OSA's Digital Library

Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 8, Iss. 9 — Oct. 2, 2013
« Show journal navigation

A three-image algorithm for hard x-ray grating interferometry

Daniele Pelliccia, Luigi Rigon, Fulvia Arfelli, Ralf-Hendrik Menk, Inna Bukreeva, and Alessia Cedola  »View Author Affiliations


Optics Express, Vol. 21, Issue 16, pp. 19401-19411 (2013)
http://dx.doi.org/10.1364/OE.21.019401


View Full Text Article

Acrobat PDF (4160 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

A three-image method to extract absorption, refraction and scattering information for hard x-ray grating interferometry is presented. The method comprises a post-processing approach alternative to the conventional phase stepping procedure and is inspired by a similar three-image technique developed for analyzer-based x-ray imaging. Results obtained with this algorithm are quantitatively comparable with phase-stepping. This method can be further extended to samples with negligible scattering, where only two images are needed to separate absorption and refraction signal. Thanks to the limited number of images required, this technique is a viable route to bio-compatible imaging with x-ray grating interferometer. In addition our method elucidates and strengthens the formal and practical analogies between grating interferometry and the (non-interferometric) diffraction enhanced imaging technique.

© 2013 OSA

1. Introduction

Hard x-ray phase-sensitive methods are becoming increasingly used [1

1. A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58(1), R1–R35 (2013). [CrossRef] [PubMed]

]. Their main advantages over standard absorption radiography are the ability to detect weakly absorbing details and to produce contrast also between features with similar index of refraction. The first approach to x-ray phase-sensitive methods dates back to 1965, when Bonse and Hart [2

2. U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6(8), 155–156 (1965). [CrossRef]

] proposed a crystal-based interferometer. Remarkably, just 30 years later this technique was systematically used for imaging, developing specific algorithms to quantitatively extract the phase shift caused by the object [3

3. A. Momose, “Demonstration of phase-contrast x-ray computed-tomography using an x-ray interferometer,” Nucl. Instrum. Meth. A 352(3), 622–628 (1995). [CrossRef]

]. More recently, a new interferometric approach based on the use of gratings has been introduced [4

4. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003). [CrossRef]

6

6. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7(2), 134–137 (2008). [CrossRef] [PubMed]

], overcoming many of the limitations of crystal-based methods, including their high requirements in terms of beam coherence and mechanical stability. Grating interferometry (GI) is based on a pair of gratings spaced by a fractional Talbot distance. The first grating produces a Talbot pattern that is analyzed by the second grating. The data acquisition and analysis methods mostly employed with GI are essentially derived from the algorithms developed for crystal interferometry: in particular, the so-called phase stepping (PS) technique [5

5. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005). [CrossRef] [PubMed]

] used in GI bears a strong resemblance to the fringe-scanning method used in crystal interferometry [3

3. A. Momose, “Demonstration of phase-contrast x-ray computed-tomography using an x-ray interferometer,” Nucl. Instrum. Meth. A 352(3), 622–628 (1995). [CrossRef]

]. More in detail, PS exploits the periodicity of the transmission function of the gratings pair: several images are acquired in different positions along one or more oscillation periods. A first-order approximation of the Fourier series of the transmission function is applied to extract three different parametric images, usually referred to as absorption, differential phase and dark field images. The latter actually depends on the scattering properties of the sample and it is measured as the decrement of the amplitude of the periodic transmission function with respect to its value in absence of the sample [6

6. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7(2), 134–137 (2008). [CrossRef] [PubMed]

].

In this paper we apply the three-image method, introduced for ABI [8

8. L. Rigon, F. Arfelli, and R.-H. Menk, “Three-image diffraction enhanced imaging algorithm to extract absorption, refraction and ultrasmall-angle scattering,” Appl. Phys. Lett. 90(11), 114102 (2007). [CrossRef]

] to a single period of the transmission function of the GI to image samples featuring absorption, refraction and scattering. We show that the novel method can be used alternatively to conventional PS even though it relies on different approximations. In both cases at least three images are required to measure absorption, differential phase and scattering width maps. Moreover, in the case of samples with negligible scattering, we show that our approach – unlike PS – can be additionally simplified in analogy with the original DEI approach [7

7. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. H. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997). [CrossRef] [PubMed]

]. In this case only two images are sufficient to extract absorption and refraction information, shortening the required measurement time and possibly reducing the dose.

2. Theory

The x-ray beam passing through the sample can be absorbed, refracted or scattered. Refraction is characterized by an angular deviation ΔθR while the deviation due to the ultra-small angle scattering is correspondingly indicated withΔθS. The latter accounts for deviations that cannot be resolved by the detector and are therefore convolved with the point spread function of the detection system [8

8. L. Rigon, F. Arfelli, and R.-H. Menk, “Three-image diffraction enhanced imaging algorithm to extract absorption, refraction and ultrasmall-angle scattering,” Appl. Phys. Lett. 90(11), 114102 (2007). [CrossRef]

, 12

12. L. Rigon, H.-J. Besch, F. Arfelli, R.-H. Menk, G. Heitner, and H. Plothow-Besch, “A new DEI algorithm capable of investigating sub-pixel structures,” J. Phys. D Appl. Phys. 36(10A), A107–A112 (2003). [CrossRef]

]. While the refraction angleΔθRis assumed to have a well-defined value at each object plane position, the ultra-small-angle scattering angleΔθSderives from a stochastic process, represented by a certain probability density function. Moreover by collecting a large number of photons, each pixel effectively integrates over this statistical distribution [12

12. L. Rigon, H.-J. Besch, F. Arfelli, R.-H. Menk, G. Heitner, and H. Plothow-Besch, “A new DEI algorithm capable of investigating sub-pixel structures,” J. Phys. D Appl. Phys. 36(10A), A107–A112 (2003). [CrossRef]

].

In the case of ABI, the angular sensitivity of a crystal rocked about its Bragg peak is used in [8

8. L. Rigon, F. Arfelli, and R.-H. Menk, “Three-image diffraction enhanced imaging algorithm to extract absorption, refraction and ultrasmall-angle scattering,” Appl. Phys. Lett. 90(11), 114102 (2007). [CrossRef]

] to decouple the refraction and scattering information from the absorption image. If the measurement is performed with a GI instead, for a certain distance L between the sample and the second grating, we can define refraction and scattering displacement at the position of the second grating asΔxR=LΔθR and ΔxS=LΔθSrespectively. When the sample is placed right before the first grating, L is the just the chosen fractional Talbot distance. Furthermore, let us indicate with f(ΔxS) the probability density that a photon would be scattered byΔxSin the horizontal plane, i.e. in the plane perpendicular to the grating lines. Therefore, adapting Eq. (1) in [8

8. L. Rigon, F. Arfelli, and R.-H. Menk, “Three-image diffraction enhanced imaging algorithm to extract absorption, refraction and ultrasmall-angle scattering,” Appl. Phys. Lett. 90(11), 114102 (2007). [CrossRef]

] to the present case, we can model the intensity transmitted through a GI as:
I(x,y,ξ)=I0(x,y)T(ξΔxRΔxs)f(ΔxS)dΔxS.
(1)
In Eq. (1) T is the transmission function of the GI and ξ is the relative displacement of the two gratings. The function T is periodical with the period p of the Talbot pattern. Equation (1) is formally identical to the corresponding expression used in ABI [8

8. L. Rigon, F. Arfelli, and R.-H. Menk, “Three-image diffraction enhanced imaging algorithm to extract absorption, refraction and ultrasmall-angle scattering,” Appl. Phys. Lett. 90(11), 114102 (2007). [CrossRef]

]. Nonetheless the difference in the transmission function between ABI and GI must be taken into account to correctly interpret this formula.

I(x,y,ξ)=I0(x,y)T(η)f(ξΔxRη)dη=T(ξ)g(ξ).
(5)

The symbol denotes the convolution operator and the last equality holds by definingg(ξ)=I0f(ξΔxR). Therefore, as in [19

19. P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the Ultrasmall-Angle X-Ray Scattering Distribution with Grating Interferometry,” Phys. Rev. Lett. 108(4), 048101 (2012). [CrossRef] [PubMed]

, 20

20. F. Scattarella, S. Tangaro, P. Modregger, M. Stampanoni, L. De Caro, and R. Bellotti, “Post-detection analysis for grating-based ultra-small angle X-ray scattering,” Phys. Med. (2013), doi:. [CrossRef] [PubMed]

], the function g is centered on the value ∆xR of the refraction shift while the scattering contribution produces a decrease in the amplitude of the oscillation of T, corresponding to the standard deviation of the distribution. Hence, while the refraction shift of the oscillatory transmission function is directly measurable, the sole information accessible for scattering is the standard deviation of its distribution, i.e. formally the there is no explicit dependence on η of I(x,y,ξ) in Eq. (5). The scattering displacement ΔxS is effectively treated as a random number that is ensemble-averaged in the convolution process and thereby, assuming that the distribution is symmetric, does not induce a net shift of the transmission function. This concept was already noted in [13

13. P. C. Diemoz, P. Coan, I. Zanette, A. Bravin, S. Lang, C. Glaser, and T. Weitkamp, “A simplified approach for computed tomography with an X-ray grating interferometer,” Opt. Express 19(3), 1691–1698 (2011). [CrossRef] [PubMed]

] where it is stated that positive and negative scattering contributions “cancel out” when the second derivative of the transmission function is zero. As we notice here, in general those contributions do not cancel out but convolute with the transmission function without shifting its center of mass. Accordingly, even when the extension of the tails of the scattering distribution is comparable to the period, one can safely assume Eq. (2) to be approximately valid, when the refraction shift is small compared to the period of T.

Therefore, we can follow the treatment shown in [8

8. L. Rigon, F. Arfelli, and R.-H. Menk, “Three-image diffraction enhanced imaging algorithm to extract absorption, refraction and ultrasmall-angle scattering,” Appl. Phys. Lett. 90(11), 114102 (2007). [CrossRef]

, 9

9. L. Rigon, F. Arfelli, and R.-H. Menk, “Generalized diffraction enhanced imaging to retrieve absorption, refraction and scattering effects,” J. Phys. D Appl. Phys. 40(10), 3077–3089 (2007). [CrossRef]

]. The key idea is that, by acquiring three images at different displacement of the grating, the three unknown quantities in Eq. (2) viz.I0(x,y), ΔθR=ΔxR/L and σS2=(σX/L)2can be obtained. The latter quantities have been introduced to deal with the angular deviation and angular scattered distribution respectively, which are independent of the chosen fractional Talbot distance. The solution is:
I0=(i,j,k=13εijkIiT˙jT¨k)(i,j,k=13εijkTiT˙jT¨k)1,ΔθR=1L(i,j,k=13εijkIiTjT¨k)(i,j,k=13εijkTiT˙jT¨k)1,σS2=2L2(i,j,k=13εijkIiTjT˙k)(i,j,k=13εijkTiT˙jT¨k)1ΔθR2.
(6)
In Eq. (6) εijk is the totally antisymmetric tensor and all quantities are to be considered function of the coordinates, i.e. Ii=I(x,y,ξi)and so forth, i.e. these relations are applied on a pixel-per-pixel basis.

3. Measurement procedure

The novel algorithm presented here uses a similar experimental approach as the PS technique, while introducing a qualitative change in the post-processing approach. The GI setup is made of two gratings: a first (phase) grating used to produce a Talbot pattern downstream of it, reproducing a periodic intensity distribution of period p [5

5. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005). [CrossRef] [PubMed]

] and a second (absorption) grating used to analyze such pattern with and without the sample. The measurement procedure consists in scanning one of the gratings with sub-period steps. In this way the absorption grating is continuously moved with respect to the Talbot pattern (or vice versa) causing a periodical variation of the transmitted intensity, with period p. The transmitted intensity in any given pixel can be written as a Fourier series:
I(x,y,ξ)=I0(x,y)+m=1am(x,y)cos(2πpξ+ϕm(x,y)).
(8)
The PS approach is to approximate Eq. (8) as a cosine graph (i.e. neglecting the m>1 terms in the Fourier expansion) and determine the parametersI0(x,y), a1(x,y) and ϕ1(x,y) via a Fourier analysis of the transmitted intensity in each pixel. In practice these quantities are measured with respect their corresponding value measured in absence of the sample, viz.I0(x,y)=Ios(x,y)/Ior(x,y), ϕ1(x,y)=ϕ1s(x,y)ϕ1r(x,y)anda1(x,y)=a1s(x,y)/a1r(x,y). The superscripts s and r refer to the measurement with the sample and the reference beam (i.e. without the sample) respectively. I0(x,y) is the absorption image of the sample, while the refraction and the visibility images can be computed from ϕ1(x,y) and a1(x,y) as:
ΔθR(x,y)=p2πLϕ1(x,y),V(x,y)V0(x,y)=a1(x,y)I0(x,y).
(9)
Absorption, refraction and visibility images of the sample can be retrieved with the PS method using Eqs. (8) and (9). The same physical quantities can be retrieved with the novel method via Eqs. (6) and (7). For both approaches the minimum number of steps required to find the three parameters is three. Both post-processing approaches are able to access the same physical quantities and can be implemented using the same experimental setup but rely on different approximations.

PS procedure works under the approximation of periodic transmission function and — in the specific case of three-image PS —under the more stringent condition that the transmission function is well described by a cosine term only. On the contrary, the method proposed here does not require a perfect cosine graph, while it needs a very precise knowledge of the reference intensity (without the sample) to guarantee sufficient stability in the numerical computation of the derivatives. These considerations lead to a slightly different experimental approach. For the PS one needs to acquire scans with the same number of steps either with or without the sample. In our alternative method instead the reference intensity must be acquired with a very large number of steps. Then only three-images at specific locations are to be acquired with the sample in the beam.

We performed GI scans and compared the results obtained with our method and with the PS method on the same samples. In Fig. 1
Fig. 1 Measured intensity after the absorption grating in a certain pixel of the detector with coordinate x0 and y0 as a function of the relative position ξ of the two gratins. The blue crosses indicate the position where the images used in Eq. (6) have been acquired. Open red circles indicate the positions used for the PS approach, Eqs. (8) and (9).
the measured reference intensity for a typical GI scan is shown. The total number of steps in one period is 48 in this case. The blue crosses mark the positions where measurements with the sample have been taken to apply Eq. (6). For an honest comparison between GI and PS the photon statistics was kept constant: the respective PS positions used are indicated by the open red circles.

Measurements have been performed on the SYRMEP beamline at the Synchrotron light source Elettra (Trieste, Italy). The incident x-ray beam was monochromatized at the energy of 25 keV by a double-bounce Si 111 monochromator. The beam size at the sample was 3.2 mm vertical and 27 mm horizontal. The GI was composed of two gratings (phase and absorption) separated by a distance L = 58 mm, corresponding to the first fractional Talbot distance for the working parameters. The phase grating had a period of 2.36 μm and Ni structures, while the absorption grating had period 2.4 μm and Au absorbing lines. The period mismatch has been designed to compensate for the beam divergence. All images were recorded using a 16-bit CCD detector (Photonic Science Ltd, Roberts-bridge, UK) with pixel size of 14x14 μm2 placed 10 cm downstream of the absorption grating.

4. Experimental results

In Fig. 2
Fig. 2 (a) Absorption and (b) refraction images obtained with PS and (c)-(d) with the novel algorithm. (e) Line profiles along the rows marked with arrows in (b) and (d), black and red line respectively. (f) Histogram of the refraction angle as obtained by both methods in a region within the sample and (g) in the background region.
absorption and refraction images of a bovine bone sample are shown. A very good agreement is clearly found in the absorption maps obtained with the two methods, shown in the panels (a) and (c), respectively. Refraction images are shown in the panels (b) and (d). The diagram in Fig. 2(e) shows the line profiles of the refraction maps at the position indicated by the arrows. The black solid line corresponds to the PS approach while the red line is obtained using the novel method. The agreement of the two line profiles is quantitatively very good, confirmed by the value of the calculated correlation coefficient R = 0.96. A minor difference is noticeable: while the image in 2(d) possesses higher dynamics, its background noise is higher. This is visually appreciable from the panels 2(f) and 2(g) showing the histograms taken in a region in the middle of the sample and in the background region respectively. As before the black line corresponds to PS and the red line to the novel method. Both distributions obtained with the novel method are consistently wider, indicating larger dynamics and larger fluctuations associated with the background region, (panel 2(g)). This effect is a direct consequence of the application of Eq. (6): the computation of the local derivatives produces higher dynamics with respect to the “non-local” Fourier analysis. On the other hand the numerical evaluation of the derivatives is also more sensitive to noise, resulting in images with noisier background. It is important to reiterate though that the differences are actually minor and the results obtained with the two methods are largely consistent.

Another quantification of the similarity of the two results is given by the histograms of the counts, from the retrieved maps in Figs. 3(a) and 3(b). The histograms are superimposed in Fig. 3(d) where the black line is the histogram of Fig. 3(a) and the red line of Fig. 3(b). The histograms represent the frequency of pixels counting within a certain range. The peaks around the zero come from the background noise, while the broad peak on the right is the scattering signal. The map obtained with the novel algorithm shows a wider distribution that, from Fig. 3(c), it is seen to correspond to the low absorbing part of the sample. Despite the small differences the results obtained with both methods are consistent, thus confirming the validity of the new approach.

It is worth remarking that the possibility of measuring the variance (and higher moments) of the scattering distribution using a GI without PS has been already shown in [19

19. P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the Ultrasmall-Angle X-Ray Scattering Distribution with Grating Interferometry,” Phys. Rev. Lett. 108(4), 048101 (2012). [CrossRef] [PubMed]

, 20

20. F. Scattarella, S. Tangaro, P. Modregger, M. Stampanoni, L. De Caro, and R. Bellotti, “Post-detection analysis for grating-based ultra-small angle X-ray scattering,” Phys. Med. (2013), doi:. [CrossRef] [PubMed]

]. In that case the scattering information has been obtained by deconvolution of the transmission function measured with the sample by the same curve obtained in absence of the sample. When more than three steps are used, a map for each position along the PS curve could be measured and higher order momenta of the scattering distribution become accessible. As we have shown in Sec. 2, the method presented here can be viewed as a special case of the one in [19

19. P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the Ultrasmall-Angle X-Ray Scattering Distribution with Grating Interferometry,” Phys. Rev. Lett. 108(4), 048101 (2012). [CrossRef] [PubMed]

, 20

20. F. Scattarella, S. Tangaro, P. Modregger, M. Stampanoni, L. De Caro, and R. Bellotti, “Post-detection analysis for grating-based ultra-small angle X-ray scattering,” Phys. Med. (2013), doi:. [CrossRef] [PubMed]

], with the important advantage of simple and numerically stable post-processing and low dose because of the need of only three images.

A very important consequence of the novel approach is the possibility of using only two images for the reconstruction of the absorption and refraction images, as in the original DEI algorithm. Mathematically this corresponds to neglecting T¨ (negligible scattering) in Eq. (2) and therefore solving it for the absorption and refraction only. To test this procedure we post-processed the bovine bone images with this simplified method. Results are displayed in Figs. 4(a)
Fig. 4 (a) Absorption and (b) refraction image of the bovine bone sample obtained with a two-image algorithm. (c) Absorption and (d) refraction images (in μrad) of the nylon wire obtained with the two-image method and (e), (f) corresponding images obtained by PS.
and 4(b) for the apparent absorption and refraction respectively. While the absorption image appears to be little affected by the reduced statistics, the refraction map shows distortions, especially in the region close to the middle of the sample where absorption and scattering are stronger.

On the other hand, an example of the same method applied to a sample producing negligible scattering – a nylon wire, with diameter 0.8 mm – is shown in Figs. 4(c)4(f). In this case the approximation leading to neglect the second derivative is well justified. Absorption and refraction images obtained with only two steps (Figs. 4(c) and 4(d)) are compatible with the corresponding images obtained by the PS method (Figs. 4(e) and 4(f)) taking into account the better statistics of the latter.

5. Discussion

PS method is based on the periodicity of the transmission function of the GI. The principle of the method presented here, initially introduced as a generalization of the DEI algorithm [8

8. L. Rigon, F. Arfelli, and R.-H. Menk, “Three-image diffraction enhanced imaging algorithm to extract absorption, refraction and ultrasmall-angle scattering,” Appl. Phys. Lett. 90(11), 114102 (2007). [CrossRef]

], stands simply on a variable transmission function such as a crystal’s rocking curve. More specifically one needs to find three points where the transmission function T(x) and its derivatives take distinct values, to have the system in Eq. (6) well posed. An ideal GI with square grating profile and fully coherent illumination is characterized by a triangular transmission function. Deviations of the gratings shape from the nominal profile, along with partially coherent illumination contribute in “smoothing” the triangular function making it well approximated with a first order Fourier series. It is worth pointing out that in our experimental case the GI transmission function was actually well approximated by a simple cosine therefore the PS approach with three steps was well suited.

However, the post-processing method introduced here is qualitatively different from PS as it relies on a Taylor expansion of the transmitted intensity. From a general point of view the experimental demonstration of the applicability of the novel algorithm to the GI setup provides a very interesting bridge between the two widely used methods of GI and ABI. The result itself is not surprising as both techniques are sensitive to the same physical quantities. Despite slightly different theoretical considerations required to use of Eq. (6) with GI, the results are quantitatively very similar to PS technique. This would potentially offer a way to quantitatively compare ABI and GI in a way that has not been done thus far.

Considering GI alone, one might envisage specific experimental situations in which one of the two post-processing methods can be preferable. For instance the application of Fourier analysis for PS requires the working positions to be equally spaced along the oscillation of the transmission function. This requirement is dropped in the new method that can perform also when the spacing between the points is not equal. Furthermore the presence of higher order harmonics in the transmission function may lead to artifact in the PS method [27

27. A. Momose, W. Yashiro, and Y. Takeda, “X-Ray Phase Imaging with Talbot Interferometry” in Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning, and Inverse Problems, Y. Censor, M. Jiang, G. Wang, ed. (Medical Physics Publishing, 2009).

], especially when only three steps are used. This constraint is removed in the novel method while, on the other hand, the numerical computation of the local derivatives of the transmission function may increase the noise of the computed images.

It is worth remarking that the measurements, for both, our method and the PS, are always taken with reference to the measurement without the sample. This means that when measuring with the sample, the actual location of the measurement positions along the curve is not crucial, while the important parameter is the relative spacing between working points. Nevertheless two main differences do exist: As already noticed in Sec. 3, the reference scan in the novel method must be acquired with a large number of steps (unlike PS) to attain a reliable numerical estimation of the derivatives needed to apply Eq. (6). Furthermore since quantities obtained with PS are always calculated relatively to the corresponding quantities measured without the sample, the two scans (with and without the sample) should be in phase. However if a small fixed phase existed between the sample scan and the reference scan this would only cause a small constant offset to ϕ1(x,y) (see Eqs. (8)(9)). On the other hand the restriction to acquire the sample scan exactly in phase with the reference scan must be enforced for the novel method. In other words the position along the transmission function where the three images are acquired has to be known with sufficiently good accuracy with respect to the reference scan to avoid reconstruction errors. Therefore particular care must be taken that both, the reference scan and the three-step scan used in the phase retrieval actually have the same starting position for an accurate estimation of the refraction shift.

The general methodology of using three or two images allows for a sensible reduction of the radiation dose, especially extending the methodology to tomography [25

25. P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based X-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. U.S.A. 107(31), 13576–13581 (2010). [CrossRef] [PubMed]

, 26

26. I. Zanette, M. Bech, A. Rack, G. Le Duc, P. Tafforeau, C. David, J. Mohr, F. Pfeiffer, and T. Weitkamp, “Trimodal low-dose X-ray tomography,” Proc. Natl. Acad. Sci. U.S.A. 109(26), 10199–10204 (2012). [CrossRef] [PubMed]

]. Therefore our approach paves the way for using GI at bio-compatible doses retaining the key advantages of this system, namely the robustness against partial spatial and spectral coherence and mechanical issues.

6. Conclusions

As alternative to conventional phase stepping we have presented a new method to record absorption, refraction and scattering from a sample using a hard x-ray grating interferometer. It utilizes three images acquired at different position along the transmission function of the interferometer. Differences in data acquisition and analysis between the novel method and the phase stepping are presented. The results obtained with the two methods are quantitatively compared and a very good agreement is found. A further approximation of the method, using only two images is presented and discussed with the potential of obtaining quantitative absorption and refraction imaging with bio compatible radiation dose. The post-processing method makes use of the same equations already derived for DEI, thus providing a good experimental verification of the formal and practical analogies between grating interferometry and analyzer-based imaging.

Acknowledgments

The authors acknowledge the support from the European Research Infrastructure EUMINAFAB Grant No. FP7-226460. D. Kunka and J. Mohr are acknowledged for the fabrication of the gratings. S. Lagomarsino and M. Fratini for providing the sample, D. Dreossi, N. Sodini and F. Scarinci for technical assistance. D.P. acknowledges travel funding provided by the International Synchrotron Access Program ISP4372 managed by the Australian Synchrotron and funded by the Australian Government. The authors acknowledge the INFN of Italy (Istituto di Fisica Nucleare) for partial financial support.

References and links

1.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58(1), R1–R35 (2013). [CrossRef] [PubMed]

2.

U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6(8), 155–156 (1965). [CrossRef]

3.

A. Momose, “Demonstration of phase-contrast x-ray computed-tomography using an x-ray interferometer,” Nucl. Instrum. Meth. A 352(3), 622–628 (1995). [CrossRef]

4.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003). [CrossRef]

5.

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005). [CrossRef] [PubMed]

6.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7(2), 134–137 (2008). [CrossRef] [PubMed]

7.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. H. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997). [CrossRef] [PubMed]

8.

L. Rigon, F. Arfelli, and R.-H. Menk, “Three-image diffraction enhanced imaging algorithm to extract absorption, refraction and ultrasmall-angle scattering,” Appl. Phys. Lett. 90(11), 114102 (2007). [CrossRef]

9.

L. Rigon, F. Arfelli, and R.-H. Menk, “Generalized diffraction enhanced imaging to retrieve absorption, refraction and scattering effects,” J. Phys. D Appl. Phys. 40(10), 3077–3089 (2007). [CrossRef]

10.

P. C. Diemoz, A. Bravin, and P. Coan, “Theoretical comparison of three X-ray phase-contrast imaging techniques: Propagation-based imaging, analyzer-based imaging and grating interferometry,” Opt. Express 20(3), 2789–2805 (2012). [CrossRef] [PubMed]

11.

F. Arfelli, D. Pelliccia, A. Cedola, A. Astolfo, I. Bukreeva, P. Cardarelli, D. Dreossi, S. Lagomarsino, R. Longo, L. Rigon, N. Sodini, and R. H. Menk, “Recent developments on techniques for differential phase imaging at the medical beamline of ELETTRA,” J. Instrum. 8(06), 06001 (2013). [CrossRef]

12.

L. Rigon, H.-J. Besch, F. Arfelli, R.-H. Menk, G. Heitner, and H. Plothow-Besch, “A new DEI algorithm capable of investigating sub-pixel structures,” J. Phys. D Appl. Phys. 36(10A), A107–A112 (2003). [CrossRef]

13.

P. C. Diemoz, P. Coan, I. Zanette, A. Bravin, S. Lang, C. Glaser, and T. Weitkamp, “A simplified approach for computed tomography with an X-ray grating interferometer,” Opt. Express 19(3), 1691–1698 (2011). [CrossRef] [PubMed]

14.

M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol. 48(23), 3875–3895 (2003). [CrossRef] [PubMed]

15.

G. Khelashvili, J. G. Brankov, D. Chapman, M. A. Anastasio, Y. Yang, Z. Zhong, and M. N. Wernick, “A physical model of multiple-image radiography,” Phys. Med. Biol. 51(2), 221–236 (2006). [CrossRef] [PubMed]

16.

L. Rigon, A. Astolfo, F. Arfelli, and R.-H. Menk, “Generalized diffraction enhanced imaging: Application to tomography,” Eur. J. Radiol. 68(3Suppl), S3–S7 (2008). [CrossRef] [PubMed]

17.

Ya. I. Nesterets, “On the origin of decoherence and extinction contrast in phase-contrast imaging,” Opt. Commun. 281(4), 533–542 (2008). [CrossRef]

18.

Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95(9), 094105 (2009). [CrossRef]

19.

P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the Ultrasmall-Angle X-Ray Scattering Distribution with Grating Interferometry,” Phys. Rev. Lett. 108(4), 048101 (2012). [CrossRef] [PubMed]

20.

F. Scattarella, S. Tangaro, P. Modregger, M. Stampanoni, L. De Caro, and R. Bellotti, “Post-detection analysis for grating-based ultra-small angle X-ray scattering,” Phys. Med. (2013), doi:. [CrossRef] [PubMed]

21.

W. Yashiro, Y. Terui, K. Kawabata, and A. Momose, “On the origin of visibility contrast in x-ray Talbot interferometry,” Opt. Express 18(16), 16890–16901 (2010). [CrossRef] [PubMed]

22.

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55(18), 5529–5539 (2010). [CrossRef] [PubMed]

23.

S. K. Lynch, V. Pai, J. Auxier, A. F. Stein, E. E. Bennett, C. K. Kemble, X. Xiao, W.-K. Lee, N. Y. Morgan, and H. H. Wen, “Interpretation of dark-field contrast and particle-size selectivity in grating interferometers,” Appl. Opt. 50(22), 4310–4319 (2011). [CrossRef] [PubMed]

24.

H. Wen, E. E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of x-ray scattering--initial results,” IEEE Trans. Med. Imaging 27(8), 997–1002 (2008). [CrossRef] [PubMed]

25.

P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based X-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. U.S.A. 107(31), 13576–13581 (2010). [CrossRef] [PubMed]

26.

I. Zanette, M. Bech, A. Rack, G. Le Duc, P. Tafforeau, C. David, J. Mohr, F. Pfeiffer, and T. Weitkamp, “Trimodal low-dose X-ray tomography,” Proc. Natl. Acad. Sci. U.S.A. 109(26), 10199–10204 (2012). [CrossRef] [PubMed]

27.

A. Momose, W. Yashiro, and Y. Takeda, “X-Ray Phase Imaging with Talbot Interferometry” in Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning, and Inverse Problems, Y. Censor, M. Jiang, G. Wang, ed. (Medical Physics Publishing, 2009).

OCIS Codes
(100.0100) Image processing : Image processing
(340.7440) X-ray optics : X-ray imaging
(340.7450) X-ray optics : X-ray interferometry

ToC Category:
X-ray Optics

History
Original Manuscript: July 5, 2013
Revised Manuscript: August 1, 2013
Manuscript Accepted: August 1, 2013
Published: August 8, 2013

Virtual Issues
Vol. 8, Iss. 9 Virtual Journal for Biomedical Optics

Citation
Daniele Pelliccia, Luigi Rigon, Fulvia Arfelli, Ralf-Hendrik Menk, Inna Bukreeva, and Alessia Cedola, "A three-image algorithm for hard x-ray grating interferometry," Opt. Express 21, 19401-19411 (2013)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-16-19401


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol.58(1), R1–R35 (2013). [CrossRef] [PubMed]
  2. U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett.6(8), 155–156 (1965). [CrossRef]
  3. A. Momose, “Demonstration of phase-contrast x-ray computed-tomography using an x-ray interferometer,” Nucl. Instrum. Meth. A352(3), 622–628 (1995). [CrossRef]
  4. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys.42(Part 2, No. 7B), L866–L868 (2003). [CrossRef]
  5. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express13(16), 6296–6304 (2005). [CrossRef] [PubMed]
  6. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, Ch. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater.7(2), 134–137 (2008). [CrossRef] [PubMed]
  7. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. H. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol.42(11), 2015–2025 (1997). [CrossRef] [PubMed]
  8. L. Rigon, F. Arfelli, and R.-H. Menk, “Three-image diffraction enhanced imaging algorithm to extract absorption, refraction and ultrasmall-angle scattering,” Appl. Phys. Lett.90(11), 114102 (2007). [CrossRef]
  9. L. Rigon, F. Arfelli, and R.-H. Menk, “Generalized diffraction enhanced imaging to retrieve absorption, refraction and scattering effects,” J. Phys. D Appl. Phys.40(10), 3077–3089 (2007). [CrossRef]
  10. P. C. Diemoz, A. Bravin, and P. Coan, “Theoretical comparison of three X-ray phase-contrast imaging techniques: Propagation-based imaging, analyzer-based imaging and grating interferometry,” Opt. Express20(3), 2789–2805 (2012). [CrossRef] [PubMed]
  11. F. Arfelli, D. Pelliccia, A. Cedola, A. Astolfo, I. Bukreeva, P. Cardarelli, D. Dreossi, S. Lagomarsino, R. Longo, L. Rigon, N. Sodini, and R. H. Menk, “Recent developments on techniques for differential phase imaging at the medical beamline of ELETTRA,” J. Instrum.8(06), 06001 (2013). [CrossRef]
  12. L. Rigon, H.-J. Besch, F. Arfelli, R.-H. Menk, G. Heitner, and H. Plothow-Besch, “A new DEI algorithm capable of investigating sub-pixel structures,” J. Phys. D Appl. Phys.36(10A), A107–A112 (2003). [CrossRef]
  13. P. C. Diemoz, P. Coan, I. Zanette, A. Bravin, S. Lang, C. Glaser, and T. Weitkamp, “A simplified approach for computed tomography with an X-ray grating interferometer,” Opt. Express19(3), 1691–1698 (2011). [CrossRef] [PubMed]
  14. M. N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N. P. Galatsanos, Y. Yang, J. G. Brankov, O. Oltulu, M. A. Anastasio, and C. Muehleman, “Multiple-image radiography,” Phys. Med. Biol.48(23), 3875–3895 (2003). [CrossRef] [PubMed]
  15. G. Khelashvili, J. G. Brankov, D. Chapman, M. A. Anastasio, Y. Yang, Z. Zhong, and M. N. Wernick, “A physical model of multiple-image radiography,” Phys. Med. Biol.51(2), 221–236 (2006). [CrossRef] [PubMed]
  16. L. Rigon, A. Astolfo, F. Arfelli, and R.-H. Menk, “Generalized diffraction enhanced imaging: Application to tomography,” Eur. J. Radiol.68(3Suppl), S3–S7 (2008). [CrossRef] [PubMed]
  17. Ya. I. Nesterets, “On the origin of decoherence and extinction contrast in phase-contrast imaging,” Opt. Commun.281(4), 533–542 (2008). [CrossRef]
  18. Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z.-Q. Chen, “Quantitative grating-based x-ray dark-field computed tomography,” Appl. Phys. Lett.95(9), 094105 (2009). [CrossRef]
  19. P. Modregger, F. Scattarella, B. R. Pinzer, C. David, R. Bellotti, and M. Stampanoni, “Imaging the Ultrasmall-Angle X-Ray Scattering Distribution with Grating Interferometry,” Phys. Rev. Lett.108(4), 048101 (2012). [CrossRef] [PubMed]
  20. F. Scattarella, S. Tangaro, P. Modregger, M. Stampanoni, L. De Caro, and R. Bellotti, “Post-detection analysis for grating-based ultra-small angle X-ray scattering,” Phys. Med. (2013), doi:. [CrossRef] [PubMed]
  21. W. Yashiro, Y. Terui, K. Kawabata, and A. Momose, “On the origin of visibility contrast in x-ray Talbot interferometry,” Opt. Express18(16), 16890–16901 (2010). [CrossRef] [PubMed]
  22. M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol.55(18), 5529–5539 (2010). [CrossRef] [PubMed]
  23. S. K. Lynch, V. Pai, J. Auxier, A. F. Stein, E. E. Bennett, C. K. Kemble, X. Xiao, W.-K. Lee, N. Y. Morgan, and H. H. Wen, “Interpretation of dark-field contrast and particle-size selectivity in grating interferometers,” Appl. Opt.50(22), 4310–4319 (2011). [CrossRef] [PubMed]
  24. H. Wen, E. E. Bennett, M. M. Hegedus, and S. C. Carroll, “Spatial harmonic imaging of x-ray scattering--initial results,” IEEE Trans. Med. Imaging27(8), 997–1002 (2008). [CrossRef] [PubMed]
  25. P. Zhu, K. Zhang, Z. Wang, Y. Liu, X. Liu, Z. Wu, S. A. McDonald, F. Marone, and M. Stampanoni, “Low-dose, simple, and fast grating-based X-ray phase-contrast imaging,” Proc. Natl. Acad. Sci. U.S.A.107(31), 13576–13581 (2010). [CrossRef] [PubMed]
  26. I. Zanette, M. Bech, A. Rack, G. Le Duc, P. Tafforeau, C. David, J. Mohr, F. Pfeiffer, and T. Weitkamp, “Trimodal low-dose X-ray tomography,” Proc. Natl. Acad. Sci. U.S.A.109(26), 10199–10204 (2012). [CrossRef] [PubMed]
  27. A. Momose, W. Yashiro, and Y. Takeda, “X-Ray Phase Imaging with Talbot Interferometry” in Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning, and Inverse Problems, Y. Censor, M. Jiang, G. Wang, ed. (Medical Physics Publishing, 2009).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited