mask, in our proposed form of x-ray compu-tational imaging. In the near future, this highly struc-tured mask might even be amenable to fabrication using
FIG. 6. Schematic setup for parallelised x-ray ghost imaging. three-dimensional printers.We close the discussion with a final remark: spatiallyrandom masks are not necessarily optimal for x-ray ghostimaging. While random masks have the virtue of easeof synthesis, structures such as the uniformly redundantarray [39] may be more efficient.
CONCLUSION
We have presented a practical realization of x-ray ghostimaging using synchrotron x-rays from an undulator.This experimental demonstration shows a practical av-enue for producing x-ray ghost images. We reported themeasurements of two samples, a stencil in a lead maskand a tungsten coil. For both samples we reported twodifferent reconstruction strategies. The first is based onthe conventional ghost imaging formula, in which theghost image is approximated by the weighted averageof the speckle illuminating images. The weights of thesuperposition are the bucket signals subtracted by theiraverage. The second approach is based on prior QR de-composition of the measured speckle reference images.In this way the ensemble of speckle images can be madeto be a better approximation to an orthogonal basis,thereby improving the resolution of the reconstruction.Next, we analyzed in more detail the resolution of ourghost imaging system, defining an effective Point SpreadFunction (PSF) which was shown to be improved uponQR decomposition of the illuminating functions. Finally,we discussed practical aspects for future applications ofx-ray ghost imaging, including its robustness against tur-bulence and improved measurement strategies using par-allelised ghost imaging and computational x-ray ghostimaging.
ACKNOWLEDGMENTS
We thank the directors of the ESRF for funding DPand DMP to visit in early 2017. S. B´erujon and E. Brunloaned us the speckle masks used in our experiment. T.E.Gureyev suggested we investigate the completeness rela-tion in the context of the present work. We acknowledgeuseful discussions with T.E. Gureyev, I.D. Svalbe, G.R.Myers, A. Kingston and D. Ceddia.
APPENDIX: ESTIMATING GHOST-IMAGINGRESOLUTION FROM A GIVEN SPECKLE BASIS
Consider an ensemble of m ≫ { I j ( x, y ) } definedover a domain Ω with area A (Ω) in the ( x, y ) plane. Eachensemble member is labelled by the integer j , with each I j ( x, y ) being non-negative on account of being an inten-sity map.Assume “many speckles” in each random speckle field.More precisely, consider each to be a distinct realisa-tion of a stochastic process with every member of theensemble of speckle fields having the same characteristictransverse length scale σ , in both x and y . Stated dif-ferently, we assume each realisation of the speckle fieldto be statistically identical. Since each speckle field has M speckles with M ≈ A (Ω) /σ ≫
1, this implies that(i) the spatially-averaged intensity of each realisation ofthe ensemble is approximately the same, (ii) the spatiallyaveraged squared intensity, of each member, is also ap-proximately the same, and (iii) the ensemble-averagedintensity at any point in Ω is independent of position,and approximately equal to the spatially averaged inten-sity in any particular realisation.The standard ghost-imaging formula considers the en-semble { I j ( x, y ) } as a speckle basis, from which the ghostimage of the object transmission function v ( x, y ) may besynthesized [11, 12]: v ( x, y ) ⊛ PSF( x, y ) = 1 m m X j =1 ( b j − b ) I j ( x, y ) . (9)Here, ⊛ denotes convolution over x and y , PSF( x, y ) isa point spread function associated with the finite spatialresolution with which v ( x, y ) is estimated, the bucket sig-nal is b j = Z Z Ω v ( x, y ) I j ( x, y ) dxdy (10)and the average bucket signal is b = 1 m m X j =1 b j . (11) Using the above definitions and assumptions, one canreadily show that b = I vA (Ω) , (12)where I = 1 m m X j =1 I j ( x, y ) (cid:18) = 1 A (Ω) Z Z Ω I j ( x, y ) dxdy for any j (cid:19) , (13)and v = 1 A (Ω) Z Z Ω v ( x, y ) dxdy. (14)Using Eqs. (10), (12) and (13), Eq. (9) can be manip-ulated into the form: v ( x, y ) ⊛ PSF( x, y )= Z Z Ω dx ′ dy ′ v ( x ′ , y ′ ) 1 m m X j =1 [ I j ( x ′ , y ′ ) − I ][ I j ( x, y ) − I ] . (15)Since v ( x, y ) ⊛ PSF( x, y ) = RR Ω v ( x ′ , y ′ ) PSF( x − x ′ , y − y ′ ) dx ′ dy ′ , the point-spread function associatedwith the ghost-imaging reconstruction is the ensemble-averaged intensity–intensity correlation between the lo-cations ( x, y ) and ( x ′ , y ′ ), namely the smoothed com-pleteness relation:PSF( x − x ′ , y − y ′ ) = 1 m m X j =1 [ I j ( x ′ , y ′ ) − I ][ I j ( x, y ) − I ] . (16)For any fixed position ( x, y ) = ( x ′ , y ′ ), and a specifiedmeasured ensemble of intensity-speckle fields, Eq. (16)can be used to estimate the local resolution associatedwith superpositions (such as Eq. (9)) that utilise thisbasis in a ghost imaging context—see Fig. 5. The aboveexpression, the right side of which is the same correlationfunction (intensity covariance) measured in classic Han-bury Brown–Twiss scenarios [8, 9], may also be used togenerate a “pixel basis” consisting of a lattice of PSFswhose centroids are separated by the full-width-at-half-maximum of each PSF. This set of PSFs is another ap-proximately orthogonal basis, which is complementary tothe approximately-orthogonal speckle basis from whichit is derived, insofar as the former is localised whereasthe latter is not. Indeed, one may loosely speak of Eq.(16) as showing how linear combinations of the spatiallydelocalized speckle probes may be formed to yield verylocalised probes.Interestingly, a variant of the above chain of reason-ing may be used to derive the standard ghost-imagingformula from first principles, in a physically transpar-ent manner. One can start with an ensemble of inten-sity speckle fields that obey the previously articulatedassumptions (see second paragraph of this Appendix).The intensity covariance on the right side of Eq. (16)will typically be a narrowly-peaked normalisable func-tion, making it natural to adopt it as a point-spreadfunction, by definition. The convolution of this point-spread-function with a well-behaved but otherwise arbi-trary function v ( x, y ), then leads directly to the standardghost-imaging formula in Eq. (9), by reversing the logicaldevelopment of this Appendix.We close by noting that the standard ghost imagingformula may be viewed in approximate terms as a form oforthogonal function expansion. Indeed, for our ensembleof speckle images, each of which are statistically identicaland each of which contain many speckles, one has1 A (Ω) Z Z Ω ˜ I j ( x, y ) ˜ I j ′ ( x, y ) dxdy ≈ δ jj ′ Var( I ) , (17)where ˜ I j ( x, y ) are background-subtracted speckle fieldswith ˜ I j ( x, y ) ≡ I j ( x, y ) − I, (18)and Var( I ) is the variance of each I j ( x, y ), which is as-sumed to be independent of j on account of the previ-ously articulated assumptions. By construction, the set { ˜ I j ( x, y ) } of speckle patterns has each member averagingto zero over Ω, with distinct members being orthogonalto one another, and all members being approximately or-thogonal to a constant offset. 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