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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 7, Iss. 9 — Aug. 28, 2012
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Phase-contrast X-ray tomography using Teague’s method

Thomas W. Baillie, Timur E. Gureyev, Jelena A. Schmalz, and Konstantin M. Pavlov  »View Author Affiliations


Optics Express, Vol. 20, Issue 15, pp. 16913-16925 (2012)
http://dx.doi.org/10.1364/OE.20.016913


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Abstract

We investigate a variant of the reconstruction technique for the in-line X-ray phase-contrast tomography data. This technique uses a newly introduced quantity, which represents a particular combination of the real and imaginary parts of the complex refractive index n. This quantity coincides with the real part of (1-n) in the case of objects having negligible absorption. The advantage of the proposed approach is in the significantly simplified form of the reconstruction algorithm for the introduced quantity. As demonstrated by our numerical experiments, the newly introduced quantity can be predictably associated with a particular refractive index.

© 2012 OSA

1. Introduction

X-ray computer tomography (CT) is a well-established technique for non-destructive three-dimensional (3D) imaging of internal structure of optically opaque samples [1

1. G. T. Herman, Image Reconstruction from Projections. The Fundamentals of Computerized Tomography (Academic Press, 1980).

3

3. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, Cambridge, 1999).

]. Conventional CT is based on the differential attenuation of transmitted X-rays by different components of the sample. This contrast mechanism is effective for distinguishing between components with significant differences in atomic number or density, e.g. between flesh and bones in the case of medical CT. However, the difference in X-ray attenuation by different types of soft tissues (e.g. healthy and malignant ones) is rather weak, which typically results in poor image contrast. It has been suggested that X-ray phase contrast can be utilized for improvement of the contrast in transmission images of non-crystalline samples consisting predominantly of light chemical elements [4

4. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995). [CrossRef]

6

6. 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]

]. Subsequently, phase-contrast X-ray CT (PCT) has been implemented in several forms, including crystal-based and grating-based X-ray interferometry [5

5. A. Momose, T. Takeda, and Y. Itai, “X-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66(2), 1434–1436 (1995). [CrossRef]

,7

7. T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE 6318, 63180S, 63180S-10 (2006). [CrossRef]

,8

8. 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(6A), 5254–5262 (2006). [CrossRef]

], analyser-based [9

9. 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]

,10

10. K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, “A variant on the geometrical optics approximation in diffraction enhanced tomography,” J. Phys. D Appl. Phys. 34(10A), A168–A172 (2001). [CrossRef]

] and in-line (also called propagation-based) phase contrast [11

11. C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, “Phase-contrast microtomography with coherent high-energy synchrotron x-rays,” Appl. Phys. Lett. 69(13), 1826–1828 (1996). [CrossRef]

15

15. M. A. Anastasio, D. Shi, F. De Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49(1), 121–144 (2004). [CrossRef] [PubMed]

] and others. The present paper investigates a particular method of object reconstruction in in-line PCT.

In-line X-ray images of most samples usually display conventional absorption contrast as well as phase contrast [4

4. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995). [CrossRef]

,6

6. 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]

]. The in-line phase contrast appears as the result of free-space propagation of the transmitted beam, which transforms phase variations in the plane located immediately after the sample into measurable intensity variations in the detector plane, provided the propagation distance is large enough for the given X-ray energy, imaging configuration and sample composition and structure. For hard X-rays (with the wavelength of around 1 Å or shorter), the complex refractive index, n(r)1Δ(r)+iβ(r), of most materials is very close to unity (e.g., the decrement, Δ(r), and the imaginary part, β(r), are typically of the order of 10−6 and 10−8, respectively), hence the refraction angles are small too and it is usually possible to approximate the X-ray trajectories through a sample by straight lines. Here we also do not consider such diffraction effects as small-angle and wide-angle scattering as it is done for instance in diffraction tomography [3

3. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, Cambridge, 1999).

,16

16. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969). [CrossRef]

18

18. K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, “A new theoretical approach to x-ray diffraction tomography,” J. Phys. D Appl. Phys. 33(13), 1596–1605 (2000). [CrossRef]

]. This simplifies the mathematical formalism of PCT and leads to relatively simple linear reconstruction algorithms [12

12. P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81(9), 5878–5886 (1997). [CrossRef]

,19

19. G. Gbur and E. Wolf, “Diffraction tomography without phase information,” Opt. Lett. 27(21), 1890–1892 (2002). [CrossRef] [PubMed]

21

21. T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006). [CrossRef]

]. The algorithms are particularly simple in the case of objects exhibiting negligible absorption [22

22. A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun. 171(4-6), 239–244 (1999). [CrossRef]

] and, more generally, in the case of so-called “homogeneous” objects [21

21. T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006). [CrossRef]

].

Recently, considerable effort has been applied to developing successful practical strategies for PCT of “general” objects, i.e. the objects that are not “homogeneous”. X-ray absorption and refractive properties of such objects are both non-trivial and “uncorrelated”. It is intuitively clear that in order to reconstruct the 3D distribution of the complex refractive index in such objects, one needs to measure 2D intensity distributions in at least two X-ray projections at each rotational position of the sample, with different values of some essential parameter, such as the sample-to-detector distance or the X-ray energy [23

23. M. R. Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983). [CrossRef]

,24

24. T. E. Gureyev and S. W. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun. 147(4-6), 229–232 (1998). [CrossRef]

]. The main difficulty in the reconstruction of the complex refractive index distribution from such data has so far been presented by the spatial and temporal instability of X-ray beams produced by laboratory sources and synchrotron beamlines. This instability often prevents one from accurate co-registration of different images collected (at different points of time) at a fixed view angle. Recently some techniques have been suggested to overcome this difficulty [25

25. M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express 18(7), 6423–6436 (2010). [CrossRef] [PubMed]

29

29. X. Guo, X. Liu, M. Gu, C. Ni, S. Huang, and B. Liu, “Polychromatic X-ray in-line phase-contrast tomography for soft tissue,” Europhys. Lett. 98(1), 14001 (2012). [CrossRef]

], but a general practical solution is yet to be found. Other reconstruction approaches successfully applied in propagation-based PCT include the ones based on the first Born approximation, rather than on the projection approximation [15

15. M. A. Anastasio, D. Shi, F. De Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49(1), 121–144 (2004). [CrossRef] [PubMed]

,30

30. M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, “Image reconstruction in spherical-wave intensity diffraction tomography,” J. Opt. Soc. Am. A 22(12), 2651–2661 (2005). [CrossRef] [PubMed]

,31

31. Y. Huang and M. A. Anastasio, “Statistically principled use of in-line measurements in intensity diffraction tomography,” J. Opt. Soc. Am. A 24(3), 626–642 (2007). [CrossRef] [PubMed]

]. These methods can account for higher-order Fresnel diffraction, but require the object to be “weak” in the sense that diffracted wave is much lower in amplitude compared to the primary transmitted one.

In the present paper we investigate the options for reconstruction of a newly introduced quantity, which represents a particular combination of the real and imaginary parts of the refractive index, from X-ray PCT data. This quantity is theoretically derived from the phase-retrieval approach which was proposed by Teague [23

23. M. R. Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983). [CrossRef]

] and further developed by Paganin and Nugent [32

32. D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998). [CrossRef]

]. We show that in the case of objects with negligible absorption the new quantity coincides with the decrement of the refractive index of the object. In the general case, the quantity is affected by the imaginary part of the refractive index as well, with the dependence displaying a relatively simple and predictable character, as demonstrated by our numerical experiments. The advantage of the proposed approach is in the significantly simplified form of the PCT algorithm that can be applied for the reconstruction of the introduced quantity. This method can be viewed as an extension of the single-step PCT algorithm originally proposed in [22

22. A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun. 171(4-6), 239–244 (1999). [CrossRef]

].

We present the theoretical overview of the proposed method in Section 2 of the paper, the results of the numerical simulations are found in Section 3, while the conclusions are discussed in Section 4.

2. Theoretical background

Consider an experiment with an object illuminated by a plane monochromatic X-ray wave with wavelength λ and intensity Iin and with the transmitted wave registered by a position-sensitive detector. The interaction of X-rays with the object is described by the spatial distribution of the complex refractive index in the object, n(r), where r=(x,y,z) are the usual Cartesian spatial coordinates (we omit the dependence of all quantities on the wavelength for the sake of brevity). We only consider here the case of fully coherent X-rays, however, the theory can be extended to the partially-coherent case as in [33

33. T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004). [CrossRef] [PubMed]

,34

34. G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76(4), 045804 (2007). [CrossRef]

].

The propagation direction of the incident X-ray wave, Iin1/2exp(ikz, where k=2π/λ, makes an angle θ with the z axis in the (x, y) plane, where π/2θ<π/2. It will be also convenient to use the angle θ=θ+π/2 (Fig. 1
Fig. 1 Generic layout of the in-line CT experimental setup.
).

We assume as usual that the projection approximation [3

3. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, Cambridge, 1999).

] can be applied to calculate the phase and intensity, respectively, of the wave after transmission through the object,
φθ(x',y)=k(PθΔ)(x',y),
(1)
Iθ(x',y)=Iinexp{2k(Pθβ)(x',y)},
(2)
where
(Pθf)(x',y)=f(x,y,z)δ(x'xsinθzcosθ)dxdz
(3)
is the projection operator and r=(x,y,z) are Cartesian coordinates rotated by the angle θ around the y axis with respect to coordinate system r (Fig. 1). Note that we formally refer the transmitted intensity and phase distribution to the plane z=0, rather than the “object plane” located immediately downstream the object along the direction of propagation of the X-ray wave. As it will be seen below, this assumption is mathematically convenient and acceptable in the case of “thin” objects, i.e. the objects whose thickness is much smaller than the distance between the object and the X-ray detector.

Teague [23

23. M. R. Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983). [CrossRef]

] suggested to solve Eq. (4) via the introduction of an auxiliary function ψθ, which satisfies
ψθ(x,y)=Iθ(x,y)φθ(x,y).
(5)
If such scalar potential ψθ exists, then substituting Eq. (5) into Eq. (4) we see that it satisfies the Poisson equation
2ψθ(x,y)=(k/R)[IθR(x,y)Iθ(x,y)].
(6)
After finding ψθ from Eq. (6), a solution for the phase function in Eq. (5) can be determined by solving another Poisson equation
2φθ(x,y)=[Iθ1(x,y)ψθ(x,y)],
(7)
which can be obtained by dividing both sides of Eq. (5) by Iθ and taking the divergence. This approach provides a method for solving Eq. (4) with respect to the unknown phase φθ(x',y) via two Poisson equations [23

23. M. R. Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983). [CrossRef]

,32

32. D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998). [CrossRef]

], if the intensity distributions Iθ(x',y) and IθR(x',y) are known. One advantage of this approach, compared to a direct numerical solution of Eq. (4), is that the former is amenable to an implementation using the Fast Fourier transform (FFT). Note however, that the phase distribution reconstructed by means of Eq. (6) and Eq. (7) may in some special cases differ from an exact solution of Eq. (4). This issue is discussed in detail in reference [35

35. J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter wave fields: Validity of Teague’s method for solution of the Transport of Intensity equation,” Phys. Rev. A 84(2), 023808 (2011). [CrossRef]

].

After the phase distribution φθ is reconstructed at each projection angle θ, it is possible to reconstruct the 3D distribution of the real part of refractive index by CT techniques. Taking the 2D Fourier transform of Eq. (3) and using the Fourier slice theorem [2

2. F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

] one can easily obtain that
f(x,y,z)=[(FPθf)(ξ,η)](x,y,z),
(8)
where
(Fh)(ξ,η)=exp[i2π(xξ+yη)]h(x,y)dxdy,
(9)
is the 2D Fourier transform and
[g](x,y,z)=0πexp{i2π[ξ(xsinθ+zcosθ)+ηy]}g(ξ,η,θ)|ξ|dξdηdθ
(10)
is the so-called filtered backprojection operator [2

2. F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

]. Therefore, if projections (Pθf)(x',y) can be measured for all view angles θ from the interval [0,π), Eq. (8) can be used to reconstruct the 3D distribution f(x,y,z). Following this approach with respect to Eq. (1), we obtain
Δ(x,y,z)=(1/k)[(Fφθ)(ξ,η)](x,y,z),
(11)
where φθ(x,y) can be computed at each angle θ e.g. by means of Eq. (6) and Eq. (7).

Note that in the case of Eq. (2) the same approach yields the conventional (“absorption”) CT reconstruction formula:
β(x,y,z)=(1/2k){F[ln(Iθ/Iin)](ξ,η)}(x,y,z).
(12)
One can see that Eq. (12) provides a “single-step” reconstruction algorithm for the quantity of interest, β(x,y,z), from a measurable quantity, Iθ(x',y). On the other hand, Eq. (11) requires a preliminary step of phase reconstruction from measurable quantities, Iθ(x',y) and IθR(x',y) at each view angle. Apart from additional computational complexity, the two-step reconstruction of the real part of refractive index suffers from well-known numerical instabilities [21

21. T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006). [CrossRef]

,22

22. A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun. 171(4-6), 239–244 (1999). [CrossRef]

]: one associated with the low spatial frequencies in Eq. (6) and the other one due to the (moderate) amplification in Eq. (8) of high-frequency noise due to the effect of the ramp filter (multiplication by |ξ| in Fourier space).

These problems can be partially eliminated in the case of objects with negligible absorption, by means of a single-step reconstruction formula due to Bronnikov [22

22. A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun. 171(4-6), 239–244 (1999). [CrossRef]

]:
Δ(x,y,z)=[1/(4π2R)]{F[ln(IθR/Iin)](ξ,η)/(ξ2+η2)}(x,y,z).
(13)
Equation (13) can be obtained by substitution into Eq. (11) of a simplified form of the TIE solution φθ(x,y)=(k/R)2[IθR(x,y)/Iin1], which one gets from Eq. (6) and Eq. (7) in the case Iθ(x',y)Iin, and using the (optional) approximation [1IθR(x',y)/Iin]ln[IθR(x',y)/Iin], which holds under the general validity conditions of the TIE [23

23. M. R. Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983). [CrossRef]

]. Note that Eq. (13) not only provides a single-step reconstruction algorithm for the decrement of the refractive index, Δ(x,y,z), but also partially eliminates the abovementioned instabilities of the reconstruction thanks to the cancellation of the noise-amplification effect of the ramp filter |ξ| by the Fourier-space kernel of the Laplace operator, 1/(ξ2+η2).

A similar (but much stronger) stabilization of the PCT reconstruction with respect to noise in the input data is achieved in the case of so-called “homogeneous” objects [21

21. T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006). [CrossRef]

,36

36. D. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Y. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc. 214(3), 315–327 (2004). [CrossRef] [PubMed]

]. The latter class of objects includes objects consisting predominantly of a single material, and, in the case of X-rays with energies between approximately 60 keV and 500 keV, any objects containing only light chemical elements with Z<10 [37

37. X. Wu, H. Liu, and A. Yan, “X-ray phase-attenuation duality and phase retrieval,” Opt. Lett. 30(4), 379–381 (2005). [CrossRef] [PubMed]

].

Using Eq. (14), the single-step reconstruction algorithm, Eq. (13), can be re-written in the following form:
Δ(x,y,z)=R12{F[ln(IθR/Iin)](ξ,η)}(x,y,z).
(15)
In Eq. (15) the “phase retrieval” step (inverse Laplacian) takes place in 3D after the conventional CT reconstruction. This approach could present an advantage in the case of general objects, where the phase retrieval in 2D involves the subtraction of two projection images collected at different object-to-detector distances (see e.g. Equation (6)). Indeed, as explained above, such subtraction often involves substantial practical problems, as it is strongly affected by the changes in the illumination conditions with time (i.e. by the temporal fluctuations of low spatial frequencies of the incident beam). One can expect that certain types of random background fluctuations will average out in the process of image summation during CT reconstruction (at the backprojection step). Moreover, some misalignment issues, such as e.g. disagreement between angular positions of the sample at different object-to-detector distances, become largely irrelevant after the CT reconstruction. Of course, systematic variations in the background illumination during CT scans will still affect the “phase retrieval” in 3D, but these variations may arguably be also easier to track and correct during or after the CT reconstruction (note that the mutual alignment of two 3D distributions is likely to be more efficient compared to pair-wise correlations of projections).

Unfortunately, in the cases with non-trivial absorption the approach which led to Eq. (15) is usually no longer valid, as, unlike the Laplace operator, the general TIE operator, DTIE[φθ](x,y)=[Iθ(x,y)φθ(x,y)], does not commute with the Radon transform. This is why we propose the following approach based on the Teague's method for solution of the TIE described above.

Let us assume that there exists a function Δ˜(x,y,z) such that
ψθ(x',y)=kIin(PθΔ˜)(x',y),
(16)
where the function ψθ(x',y) is defined by Eq. (6). Then, according to Eq. (6) and Eq. (8), we obtain
Δ˜(x,y,z)=14π2R{F[(IθRIθ)](ξ,η)Iin(ξ2+η2)}(x,y,z).
(17)
Therefore, the 3D distribution of the quantity Δ˜(x,y,z) can be obtained by a single-step CT reconstruction algorithm from the transmitted intensity distributions measured at two different object-to-detector distances for each view angle.

By an exact analogy with the derivation of Eq. (15) from Eq. (13), one can now also derive the following reconstruction formula from Eq. (17):
Δ˜(x,y,z)=R12{[F(IθR/Iin)(ξ,η)][F(Iθ/Iin)(ξ,η)]}(x,y,z).
(18)
Thus, unlike the function Δ(x,y,z), the new quantity Δ˜(x,y,z) can be computed by applying the 3D inverse Laplacian to the difference between the two conventional CT reconstructions (only without the logarithm operation) obtained from projections collected at different object-to-detector distances. Note that in experiments the incident intensity Iin corresponding to the measurements of Iθ(x',y) and IθR(x',y) can be different, which can be easily taken into account in Eq. (18).

An important question that needs to be answered with respect to quantity Δ˜(x,y,z) is that about its physical nature. Firstly, we note that in the case of objects with negligible absorption, i.e. when it is possible to use the approximation Iθ(x',y)Iin, Eq. (17) reduces to Eq. (13), and hence Δ˜(x,y,z)Δ(x,y,z) in this case. In the general case, when absorption inside the object cannot be ignored, the quantity Δ˜(x,y,z) depends both on the real and imaginary parts of the complex refractive index. The following formal expression can be derived from Eqs. (17), (6), (4), (1) and (2):
Δ˜(x,y,z)=14π2{F([exp{2k(Pθβ)(x',y)}(PθΔ)(x,y)])(ξ,η)ξ2+η2}(x,y,z).
(19)
Equation (19) establishes an explicit relationship between the introduced quantity Δ˜(x,y,z) and the complex refractive index. It also shows that the quantity Δ˜(x,y,z) always exists and is unique as long as the complex refractive index is amenable to the mathematical operations involved in Eq. (19). However, as far as the physical nature of quantity Δ˜(x,y,z) is concerned, Eq. (19) does not appear particularly helpful, rather than demonstrating again that in the case of negligible absorption, when exp{2k(Pθβ)(x',y)}1, one gets Δ˜(x,y,z)Δ(x,y,z). To obtain a simplified relationship between Δ˜(x,y,z) and the complex refractive index, we assume that all projections of the imaginary part of the refractive index, (Pθβ)(x',y), are slowly varying functions with respect to all its arguments. Then we can apply the divergence operator to both parts of Eq. (5) and move the gradient operator out of the square brackets (neglecting the terms that contain derivatives of exp{2k(Pθβ)(x',y)}):
2(PθΔ˜)(x',y)=[exp{2k(Pθβ)(x',y)}(PθΔ)(x,y)]2[exp{2k(Pθβ)(x',y)}(PθΔ)(x,y)].
(20)
Hence, assuming e.g. the Dirichlet boundary conditions, we conclude from Eq. (20) that (PθΔ˜)(x',y)=exp{2k(Pθβ)(x',y)}(PθΔ)(x,y) due to the uniqueness of the solution to the boundary value problem for the Laplace equation. Using again the assumption about the slow variation of (Pθβ)(x',y) (this time with respect to variables x and θ), we can now obtain:
(PθΔ˜)(x',y)exp{2k<Pβ>(y)}(PθΔ)(x,y)=(Pθ[exp{2k<Pβ>(y)}Δ])(x,y),
(21)
where <> denotes averaging over x and θ for a fixed value of y. As a complete set of parallel projections over 180 degrees uniquely defines a 2D distribution, Eq. (21) allows one to obtain the following approximate relationship:
Δ(x,y,z)Δ˜(x,y,z)exp{2k<Pβ>(y)}.
(22)
Equation (22) can be used to find the approximate values of the real part of the decrement of refractive index, Δ(x,y,z), from the recovered distributions of Δ˜(x,y,z) and β(x,y,z). It might seem that the exact (non-averaged) values of (Pθβ)(x',y), or even β(x,y,z) itself, could be used in Eq. (22) instead of the projection values averaged over the axial slices, producing a more accurate variant of the equation. However, we are not aware of a method that would allow such a modified version of Eq. (22) to be obtained within the framework of the approach described above.

3. Methods and results

The simulated object, as depicted in Fig. 2
Fig. 2 The structure of the object—and its bounding cube—that was simulated. It consisted of twelve spheres arranged in three layers of four, each parallel to the base. The bottom and top layers had the spheres situated in the corners, and the middle layer had them in the centre of each face. The lightness of the spheres decreases with increasing β for the constant Δ objects, Δfor the constant β objects, and β and Δ for the objects with both varying.
, was a collection of 12 non-overlapping homogeneous spheres of radius 100 µm, one in each of the eight corners of the bounding cube and one on each face excluding the top and bottom. It was represented as two 1024 × 1024 × 1024 arrays of 32-bit floating-point numbers, corresponding to the β and Δ distributions of the complex refractive index n=1Δ+iβ, which was different for each of the spheres. Both arrays were stored and processed as 1024 square planar cross-sections parallel to the x-z plane. The voxel size was 1 × 1 × 1 μm3.

The simulations were performed using X-TRACT [39

39. T. E. Gureyev, Y. Nesterets, D. Ternovski, D. Thompson, S. W. Wilkins, A. W. Stevenson, A. Sakellariou, and J. A. Taylor, “Toolbox for advanced X-ray image processing,” Proc. SPIE 8141, 81410B, 81410B-14 (2011). [CrossRef]

], a program that amongst other things allows one to calculate parallel-beam X-ray projections of objects given the object's β and Δ slices. X-TRACT allows one to perform a variety of useful image operations such as pixel-wise addition and inverse Laplacian, to process image files into sinograms, and to perform CT reconstructions by calculating the inverse Radon transform of sinogram images. All of these operations were required for the simulations.

In order to explore the relationship between the complex refractive index n=1Δ+iβand Δ˜ with regards to CT reconstruction, six different objects were used: three with constant Δ and three with constant β. Then, to more closely model experimental results, the constant β simulations were performed again with Poisson noise being added to the projections, with a relative standard deviation, σrel, of 5% calculated for the average intensity.

The three constant β objects had β values of 1.5 × 10−9, 1 × 10−8, and 2 × 10−8, each with a Δ range of 5 × 10−8 to 1 × 10−6. The three constant Δ objects had Δ values of 5 × 10−7, 1 × 10−6, and 2 × 10−6, each with a β range of 1 × 10−9 to 2 × 10−8. These values were chosen close to refractive indices of certain real materials; at a wavelength of 1 Å [40

40. X-ray complex refraction coefficient, CSIRO. Retrieved March 15, 2012, from www.ts-imaging.net/Services/Simple/ICUtilXdata.aspx.

] are shown in Table 1

Table 1. The Components of Refractive Indices of Certain Real Materials at a Wavelength of 1 Å [35].

table-icon
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. The results are shown in Figs. 3
Fig. 3 The reconstructed values using Δ˜ for the three objects with constant β (closed circles, squares and diamond symbols). The dashed line is Δ˜=Δ. The uncertainty is taken to be 1 standard deviation of the values in the centre of the spheres in the reconstructed slices. Also shown are the curves of best fit Δ˜=Δexp{2k<Pβ>ββ} for closed circles, squares and diamond symbols, which parameters are shown in Table 2. The triangular symbols show the result of a “homogenous” TIE reconstruction.
, 4
Fig. 4 The reconstructed values using Δ˜ for the three objects with constant β and 5% Poisson noise. The dashed line is Δ˜=Δ. The uncertainty is taken to be 1 standard deviation of the values in the centre of the spheres in the reconstructed slices. Also shown are the same curves of best fit as in Fig. 3.
and 5
Fig. 5 Δ˜ (triangular, open circles and inverted triangular symbols) and the reconstructed values of Δ (see Eq. (22)) (closed circles, diamond and square symbols) using Δ˜ for the three objects with constant Δ. The uncertainty is taken to be 1 standard deviation of the values in the centre of the spheres in the reconstructed slices. Also shown are the curves of best fit Δ˜=Δexp{2k<Pβ>ββ}, which parameters are given in Table 3.
, and all confirmed the theoretical prediction that Δ˜ would approach Δ as β approached 0.

Figure 3 suggests that Δ˜ can be approximated by a linear function Δ˜=mΔ, with 0<m<1, in agreement with Eq. (22) for constant β. This is also supported by Fig. 5, and both figures show the curves of best fit for each simulation's data series. Tables 2

Table 2. The Best-Fit Parameters Obtained from Figs. 3 and 4.

table-icon
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and 3

Table 3. The Best-Fit Parameters Obtained from Fig. 5.

table-icon
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show the values of <Pβ>β corresponding to each curve of best fit. Figure 5 also demonstrates that Eq. (22) may allow one to accurately obtain the function Δ(x,y,z) from the reconstructed functions Δ˜(x,y,z) and β(x,y,z) in the cases with significant spatial variation of both β and Δ/β. It should be noted that while the slow variation of (Pθβ)(x',y) was demonstrated to be sufficient for the validity of Eq. (22), it is not a necessary condition. As our numerical results show, Eq. (22) remains a good approximation for a broader range of objects than that satisfying the condition of slow variation, including certain types of objects with localized areas of rapid variation of (Pθβ)(x',y).

For comparison, we have also performed the reconstruction of the real decrement of the refractive index Δ(x,y,z) in the sample with constant β=1.5×109 using the PCT reconstruction formulae for homogeneous objects [21

21. T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006). [CrossRef]

] with Δ=5×107 (which approximately corresponded to an average Δ value for this sample) . The appropriate results are shown in Fig. 3 by green triangular symbols. This data demonstrates that the “homogeneous” TIE method, although producing reasonable looking CT reconstructions, cannot quantitatively reconstruct with satisfying accuracy the distribution of the real decrement of the refractive index in objects with significant spatial variability of the Δ/β ratio.

Figure 4 shows the constant β reconstructions with 5% noise. Whilst being visibly more unstable than their noiseless counterparts, the quantity Δ˜(x,y,z) is still reconstructed with remarkable stability and closely followed the same trend lines.

The graphs of Fig. 5, whilst being largely linear, are all marred by a small but systematic 'jump' between the fourth and fifth data points. This jump is likely caused by the geometry of the simulated object, as the four spheres in the middle layer are closer together than those in the top and bottom layers.

Figure 6
Fig. 6 (a) Typical projection with an object-to-detector projection distance of 10 cm; (b) reconstructed slice of Δ through the bottom layer of spheres and (c) a cross-section indicated by the solid line through the bottom row in (a). This particular object had spheres with constant Δ=5×107 and β varying from 1 × 10−9 to 2 × 10−8.
shows the projection at θ=26 of the object with Δ=5×107 at an object-to-detector propagation distance of 10 cm, along with a reconstructed slice of Δ through the bottom layer of spheres. Visible are the bright diffraction fringes encircling each sphere (Fig. 6(a)), as typical for the in-line phase contrast images.

4. Conclusion

Acknowledgment

KMP acknowledges financial support from University of New England.

References and links

1.

G. T. Herman, Image Reconstruction from Projections. The Fundamentals of Computerized Tomography (Academic Press, 1980).

2.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

3.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, Cambridge, 1999).

4.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995). [CrossRef]

5.

A. Momose, T. Takeda, and Y. Itai, “X-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum. 66(2), 1434–1436 (1995). [CrossRef]

6.

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]

7.

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE 6318, 63180S, 63180S-10 (2006). [CrossRef]

8.

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(6A), 5254–5262 (2006). [CrossRef]

9.

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]

10.

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, “A variant on the geometrical optics approximation in diffraction enhanced tomography,” J. Phys. D Appl. Phys. 34(10A), A168–A172 (2001). [CrossRef]

11.

C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, “Phase-contrast microtomography with coherent high-energy synchrotron x-rays,” Appl. Phys. Lett. 69(13), 1826–1828 (1996). [CrossRef]

12.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81(9), 5878–5886 (1997). [CrossRef]

13.

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175(4–6), 329–336 (2000). [CrossRef]

14.

S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11(19), 2289–2302 (2003). [CrossRef] [PubMed]

15.

M. A. Anastasio, D. Shi, F. De Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49(1), 121–144 (2004). [CrossRef] [PubMed]

16.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969). [CrossRef]

17.

G. Harding, J. M. Kosanetzky, and U. Neitzel, “Elastic scatter computed tomography,” Phys. Med. Biol. 30(2), 183–186 (1985). [CrossRef] [PubMed]

18.

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, “A new theoretical approach to x-ray diffraction tomography,” J. Phys. D Appl. Phys. 33(13), 1596–1605 (2000). [CrossRef]

19.

G. Gbur and E. Wolf, “Diffraction tomography without phase information,” Opt. Lett. 27(21), 1890–1892 (2002). [CrossRef] [PubMed]

20.

A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19(3), 472–480 (2002). [CrossRef] [PubMed]

21.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89(3), 034102 (2006). [CrossRef]

22.

A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun. 171(4-6), 239–244 (1999). [CrossRef]

23.

M. R. Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983). [CrossRef]

24.

T. E. Gureyev and S. W. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun. 147(4-6), 229–232 (1998). [CrossRef]

25.

M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express 18(7), 6423–6436 (2010). [CrossRef] [PubMed]

26.

M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process. 19(9), 2428–2436 (2010). [CrossRef] [PubMed]

27.

M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol. 56(23), 7353–7369 (2011). [CrossRef] [PubMed]

28.

M. Langer, P. Cloetens, A. Pacureanu, and F. Peyrin, “X-ray in-line phase tomography of multimaterial objects,” Opt. Lett. 37(11), 2151–2153 (2012). [CrossRef] [PubMed]

29.

X. Guo, X. Liu, M. Gu, C. Ni, S. Huang, and B. Liu, “Polychromatic X-ray in-line phase-contrast tomography for soft tissue,” Europhys. Lett. 98(1), 14001 (2012). [CrossRef]

30.

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, “Image reconstruction in spherical-wave intensity diffraction tomography,” J. Opt. Soc. Am. A 22(12), 2651–2661 (2005). [CrossRef] [PubMed]

31.

Y. Huang and M. A. Anastasio, “Statistically principled use of in-line measurements in intensity diffraction tomography,” J. Opt. Soc. Am. A 24(3), 626–642 (2007). [CrossRef] [PubMed]

32.

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998). [CrossRef]

33.

T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett. 93(6), 068103 (2004). [CrossRef] [PubMed]

34.

G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A 76(4), 045804 (2007). [CrossRef]

35.

J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter wave fields: Validity of Teague’s method for solution of the Transport of Intensity equation,” Phys. Rev. A 84(2), 023808 (2011). [CrossRef]

36.

D. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Y. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc. 214(3), 315–327 (2004). [CrossRef] [PubMed]

37.

X. Wu, H. Liu, and A. Yan, “X-ray phase-attenuation duality and phase retrieval,” Opt. Lett. 30(4), 379–381 (2005). [CrossRef] [PubMed]

38.

G. R. Myers, T. E. Gureyev, and D. M. Paganin, “Stability of phase-contrast tomography,” J. Opt. Soc. Am. A 24(9), 2516–2526 (2007). [CrossRef] [PubMed]

39.

T. E. Gureyev, Y. Nesterets, D. Ternovski, D. Thompson, S. W. Wilkins, A. W. Stevenson, A. Sakellariou, and J. A. Taylor, “Toolbox for advanced X-ray image processing,” Proc. SPIE 8141, 81410B, 81410B-14 (2011). [CrossRef]

40.

X-ray complex refraction coefficient, CSIRO. Retrieved March 15, 2012, from www.ts-imaging.net/Services/Simple/ICUtilXdata.aspx.

OCIS Codes
(100.3190) Image processing : Inverse problems
(100.5070) Image processing : Phase retrieval
(100.6950) Image processing : Tomographic image processing
(340.7440) X-ray optics : X-ray imaging
(110.3010) Imaging systems : Image reconstruction techniques

ToC Category:
Image Processing

History
Original Manuscript: March 30, 2012
Revised Manuscript: July 1, 2012
Manuscript Accepted: July 3, 2012
Published: July 11, 2012

Virtual Issues
Vol. 7, Iss. 9 Virtual Journal for Biomedical Optics

Citation
Thomas W. Baillie, Timur E. Gureyev, Jelena A. Schmalz, and Konstantin M. Pavlov, "Phase-contrast X-ray tomography using Teague’s method," Opt. Express 20, 16913-16925 (2012)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-15-16913


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References

  1. G. T. Herman, Image Reconstruction from Projections. The Fundamentals of Computerized Tomography (Academic Press, 1980).
  2. F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).
  3. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, Cambridge, 1999).
  4. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum.66(12), 5486–5492 (1995). [CrossRef]
  5. A. Momose, T. Takeda, and Y. Itai, “X-ray computed tomography for observing biological specimens and organic materials,” Rev. Sci. Instrum.66(2), 1434–1436 (1995). [CrossRef]
  6. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature384(6607), 335–338 (1996). [CrossRef]
  7. T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE6318, 63180S, 63180S-10 (2006). [CrossRef]
  8. 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(6A), 5254–5262 (2006). [CrossRef]
  9. 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]
  10. K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, “A variant on the geometrical optics approximation in diffraction enhanced tomography,” J. Phys. D Appl. Phys.34(10A), A168–A172 (2001). [CrossRef]
  11. C. Raven, A. Snigirev, I. Snigireva, P. Spanne, A. Souvorov, and V. Kohn, “Phase-contrast microtomography with coherent high-energy synchrotron x-rays,” Appl. Phys. Lett.69(13), 1826–1828 (1996). [CrossRef]
  12. P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys.81(9), 5878–5886 (1997). [CrossRef]
  13. A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun.175(4–6), 329–336 (2000). [CrossRef]
  14. S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express11(19), 2289–2302 (2003). [CrossRef] [PubMed]
  15. M. A. Anastasio, D. Shi, F. De Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol.49(1), 121–144 (2004). [CrossRef] [PubMed]
  16. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun.1(4), 153–156 (1969). [CrossRef]
  17. G. Harding, J. M. Kosanetzky, and U. Neitzel, “Elastic scatter computed tomography,” Phys. Med. Biol.30(2), 183–186 (1985). [CrossRef] [PubMed]
  18. K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morgan, “A new theoretical approach to x-ray diffraction tomography,” J. Phys. D Appl. Phys.33(13), 1596–1605 (2000). [CrossRef]
  19. G. Gbur and E. Wolf, “Diffraction tomography without phase information,” Opt. Lett.27(21), 1890–1892 (2002). [CrossRef] [PubMed]
  20. A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A19(3), 472–480 (2002). [CrossRef] [PubMed]
  21. T. E. Gureyev, D. M. Paganin, G. R. Myers, Ya. I. Nesterets, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett.89(3), 034102 (2006). [CrossRef]
  22. A. V. Bronnikov, “Reconstruction formulas in phase-contrast tomography,” Opt. Commun.171(4-6), 239–244 (1999). [CrossRef]
  23. M. R. Teague, “Deterministic phase retrieval: a Green's function solution,” J. Opt. Soc. Am.73(11), 1434–1441 (1983). [CrossRef]
  24. T. E. Gureyev and S. W. Wilkins, “On X-ray phase retrieval from polychromatic images,” Opt. Commun.147(4-6), 229–232 (1998). [CrossRef]
  25. M. A. Beltran, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance,” Opt. Express18(7), 6423–6436 (2010). [CrossRef] [PubMed]
  26. M. Langer, P. Cloetens, and F. Peyrin, “Regularization of phase retrieval with phase-attenuation duality prior for 3D holotomography,” IEEE Trans. Image Process.19(9), 2428–2436 (2010). [CrossRef] [PubMed]
  27. M. A. Beltran, D. M. Paganin, K. K. W. Siu, A. Fouras, S. B. Hooper, D. H. Reser, and M. J. Kitchen, “Interface-specific x-ray phase retrieval tomography of complex biological organs,” Phys. Med. Biol.56(23), 7353–7369 (2011). [CrossRef] [PubMed]
  28. M. Langer, P. Cloetens, A. Pacureanu, and F. Peyrin, “X-ray in-line phase tomography of multimaterial objects,” Opt. Lett.37(11), 2151–2153 (2012). [CrossRef] [PubMed]
  29. X. Guo, X. Liu, M. Gu, C. Ni, S. Huang, and B. Liu, “Polychromatic X-ray in-line phase-contrast tomography for soft tissue,” Europhys. Lett.98(1), 14001 (2012). [CrossRef]
  30. M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, “Image reconstruction in spherical-wave intensity diffraction tomography,” J. Opt. Soc. Am. A22(12), 2651–2661 (2005). [CrossRef] [PubMed]
  31. Y. Huang and M. A. Anastasio, “Statistically principled use of in-line measurements in intensity diffraction tomography,” J. Opt. Soc. Am. A24(3), 626–642 (2007). [CrossRef] [PubMed]
  32. D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80(12), 2586–2589 (1998). [CrossRef]
  33. T. E. Gureyev, D. M. Paganin, A. W. Stevenson, S. C. Mayo, and S. W. Wilkins, “Generalized eikonal of partially coherent beams and its use in quantitative imaging,” Phys. Rev. Lett.93(6), 068103 (2004). [CrossRef] [PubMed]
  34. G. R. Myers, S. C. Mayo, T. E. Gureyev, D. M. Paganin, and S. W. Wilkins, “Polychromatic cone-beam phase-contrast tomography,” Phys. Rev. A76(4), 045804 (2007). [CrossRef]
  35. J. A. Schmalz, T. E. Gureyev, D. M. Paganin, and K. M. Pavlov, “Phase retrieval using radiation and matter wave fields: Validity of Teague’s method for solution of the Transport of Intensity equation,” Phys. Rev. A84(2), 023808 (2011). [CrossRef]
  36. D. Paganin, T. E. Gureyev, S. C. Mayo, A. W. Stevenson, Y. I. Nesterets, and S. W. Wilkins, “X-ray omni microscopy,” J. Microsc.214(3), 315–327 (2004). [CrossRef] [PubMed]
  37. X. Wu, H. Liu, and A. Yan, “X-ray phase-attenuation duality and phase retrieval,” Opt. Lett.30(4), 379–381 (2005). [CrossRef] [PubMed]
  38. G. R. Myers, T. E. Gureyev, and D. M. Paganin, “Stability of phase-contrast tomography,” J. Opt. Soc. Am. A24(9), 2516–2526 (2007). [CrossRef] [PubMed]
  39. T. E. Gureyev, Y. Nesterets, D. Ternovski, D. Thompson, S. W. Wilkins, A. W. Stevenson, A. Sakellariou, and J. A. Taylor, “Toolbox for advanced X-ray image processing,” Proc. SPIE8141, 81410B, 81410B-14 (2011). [CrossRef]
  40. X-ray complex refraction coefficient, CSIRO. Retrieved March 15, 2012, from www.ts-imaging.net/Services/Simple/ICUtilXdata.aspx .

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