Phase-sensitive x-ray ghost imaging

Phase-sensitiv e x-ra y ghost imaging Margie P . Olbinado ∗ The Eur op e an Synchr otr on – ESRF, CS40220, 38043 Gr enoble, F r anc e Da vid M. Paganin Scho ol of Physics and Astr onomy, Monash University, Victoria 3800, A ustr alia Yin Cheng and Alexander Rac k The Eur op e an Synchr otr on – ESRF, CS40220, 38043 Gr enoble, F r anc e (Dated: Marc h 26, 2019) Imaging with hard x-ra ys is an in v aluable tool in medicine, biology , materials science, and cultural heritage. Propagation-based x-ray phase-con trast imaging [1–3] and tomography hav e b een mostly used to resolve micrometer-scale structures inside weakly absorbing ob jects as well as inside dense sp ecimens. Indirect x-ray detection has b een the key technology to achiev e up to sub-micrometer spatial resolutions [4], albeit inefficiently and hence at the exp ense of increased radiation dose to the sp ecimen. A promising approach to low-dose imaging and high spatial resolution ev en at high x-ray energies is ghost imaging [5–11], whic h could use single-pixel, yet efficient direct x-ray detectors made of high-density materials. Ho wev er, phase contrast has not yet b een realised with x-ra y ghost imaging. W e presen t an approac h whic h exploits b oth the adv antages of x-ray ghost imaging and the high sensitivity of phase-contrast imaging. In comparison with existing techniques, our metho d is efficient and achiev es high-fidelity x-ray ghost images with phase contrast, accurate densit y resolution and dramatically higher spatial resolution. The metho d is scalable to practical tomograph y with large fields of view, micrometer spatial resolution, and with high-energy x-ra ys ab o v e 100 keV. It is also applicable to other phase-sensitiv e imaging techniques [12–15] and with other prob es such as neutrons, alpha rays, and muons, for which high spatial resolution detectors are limited or even not av ailable. X-ra y ghost imaging [5–11] is a newly developed imaging tec hnique, deriv ed from visible light optics [16–19], whic h has the potential to achiev e ultra-lo w radiation dose imaging and high spatial resolution. It utilises optical correlations betw een spatially resolved photons that never pass through an ob- ject of in terest and non-spatially resolved photons that do pass through the ob ject. Similarly to classical ghost imaging [18, 19], exp erimen tal x-ray ghost imaging [5–10] has b een realised with sp ec kled illumination. This has b een done by using intrinsic noise of a synchrotron x-ray source [5], and b y phase-con trast-generated [6–9] or attenuation-con trast- generated [10] sp eckle patterns. A ghost image is retrieved from in tensity correlations b et ween a series of sp eckle fields that illuminate an ob ject and the total intensities transmit- ted b y the ob ject. Remark ably , the photons passing through the ob ject are detected using only a so-called single-pixel ‘buc ket’ detector. Since single-pixel, high-Z (atomic n umber), direct x-ra y detectors are significan tly more efficient than t w o- dimensional (2D) indirect x-ray detectors commonly used in x-ra y phase-contrast imaging, the p oten tial reduction of the radiation dose to the sp ecimens is highly appealing [5, 10]. The efficiency of indirect detectors is worse at high x-ray en- ergies. Consequently , resolutions of only several micrometers ha ve been ac hieved abov e 100 k eV. In con trast, since the spa- tial resolution of a ghost image is determined by the speckle size and not by the detector [8, 20], ghost imaging may achiev e the muc h-co veted high spatial resolution with high x-ray en- ergies. F or example, large, single-pixel detectors made with the high-Z material CdT e hav e a quantum efficiency close to 100% up to or even ab o v e 100 keV x-ray energy . ∗ margie.olbinado@esrf.fr; Corresponding author In essence, classical ghost imaging expresses the spatial dis- tribution of an ob ject’s transmission function as a linear com- bination of sp ec kle fields [18, 19]. Each speckle field is consid- ered as a linearly indep enden t basis vector. This is concep- tually similar to building functions as a sum of sin usoids in a F ourier series. A classical ghost imaging setup is comp osed of a b eam path (or reference arm) in whic h the sp ec kle fields are detected by a 2D detector, and a b eam path (or ob ject arm) in which the ob ject’s transmitted in tensities are detected by a buck et detector. In computational x-ra y ghost imaging, the reference images are pre-determined or pre-c haracterised. Figure 1a sho ws a sc hematic diagram of a computational x-ra y ghost imaging setup using sp ec kle-generating masks, whic h are either x-ray absorbing or phase-shifting. A series of sp ec kled illuminating fields is obtained by raster scanning the mask in the transverse plane. Let I in b e a uniform, incident illumination in tensity and ˆ T M ,j ( r ⊥ , z = R M ) b e the transmission function of the j th sp ec kle mask (M) at a mask-to-sample propagation distance R M . Here, r ⊥ = ( x, y ) denotes co ordinates in the planes p erpendicular to the optical axis z . The j th illumination in- tensit y on to 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 ) b e the transmission function of the sample (S) at the sample-to-buck et detector distance R S , the signal collected by the buck et detector may be written as: b j = Z Z Ω ˆ T S ( r ⊥ , z = R S ) I j ( r ⊥ ) d r ⊥ , (2) where Ω is the surface o v er which the b eam intensit y is recorded and within which the sample is entirely contained. 2 FIG. 1. Computational x-ra y ghost imaging (XGI). a , A conv en tional XGI set-up with a structured illumination approach using speckles. Intensit y correlations betw een a series of known illuminating sp ec kle fields (reference images) and the corresp onding total intensit y transmitted by the sample (buck et signals) collected by a bucket detector are utilised to synthesise an attenuation-con trast x-ray ghost image. b , Our phase-con trast XGI experimental set-up with a structured detection approach using p eriodic structures instead of speckles. By interchanging the mask and the sample in the sequence, the buck et signal is sensitive to x-ray phase shifts from the sample and a phase-contrast x-ra y ghost image can b e synthesised. 1D gratings are used as masks in combination with a 1D buck et detector, which is a collection of ‘mailb o x’ detectors. The ghost image reconstruction formula is applied for each mailbox detector ( c ). d , Reference x-ray images of N = 10 grating patterns with 1 to 10 lines p er mm. e , Calculated p oint-spread-function PSF( x, x 0 ) of the phase-contrast x-ray ghost image without the sample for one of the mailb o x detec tors ( l = 1 mm, N = 10). The spatial resolution (FWHM of the PSF) is 10 × the mailb o x detector length. The hat ( ˆ ) indicates that the transmission function is an op erator and that the order of op eration is crucial. Using N sp eckle masks, an x-ray ghost image may b e syn- thesised using [18, 19]: G ( r ⊥ ) = 1 N N X j =1 ( b j − ¯ b ) I j ( r ⊥ ) . (3) The buck et signal b j subtracted by the mean ¯ b acts as a w eighting co efficient for the corresp onding reference sp eckle field I j in the sup erposition. Interestingly , even though nei- ther of the detectors used in the mea surement yields an image of the ob ject, and though no photons passing through the ob- ject are ev er registered b y a position-sensitive detector, an image can b e obtained b y harnessing intensit y c orr elations . This sc heme, named classical ghost imaging with x-ra ys [5, 7–10] only retrieves the atten uation-contrast component of the ob ject’s x-ra y transmission image independent of R S , th us losing the m uch-enhanced contrast or sensitivity to densit y v ariations in the sample that could b e obtained by exploiting propagation-induced x-ra y phase con trast. This inability to exploit phase con trast is the case with the usual mask–sample sequence. Here we show that the k ey to ac hiev e phase con trast in x-ray ghost imaging is to rev erse the sequence of sample and mask: a structured detection approach [21–23] instead of a structured illumination approach. The origin of contrast in an x-ra y image is related to the ob ject’s complex refractive index: n ( r ⊥ , z , λ ) = 1 − δ ( r ⊥ , z , λ ) + iβ ( r ⊥ , z , λ ) . (4) F or simplicity , consider a homogeneous ob ject of pro jected thic kness t ( r ⊥ ), and quasi-mono c hromatic, plane-wa v e x-ray radiation with wa velength λ and initial intensit y I in . Under the pro jection approximation, the spatial v ariations of the optical density , D ( r ⊥ ) = 4 π λ β t ( r ⊥ ) (5) 3 FIG. 2. X-ra y transmission images of interconnected aluminium lamellae cut from a sp onge sample. a , Direct x-ra y phase-contrast image ( I S /I flat field ): calculated from radiographs directly recorded using a 2D imaging detector. b , Ghost x-ra y phase-con trast image ( G S /G flat field ): calculated from syn thesised ghost images b oth with the sample ( G S ) and without ( G flat field ). Scale bars in a and b represent 1 mm. c , Horizontal line profiles. d , V ertical line profiles. e, f Magnified views of the insets in a and b showing phase-contrast enhancement at the edges of representativ e thick and thin region of the lamellae. Scale bars in e and f represen t 250 µ m. and the x-ray phase shift, φ ( r ⊥ ) = − 2 π λ δ t ( r ⊥ ) (6) generate the attenuation con trast and phase contrast in the x- ra y image, resp ectiv ely [24]. In the hard x-ray regime, phase- con trast imaging is up to three orders of magnitude more sensitiv e than attenuation-con trast imaging. This is b ecause absorption decreases with the fourth p o w er of photon energy while phase contrast decreases with the square of the energy . A phase-contrast image is readily achiev ed through free- space propagation or F resnel diffraction. At an ob ject-to- detector propagation distance R ( R ≤ d 2 λ , where d is the c haracteristic length scale ov er which the ob ject appreciably c hanges), a near-field phase-contrast image or F resnel image ma y be deriv ed from the so-called transport-of-intensit y equa- tion [25] and is given by [24]: ˆ T ( r ⊥ , z = R ) I in =  − Rδ µ ∇ 2 ⊥ + 1  e − µt ( r ⊥ ) I in , (7) where the linear atten uation co efficient µ = 4 π β /λ and ∇ ⊥ is the gradient op erator in the x - y plane. The transmis- sion function contains b oth the the attenuation contrast and propagation-induced phase-contrast comp onents. By conser- v ation of energy , the av erage intensit y of the F resnel image o ver 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 in voking the conserv ation 4 of energy , the resulting buck et signal is giv en b y: b j = Z Z Ω  − R S δ S µ S ∇ 2 ⊥ + 1  e − µ S t S ( r ⊥ ) I j ( r ⊥ ) d r ⊥ = Z Z Ω e − µ S t S ( r ⊥ ) I j ( r ⊥ ) d r ⊥ . (8) Therefore, with the usual setup, where the mask is upstream of the sample, the buck et signal (Eqn. 8) is not sensitiv e to the propagation-induced phase-contrast comp onen t of ˆ T S . Only the attenuation-con trast comp onen t exp[ − µ S t S ( r ⊥ )] is recov- ered in the ghost image. The key to ac hieving phase con trast in x-ray ghost imaging is to interc hange the sequence of the sample and the mask in the b eam. The mask is placed at the desired phase-contrast image plane, a distance R S do wnstream from the sample. The buc ket signals, which measure the weigh ting coefficients of the sup erposition in the ghost image reconstruction (Eqn. 3), do not register a constan t signal that is insensitive to phase con- trast. W e emphasise the crucial order of the sample and the mask by the order of operation of the transmission function in the buck et signal equation, which we write as: b j = Z Z Ω ˆ T M ( r ⊥ , z = R M ) ˆ T S ( r ⊥ , z = R S ) I in ( r ⊥ ) d r ⊥ . (9) In this wa y , the buck et signal is sensitiv e to the phase- con trast component of the sample’s transmission image, ˆ T S ( r ⊥ , z = R S ) I in ( r ⊥ ). Notice also that dep ending on R S , a phase-contrast x-ray ghost image in the F resnel (near-field) or F raunhofer (far-field) regime can b e synthesised. F urther- more, the ghost image synthesis can even b e extended to any x-ra y phase-contrast imaging approac hes such as T albot in- terferometry [12, 13], and near-field sp ec kle-tracking [14, 15]. This approach is also compatible with ghost imaging com- bined with x-ray diffraction top ography and crystallograph y . Sim ulations ha ve indeed shown that our detection approac h is a means to ac hieve analyser-based x-ray phase-contrast ghost imaging [26]. Equation 9 also clearly indicates that only the atten uation-contrast comp onen t of ˆ T M matters, hence ampli- tude masks should b e used. Our setup for x-ra y phase-contrast ghost imaging is de- picted in Fig. 1b. Instead of using sp ec kle patterns, whic h form a non-orthogonal basis and require N  p measure- men ts in order to synthesise a ghost image consisting of p pixels, w e employ a linear combination of p eriodic fields with v arying frequencies, whic h form a nearly-orthogonal set of ba- sis patterns. This eliminates the inheren t redundancy of the sp ec kle-based approac h. W e implemented this using trans- mission gratings, whic h is practical since the fabrication of suc h gratings for hard x-rays is w ell-established. F or exam- ple, high-aspect-ratio gratings with 2 µ m pitch and 160 µ m thic kness used for up to 180 keV x-ray energy hav e b een re- p orted [27]. Deterministic orthogonal basis patterns such as the Hadamard and F ourier basis patterns [21–23] hav e b een used instead of non-orthogonal random patterns or sp ec k- les. Compressive sensing concepts [18], orthogonalisation of sp ec kle fields [8] and iterative refinement [9] may also b e em- plo yed to reduce N . W e opted not to use 2D gratings. Instead, we considered that a combination of 1D gratings with a 1D buck et detector is equiv alent to using 2D gratings with a single-pixel (0D) detector. A 1D buck et detector constitutes a set of what we call mailb o x detectors. Recen tly , high-Z, 2D direct detectors suc h as the EIGER2 CdT e (Dectris Ltd., Switzerland) with pixel size of 75 µ m hav e b ecome commercially av ailable. Line arra ys or even pixels of such 2D detectors ma y b e used as mailb o x detectors. The ghost image reconstruction formula is applied for eac h mailbox detector (Fig. 1c). The adv antages are t w o-fold: (1) N is reduced b y a factor equal to the n umber of mailb o x detectors used. (2) The spatial resolution may b e tuned indep enden tly in tw o directions: one with the smallest grating line width w , and the other with the mailb o x detector heigh t h . The p osition-dependent p oint-spread function (PSF) of an x-ra y ghost image without sample was calculated for a given mailb o x detector in order to chec k whether a set of gratings (Fig. 1d) constitute a complete set of basis elements. The corresp onding completeness relation [8, 20] that needs to b e satisfied is given by: PSF( x − x 0 ) = 1 N N X j =1 [ I j ( x 0 ) − ¯ I ][ I j ( x ) − ¯ I ] , (10) where ¯ I is the av erage intensit y of the j th illumination in- stead of an av erage o ver all illuminations and x runs ov er pixels equal to the mailb o x detector length l . By wa y of ex- ample w e sho w the calculated PSF( x − x 0 ) for l = 1 mm and N = 10 (Fig. 1e) with a near-diagonal matrix, proving that the set of gratings is nearly orthogonal up to a resolution given b y the width of the diagonal. The measured av erage full- width-at-half-maxim um of the PSF( x − x 0 ), which represents the spatial resolution of the system, was 100 µ m. This was exp ected and is equal to the smallest grating width w N =10 . With N = p , where p = l/w 10 , the resulting p eriodic illumi- nating fields indeed constitute a nearly-orthogonal set. Note that neither orthogonalisation nor compressive sensing meth- o ds were applied prior to the calculation of the PSF. Figure 2 shows a comparison of x-ray transmission images of interconnected aluminium lamellae cut from a sponge sam- ple. The sample’s transmission image ( I S /I flat field ) in Fig. 2a w as calculated from images directly recorded by a 2D imag- ing detector. The sample’s transmission image in Fig. 2b w as calculated from synthesised ghost images b oth with the sample ( G S ) and without ( G flat field ). The phase-contrast- enhanced edges c haracteristic of a near-field F resnel image [1–3] are clearly visible in the ghost image just as in the direct image. The ghost image shows quantitativ e accuracy compa- rable to the direct image as illustrated in the line profiles sho wn in Figs. 2c-d. The transmission is unity at the p ores, greater than unity at the material edges (phase contrast) and decreases at regions with increasing material density (atten u- ation con trast). Strong phase con trast, which is expected at a thic k edge where a large x-ray phase gradient o ccurs, can b e resolv ed along the horizon tal (Fig. 2e). Due to the better spa- tial resolution along the vertical, a fringe pattern at the thin edge can b e resolved (Fig. 2f ). The white-blac k-white contrast is the familiar F resnel diffraction phenomenon. The central minim um is due to destructive in terference of wa ves symmet- rical with the edge and the oscillations are F resnel fringes [2]. W e emphasise that with attenuation contrast alone, this edge w ould hav e b een invisible as the x-ray transmissions b y the air and near the thin edge are essentially unit y (see line profile in Fig. 2f ). More imp ortantly , this quantitativ e accuracy was ac hieved despite the fact that the resolution along the hori- zon tal is an order of magnitude less than along the v ertical ( w = 100 µ m and h = 6.5 µ m). This ensures that the same high fidelit y and quantitativ e accuracy can b e achiev ed with large- pixel direct detectors when combined with micrometer-pitc h 5 gratings. F or example, a combination of w = 6.5 µ m and h = 100 µ m should achiev e a similar ghost image. The quan- titativ e accuracy is a consequence of using a near-complete, near-orthogonal set of basis patterns versus a non-orthogonal set. Finally , w e note that, similar to standard propagation- based phase-contrast imaging, phase-sensitive ghost imaging only requires sufficient spatial coherence via a small source size but essentially no temp oral coherence. On the basis of these results w e conclude that the presen ted metho d is a significan t step forward in x-ray ghost imaging. The sp ecific merit of our structured detection approach is the ac hievemen t of phase-contrast images with a resolution that go es b ey ond the spatial resolution of a buck et or mail- b o x detector. F urthermore, the use of gratings instead of sp ec kles produces x-ray ghost images with high fidelity and quan titative accuracy . As the fabrication of large-area, high- asp ect ratio transmission gratings is already well-established, the technique is easily scalable to large fields of view and micrometer spatial resolutions with high energy x-rays. Our metho d of x-ray phase-con trast ghost imaging can b e imple- men ted, and combined with phase retriev al and computed to- mograph y . Our computational ghost imaging approach using gratings is also applicable with other prob es suc h as neutrons, alpha rays, and muons, for which high spatial resolution de- tectors are limited or do not ev en exist. Single-pixel detectors w ould need to be com bined with 2D gratings, or could be used with slits to form mailbox detectors that can be used with 1D gratings. Metho ds The exp erimen t was carried out at b eamline ID19 of The Europ ean Synchrotron - ESRF (Grenoble, F rance). A U-17 t yp e undulator was used, with the gap tuned to generate 19 k eV pink x-ra y b eam. The vertical and horizontal x-ra y source sizes (full-width-at-half-maximum) w ere 25 µ m and 150 µ m, resp ectiv ely . The sample was lo cated 140 m from the source, while the mask and the detector were located 13 m from the sample. The sample w as a metal foam (Mayser GmbH & Co. K G, 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 thick lead patterns on 1 mm plexiglass w as used for the gratings. The x-ra y transmissions through the plexiglass and the lead w ere 96% and 2%, resp ectively . The 10 grating patterns used ha ve 1 to 10 lines p er mm. The set of 10 reference images (Fig. 1d) was obtained by scanning the test pattern in steps of 65 µ m parallel with the middle grating line. This scanning w as rep eated for the measurements with the sample. An in- direct x-ray image detector comp osed of an sCMOS camera (p co.edge; pixel size: 6.5 µ m, PCO A G, Germany) coupled with a 100 µ m-thick LuAG:Ce scintillator using a tandem of lenses (Hasselblad, Sweden) with 100 mm fo cal lengths w as used. The mailbox detectors used were line arra ys 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 presen ted ghost image (Fig. 2b). The 11-mm horizontal field of view w as achiev ed by stitching 21 images with 500 µ m width each, that were cropp ed from 1 mm width ghost images. Both the scanning and the stitching would not hav e been necessary if w e had a large rectangular grating. [1] A. Snigirev, I. 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