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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 25 — Dec. 6, 2010
  • pp: 25771–25785
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Nonlinear phase retrieval from single-distance radiograph

Julian Moosmann, Ralf Hofmann, Andrei V. Bronnikov, and Tilo Baumbach  »View Author Affiliations


Optics Express, Vol. 18, Issue 25, pp. 25771-25785 (2010)
http://dx.doi.org/10.1364/OE.18.025771


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Abstract

Phase contrast in the object plane of a phase object is retrieved from intensity contrast at a single object-detector distance. Expanding intensity contrast and phase shift in the detector plane in powers of object-detector distance, phase retrieval is performed beyond the solution to the linearized transport-of-intensity equation. The expansion coefficients are determined by the entire paraxial wave equation. The Laplacian of the phase shift in the object plane thus is written as a local expression linear in the intensity contrast and nonlinear in the phase shift in the object plane. A perturbative approach to this expression is proposed and tested with simulated phantom data.

© 2010 Optical Society of America

1. Introduction

With the advent of third-generation synchrotron light sources producing brilliant and partially coherent radiation, whose spectrum contains highly energetic X-ray components, the application of phase sensitive imaging methods became feasible. Within samples, whose atoms are of low Z and of low number density ρ̂ (polymers, soft biological tissue, etc.), the attenuation of highly energetic X-rays is weak yielding a poor absorption contrast. Sizable phase shifts are, however, induced to the coherent X-ray wave fronts. The latter induce intensity contrast that can be used for 2D and 3D imaging.

The motivation of this paper is to elaborate on the optics-free in-line propagation technique where phase contrast in the object plane is retrieved from intensity contrast measured at a single object-detector distance z. Retrieving phase information from intensity at a single distance is a precursor for 2D and 3D in-situ imaging. Specifically, we aim at a higher flexibility in setting z to values beyond the edge-enhancement regime (from typically a few centimeters for X-ray energies above 20 keV to several ten centimeters). This flexibility is required in experiments where the sample is suspended by a large environment not accessible to the detector. As can be argued on the basis of the contrast-transfer function [1

1. A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779 (1974). [CrossRef]

3

3. J.-P. Guigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136 (1978). [CrossRef]

], at larger z small object scales, such as the fine-structure of edges, increasingly contribute to the fringe pattern discerned by the detector. The promise to resolve such structures at larger z represents another motivation for our work.

The interaction of radiation with matter is effectively described by the complex refractive index
n(x1,x2,x3)=1δ(x1,x2,x3)+iβ(x1,x2,x3),
(1)
where δ = 1 – Re(n) is the real refractive index decrement introduced by elastic scattering, and β refers to the real absorption index. Assuming the projection approximation to be valid (negligible scattering away from the ray path), the effects induced by the object onto a monchromatic, coherent wave field propagating along, say, the x2 direction, see Fig. 1, is described by the transmission function ψz=0(x, y) as
ψz=0(x,y)=exp(kdx2β(x,x2,y))exp(ikdx2δ(x,x2,y))exp(B(x,y))exp(iϕz=0(x,y)),
(2)
where k2πλ, λ denotes the wavelength, the integration over x2 is terminated at the object boundaries, and x,y are coordinates transverse to the ray path along x2. The 2D phase map ϕz=0(x, y) is interpreted as the projection of the electron density. Functions B and ϕ are radiographs of the object which contain some information on the object that can be interpreted. Since –B(x,y) + z=0(x, y) essentially is the Radon transform of n(x1, x2, x3) a stack of such radiographs, obtained by a stepwise rotation of the object about axis x3, is input for computed tomographic 3D reconstruction of n(x1, x2, x3), see for example [4

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

].

Fig. 1 The experimental set-up and coordinate conventions for single-distance phase-contrast tomography employing the in-line propagation method.

For the retrieval of the phase-sensitive part of the transmission function from an intensity map several methods have been explored. Interferometric techniques exploit the intensity pattern I generated by the phase-distorted wavefront through interference with an undistorted reference beam [5

5. M. Born and E. Wolf, Principles of Optics, 6th ed., Pergamon Press, Oxford, New York (1980).

9

9. F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part I,” Physica 9, 686 (1942). [CrossRef]

]. Crystal based Bragg-type interferometry requires an extreme stability of the environment to prevent spatial displacements down to the atomic-level within the interferometric set-up. Interferometric phase-contrast imaging employing diffraction gratings is much less sensitive to the positioning and the stability of the experimental setup [10

10. C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287 (2002). [CrossRef]

, 11

11. 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, 258 (2006). [CrossRef]

]. Appealing to the relation Δα = −1/k∂xϕ, the Schlieren technique (or crystal diffraction enhanced imaging) analyses the angular deviation Δα of the transmitted X-rays. The quantitative analysis of the images requires input from dynamical X-ray diffraction theory (finite Darwin width) [12

12. E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 1, 937 (1980). [CrossRef]

14

14. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595 (1995). [CrossRef]

].

The instrumentally simple and robust in-line or free-space propagation technique is based on Fresnel diffraction theory [15

15. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of xray phase contrast microimaging by coherent highenergy synchrotron radiation,” Rev. Sci. Instrum. 66 (12), 5486 (1995). [CrossRef]

, 16

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

]: An intensity pattern Iz, which develops due to interference of different, directionally displaced parts of the transmitted and propagated wavefront ψz(x, y), orginating by plane-wave illumination of the object, is observed by a detector placed at distance z behind the object. Within any transverse plane z ≥ 0 only a non-homogeneous gradient of the phase (xϕz ≠ const and yϕz ≠ const) generates an intensity contrast gz(x,y)Iz>0Iz=01 [17

17. D. Paganin, Coherent X-Ray Optics, Oxford University Press (2006). [CrossRef]

].

In some free-space propagation techniques [18

18. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1942 (1995).

, 19

19. D. Paganin and K. A. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phys. Rev. Lett. 80, 2586 (1998). [CrossRef]

] one directly appeals to the transport-of-intensity equation (TIE):
kzIz=(Izϕz),
(3)
where ∇ denotes the two-dimensional Nabla operator in the tranverse plane. In the limit of constant absorption (Iz=0 ≡ const) Eq. (3) takes the following form
kzgz=((gz+1)ϕz).
(4)
Relaxing the assumption of constant absorption, the contrast-transfer function (CTF) considers a linearized version of the transmission function of Eq. (2) in B and ϕz=0. In [20

20. L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: Demonstration of extended conditions for homogeneous objects,” Opt. Express 12, 2960 (2004). [CrossRef] [PubMed]

22

22. 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, 034102 (2006). [CrossRef]

] single-distance phase retrieval based on the CTF is considered and successfully applied. In particular, in [20

20. L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: Demonstration of extended conditions for homogeneous objects,” Opt. Express 12, 2960 (2004). [CrossRef] [PubMed]

] an interesting discussion of the extended regime of validity of CTF-based retrieval of the thickness function for a homogeneous object as compared to the TIE-based approach was given using a single-distance radiograph. In [21

21. T. E. Gureyev, Ya.I. Nesterets, D.M. Paganin, A. Pogany, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569 (2006). [CrossRef]

, 22

22. 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, 034102 (2006). [CrossRef]

] the usefulness of a combination of the TIE- and CTF-based approach (both approaches employing a linear relation between phase in the object plane and intensity a distance z away from the object) appealing to a multiplicative separation of a slowly varying factor from a small factor in the transmission function was shown. In Sec. 3.3 we point out the differences of TIE- and CTF-based phase retrieval to the nonlinear method proposed in the present paper.

In [23

23. M. Op de Beeck, D. Van Dyck, and W. Coene, “Wave function reconstruction in HRTEM: the parabola method,” Ultramicroscopy 64, 167 (1996) and refs. therein. [CrossRef]

, 24

24. P. Cloetens, Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation, PhD thesis at Vrije Universiteit Brussel (1999) and references therein. [PubMed]

] the CTF is used to overdetermine B and ϕz=0 by intensity measurements performed at several distances zi yielding remarkably accurate results with nonlinear effects taken into account iteratively. Judging by the linear approximation only, large spatial object scales are poorly retrieved by the CTF based approach because the Fourier transform of ϕz=0 has a sine-function coefficient that vanishes at the origin (zeros of the sine function at nonvanishing frequencies are regularized by summation over various values of z). A mixed TIE and CTF based approach is also applied, see [25

25. X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378 (2004). [CrossRef] [PubMed]

27

27. M. Langer, F. Peyrin, P. Cloetens, and J.-P. Guigay, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556 (2008). [CrossRef] [PubMed]

].

For sufficiently small z function gz varies in an approximately linear way in z, and Eq. (4) simplifies as: [Substituting gz(x, y) = g1(x, y) z and ϕz(x, y) = ϕz=0(x, y) + ϕ1(x, y) z into Eq. (4), consistently ignoring nonvanishing powers in z on the right-hand side, and multiplying by z yields Eq. (5).]
gz=λz2π2ϕz=0.
(5)
In [4

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

] the approximation of Eq. (5) is used to tomographically reconstruct the object function δ(x1, x2, x3) directly in terms of the data gz;θ where θ refers to an angle at which the object is rotated about the x3 direction, see Fig. 1. To retrieve ϕz=0 radiographs need to be collected at a single distance z only which is a prerequisite for the characterization of processes [27

27. M. Langer, F. Peyrin, P. Cloetens, and J.-P. Guigay, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556 (2008). [CrossRef] [PubMed]

29

29. M. Boone, Y. De Witte, M. Dierick, J. Van den Bulcke, J. Vlassenbroeck, and L. Van Hoorebeke, “Practical use of the modified Bronnikov algorithm in micro-CT,” Nucl. Instrum. Methods Phys. Res. B 267, 1182 (2009). [CrossRef]

].

The applicability of Eq. (5) is, however, restricted to the edge-enhancement regime thus preventing the reconstruction of object information on larger spatial scales whose effect emerges in terms of conspicuous fringes in the intensity pattern at larger values of z only. The edge-enhancement regime with increasing distance z emerges first in the intensity contrast due to a large density of interference points introduced by the strongly curved wavefront in the vicinity of boundaries between regions within the object that exhibit a large difference in δ.

In the present work we propose an extension of the phase-retrieval algorithm for pure-phase, nonperiodic objects relying on Eq. (5). This extension maintains the efficiency of that algorithm (single distance z, simple algorithm for phase retrieval) but works systematically beyond the linear approximation of Eq. (5). We demonstrate the potential usefulness of the extended algorithm by analysing simulated phantom data.

The paper is organized as follows: To set conventions we briefly outline the tomographic reconstruction algorithm of Ref. [4

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

] in Sec. 2. A systematic extension is developed in Sec. 3 by appeal to a model where gz as well as ϕz are expanded in powers series in z. The coefficients of the power series expansion are determined by the paraxial wave equation [Eq. (11)]. We give explicit expressions up to the next-to-leading order and discuss how they can be treated perturbatively. In Sec. 4 we perform first tests of our phase-retrieval algorithm applying it to simulated phantom data. Section 5 summarizes our work. An appendix contains the relation between gz and ϕz=0 at next-to-next-to leading order.

2. Tomographic reconstruction algorithm

The quantity required for tomographic reconstruction of function δ(x1, x2, x3) by inverse Radon transformation is 2ϕz=0;θ in the case of a pure-phase object. The main purpose of our work is to extend the expression of Eq. (5) for 2ϕz=0;θ in terms of intensity contrast gz;θ. Although we do not perform tomographic reconstruction in the present paper we would like to sketch conventions for the tomographic set-up as used in [4

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

] and to point out how 2ϕz=0;θ enters in the reconstruction algorithm.

The algorithm of [4

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

] relates the 3D Radon transform δ̂(s,θ,ω) of the object function δ(x1, x2, x3) to the 2D Radon transform 2ϕz=0;θ of 2ϕz=0;θ as
s2δ^(s,θ,ω)=λ2π(2ϕz=0;θ)(s,ω).
(6)
Here, x1, x2 and x3 denote the Cartesian coordinates in the object frame. The coordinates in the plane transverse to the direction of propagation are x and y. In this plane the 2D Radon transformation is understood as an integration over lines parameterized as xsinω + ycosω = s (see Fig. 1). In order to apply the Radon inversion formula
δ(x1,x2,x3)=14π20πdωsinω0πdθs2δ^(s,θ,ω)|s=(x1cosθ+x2sinθ)sinω+x3cosω
(7)
for the reconstruction of object function δ(x1, x2, x3) from the data gz;θ(x,y), Eqs. (6) and (7) thus requires a one-to-one relation between 2ϕz=0;θ and gz;θ. To leading (linear) order in a Taylor expansion of gz;θ in z this relation is provided by Eq. (5). Notice that Eq. (4) continues to hold approximately for weakly absorbing objects, that is, if |∇Iz=0;θ| is finite but |∇Iz=0;θ| ≪ |∇Iz>0;θ|. In this case two measurements, Iz=0;θ and Iz>0;θ, are required to determine gz;θ and hence, by virtue of Eqs. (6), (5), and (7), object function δ(x1, x2, x3).

3. Intensity contrast beyond linearity in z

3.1. Powers in z, paraxial wave equation, and perturbation theory

To characterize larger object scales than achieved within the edge-enhancement regime the object-detector distance z should be increased when applying the single-distance in-line propagation technique: As discussed in the Introduction, the intensity pattern Iz;θ then contains discernable information on the exiting phase ϕz=0;θ (x,y) and thus on the object function δ(x1, x2, x3). The model of Eq. (5), assuming a linear z dependence of gz;θ (x,y), however, breaks down at larger z, and thus a more general ansatz is needed:
gz;θ(x,y)=l=1lmaxgl;θ(x,y)zn.
(8)
Also, function ϕz;θ(x,y) is expanded in powers of z:
ϕz;θ(x,y)=m=0mmaxϕm;θ(x,y)zm.
(9)
Note that {gl;θ} = {g1;θ, g2;θ,...} and {ϕm;θ} = {ϕ0;θ, ϕ1;θ,..} denote the z-independent coefficients of the expansion of the z-dependent functions gz;θ and ϕz;θ, respectively. Substituting ansatz (8) and ansatz (9) into Eq. (4) and comparing coefficients up to linear order in z yields
g1;θ=1k2ϕ0;θ,g2;θ=12k[1k((2ϕ0;θ)ϕ0;θ+(2ϕ0;θ)2)2ϕ1;θ].
(10)
Equation (4) follows from the imaginary part of the paraxial equation
(2ikz+2)ψz;θ=0
(11)
for the wavefield ψz;θ=Iz;θexp(iϕz;θ). Again considering the absorption-free limit Iz=0;θ ≡ const, the real part of Eq. (11) is given as
zϕz;θ=12k(2gz;θ+1gz;θ+1(ϕz;θ)2).
(12)
In the limit z → 0 we have from Eq. (12) for the coefficient ϕ1;θ in Eq. (9)
ϕ1;θ=limz0zϕz;θ=12k(ϕ0;θ)2.
(13)
Combining Eqs. (8), (9), (10), and (13) (truncating at lmax = 2 and mmax = 1, next-to-leading-order) finally yields
2ϕz=0;θ2ϕ0;θ=kzgz;θ+z2k[(2ϕ0;θ)ϕ0;θ+(2ϕ0;θ)2+122(ϕ0;θ)2].
(14)
The result in Eq. (14) can also be obtained along the lines of [4

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

] by computing intensity Iz from the wavefield ψz, Fresnel-propagated from z = 0, and by neglecting variations in x and y of the attenuation factor. The counting in powers of z then is due to a Taylor expansion of the Fourier transform of the Fresnel propagator. In such a derivation of Eq. (14) it is, however, impossible to keep track of how the nonlinear corrections arise. This becomes clear when directly appealing to Eq. (10). Namely, the first two terms in the square brackets arise from the unpropagated phase while the third term is due to phase-propagation.

The formal expansion of 2ϕz=0;θ in powers of zk can be carried to higher orders, see the Appendix for an explicit listing of the next-to-next-to-leading-order correction. In doing so, Eq. (4) and lmax – 2 derivatives of Eq. (12) with respect to z are needed to express the coefficients in the expansions of Eqs. (8) and (9) in terms of polynomials of transverse derivatives acting on ϕ0;θ. For later discussion we notice that the nonlinear correction in Eq. (14) represents a sum of total derivatives in x and y. The expansion of 2ϕz=0;θ on the right-hand side of Eq. (14) formally is in ‘powers’ of derivatives of ϕ0;θ. We thus are required to demand a weak variation of ϕ0;θ in dependence of x,y. (Otherwise the coefficients of powers in z would be unacceptably large when increasing the order of the expansion in powers of z.)

Equation (14) represents a nonlinear partial differential equation for ϕz=0;θ linking the latter to the data gz;θ. Let us now discuss how the solution to Eq. (14) can be approached. In principle, one could try to solve Eq. (14) numerically subjecting it to appropriate boundary conditions. Notice that due to the presence of additional powers of ∇ϕ0;θ in Eq. (14) as compared to Eq. (5) an ambiguity ∇ϕ0;θ → ∇ϕ0;θ + v⃗, v⃗ a constant 2D vector no longer is present. Also, a reduction of the influence of optical vortices [18

18. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1942 (1995).

, 19

19. D. Paganin and K. A. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phys. Rev. Lett. 80, 2586 (1998). [CrossRef]

] should occur in ϕ0;θ: Phase retrieval is not only constrained by the transport-of-intensity equation (imaginary part of the paraxial wave equation representing Fresnel diffraction theory incompletely) but also by the real part of the paraxial wave equation. Thus, expanding beyond leading order increasingly accounts for the constraints of the full Fresnel theory.

An alternative to a brute-force treatment of Eq. (14) is to approach the right-hand side by appeal to the leading-order situation (setting lmax = 1 and mmax = 0 in Eqs. (8) and (9), respectively) to find a useful approximation. The nonlinear terms in square brackets are then approximated in terms of the solution ϕ0;θ of Eq. (5) easily obtained by Fourier transformation (perturbation theory).

In general, the right-hand side of Eq. (14) is expressed in terms of the following power series
2ϕ0;θ=l=0lmax+1cl(x,y)zl.
(15)
Ideally, all coefficients other than c0(x, y) vanish in Eq. (15). For example, setting lmax = 2, as assumed in deriving Eq. (14), the term kzgz;θ generates a contribution –k(g1;θ (x,y) + g2;θ (x,y)z) while the terms in square brackets is by definition linear in z. Ideally, this linear term cancels the linear term arising in kzgz;θ. When increasing lmax = 2 → lmax = 3 the quadratic term in kzgz;θ is cancelled by the quadratic correction on the right-hand side of Eq. (34) in the Appendix and so on.

Perturbation theory respects this counting in powers of z since by inverting the Laplacian in a truncation of the right-hand side of Eq. (14) at the leading order one, indeed, ends up with an estimate for ϕ0;θ that is independent of z. Due to its perturbative nature, this estimate, when substituted in the corrections to next-to-leading and next-to-next-to-leading order of Eq. (34), does, however, not completely cancel the respective powers in z in the term kzgz;θ. Still, it reduces the modulus of their coefficients.

3.2. Numerical implementation

Technically, it is advantageous to rescale coordinates x,y dimensionless as x,yx˜kzx,y˜kzy. In these variables, the right-hand side of the equivalent of Eq. (14) does not exhibit prefactors kz or zk of the leading order (LO) and the next-to-leading order (NLO), respectively. Since gz;θ is dimensionless its z dependence is determined by the dimensionless variables kz and α=za, β=zb, ··· where a, b, ··· are object scales.

Let us now abbreviate the NLO terms as
NLO1(x,y)(2ϕ0;θ)ϕ0;θNLO2(x,y)(2ϕ0;θ)2NLO3(x,y)122(ϕ0;θ)2.
(16)
To express the leading-order estimate for ϕ0;θ in terms of the data we appeal to Fourier transformation. To leading order we have g1;θ = gz/z. Thus in perturbation theory NLO1 reads as follows:
(2ϕ0;θ)ϕ0;θ=k21(ξig1;θ)1(ξiξ2g1;θ),
(17)
where ℱ (ℱ−1) denotes (inverse) Fourier transformation, ξ1, ξ2 are the Fourier conjugates to x,y, respectively, and a summation convention for repeated indices is understood (i = 1,2). NLO2 simply is
(2ϕ0;θ)2=k2g1;θ2,
(18)
and NLO3 is given as
122(ϕ0;θ)2=12k21{ξ2[1(ξiξ2g1;θ)1(ξiξ2g1;θ)]}.
Let us now discuss the expressions NLO1(x, y), NLO2(x, y), and NLO3(x, y) in more detail. Up to the factor ∇ϕ0;θ in NLO1, which relates the Laplacian of ϕ0;θ to the data g1;θ in a nonlocal way, the right-hand side of Eq. (17) is local in g1;θ. Loosely speaking, ∇ϕ0;θ is obtained by integrating the leading-order equation 2ϕz=0;θ=2ϕ0;θ=kzgz;θ once. Moreover, NLO2 is completely local in g1;θ while NLO3 is nonlocal. Each of the terms NLO1(x, y), NLO 2(x, y), and NLO3(x, y), can be decomposed as
NLOi(x,y)=NLOi+δNLOi(x,y),(i=1,2,3),
(19)
where 〈NLOi〉 ≡ ∫dxdyNLOi(x, y). Since NLO1(x, y) + NLO2(x, y) and NLO3(x, y) separately are sums of partial derivatives we have
i=13NLOi=0.
(20)
Since NLO3 is nonlocal in gz;θ, this term smeares g1;θ and averages out large amplitudes.

A pragmatic approach to inverting the Laplacian appealing to discrete Fast Fourier Transforms is to simply let
1ξ21ξ2+α2,
(21)
where α is a real constant. Requiring that the retrieved phase is insensitive to a variation of α fixes α’s value. It is straightforward to show that using the regularization of Eq. (21) in applying 2 to a function F(x,y) one has
,α2F(x,y)=α2F,
(22)
where ,α2 refers to a version of 2 regularized by virtue of (21).

From Eqs. (22), (20) and the fact that
gz;θ=0
(23)
(energy conservation for pure-phase objects) it follows that the phase retrieved by applying ,α2 to the right-hand side of Eq. (14) has zero mean (〈NLO1〉 and 〈NLO2〉 individually can be large but their sum is nil.). Deviations from this situation are introduced by numerical artifacts introduced through discrete Fourier transformations and thus should be subtracted. The exact phase, which is expressed by a line integral over the object function δ(x1, x2, x3), however has a nonvanishing mean (δ(x1, x2, x3) is sign definite). This ambiguity is due to the fact that an arbitrary constant can be added to
ϕ0;θ(x,y)ϕ0;θ+δϕ0;θ(x,y)
(24)
without changing Eq. (14) for ϕ0;θ. To compare exact with retrieved phase we thus subtract means in both cases.

3.3. Comparison with CTF-based single-distance retrieval

An important result due to Guigay [30

30. J.-P. Guigay, R. H. Wade, and C. Delpha, “Optical diffraction of Lorentz microscope images,” Proceedings of the 25th meeting of the Electron Microscopy and Analysis Group, W.C. Nixon ed. (The Institute of Physics, London, 1971), pp. 238–239.

] is an expression linking the autoproduct of the object transmission ψ0,θ to the Fourier transform of intensity ℱIz,θ at distance z:
Iz,θ(ξ)=dxdyexp(2πirξ)ψ0,θ*(r+πkzξ)ψ0,θ(rπkzξ),
(25)
where r⃗ ≡ (x,y). Assuming a pure-phase object, one has ψ0,θ=I0,θexp(iϕ0;θ) where I0,θ is homogeneous. In exploiting Eq. (25) one makes the assumption that ϕ0;θ(r+πkzξ)ϕ0;θ(rπkzξ)1 for all r⃗ and for all ξ⃗ at fixed z and fixed k. If this assumption is met then one may write
ψ0;θ=I0,θ(1+iϕ0;θ).
(26)
Substituting Eq. (26) into Eq. (25), one has [30

30. J.-P. Guigay, R. H. Wade, and C. Delpha, “Optical diffraction of Lorentz microscope images,” Proceedings of the 25th meeting of the Electron Microscopy and Analysis Group, W.C. Nixon ed. (The Institute of Physics, London, 1971), pp. 238–239.

]
Iz,θ(ξ)=I0,θ(δ(ξ)2sin(2π2kzξ2)ϕ0;θ),
(27)
Eq. (27) provides for an algebraic solution of the phase-retrieval problem in Fourier space provided the zeros of the sine function are regularized in a natural way. Notice that going beyond Eq. (26) in taking higher powers of ϕ0;θ into account algebraic inversion in the sense of Eq. (27) no longer is possible. This is because in Fourier space quadratic and higher orders enter in Eq. (25) in a highly nonlocal way.

Since
sin(2π2kzξ2)=i=1(1)i1(2i1)!(2π2zkξ2)2i1
(28)
(with an infinite radius of convergence of the series on the right-hand side) we deduce, upon letting ℱ−1 act on Eq. (27), that
gz;θ(r)=Iz,θ(r)I0,θI0,θ=2i=11(2i1)!(z2k)2i1(2)2i1ϕ0;θ(r).
(29)
Therefore, by assuming the validity of Eq. (26) the inversion of Eq. (27) corresponds to an infinite summation of powers of 2 acting on ϕ0;θ(r⃗). We stress that our expansion (8) contains these powers of 2 in a truncated way, see left-hand side of z× Eq. (14) at order z and the term (2)3ϕ0;θ in the sixth line of z× Eq. (34) at order z3. Notice that in addition to these terms nonlinear corrections in ϕ0;θ enter at these orders. This is the reason why the above assumption can increasingly be relaxed taking increasing powers of z into account in Eq. (8). The series Eq. (8) expands in powers of transverse derivatives including nonlinearities in ϕ0;θ whereas expansion (29) takes into account all powers of derivatives at linear order in ϕ0;θ only. For weak phase objects expansion (29) is an excellent approximation in treating the case where all frequencies contribute to the overall phase shift such that the latter is smaller than unity but breaks down as soon as a part of the spectrum contributes relative phase shifts comparable or larger than unity. Since the nonlinear expansion of Eq. (8) for pragmatic reasons, needs to be truncated at a finite order in z large frequencies are retrieved in a worse way than they are by inverting Eq. (27) within that equation’s regime of validity.

4. Phase retrieval for simulated phantom data

Let us now apply the phase retrieval algorithm of Sec. 3 to simulated phantom data. We use the regularization of Eq. (21) to invert the Laplacian in Eqs. (14), both in the perturbative estimate and the final expression for ϕ0;θ.

To see how sensitive the retrieved phase depends on the regularization parameter α we define a function Φ(θ,α) as follows
Φ(θ,α)x,y|ϕ0;θ(x,y)|,
(30)
where ϕ0;θ(x,y) is the retrieved phase at regularization parameter value α. The value of α is chosen such that Φ(θ,α) is least sensitive to a variation in α. For the phantoms considered below there occurs a wide region (several orders of magnitude with central value αc ∼ 10−6) of insensitivity for Φ(θ,α).

To measure the improvement achieved by the inclusion of the perturbatively evaluated correction terms in Eq. (14) compared to the leading-order linear term we define a quantity Ψθ,αc as follows:
Ψθ|ϕ0;θϕ0;θexact|,
(31)
where ϕ0;θ either is the leading-order (LO) or includes the next-to-leading-order (LO + NLO) result. The quantity 〈Ψθ〉 is defined as the mean of Ψθ over all pixels.

Let us now test the algorithm for a pure-phase phantom (Siemens star) with phase retrieval indicated in Fig. 2. All simulations are performed for a central projection (θ = 0) at an X-ray energy of E = 30keV, at δ = 10−7 on a δ = 0 background, and for thickness d = 0.256mm. A central disk with constant phase shift was introduced to avoid numerical problems arising from unresolved spokes segments. To avoid extreme phase jumps we have applied a normalized Gaussian blurring to the Siemens star to yield an exact phase map from which intensity is computed by free-space propagation. As Figs. 2(e) and 2(f) indicates, function Ψθ is smaller when taking next-to-leading corrections into account. To investigate the dependence on distance and resolution of the retrieved phase we perform line cuts through the phase map as indicated in Fig. 2(a). The results are shown in Fig. 3. Keeping the standard deviation of the Gaussian blurring constant at 1.5 pixels, results do not change substantially for smaller values of δ than 10−7. (This means that at a higher resolution the averaged-over length scale becomes shorter.) For δ > 10−6, however, strong deviations of retrieved from exact phase maps occur both in the leading-order and the next-to-leading-order result suggesting that a violation of our assumption on weak phase variation takes place. Recall that strong phase variations yield large coefficients in the power series of Eq. (15) and thus worsen the convergence properties of the expansion.

Fig. 2 Phase and intensity maps for the Siemens star in central projection at a resolution of 2048 × 2048. In all cases mean values are subtracted. (a) exact phase map at z = 0, (b) intensity contrast map at z = 0.3m generated from (a) by Fresnel propagation, (c) ϕLO for leading-order retrieval based on map (b), (d) the correction ϕNLO at next-to-leading-order retrieval, (e) ΨLO (see Eq. (31) for definition), and (f) ΨLO+NLO. A Gaussian blurring with standard deviation of 1.5 pixels was applied to the Siemens star before propagation. The green line in (a) indicates the trajectory along which cuts of the phase maps are presented in Fig. 3.
Fig. 3 Line cuts as indicated in Fig. 2 (a) through phase maps for the Siemens star with 32 spokes in dependence of distance z (top to bottom) and pixel resolution (left to right). Black (blue, red) lines refer to the exact (leading-order, including next-to-leading-order) result for phase retrieval.

Fig. 4 Phase maps for Siemens star with 256 spokes in central projection at a resolution of 2048 × 2048. (a) exact phase map at z = 0, (b) line cut through exact phase map (black), leading-order result for phase retrieval (blue), and for phase retrieval including next-to-leading order result at z = 0.3m. One has 〈ΨLO〉 = 0.1535 and 〈ΨLO+NLO〉 = 0.0347, respectively.

5. Summary and outlook

We have presented a systematic extension of a linear phase retrieval algorithm for the in-line propagation technique which is applicable to pure phase objects (soft tissue, polymers, etc.). This algorithm could be of advantage for computed tomography applied to processes because it appeals to a stack of radiographs taken at a single object-detector distance z only.

Specifically, we have proposed a model where intensity and phase contrast are expanded in powers of z. In Fresnel theory the expansion coefficients are determined in terms of increasing numbers of transverse derivatives of the phase map in the object plane. The resulting partial differential equation starts to be nonlinear at next-to-leading order and can be approached in a perturbative way. Using simulated phantom data, we have demonstrated the promise of this method.

Because the method is applicable to the retrieval of larger relative phase-shifts over the entire projection (nonlinearity) it should yield more satisfactory results for thick objects (ratio between thickness and linear size of field of view close to unity). Moreover, since the accumulated phase shift along a ray path through a thicker sample is less prone to small-scale variations of the projection in dependence of transverse coordinates (averaging out large variations in δ) the requirement of small values of derivatives (convergence of the expansion in Eq. (8)) is better met in this case (less nervousness on small transverse scales). Therefore, we expect the present approach to have applications in the tomography of samples introducing large phase shifts.

It is conceivable that the use of different basis sets than just powers in z (Eq. (8)) yield even better convergence properties of the expansion.

Finally, due to the linear relation (5) between phase and intensity contrast in the leading-order estimate the perturbative treatment of the nonlinearities in the next-to-leading and in the next-to-next-to-leading order corrections of the right-hand sides of Eqs. (14) and (34) extents consistently to the case of polychromatic irradiation: At leading order intensity contrast and effective phase both are understood as an integral over the spectrum involving their monochromatic components. Since in perturbation theory it is the thus defined leading-order estimate for the effective phase that enters into the evaluation of the nonlinearities of Eqs. (14) and (34) one arrives at a consistent generalization of the concept of an effective phase for the polychromatic case when invoking nonlinear corrections.

Appendix: Next-to-next-to leading corrections to 2ϕ0;θ

Here we give the next-to-next-to-leading order in the expansion x,y2ϕz=0;θ. To obtain these corrections ansatz Eqs. (8) and ansatz (9) are truncated at lmax = 3 and mmax = 2, respectively. Inserting this truncation into Eq. (4) and comparing coefficients up to quadratic order in z yields
g1;θ=1k2ϕ0;θg2;θ=12k[g1;θϕ1;θ+2ϕ1;θ+g1;θ2ϕ0;θ]g3;θ=13k[g1;θϕ1;θ+g2;θϕ0;θ+2ϕ2;θ+g1;θ2ϕ1;θ+g2;θ2ϕ0;θ].
(32)
From Eq. (12) and its derivative w.r.t. z we obtain for z → 0:
ϕ1;θ=12k(ϕ0;θ)2ϕ2;θ=14k[122g1;θ2ϕ0;θϕ1;θ]=14k2[124ϕ0;θ+ϕ0;θ(ϕ0;θ)2].
(33)
Including the next-to-next-to-leading order, we thus have
2ϕ0;θ=(zk)1gz;θ12(zk)[(2ϕ0;θ)ϕ0;θ+(2ϕ0;θ)2122(ϕ0;θ)2]+112(zk)2[2(2ϕ0;θ)(ϕ0;θ)2]+2ϕ0;θ(ϕ0;θ(2ϕ0;θ))+2ϕ0;θ(2ϕ0;θ)2+ϕ0;θ(2(ϕ0;θ)2)12(2)3ϕ0;θ+2(ϕ0;θ(ϕ0;θ)2)+32ϕ0;θ2(ϕ0;θ)2+22ϕ0;θϕ0;θ(x,y2ϕ0;θ)+2(2ϕ0;θ)2].
(34)

Acknowledgments

JM and RH have benefited from the discussion of available techniques in phase-contrast imaging provided by Peter Cloeten’s and Max Langer’s doctoral thesis [24

24. P. Cloetens, Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation, PhD thesis at Vrije Universiteit Brussel (1999) and references therein. [PubMed]

, 31

31. M. Langer, Phase Retrieval in the Fresnel Region for Hard X-ray Tomography, PhD thesis at L’Insitut National des Sciences Appliquees de Lyon (2008) and references therein.

]. In addition, the queries of a Reviewer, which have led to the inclusion of Sec. 3.3., were perceived to be very helpful and have led to an improvement of the manuscript. Useful conversations with Daniel Hänschke, Lukas Helfen, Tomy dos Santos Rolo, Heikki Suhonen, and Feng Xu are gratefully acknowledged.

References and links

1.

A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779 (1974). [CrossRef]

2.

J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik 49, 121 (1977).

3.

J.-P. Guigay, “The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams,” Opt. Commun. 26, 136 (1978). [CrossRef]

4.

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

5.

M. Born and E. Wolf, Principles of Optics, 6th ed., Pergamon Press, Oxford, New York (1980).

6.

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

7.

F. Beckmann, U. Bonse, F. Busch, O. Günnewig, and T. Biermann, “A novel system for X-ray phase-contrast microtomography,” HASYLAB Annual Report II,691 (1995).

8.

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

9.

F. Zernike, “Phase-contrast, a new method for microscopic observation of transparent objects. Part I,” Physica 9, 686 (1942). [CrossRef]

10.

C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287 (2002). [CrossRef]

11.

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, 258 (2006). [CrossRef]

12.

E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractometry for the characterization of targets for laser fusion experiments,” Krist. Tech. 1, 937 (1980). [CrossRef]

13.

V. A. Bushuev, V. N. Ingal, and E. A. Belyaevskaya, “Dynamical Theory of Images Generated by Noncrystalline Objects for the Method of Phase-Dispersive Introscopy,” Kristallografiya 41, 808 (1996).

14.

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373, 595 (1995). [CrossRef]

15.

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

16.

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

17.

D. Paganin, Coherent X-Ray Optics, Oxford University Press (2006). [CrossRef]

18.

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1942 (1995).

19.

D. Paganin and K. A. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phys. Rev. Lett. 80, 2586 (1998). [CrossRef]

20.

L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: Demonstration of extended conditions for homogeneous objects,” Opt. Express 12, 2960 (2004). [CrossRef] [PubMed]

21.

T. E. Gureyev, Ya.I. Nesterets, D.M. Paganin, A. Pogany, and S.W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259, 569 (2006). [CrossRef]

22.

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, 034102 (2006). [CrossRef]

23.

M. Op de Beeck, D. Van Dyck, and W. Coene, “Wave function reconstruction in HRTEM: the parabola method,” Ultramicroscopy 64, 167 (1996) and refs. therein. [CrossRef]

24.

P. Cloetens, Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation, PhD thesis at Vrije Universiteit Brussel (1999) and references therein. [PubMed]

25.

X. Wu and H. Liu, “A new theory of phase-contrast x-ray imaging based on Wigner distributions,” Med. Phys. 31, 2378 (2004). [CrossRef] [PubMed]

26.

T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53 (2004). [CrossRef]

27.

M. Langer, F. Peyrin, P. Cloetens, and J.-P. Guigay, “Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography,” Med. Phys. 35, 4556 (2008). [CrossRef] [PubMed]

28.

A. Groso, R. Abela, and M. Stampanoni, “Implementation of a fast method for high resolution phase contrast tomography,” Opt. Express 14, 8103 (2006). [CrossRef] [PubMed]

29.

M. Boone, Y. De Witte, M. Dierick, J. Van den Bulcke, J. Vlassenbroeck, and L. Van Hoorebeke, “Practical use of the modified Bronnikov algorithm in micro-CT,” Nucl. Instrum. Methods Phys. Res. B 267, 1182 (2009). [CrossRef]

30.

J.-P. Guigay, R. H. Wade, and C. Delpha, “Optical diffraction of Lorentz microscope images,” Proceedings of the 25th meeting of the Electron Microscopy and Analysis Group, W.C. Nixon ed. (The Institute of Physics, London, 1971), pp. 238–239.

31.

M. Langer, Phase Retrieval in the Fresnel Region for Hard X-ray Tomography, PhD thesis at L’Insitut National des Sciences Appliquees de Lyon (2008) and references therein.

OCIS Codes
(100.2960) Image processing : Image analysis
(100.3010) Image processing : Image reconstruction techniques
(100.5070) Image processing : Phase retrieval
(340.7440) X-ray optics : X-ray imaging

ToC Category:
Image Processing

History
Original Manuscript: September 16, 2010
Revised Manuscript: October 11, 2010
Manuscript Accepted: October 14, 2010
Published: November 23, 2010

Citation
Julian Moosmann, Ralf Hofmann, Andrei Bronnikov, and Tilo Baumbach, "Nonlinear phase retrieval from single-distance radiograph," Opt. Express 18, 25771-25785 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25771


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References

  1. A. Papoulis, "Ambiguity function in Fourier optics," J. Opt. Soc. Am. 64, 779 (1974). [CrossRef]
  2. J.-P. Guigay, "Fourier transform analysis of Fresnel diffraction patterns and in-line holograms," Optik 49, 121 (1977).
  3. J.-P. Guigay, "The ambiguity function in diffraction and isoplanatic imaging by partially coherent beams," Opt. Commun. 26, 136 (1978). [CrossRef]
  4. A. V. Bronnikov, "Theory of quantitative phase-contrast computed tomography," J. Opt. Soc. Am. A 19, 472 (2002). [CrossRef]
  5. M. Born, and E. Wolf, Principles of Optics, 6th ed., Pergamon Press, Oxford, New York (1980).
  6. A. Momose, "Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer," Nucl. Instrum. Methods Phys. Res. A 352, 622 (1995). [CrossRef]
  7. F. Beckmann, U. Bonse, F. Busch, O. Günnewig, and T. Biermann, "A novel system for X-ray phase-contrast microtomography," HASYLAB Annual Report II, 691 (1995).
  8. U. Bonse, and M. Hart, "An X-ray interferometer," Appl. Phys. Lett. 6, 155 (1965). [CrossRef]
  9. F. Zernike, "Phase-contrast, a new method for microscopic observation of transparent objects. Part I," Physica 9, 686 (1942). [CrossRef]
  10. C. David, B. Nöhammer, H. H. Solak, and E. Ziegler, "Differential x-ray phase contrast imaging using a shearing interferometer," Appl. Phys. Lett. 81, 3287 (2002). [CrossRef]
  11. 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, 258 (2006). [CrossRef]
  12. E. Förster, K. Goetz, and P. Zaumseil, "Double crystal diffractometry for the characterization of targets for laser fusion experiments," Krist. Tech. 1, 937 (1980). [CrossRef]
  13. V. A. Bushuev, V. N. Ingal, and E. A. Belyaevskaya, "Dynamical Theory of Images Generated by Noncrystalline Objects for the Method of Phase-Dispersive Introscopy," Kristallografiya 41, 808 (1996).
  14. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and W. Wilkins, "Phase-contrast imaging of weakly absorbing materials using hard X-rays," Nature 373, 595 (1995). [CrossRef]
  15. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, "On the possibilities of xray phase contrast microimaging by coherent high energy synchrotron radiation," Rev. Sci. Instrum. 66(12), 5486 (1995). [CrossRef]
  16. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays," Nature 384, 335 (1996). [CrossRef]
  17. D. Paganin, Coherent X-Ray Optics, Oxford University Press (2006). [CrossRef]
  18. T. E. Gureyev, A. Roberts, and K. A. Nugent, "Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials," J. Opt. Soc. Am. A 12, 1942 (1995).
  19. D. Paganin, and K. A. Nugent, "Noninterferometric Phase Imaging with Partially Coherent Light," Phys. Rev. Lett. 80, 2586 (1998). [CrossRef]
  20. L. D. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, "X-ray phase imaging: Demonstration of extended conditions for homogeneous objects," Opt. Express 12, 2960 (2004). [CrossRef] [PubMed]
  21. T. E. Gureyev, Ya. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination," Opt. Commun. 259, 569 (2006). [CrossRef]
  22. 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, 034102 (2006). [CrossRef]
  23. M. Op de Beeck, D. Van Dyck, and W. Coene, "Wave function reconstruction in HRTEM: the parabola method," Ultramicroscopy 64, 167 (1996) and refs. therein. [CrossRef]
  24. P. Cloetens, Contribution to Phase Contrast Imaging, Reconstruction and Tomography with Hard Synchrotron Radiation, PhD thesis at Vrije Universiteit Brussel (1999) and references therein. [PubMed]
  25. X. Wu, and H. Liu, "A new theory of phase-contrast x-ray imaging based on Wigner distributions," Med. Phys. 31, 2378 (2004). [CrossRef] [PubMed]
  26. T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53 (2004). [CrossRef]
  27. M. Langer, F. Peyrin, P. Cloetens, and J.-P. Guigay, "Quantitative comparison of direct phase retrieval algorithms in in-line phase tomography," Med. Phys. 35, 4556 (2008). [CrossRef] [PubMed]
  28. A. Groso, R. Abela, and M. Stampanoni, "Implementation of a fast method for high resolution phase contrast tomography," Opt. Express 14, 8103 (2006). [CrossRef] [PubMed]
  29. M. Boone, Y. De Witte, M. Dierick, J. Van den Bulcke, J. Vlassenbroeck, and L. Van Hoorebeke, "Practical use of the modified Bronnikov algorithm in micro-CT," Nucl. Instrum. Methods Phys. Res. B 267, 1182 (2009). [CrossRef]
  30. J.-P. Guigay, R. H. Wade, and C. Delpha, "Optical diffraction of Lorentz microscope images," Proceedings of the 25th meeting of the Electron Microscopy and Analysis Group, W. C. Nixon ed. (The Institute of Physics, London, 1971), pp. 238-239.
  31. M. Langer, Phase Retrieval in the Fresnel Region for Hard X-ray Tomography, PhD thesis at L’Insitut National des Sciences Appliquees de Lyon (2008) and references therein.

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