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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 12 — Jun. 7, 2010
  • pp: 12960–12970
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Small-angle scattering computed tomography (SAS-CT) using a Talbot-Lau interferometer and a rotating anode x-ray tube: theory and experiments

Guang-Hong Chen, Nicholas Bevins, Joseph Zambelli, and Zhihua Qi  »View Author Affiliations


Optics Express, Vol. 18, Issue 12, pp. 12960-12970 (2010)
http://dx.doi.org/10.1364/OE.18.012960


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Abstract

X-ray differential phase contrast imaging methods, including projection imaging and the corresponding computed tomography (CT), have been implemented using a Talbot interferometer and either a synchrotron beam line or a low brilliance x-ray source generated by a stationary-anode x-ray tube. From small-angle scattering events which occur as an x-ray propagates through a medium, a signal intensity loss can be recorded and analyzed for an understanding of the micro-structures in an image object. This has been demonstrated using a Talbot-Lau interferometer and a stationary-anode x-ray tube. In this paper, theoretical principles and an experimental implementation of the corresponding CT imaging method are presented. First, a line integral is derived from analyzing the cross section of the small-angle scattering events. This method is referred to as small-angle scattering computed tomography (SAS-CT). Next, a Talbot-Lau interferometer and a rotating-anode x-ray tube were used to implement SAS-CT. A physical phantom and human breast tissue sample were used to demonstrate the reconstructed SAS-CT image volumes.

© 2010 OSA

1. Introduction

In recent years, important progress has been made in x-ray differential phase contrast CT imaging [1

1. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), 866–868 (2003). [CrossRef]

11

11. Z.-F. Huang, K.-J. Kang, L. Zhang, Z. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, “Alternative method for differential phase contrast imaging with weakly coherent hard x-rays,” Phys. Rev. A 79(1), 013815 (2009). [CrossRef]

] using a Talbot interferometer. The method was implemented using either synchrotron beam lines [1

1. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), 866–868 (2003). [CrossRef]

,2

2. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45(No. 6A), 5254–5262 (2006). [CrossRef]

,12

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

16

16. A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express 17(15), 12540–12545 (2009). [CrossRef] [PubMed]

] with high brilliance and high spatial and spectral coherence or a conventional x-ray tube [4

4. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006). [CrossRef]

7

7. M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High resolution differential phase contrast imaging using a magnifying projection geometry with micro-focus x-ray source,” Appl. Phys. Lett. 90(22), 224101 (2007). [CrossRef]

,11

11. Z.-F. Huang, K.-J. Kang, L. Zhang, Z. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, “Alternative method for differential phase contrast imaging with weakly coherent hard x-rays,” Phys. Rev. A 79(1), 013815 (2009). [CrossRef]

,17

17. T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance source,” Proc. SPIE 6318, 631828 (2006).

,18

18. A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-Based X-ray Phase Imaging Using Multiline X-ray Source,” Jpn. J. Appl. Phys. 48(7), 076512 (2009). [CrossRef]

] with low brilliance. When this method is compared with other phase contrast imaging methods, such as diffraction enhanced imaging [19

19. F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45(4), 933–946 (2000). [CrossRef] [PubMed]

24

24. P. P. Zhu, J. Y. Wang, Q. X. Yuan, W. X. Huang, H. Shu, B. Gao, T. D. Hu, and Z. Y. Wu, “Computed tomography algorithm based on diffraction-enhanced imaging setup,” Appl. Phys. Lett. 87(26), 264101 (2005). [CrossRef]

] or in-line holography [25

25. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996). [CrossRef]

28

28. D. Zhang, M. Donovan, L. L. Fajardo, A. Archer, X. Wu, and H. Liu, “Preliminary feasibility study of an in-line phase contrast X-ray imaging prototype,” IEEE Trans. Biomed. Eng. 55(9), 2249–2257 (2008). [CrossRef] [PubMed]

], the requirements of spatial coherence and spectral coherence are reduced. A conventional x-ray tube with a focal spot on the order of a millimeter does not provide sufficient spatial coherence length for Talbot interferometry. However, this technical hurdle can be easily overcome [3

3. 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]

,4

4. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006). [CrossRef]

] by additional beam collimation using an absorption linear grating with slit opening of several microns. As a result, a conventional medical x-ray tube can be used to implement phase contrast imaging. The result is a reduction in the distance from the x-ray focal spot to detector compared with in-line holography. A micro-focus x-ray tube [29

29. M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microsc. 232(1), 145–157 (2008). [CrossRef] [PubMed]

] can also be used to implement differential phase contrast CT with a compact system design.

When an x-ray beam interacts with a medium, x-ray photons may be absorbed by the medium via photoelectric effect. X-ray photons may also be scattered by electrons via inelastic Compton scattering. These scattered photons have a wide distribution across the entire image object. The above two processes are the physical foundation of traditional absorption x-ray CT imaging method and have been well studied [30

30. J. R. Bushberg, J. A. Seibert, M. E. Leidholdt, Jr., and J. M. Boone, The essential physics of medical imaing (Lippincott Wilkiams & Wilkins, Philadelphia, 2001).

,31

31. A. C. Kak, and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

]. Due to the wave nature of the x-ray beam, a phase shift is also induced when the beam propagates through the medium. As a result, the incident x-ray beam will deviate from its incident direction, leading to refraction of the beam. This phenomenon is the physical foundation of and has been explored in diffraction enhanced imaging and differential phase contrast imaging methods.

In differential phase contrast imaging, an implicit assumption is that the medium can be considered as many uniform sub-regions such that wave front distortion is the primary effect when the beam propagates through the medium. This condition is challenged by small-angle scattering (SAS) events in the above piece-wise uniform assumption. In fact, when the mean size of the “particulate scatter” [32

32. A. Guinier, “Diffraction of X-rays at Small Angles: Application to the Study of Microscopic Phenomena,” Ann. Phys. 12, 161 (1939).

,33

33. O. Glatter, and O. Kratky, eds., Small Angle X-ray Scattering (Academic Press, New York, 1982).

] is much greater than the wavelength of the coherent x-ray beam, x-ray small-angle coherent scattering events will cause an intensity reduction in a small cone shape region around the incident beam direction [33

33. O. Glatter, and O. Kratky, eds., Small Angle X-ray Scattering (Academic Press, New York, 1982).

]. The scattering angle Δθ~(ka)1, for a given x-ray energy, and thus wave number, k, is inversely proportional to the size of the scatter, a [33

33. O. Glatter, and O. Kratky, eds., Small Angle X-ray Scattering (Academic Press, New York, 1982).

,34

34. L. D. Landau, and E. M. Lifshitz, Quantum Mechanics, 3rd Edition ed. (Pergamon Press, New York, 1977).

].

In the hard x-ray energy regime, the x-ray wavelength is shorter than 0.1 nm. Thus, the scatterers with a mean size from tens of nanometers to tens or hundreds of micrometers will contribute to the SAS cross section. As a result, analysis of SAS may provide a powerful tool to understand the micro-structure of the medium. In fact, small-angle x-ray scattering techniques have been well-established for studying features of colloidal size [33

33. O. Glatter, and O. Kratky, eds., Small Angle X-ray Scattering (Academic Press, New York, 1982).

]. It has been widely used in determining the structure of biological samples such as proteins, lipoproteins and membranes, and nucleic acids.

In Talbot-Lau interferometer based differential phase contrast imaging experiments, the above SAS effect is encoded in the detected intensity profile [5

5. 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]

,8

8. F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. 105(10), 102006 (2009). [CrossRef]

]. Due to the nature of SAS, it is expected that the intensity reduction caused by SAS is manifested as a change in the amplitude of the intensity modulation profile:
I(x,y)I0+I1cos[2πp2xg+φ(x,y)]  ,
(1)
where I 0 represents mean intensity measured at the given detector element (x,y), I 1 is the amplitude of the intensity modulation, φ(x,y)is the phase shift, p 2 is the period of the analyzer grating, and xg is the position of the G2 grating in the phase stepping procedure [1

1. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), 866–868 (2003). [CrossRef]

4

4. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006). [CrossRef]

]. It has been argued [5

5. 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]

] that the ratio of the modulation visibility, V=I1/I0, measured with and without an image object can be defined as the dark field imaging signal caused by the SAS events. Indeed, experimental results demonstrate that some micro-structures can be observed in the dark field signal defined above, but not in the absorption or differential phase contrast signals. These findings indicate the rich contrast information provided by a single data acquisition process with absorption contrast, phase contrast, and SAS-induced dark field contrast all present. The above three contrast mechanisms may potentially provide complimentary information about the image object.

In this paper, we present a theoretical and experimental study to address the above two issues. A microscopic model is developed to demonstrate that the modulation visibility, V=I1/I0, is indeed the intensity reduction factor caused by the small-angle scatterers. The logarithmic of the final intensity reduction factor is a line integral of a combination of physical quantities used to characterize the scatterer size and number density. As a result, a tomographic reconstruction algorithm for absorption CT can be used to directly reconstruct the physical parameters to understand the local distribution of the small-angle scatterers. Finally, we conducted experiments using both a physical phantom and a breast tissue sample to demonstrate the above imaging models.

2. Small-angle scattering imaging model

2.1 Notations and small-angle approximations

Assume that the incident x-ray beam is described by a wave vector k with amplitude |k|=k=2π/λ, where λ is the wavelength. After the incident wave is scattered once, the exit wave vector is determined by the wave vector transfer q:

k'=q+k  .
(2)

Since the SAS event under consideration is elastic, |k'|=|k|=k. The angle between k and k' is the scattering angle θ. Using a simple geometric analysis and small-angle approximation,

q=2ksinθ2kθ  .
(3)

Under the same approximation, the solid angle dΩ=sinθdθdϕcan be approximated:

dΩqdqdϕk2=λ24π2qdqdϕ  .
(4)

Namely, an integral over a solid angle also represents an integral of all possible wave vector transfers q under the small-angle approximation.

2.2 Recursive relation of intensity change in the small-angle scattering process

According to the scattering theory, after a scattering event, the probability of finding one photon in a solid angle dΩ=sinθdθdϕ in the direction (θ,ϕ) is given byP1(q)dΩ. Here P1(q)=r02|f(q)|2is determined by the form factor f(q)of the scatterer, where r 0 is the classical electron radius.

When an x-ray wave propagates through a medium along the z-direction, it experiences many SAS events. Assume the x-ray wave experiences (n-1) SAS events by the time it reaches position z along the propagation direction (Fig. 1
Fig. 1 Illustration of geometry used in deriving the basic equations in SAS-CT.
). The wave is then scattered one more time within line element dz such that the wave experiences n SAS events at the z+dz position.

The angular distribution of photons elastically scattered (n-1) times is Pn1(q1), the probability of finding one photon in a unit solid angle with a wave vector transfer q 1. Suppose q 2 is the wave vector transfer due to the single SAS event which occurred in the distance dz. The probability distribution of finding one scattered photon in a unit solid angle is given by [35

35. T. M. Sabine and W. K. Bertram, “The use of multiple-scattering data to enhance small-angle neutron scattering experiments,” Acta Crystallogr. A 55(3), 500–507 (1999). [CrossRef]

]

Pn(q)=dΩ1Pn1(q1)P1(qq1)  .
(5)

Although the integral is formally written as over all possible directions, as shown in Eq. (4), the above integral also implies that the integral is performed over the amplitude of the vector q 1 when small scattering angle approximation is used. Equation (5) therefore implies that the probability of scattering n times is a convolution between probability of scattering (n-1) times events and a consecutive single scattering event.

As is well known in scattering theory, the Fourier transform of scattering intensity distribution Pn(q)gives the corresponding normalized signal intensity Sn(r)measured at the detector plane. Therefore, if we define a Fourier transform,
Sn(r)=dΩPn(q)eiqr  ,
(6)
then Eq. (5) can be rewritten into the following recursive relation,

Sn(r)=Sn1(r)S1(r)  .
(7)

Using Eq. (7), the following important relation can be derived:

Sn(r)=[S1(r)]n  .
(8)

Namely, the final effect of the SAS events on signal intensity is a simple product of the effect of all individual scattering events [35

35. T. M. Sabine and W. K. Bertram, “The use of multiple-scattering data to enhance small-angle neutron scattering experiments,” Acta Crystallogr. A 55(3), 500–507 (1999). [CrossRef]

]. Note that this is significantly different from intuitive arguments [36

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

,37

37. M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, “Neutron dark-field tomography,” Phys. Rev. Lett. 101(12), 123902 (2008). [CrossRef] [PubMed]

] where the effect of SAS is treated as a convolution between a broadening kernel and the signal intensity without SAS.

2.3 Gaussian approximation and signal intensity reduction factor

For a single scattering event, the resulting angular distribution can be approximated as a Gaussian function [35

35. T. M. Sabine and W. K. Bertram, “The use of multiple-scattering data to enhance small-angle neutron scattering experiments,” Acta Crystallogr. A 55(3), 500–507 (1999). [CrossRef]

]:
P1(q)=(2πλ)2R2πeq2R2  ,
(9)
where R is the characteristic size of the scatters. When the condition qR<<1 is satisfied, which is often true in SAS processes, the above Gaussian approximation can be shown to be rather robust.

A straightforward calculation of the Fourier integral gives the corresponding spatial distribution of the signal intensity,

S1(r)=e(r2R)2  .
(10)

Therefore, when an individual small-angle scattering event occurs, based on the characteristic size of the scatterers, the final signal intensity is reduced by a Gaussian factor.

2.4 Fundamental imaging equation in SAS-CT

In this subsection, based on the above theoretical framework, the basic imaging equation for SAS-CT is derived. Note that the probability of one SAS event within a propagation distance dz is proportional to
dzSAS  ,
(11)
where SAS is the mean free path of SAS events. The mean free path is defined as
SAS=1σSASρSAS(x,y,z),
(12)
where σSAS is the small-angle scattering cross section and ρ SAS(x, y, z) is number density of the small-angle scatterers. Using the derived relation in Eq. (8) and the Gaussian approximation in Eq. (10), the final Gaussian reduction factor due to small-angle scattering events is

U(r)=exp[r24dzσSASρSASR2(z)]  .
(13)

Equation (13) states that after all single SAS events, within the Gaussian approximation, the final signal intensity is reduced by an Gaussian factor compared to the signal intensity which has not been affected by SAS processes. The final Gaussian factor with the effective width, R eff, is given by

U(r)=exp[r24Reff2]  , and
(14)
1Reff2=dzσSASρSASR2(z)  .
(15)

Based on Eq. (1), the modulation visibility with an object in place is defined by
Vobj(m,n)I1obj(m,n)I0obj(m,n)  ,
(16)
which is then normalized to the visibility of the background to determine the visibility reduction due to small-angle scattering events,

VSAS(m,n)Vobj(m,n)Vbkgd(m,n)=I1obj(m,n)I0obj(m,n)I0bkgd(m,n)I1bkgd(m,n)  .
(17)

Since the spatial variable r represents the transverse distance measured from the incident wave direction, Eq. (13) dictates that there is no signal reduction at r = 0. This corresponds to the DC component in signal intensity Eq. (1) for the differential phase contrast CT data acquisition. Therefore, the normalized modulation visibility, V SAS, is nothing but the signal intensity reduction factor given by Eq. (13):

VSAS=U(r)=exp[r24dzσSASρSASR2(z)].
(18)

From this, the fundamental imaging equation for SAS-CT is obtained:
lnVSAS=r24dzσSASρSASR2(z)=dzσSASρSASR˜2(z),
(19)
where the characteristic size of the scatterer is measured by the dimensionless quantity, R˜(z), defined as

R˜(z)=2R(z)r.
(20)

Equation (19) gives the relation of a physically measurable quantity, lnV SAS, and the microscopic nature of the small-angle scatterers, in terms of both their characteristic size and local density.

2.5 Image reconstruction algorithm in SAS-CT

The fundamental imaging equation of SAS-CT, Eq. (19), is of the same form as that for absorption CT. Therefore, the well-known filtered backprojection (FBP) algorithm with a ramp filter can be used to reconstruct SAS-CT images. By extension, an absorption cone-beam CT image reconstruction algorithm can be directly applied to SAS-CT without modifications. In this paper, the FDK reconstruction algorithm [38

38. L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-bam algorithms,” J. Opt. Soc. Am. A 1(6), 612–619 (1984). [CrossRef]

] is used to reconstruct images in the experimental studies described below.

3. Experimental methods

All data for the results presented in this paper were collected from a Talbot-Lau interferometer experimental setup constructed at the University of Wisconsin-Madison, as shown in Fig. 2
Fig. 2 Photographs of the grating interferometer x-ray system. Subfigure (a) shows the x-ray tube and the G0 grating, while (b) shows the detector, G1, and G2 gratings.
. The data acquisition system is similar to those previously reported [6

6. F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98(10), 108105 (2007). [CrossRef] [PubMed]

], except for the use of a rotating-anode x-ray tube, compared with a stationary-anode tube used by other groups.

The system subcomponents used in the experimental setup include x-ray gratings, a rotating-anode x-ray tube, a flat-panel x-ray detector, and a rotating motion stage to enable tomographic acquisitions. The grating portion of the setup is comprised of three linear gratings fabricated using techniques described in literature [39

39. C. David, J. Bruder, T. Rohbeck, C. Grunzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard x-ray phase contrast imaging,” Microelectron. Eng. 84(5-8), 1172–1177 (2007). [CrossRef]

]. The first grating, labeled G0, is an absorption grating, which divides the x-ray beam exiting from the tube into an array of spatially coherent line sources, allowing the use of large focal spot (~1 mm) x-ray sources. The second grating, G1, is a phase grating, fabricated using a wet-etch procedure in Si. G1 is designed to introduce a π-phase shift at the mean energy of the x-ray beam for half of the incident x-rays. The final grating, G2, is an absorption grating, fabricated the same way as G1, but with the additional step of electroplating gold into the grating slits. This grating acts as an analyzer so that the replicated fringe pattern from G1 is converted to an intensity distribution at the detector plane. The fringe pattern occurs with a spatial period of 4.5 μm, which is too small to be resolved directly. In order to measure intensity modulation at the detector plane, a phase stepping approach is used [1

1. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), 866–868 (2003). [CrossRef]

4

4. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006). [CrossRef]

]. In this work, 8 phase steps were used, sampled over the 4.5 μm period.

The x-ray tube used is a Varian G1592 with a 0.3 mm nominal focal spot size, which is connected to a CPI Indico 100 generator. Note that the demonstration of the use of a conventional rotating-anode x-ray tube is important if the technique is to prove useful outside the laboratory. As this is an interferometer-based design, it is sensitive to displacements of the system elements, with relative position changes as small as 10 nm between some components being detectable. This results in stability requirements that are orders of magnitude more demanding than conventional absorption-based imaging systems. The rotating-anode tube allows for much higher x-ray outputs than would be possible with stationary-anode tubes, thus allowing faster scans.

Two different image objects were studied: a calcification phantom and diseased human breast tissue. An illustration of the calcification phantom is shown in Fig. 3
Fig. 3 Calcification phantom made by filling a PMMA tube with gelatin and placing various sizes of calcifications (indicated on the right) in different layers. A cross section of the phantom is shown on the left, with the approximate arrangement of the calcifications within each layer.
. The calcification phantom is constructed from a PMMA cylinder with an inner diameter of 22.2 mm and wall thickness of 1.65 mm. The cylinder is filled with 15 g/100 mL-H2O of beef-hide gelatin and contains discrete layers of calcifications of different sizes. The calcifications are composed of calcium hydroxylapatite, and their arrangement in each layer is also shown in the illustration. The human breast tissue was acquired from a full mastectomy which was fixed in a formalin solution. A sample of the tissue containing known-diseased portions was removed by a pathologist to be scanned with the grating interferometer system.

For the CT acquisition, 360 projections were taken at 1° increments. Each projection had an exposure time of 40 seconds. The detector is a Rad-icon Shad-o-Box 2048, with 48 × 48 μm2 pixels across a 1024 × 2048 array. The tube potential was 40 kVp, with a continuous tube current of 20 mA. Once the intensity modulation is recorded, the data is processed to extract the SAS-CT and absorption projections.

In general, objects which have large amounts of small-angle scattering, regardless of x-ray attenuation, will have a substantial SAS-CT component. Using the relationship in Eq. (16), Eq. (1) can be rewritten:

I(x,y)I0{1+Vcos[2πp2xg+φ(x,y)]},
(21)

so that the contributions of x-ray attenuation, small-angle scatter, and phase shift can all be separated. In addition to the known absorption contrast CT, SAS-CT image slices can be reconstructed using Eq. (19) as the fundamental imaging equation.

In order to reconstruct the projection data, the pixels are binned 2 × 2 for an effective pixel size of 96 × 96 μm2. As a result of the cone beam geometry of the imaging system, the absorption and SAS-CT images are reconstructed by the FDK reconstruction algorithm commonly used in cone-beam CT [38

38. L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-bam algorithms,” J. Opt. Soc. Am. A 1(6), 612–619 (1984). [CrossRef]

]. The reconstructed image matrix is 500 × 500 × 500, with a voxel size of (80 μm)3. The resulting images show two different quantities, with the absorption image showing the usual linear attenuation, and the SAS-CT image showing a map of the local small-angle scattering characteristics.

4. Experimental results

The SAS-CT and attenuation contrast reconstructions of the calcification phantom are shown in Fig. 4
Fig. 4 SAS-CT (a) and absorption contrast CT (b) reconstructions of the calcification phantom. The images shown are both maximal intensity projections (MIP) over the same image volume, with a pixel size of (80 μm)2 and a thickness of 1.12 mm. A MIP was used in order to visualize all calcifications within one size layer.
. The reconstructed layer shown in the figure is the 150-212 μm layer. Both the attenuation contrast and SAS-CT image show good visibility of the calcifications, though the contrast is higher in the SAS-CT image due to the removal of the background gelatin material. The attenuation image shows contributions from all parts of the phantom, including the gelatin background. This demonstrates a possible advantage of the SAS-CT contrast mechanism, the removal of relatively homogeneous material from an image object while leaving high contrast scattering objects.

SAS-CT and attenuation contrast reconstructions of human breast tissue are shown in Fig. 5
Fig. 5 SAS-CT (a) and absorption contrast CT (b) reconstructions of human breast tissue. Each image shows the same slice from the same data set. An oil cyst is clearly visible in both of the images, with additional contrast from background structure present in the attenuation contrast reconstruction.
. An oil cyst is present in both of the reconstructions. This type of cyst is categorized by its spherical shape and hollow center. A large portion of the cyst is composed of calcium, which contributes significantly to the attenuation image. The structure of the cyst along with attenuation contributes to the SAS-CT contrast seen in the left hand image. As was the case with the calcification phantom, both contrast mechanisms allow for easy visualization of the object, though again the attenuation image has signal contribution from the background material which is not present in the SAS-CT image. This reconstruction shows the potential for SAS-CT imaging in breast imaging and the ability of the SAS-CT contrast mechanism to isolate signal presence from small-angle scatters like calcifications.

5. Discussion and conclusions

Using the derived imaging equation, SAS-CT images were reconstructed for comparison with corresponding phase contrast and absorption contrast CT images. The SAS-CT reconstructions provided unique and complementary information, enhancing the overall evaluation of the image object. Objects which contain small-angle scattering components do not necessarily provide enough attenuation contrast for easy detection in absorption-based imaging. SAS-CT imaging may be able to resolve small-scale, low-attenuation scattering structures in an otherwise low-contrast, relatively-uniform material. The potential for this was shown with both physical phantom and breast tissue results, where the SAS-CT contrast mechanism removed signal contributions from a relatively homogeneous background.

The technique used in this study allows for the simultaneous acquisition of SAS-CT, differential phase, and absorption contrast CT images, due to the fact that reconstructions of each type can be completed from a single data set. This would be critical in future clinical applications, where repeated scans for better disease detection would not be feasible.

Further work is planned to quantify the SAS-CT contrast mechanism in order to determine the physical aspects of the image object and to further investigate the potential for SAS-CT reconstructions in medical and industrial applications.

References and links

1.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), 866–868 (2003). [CrossRef]

2.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45(No. 6A), 5254–5262 (2006). [CrossRef]

3.

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]

4.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006). [CrossRef]

5.

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]

6.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98(10), 108105 (2007). [CrossRef] [PubMed]

7.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High resolution differential phase contrast imaging using a magnifying projection geometry with micro-focus x-ray source,” Appl. Phys. Lett. 90(22), 224101 (2007). [CrossRef]

8.

F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. 105(10), 102006 (2009). [CrossRef]

9.

M. Bech, T. H. Jensen, R. Feidenhans, O. Bunk, C. David, and F. Pfeiffer, “Soft-tissue phase-contrast tomography with an x-ray tube source,” Phys. Med. Biol. 54(9), 2747–2753 (2009). [CrossRef] [PubMed]

10.

M. Bech, O. Bunk, C. David, R. Ruth, J. Rifkin, R. Loewen, R. Feidenhans’l, and F. Pfeiffer, “Hard X-ray phase-contrast imaging with the Compact Light Source based on inverse Compton X-rays,” J. Synchrotron Radiat. 16(1), 43–47 (2009). [CrossRef]

11.

Z.-F. Huang, K.-J. Kang, L. Zhang, Z. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, “Alternative method for differential phase contrast imaging with weakly coherent hard x-rays,” Phys. Rev. A 79(1), 013815 (2009). [CrossRef]

12.

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

13.

T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. 68(3Suppl), S13–S17 (2008). [CrossRef] [PubMed]

14.

S. A. McDonald, F. Marone, C. Hintermüller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiat. 16(4), 562–572 (2009). [CrossRef] [PubMed]

15.

F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52(23), 6923–6930 (2007). [CrossRef] [PubMed]

16.

A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express 17(15), 12540–12545 (2009). [CrossRef] [PubMed]

17.

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance source,” Proc. SPIE 6318, 631828 (2006).

18.

A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-Based X-ray Phase Imaging Using Multiline X-ray Source,” Jpn. J. Appl. Phys. 48(7), 076512 (2009). [CrossRef]

19.

F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45(4), 933–946 (2000). [CrossRef] [PubMed]

20.

C. Muehleman, J. Li, D. Connor, C. Parham, E. Pisano, and Z. Zhong, “Diffraction-enhanced imaging of musculoskeletal tissues using a conventional x-ray tube,” Acad. Radiol. 16(8), 918–923 (2009). [CrossRef] [PubMed]

21.

I. Koyama, A. Momose, J. Wu, T. T. Lwin, and T. Takeda, “Biological imaging by X-ray phase tomography using diffraction-enhanced imaging,” Jpn. J. Appl. Phys. 44(11), 8219–8221 (2005). [CrossRef]

22.

C. Parham, Z. Zhong, D. M. Connor, L. D. Chapman, and E. D. Pisano, “Design and implementation of a compact low-dose diffraction enhanced medical imaging system,” Acad. Radiol. 16(8), 911–917 (2009). [CrossRef] [PubMed]

23.

M. O. Hasnah, Z. Zhong, O. Oltulu, E. Pisano, R. E. Johnston, D. Sayers, W. Thomlinson, and D. Chapman, “Diffraction enhanced imaging contrast mechanisms in breast cancer specimens,” Med. Phys. 29(10), 2216–2221 (2002). [CrossRef] [PubMed]

24.

P. P. Zhu, J. Y. Wang, Q. X. Yuan, W. X. Huang, H. Shu, B. Gao, T. D. Hu, and Z. Y. Wu, “Computed tomography algorithm based on diffraction-enhanced imaging setup,” Appl. Phys. Lett. 87(26), 264101 (2005). [CrossRef]

25.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996). [CrossRef]

26.

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with micro-focus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997). [CrossRef]

27.

T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, and S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74(16), 3173–3176 (1995). [CrossRef] [PubMed]

28.

D. Zhang, M. Donovan, L. L. Fajardo, A. Archer, X. Wu, and H. Liu, “Preliminary feasibility study of an in-line phase contrast X-ray imaging prototype,” IEEE Trans. Biomed. Eng. 55(9), 2249–2257 (2008). [CrossRef] [PubMed]

29.

M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microsc. 232(1), 145–157 (2008). [CrossRef] [PubMed]

30.

J. R. Bushberg, J. A. Seibert, M. E. Leidholdt, Jr., and J. M. Boone, The essential physics of medical imaing (Lippincott Wilkiams & Wilkins, Philadelphia, 2001).

31.

A. C. Kak, and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

32.

A. Guinier, “Diffraction of X-rays at Small Angles: Application to the Study of Microscopic Phenomena,” Ann. Phys. 12, 161 (1939).

33.

O. Glatter, and O. Kratky, eds., Small Angle X-ray Scattering (Academic Press, New York, 1982).

34.

L. D. Landau, and E. M. Lifshitz, Quantum Mechanics, 3rd Edition ed. (Pergamon Press, New York, 1977).

35.

T. M. Sabine and W. K. Bertram, “The use of multiple-scattering data to enhance small-angle neutron scattering experiments,” Acta Crystallogr. A 55(3), 500–507 (1999). [CrossRef]

36.

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

37.

M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, “Neutron dark-field tomography,” Phys. Rev. Lett. 101(12), 123902 (2008). [CrossRef] [PubMed]

38.

L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-bam algorithms,” J. Opt. Soc. Am. A 1(6), 612–619 (1984). [CrossRef]

39.

C. David, J. Bruder, T. Rohbeck, C. Grunzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard x-ray phase contrast imaging,” Microelectron. Eng. 84(5-8), 1172–1177 (2007). [CrossRef]

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(110.6760) Imaging systems : Talbot and self-imaging effects
(290.0290) Scattering : Scattering
(340.7440) X-ray optics : X-ray imaging
(340.7450) X-ray optics : X-ray interferometry
(110.3010) Imaging systems : Image reconstruction techniques

ToC Category:
X-ray Optics

History
Original Manuscript: March 22, 2010
Revised Manuscript: May 3, 2010
Manuscript Accepted: May 30, 2010
Published: June 2, 2010

Virtual Issues
Vol. 5, Iss. 10 Virtual Journal for Biomedical Optics

Citation
Guang-Hong Chen, Nicholas Bevins, Joseph Zambelli, and Zhihua Qi, "Small-angle scattering computed tomography (SAS-CT) using a Talbot-Lau interferometer
and a rotating anode x-ray tube:
theory and experiments," Opt. Express 18, 12960-12970 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-12-12960


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References

  1. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, H. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), 866–868 (2003). [CrossRef]
  2. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45(No. 6A), 5254–5262 (2006). [CrossRef]
  3. 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]
  4. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006). [CrossRef]
  5. 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]
  6. F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98(10), 108105 (2007). [CrossRef] [PubMed]
  7. M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High resolution differential phase contrast imaging using a magnifying projection geometry with micro-focus x-ray source,” Appl. Phys. Lett. 90(22), 224101 (2007). [CrossRef]
  8. F. Pfeiffer, M. Bech, O. Bunk, T. Donath, B. Henrich, P. Kraft, and C. David, “X-ray dark-field and phase-contrast imaging using a grating interferometer,” J. Appl. Phys. 105(10), 102006 (2009). [CrossRef]
  9. M. Bech, T. H. Jensen, R. Feidenhans, O. Bunk, C. David, and F. Pfeiffer, “Soft-tissue phase-contrast tomography with an x-ray tube source,” Phys. Med. Biol. 54(9), 2747–2753 (2009). [CrossRef] [PubMed]
  10. M. Bech, O. Bunk, C. David, R. Ruth, J. Rifkin, R. Loewen, R. Feidenhans’l, and F. Pfeiffer, “Hard X-ray phase-contrast imaging with the Compact Light Source based on inverse Compton X-rays,” J. Synchrotron Radiat. 16(1), 43–47 (2009). [CrossRef]
  11. Z.-F. Huang, K.-J. Kang, L. Zhang, Z. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, “Alternative method for differential phase contrast imaging with weakly coherent hard x-rays,” Phys. Rev. A 79(1), 013815 (2009). [CrossRef]
  12. A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer,” Nucl. Instrum. Methods Phys. Res. A 352(3), 622–628 (1995). [CrossRef]
  13. T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol. 68(3Suppl), S13–S17 (2008). [CrossRef] [PubMed]
  14. S. A. McDonald, F. Marone, C. Hintermüller, G. Mikuljan, C. David, F. Pfeiffer, and M. Stampanoni, “Advanced phase-contrast imaging using a grating interferometer,” J. Synchrotron Radiat. 16(4), 562–572 (2009). [CrossRef] [PubMed]
  15. F. Pfeiffer, O. Bunk, C. David, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional x-ray phase contrast tomography,” Phys. Med. Biol. 52(23), 6923–6930 (2007). [CrossRef] [PubMed]
  16. A. Momose, W. Yashiro, H. Maikusa, and Y. Takeda, “High-speed X-ray phase imaging and X-ray phase tomography with Talbot interferometer and white synchrotron radiation,” Opt. Express 17(15), 12540–12545 (2009). [CrossRef] [PubMed]
  17. T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance source,” Proc. SPIE 6318, 631828 (2006).
  18. A. Momose, W. Yashiro, H. Kuwabara, and K. Kawabata, “Grating-Based X-ray Phase Imaging Using Multiline X-ray Source,” Jpn. J. Appl. Phys. 48(7), 076512 (2009). [CrossRef]
  19. F. A. Dilmanian, Z. Zhong, B. Ren, X. Y. Wu, L. D. Chapman, I. Orion, and W. C. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45(4), 933–946 (2000). [CrossRef] [PubMed]
  20. C. Muehleman, J. Li, D. Connor, C. Parham, E. Pisano, and Z. Zhong, “Diffraction-enhanced imaging of musculoskeletal tissues using a conventional x-ray tube,” Acad. Radiol. 16(8), 918–923 (2009). [CrossRef] [PubMed]
  21. I. Koyama, A. Momose, J. Wu, T. T. Lwin, and T. Takeda, “Biological imaging by X-ray phase tomography using diffraction-enhanced imaging,” Jpn. J. Appl. Phys. 44(11), 8219–8221 (2005). [CrossRef]
  22. C. Parham, Z. Zhong, D. M. Connor, L. D. Chapman, and E. D. Pisano, “Design and implementation of a compact low-dose diffraction enhanced medical imaging system,” Acad. Radiol. 16(8), 911–917 (2009). [CrossRef] [PubMed]
  23. M. O. Hasnah, Z. Zhong, O. Oltulu, E. Pisano, R. E. Johnston, D. Sayers, W. Thomlinson, and D. Chapman, “Diffraction enhanced imaging contrast mechanisms in breast cancer specimens,” Med. Phys. 29(10), 2216–2221 (2002). [CrossRef] [PubMed]
  24. P. P. Zhu, J. Y. Wang, Q. X. Yuan, W. X. Huang, H. Shu, B. Gao, T. D. Hu, and Z. Y. Wu, “Computed tomography algorithm based on diffraction-enhanced imaging setup,” Appl. Phys. Lett. 87(26), 264101 (2005). [CrossRef]
  25. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996). [CrossRef]
  26. A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with micro-focus x-ray source,” Rev. Sci. Instrum. 68(7), 2774–2782 (1997). [CrossRef]
  27. T. J. Davis, T. E. Gureyev, D. Gao, A. W. Stevenson, and S. W. Wilkins, “X-ray image contrast from a simple phase object,” Phys. Rev. Lett. 74(16), 3173–3176 (1995). [CrossRef] [PubMed]
  28. D. Zhang, M. Donovan, L. L. Fajardo, A. Archer, X. Wu, and H. Liu, “Preliminary feasibility study of an in-line phase contrast X-ray imaging prototype,” IEEE Trans. Biomed. Eng. 55(9), 2249–2257 (2008). [CrossRef] [PubMed]
  29. M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microsc. 232(1), 145–157 (2008). [CrossRef] [PubMed]
  30. J. R. Bushberg, J. A. Seibert, M. E. Leidholdt, Jr., and J. M. Boone, The essential physics of medical imaing (Lippincott Wilkiams & Wilkins, Philadelphia, 2001).
  31. A. C. Kak, and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
  32. A. Guinier, “Diffraction of X-rays at Small Angles: Application to the Study of Microscopic Phenomena,” Ann. Phys. 12, 161 (1939).
  33. O. Glatter, and O. Kratky, eds., Small Angle X-ray Scattering (Academic Press, New York, 1982).
  34. L. D. Landau, and E. M. Lifshitz, Quantum Mechanics, 3rd Edition ed. (Pergamon Press, New York, 1977).
  35. T. M. Sabine and W. K. Bertram, “The use of multiple-scattering data to enhance small-angle neutron scattering experiments,” Acta Crystallogr. A 55(3), 500–507 (1999). [CrossRef]
  36. Z.-T. Wang, K.-J. Kang, Z.-F. Huang, and Z. Chen, “Quantitative grating based x-ray dark-field computed tomography,” Appl. Phys. Lett. 95(9), 094105 (2009). [CrossRef]
  37. M. Strobl, C. Grünzweig, A. Hilger, I. Manke, N. Kardjilov, C. David, and F. Pfeiffer, “Neutron dark-field tomography,” Phys. Rev. Lett. 101(12), 123902 (2008). [CrossRef] [PubMed]
  38. L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-bam algorithms,” J. Opt. Soc. Am. A 1(6), 612–619 (1984). [CrossRef]
  39. C. David, J. Bruder, T. Rohbeck, C. Grunzweig, C. Kottler, A. Diaz, O. Bunk, and F. Pfeiffer, “Fabrication of diffraction gratings for hard x-ray phase contrast imaging,” Microelectron. Eng. 84(5-8), 1172–1177 (2007). [CrossRef]

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