## Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging

Optics Express, Vol. 16, Issue 5, pp. 3223-3241 (2008)

http://dx.doi.org/10.1364/OE.16.003223

Acrobat PDF (548 KB)

### Abstract

Simple analytical expressions are derived for the spatial resolution, contrast and signal-to-noise in X-ray projection images of a generic phase edge. The obtained expressions take into account the maximum phase shift generated by the sample and the sharpness of the edge, as well as such parameters of the imaging set-up as the wavelength spectrum and the size of the incoherent source, the source-to-object and object-to-detector distances and the detector resolution. Different asymptotic behavior of the expressions in the cases of large and small Fresnel numbers is demonstrated. The analytical expressions are compared with the results of numerical simulations using Kirchhoff diffraction theory, as well as with experimental X-ray measurements.

© 2008 Optical Society of America

## 1. Introduction

1. R. Fitzgerald, “Phase-sensitive x-ray imaging,” Physics Today **53**(7), 23–26 (2000). [CrossRef]

2. 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**, 5486–5492 (1995). [CrossRef]

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

2. 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**, 5486–5492 (1995). [CrossRef]

4. P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. J. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. **29**, 133–146 (1996). [CrossRef]

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

5. A. Krol, J. C. Kieffer, and E. Foerster, “Laser-driven x-ray source for diagnostic radiology. Applications of X-rays Generated from Lasers and Other Bright Sources,” Proc. SPIE **3157**, 156–163 (1997). [CrossRef]

6. T. A. Shelkovenko, D. B. Sinars, S. A. Pikuz, K. M. Chandler, and D. A. Hammer, “Point-projection x-ray radiography using an X pinch as the radiation source,” Rev. Sci. Instrum. **72**, 667–670 (2001). [CrossRef]

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

**384**, 335–338 (1996). [CrossRef]

9. K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. **77**, 2961–2964 (1996). [CrossRef] [PubMed]

10. Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. **76**, 093706 (2005). [CrossRef]

## 2. Analytical formulae for in-line phase contrast

*z*=0 transverse to the optic axis

*z*(Fig. 1). The sample is illuminated by an X-ray beam emanating from an extended spatially incoherent source located near the point

*z*=-

*R*

_{1}. We assume that the X-ray transmission through the sample can be characterised by the complex transmission function

*Q*(

*x*,

*y*,ν),

*Q*≡exp[

*iφ*–

*µ*], where (

*x*,

*y*) are the Cartesian coordinates in the object plane and ν is the radiation frequency (ν=

*c*/

*λ*=

*kc*/(2

*π*), where

*k*is the wavenumber). The transmitted beam is registered by a position-sensitive detector located immediately after the “detector” plane

*z*=

*R*

_{2}.

*µ*=0), in a vicinity of which the distribution of transmitted phase can be modeled as (Fig. 2)

*y*coordinate, |

*φ*|

_{max}(ν) is a constant representing the local maximum of the absolute value of the phase shift,

*H*(

*x*) is the Heaviside “step” function (which is equal to 0 for negative

*x*and is equal to 1 for positive

*x*),

*P*

_{obj}(

*x*,

*y*)=(2

*πσ*

^{2}

_{obj})

^{-1}exp[-(

*x*

^{2}+

*y*

^{2})/(2

*σ*

^{2}

_{obj})] is a function describing the “sharpness” of the edge and the asterisk denotes two-dimensional convolution.

*P*

_{sys}(

*x*,

*y*,

*M*)=[2

*πσ*

^{2}

_{sys}(

*M*)]

^{-1}exp{-(

*x*

^{2}+

*y*

^{2})/[2

*σ*

^{2}

_{sys}(

*M*)]} be the point-spread function (PSF) of the imaging system referred to the object plane (we assume for simplicity that the PSF is the same at all X-ray energies),

*σ*

^{2}

_{sys}(

*M*)=(

*M*-1)

^{2}

*M*

^{-2}

*σ*

^{2}

_{src}+

*M*

^{-2}

*σ*

^{2}

_{det}is the variance of the PSF,

*σ*

_{src}and

*σ*

_{det}are the standard deviations of the source intensity distribution and the detector PSF, respectively, and

*M*=(

*R*

_{1}+

*R*

_{2})/

*R*

_{1}is the geometric magnification. The particular form of the above expression for σsys is a direct consequence of the projection imaging geometry (Fig. 1).

### 2.1. Monochromatic near-field in-line contrast for a phase edge

*S*(

*x*,

*y*,

*z*, ν), in in-line images can be described by the Transport of Intensity equation (TIE) [8, 11

11. M. R. Teague, “Deterministic Phase Retrieval: a Green’s Function Solution,” J. Opt. Soc. Am. **73**, 1434–1441 (1983). [CrossRef]

12. T. E. Gureyev and S. W. Wilkins, “On X-ray phase imaging with a point source,” J. Opt. Soc. Am. A **15**, 579–585 (1998). [CrossRef]

**384**, 335–338 (1996). [CrossRef]

12. T. E. Gureyev and S. W. Wilkins, “On X-ray phase imaging with a point source,” J. Opt. Soc. Am. A **15**, 579–585 (1998). [CrossRef]

*S*

_{in}(ν) is the spectral density of the incident beam,

*S*(

*x*,

*y*,

*R*

_{2}, ν) is the spectral density distribution in the image plane

*z*=

*R*

_{2}and

*R*′=

*R*

_{2}/

*M*is the effective propagation (“defocus”) distance. As the derivative of the Heaviside function is the Dirac delta-function, we can write:

*σ*

^{2}

_{M}=

*σ*

^{2}

_{obj}+

*σ*

^{2}

_{sys}(

*M*). Substituting this expression into Eq. (2), we obtain

*x*=±

*σ*, of local maximum and minimum of the spectral density in the vicinity of an edge. These positions correspond to the centre of the “positive” and “negative” Fresnel fringes near the geometric shadow of the edge (Fig. 3) (note that the TIE allows for only one positive and one negative Fresnel fringe).

_{M}*A*), as the absolute value of the difference between the image spectral density distribution and the corresponding spectral density,

*S*

_{0}(ν)≡

*M*

^{-2}

*S*

_{in}(ν), in the background image (without the edge feature), integrated over the area

*A*=2

*Ma*×

*ML*in the image plane, where

_{y}*L*is the “length” of the edge feature in the object plane in the direction parallel to the edge and (-

_{y}*a*,

*a*) is a vicinity of the edge along the

*x*coordinate. Note that

*L*is assumed to be sufficiently large (in particular, it is much larger than

_{y}*σ*

_{sys}and

*σ*

_{obj}). Using Eq. (3) we obtain:

*D*(ν,

*A*)=4

*S*

_{in}(ν)

*aL*is the sum of the total X-ray spectral densities incident on the region of interest in the images with and without the edge feature. The SNR is then equal to,

_{y}*SNR*(ν,

*A*) as a function of a reaches its maximum at

*a*≅2.162

*σ*

_{M}. As the quantity 2

*σ*is approximately equal to the width of the first Fresnel fringes in the TIE regime, it is natural to choose

_{M}*a*=2

*σ*for the calculation of the SNR corresponding to the phase contrast generated by the edge. Such a choice leads to almost maximal SNR, while also being convenient for practical evaluation of SNR in experimental phase-contrast images. We therefore define

_{M}*SNR*

^{TIE}(ν)≡

*SNR*(ν, 4

*Mσ*×

_{M}*ML*) and obtain:

_{y}*D*

^{TIE}(ν)=8

*S*

_{in}(ν)

*σ*is the corresponding total incident X-ray spectral density,

_{M}L_{y}*N*=

_{F}*σ*

^{2}

*/*

_{M}k*R*′ is the minimal Fresnel number (which corresponds to the size, 2

*σ*, of the smallest resolvable detail in the image [13

_{M}13. T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. **231**, 53–70 (2004). [CrossRef]

*C*

_{1}=(1-e

^{-2})/[4(2

*π*)

^{1/2}]≅0.0862 is an absolute constant.

*C*

_{max}(ν,

*A*)=[

*S*

_{max}(ν)-

*S*

_{min}(ν)]/[

*S*

_{max}(ν)+

*S*

_{min}(ν)], where

*S*

_{max}(ν) and

*S*

_{min}(ν) are the maximum and minimum of the image spectral density in region

*A*. Given the locations,

*x*=±

*σ*, of local maximum and minimum of spectral density in the vicinity of an edge and using Eq. (3

_{M}**384**, 335–338 (1996). [CrossRef]

*C*

^{TIE}

_{max}(ν)≡

*C*

_{max}(ν, 4

*Mσ*×

_{M}*ML*) inside the area

_{y}*A*=4

*Mσ*×

_{M}*ML*:

_{y}*C*

_{2}=(2

*π*e)

^{-1/2}≅0.2420 and

*C*

_{3}=4/[e

^{1/2}(1-e

^{-2})]≅2.806 are absolute constants. Equation (5) describes the dependence of the contrast on such parameters as the wavelength of the incident radiation, effective defocus distance and the spatial resolution of the imaging system. This equation indicates a very simple behavior of the image contrast in the TIE regime, i.e. the contrast is directly proportional to the maximum phase shift and to the inverse of the Fresnel number.

**384**, 335–338 (1996). [CrossRef]

12. T. E. Gureyev and S. W. Wilkins, “On X-ray phase imaging with a point source,” J. Opt. Soc. Am. A **15**, 579–585 (1998). [CrossRef]

*C*

^{TIE}

_{max}(ν)≪1, or, according to Eq. (5),

*N*≫|

_{F}*φ*|

_{max}(ν). This necessary TIE validity condition is complementary to another commonly used necessary condition,

*N*≫1, which does not depend on the magnitude of the phase shift. It was demonstrated in recent numerical simulations [13

_{F}13. T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. **231**, 53–70 (2004). [CrossRef]

*N*≫1 alone cannot be sufficient for the validity of the TIE. It is also easy to show that when |

_{F}*φ*|

_{max}(ν)≪1, the condition

*N*≫|

_{F}*φ*|

_{max}(ν) is not sufficient for the validity of the TIE (see section 2.4 below). On the other hand, it is known that the TIE approximation is valid if and only if the transmission function satisfies the following condition: |∇

^{2}

*Q*(

**r**, ν)|

*R*′

*λ*/

*σ*≪|∇

_{M}*Q*(

**r**, ν)|, which allows one to use the first-order Taylor approximation for

*Q*[13

13. T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. **231**, 53–70 (2004). [CrossRef]

*R*′

*λ*max{1,|

*φ*|

_{max}(ν)}≪

*σ*

_{M}σ_{obj}. Therefore, the condition

*N*′

*≫max{1, |*

_{F}*φ*|

_{max}(ν)}, where

*N*′

*=2*

_{F}*πσ*

_{M}σ_{obj}/(

*R*′

*λ*), is necessary and sufficient for the validity of the TIE, Eq. (2). As,

*σ*≡[

_{M}*σ*

^{2}

_{obj}+

*σ*

^{2}

_{sys}(

*M*)]

^{1/2}≥

*σ*

_{obj}, then

*N*≥

_{F}*N*′

*, and the condition*

_{F}*N*≫max{1, |

_{F}*φ*|

_{max}(ν)} is necessary, but not always sufficient for the validity of the TIE. In particular, an in-line image of a sharp edge (with a small

*σ*

_{obj}) may not be possible to adequately describe using the TIE, even if the spatial resolution of the imaging system is very low (

*σ*

_{sys}(

*M*) is large). Numerical simulations suggest that in such cases the convolution with the broad PSF of the imaging system may wash out high-order Fresnel fringes in the image, but the remaining first Fresnel fringe may become asymmetric, which obviously cannot be described by the TIE [10

10. Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. **76**, 093706 (2005). [CrossRef]

*σ*

_{obj}≪

*σ*

_{sys}). For large Fresnel numbers

*N*′

*F*the in-line image is described by Eq. (2), where the lateral spreading is obviously determined by the convolution with the PSF of the imaging system. If we exclude the influence of the object properties, then

*σ*=

_{M}*σ*(

_{sys}*M*) and we obtain the following expression for the finest achievable spatial resolution

*N*′

*F*≫max{1, |

*φ*|

_{max}(ν)}, imposes a limit on the spatial resolution consistent with the use of the TIE, Eq. (2). Indeed, it implies in particular that if

*σ*

_{obj}≪

*σ*

_{sys}, then

*σ*

_{sys}>0, the width of Fresnel fringes depends on

*σ*

_{sys}as well. Note also that the TIE approximation allows for existence of only a single Fresnel fringe near the geometric image of an edge (this can be easily seen from the mathematical structure of Eq. (2)), so the spatial resolution of the TIE-based imaging is naturally associated with the width of the first Fresnel fringe. As will be shown explicitly in section 2.4 below, the width of the first Fresnel fringe reduces to the value given by Eq. (6) under the condition

*N*≫1.

_{F}### 2.2. Polychromatic near-field in-line contrast for a phase edge

*I*(

*x*′,

*y*′)=∫

*S*(

*x*′,

*y*′, ν)dν, where

*I*is the corresponding time-averaged intensity. The polychromatic TIE for pure phase objects is [14

14. 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**, 068103-1–068103-4 (2004). [CrossRef] [PubMed]

*σ*×

_{M}*L*we obtain the following expression for the signal-to-noise ratio in the polychromatic case:

_{y}*D*

^{TIE}=8

*I*

_{in}

*σ*is the corresponding total incident X-ray intensity,

_{M}L_{y}*≡*ψ ¯

*I*

^{-1}

_{in}∫

*S*

_{in}(ν)

*φ*(ν)

*k*

^{-1}dν is the “generalized eikonal” [14

14. 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**, 068103-1–068103-4 (2004). [CrossRef] [PubMed]

*|*ψ ¯

_{max}≡

*I*

^{-1}

_{in}∫

*S*

_{in}(ν)|

*φ*|

_{max}(ν)

*k*

^{-1}dν is the spectrally averaged maximum eikonal and

*C*

_{1}=(1-e

^{-2})/[4(2π)

^{1/2}]≅0.0862 is an absolute constant.

*x*=±

*σ*, of the centre of the first Fresnel fringes are independent from the X-ray frequency, we can find an expression for the maximum image contrast from Eq. (7):

_{M}*C*

_{2}=(2

*π*e)

^{-1/2}≅0.2420 and

*C*

_{3}=4/[e

^{1/2}(1-e

^{-2})]≅2.806 are absolute constants. The corresponding TIE validity condition is

*σ*

_{M}σ_{obj}/

*R*′≫max{

*/(2*λ ¯

*π*), |

*|*ψ ¯

_{max}}, where

*≡*λ ¯

*cI*

^{-1}

_{in}∫

*S*

_{in}(ν)ν

^{-1}dν.

### 2.3. Optimization of SNR in near-field in-line imaging

*γ*≡(1-e

^{-2})(4

*πI*

_{in})

^{-1/2}is a constant. We should emphasize that this formulation of the SNR optimization problem assumes that the incident intensity in the region of interest of the object plane is kept constant (this is generally consistent with the assumption of a fixed dose). This implies, for example, that if the source-to-object distance is varied from

*R*

^{(0)}

_{1}to

*R*

^{(1)}

_{1}=

*R*

^{(0)}

_{1}+Δ

*R*

_{1}, then the exposure needs to be increased by the factor (

*R*

^{(1)}

_{1}/

*R*

^{(0)}

_{1})

^{2}provided that the source intensity stays the same, etc. For a given feature with fixed parameters

*σ*

_{obj}and

*L*, the maximization can be achieved by improving the spatial resolution of the imaging system (decreasing

_{y}*σ*

_{sys}(

*M*)), increasing the defocus distance

*R*′ and adjusting the incident spectrum in favor of the energies with larger values of |

*φ*|

_{max}(ν)/

*k*. If the incident spectrum is fixed as well and only the geometric parameters of the imaging system can be varied, then the quantity to be maximized is

*R*=

*R*

_{1}+

*R*

_{2}=

*R*

_{2}

*M*/(

*M*-1) is the total source-to-detector distance and

*γ*′=

*γL*

^{1/2}

_{y}∫

*S*

_{in}(ν)|

*φ*|

_{max}(ν)

*k*

^{-1}dν is a constant. Obviously, in the TIE regime this SNR monotonically increases as the source-to-detector distance

*R*increases, or as the source size and detector PSF decrease. The only non-trivial dependence is that on magnification. Note that

*SNR*

^{TIE}(

*M*)=0 at both limits,

*M*=1 and

*M*=∞, according to Eq. (11). Therefore, the function

*SNR*

^{TIE}(

*M*) has a maximum at some intermediate value of magnification,

*M*=

*M*

_{opt}, which can be found using Eq. (11).

*σ*

_{obj},

*σ*

_{src}and

*σ*

_{det}, i.e. by the sharpness of the edge, the X-ray source size and the spatial resolution of the detector system. It can be shown that when

*σ*

_{src}=

*σ*

_{det},

*M*

_{opt}is always equal to 2. However, when

*σ*

_{src}≠

*σ*

_{det},

*M*

_{opt}can take different values. Consider, for example, the parameters used in our numerical and experimental tests later in this paper (these parameters correspond to an in-line system with a laboratory microfocus X-ray source and Imaging Plates as a detector), where

*σ*

_{src}=1.7

*µm*and

*σ*

_{det}=42.5

*µm*. Then, taking e.g.

*σ*

_{obj}=0.7

*µm*, one obtains from eq.(11) that

*M*

_{opt}=18. On the other hand, under conditions more typical for synchrotron experiments, where one may have

*σ*

_{src}=42.5

*µm*,

*σ*

_{det}=1.7

*µm*, and assuming the same

*σ*

_{obj}=0.7

*µm*, one obtains

*M*

_{opt}=1.059. These values of the optimal magnification are fairly consistent with typical experimental experience.

15. 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**, 569–580 (2006). [CrossRef]

*µ*

_{max}(ν)]≅1–2

*µ*

_{max}(ν) can be used at all frequencies ν present in the incident X-ray spectrum, where 2

*µ*

_{max}(ν) is the maximum X-ray attenuation in the feature [16

16. A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman “Initial experimentation with in-line holography x-ray phasecontrast imaging with ultrafast laser-based x-ray source,” Proc. SPIE **6510**, 65100L (2007). [CrossRef]

### 2.4. Fresnel-region in-line contrast for a phase edge

*φ*satisfy the Guigay condition [17]:

**r**=(

*x*,

*y*) in the object plane and all

**r**

_{±}=

**r**±(

*R*′

*λ*/2)

**ρ**with

**ρ**=(

*ξ*,

*η*) from the circle

*B*with the centre at the origin of coordinates in the Fourier space and radius

_{ρ}*ρ*,

*ρ*=min(

*ρ*

_{sys}, 2

*ρ*

_{obj}), where

*ρ*

_{sys}and

*ρ*

_{obj}are the respective radii of the smallest circles outside which the Fourier transform of the system’s PSF and the transmission function are negligibly small in magnitude. In the case of a plane monochromatic incident wave and the object plane phase satisfying Eq. (12), the 2D spatial Fourier transform of the spectral density in the detector plane,

*S*̂(

*ξ*,

*η*,

*R*

_{2}, ν)=∬exp[-

*i*2

*π*(

*xξ*+

*yη*)]

*S*(

*x*,

*y*,

*R*

_{2}, ν)d

*x*d

*y*, can be expressed in the following form [15

15. 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**, 569–580 (2006). [CrossRef]

*P̂*

_{sys}(0,0,

*M*)=1. This equation represents a generalization of Eq. (2), with Eq. (2) formally obtainable from Eq. (13) by means of replacing the sine function by its argument followed by the inverse Fourier transform. The main advantage of Eq. (13) over Eq. (2) is that Eq. (13) is not limited to the “near-field”, i.e. the condition

*N*′

*≫max{1, |*

_{F}*φ*|

_{max}(ν)} is not necessary for the validity of Eq. (13). Instead, Eq. (13) can be derived under condition (12) which effectively allows the phase function to consist of two components, one of them being large in magnitude, but slowly varying, with the second one being small in magnitude, but possibly rapidly varying [13

**231**, 53–70 (2004). [CrossRef]

*φ*

_{0}(

*x*,

*y*)=

*ε*sin(

*x*/

*σ*), with such parameters

*ε*and

*σ*that

*ε*≪

*σ*

^{2}/(

*R*′

*λ*)≤

*N*′

*<1, obviously satisfies condition (12), because |*

_{F}*φ*

_{0}(

*x*,

*y*)|≤

*ε*≪1. Therefore, the in-line phase contrast, including the Talbot effect, for the latter function can be described by Eq. (13) [19

19. P. Cloetens, J.-P. Guigay, C. De Martino, and J. Baruchel, “Fractional Talbot imaging of phase gratings with hard x rays,” Opt. Lett. **22**, 1059–1061 (1997). [CrossRef] [PubMed]

*φ*|

_{max}(ν)≪1, the condition

*N*′

*≫|*

_{F}*φ*|

_{max}(ν) is not sufficient for the validity of the TIE. Now consider the large slowly varying component of a phase function satisfying condition (12). For such functions condition (12) implies that

*N*′

*≫|*

_{F}*φ*|

_{max}(ν). Moreover, by definition the spatial Fourier spectrum of the slowly varying component is confined to a small circle,

*ρ*

^{2}

_{obj}≪1/(

*R*′

*λ*), hence

*N*′

*≫1. In this case the sine function in Eq. (13) can be replaced by its argument, and Eq. (13) reduces to the TIE, Eq. (2). These examples agree well with the statement presented earlier in this paper that the necessary and sufficient condition for the validity of the TIE can be expressed as*

_{F}*N*′

*≫*

_{F}_{max}{1, |

*φ*|

_{max}(ν)}.

*F*(∞,

*N*)=0 (the latter fact can be easily understood in the context of Eq. (14), if one considers the energy conservation requirement together with the obvious property

_{F}*F*(

*x*,

*N*)=

_{F}*-F*(

*-x*,

*N*)). Typical profiles of the spectral density in the vicinity of a geometric image of the edge calculated in accordance with Eqs. (14)-(15) are shown in Fig. 3 for the following values of the relevant parameters:

_{F}*M*=1, |

*φ*|

_{max}=1,

*σ*=1 µm,

_{M}*N*=0.2, 1 and 5.

_{F}*x*, of local maxima and minima of the spectral density are defined by the equation (σ

_{m}*)[*

_{x}F*x*/(

_{m}*σ*),

_{M}n_{F}*N*]=0, i.e.

_{F}*m*=0,±1,±2,…. The locations of the first extrema to the left and right of the edge are

*P*

_{obj}, we obtain an expression for the limit of spatial resolution,

*N*≫1, then arctan

_{F}*N*

^{-1}

*≅*

_{F}*N*

^{-1}

*and we obtain from Eq. (16) that (Δ*

_{F}*x*)

_{min}≅2

*σ*

_{sys}(

*M*), which gives the limit of the spatial resolution for a sharp edge as defined by Eq. (6). At the opposite limit, when

*N*≪1, then arctan

_{F}*N*

^{-1}

*=*

_{F}*π*/2 and we obtain another well-known result [20

20. V. E. Cosslett and W. C. Nixon, “The X-Ray Shadow Microscope,” J. Appl. Phys. **24**, 616–623 (1953). [CrossRef]

18. A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus X-ray source,” Rev. Sci. Instrum. **68**, 2774–2782 (1997). [CrossRef]

**231**, 53–70 (2004). [CrossRef]

15. 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**, 569–580 (2006). [CrossRef]

*x*

^{±}

_{0}given above, we obtain the following expression for the contrast:

*N*arctan

_{F}*N*

^{-1}

*→1, sin[(1/2) arctan*

_{F}*N*

^{-1}

*-*

_{F}*t*

^{2}/(2

*N*)]≅(1-

_{F}*t*

^{2})/(2

*N*), and taking into account the value of the definite integral ∫

_{F}^{1}

_{0}exp(-

*t*

^{2}/2)(1-

*t*

^{2})d

*t*=e

^{-1/2}, we obtain that the expression given by Eq. (18) for

*N*≫1, coincides with that given by Eq. (5). At the opposite end, when

_{F}*N*≪1, we have

_{F}*N*

_{F}arctan

*N*

^{-1}

*→*

_{F}*πN*/2, and taking into account the value of the definite integral ∫

_{F}^{1}

_{0}sin[(

*π*/4)(1-

*t*

^{2})]d

*t*≅0.4876, we obtain that when

*N*→0, the contrast asymptotically tends to the constant value

_{F}*S*

_{0}(ν)≡

*M*

^{-2}

*S*

_{in}, integrated over the area corresponding to the first Fresnel fringes:

*D*(ν)=8

*x*

^{+}

_{0}

*L*

*y*S_{in}(ν) is the incident X-ray spectral density integrated over the area corresponding to the first Fresnel fringe in the images with and without the edge feature. The signal-to-noise,

*SNR*≡∑/

*N*, is then equal to

*N*

_{F}→ 0, then

*N*→0. One can see from Eq. (5) that

_{F}*γ*(

*N*)→e

_{F}^{1/2}(1-e

^{-2})/4≅0.3564 when

*N*→∞. We have calculated numerically the value of

_{F}*γ*(

*N*) for a wide range of Fresnel numbers and found that

_{F}*γ*(

*N*) varies slowly in between the two limits, 0.2717 and 0.3564.

_{F}*I*(

*x*,

*y*)=∫

*S*(

*x*,

*y*, ν)dν. The polychromatic analogue of Eq. (14) is the following equation for the time-averaged intensity of a projection image of a pure-phase edge feature:

*λ*=

*c*/ν) cannot be generally factored out, unless the spectrum is so narrow that the dependence of

*N*=2

_{F}*πσ*

^{2}

_{M}/(

*R*′

*λ*) on ν can be neglected. In the latter case it is trivial to obtain exact analogous of Eqs. (16)-(21) with the maximum monochromatic phase shift, |

*φ*|

_{max}(ν), replaced by its average value over the spectrum, |

*φ*|

_{max}=∫

*S*

_{in}(ν)|

*φ*|

_{max}(ν) dν/

*I*

_{in}. If such a simplification is impossible, the dependence of contrast and other image parameters on the wavelength spectrum may become rather complicated [18

18. A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus X-ray source,” Rev. Sci. Instrum. **68**, 2774–2782 (1997). [CrossRef]

## 3. Simple rules for estimation of contrast, SNR and spatial resolution

*x*)

_{min}/(2

*σ*

_{sys}) as a function of the inverse of Fresnel number calculated in accordance with Eq. (16) is presented in Fig. 4. As one can see from Fig. 4, the resolution values tend to a finite limit equal to 2

*σ*

_{sys}(

*M*) for large Fresnel numbers in agreement with Eq. (6), while for small Fresnel numbers the spatial resolution becomes proportional to the square root of the inverse Fresnel number, eq.(17), which corresponds to a straight line in the logarithmic plot. Furthermore, one can see that the following “rule of thumb”, which simply combines Eq. (6) and Eq. (17) and gives a reasonably good approximation for the spatial resolution of in-line imaging across a large range of Fresnel numbers.

*Rule 1 (spatial resolution). The spatial resolution in in-line imaging of a phase edge-like feature satisfying Eq. (12) can be estimated as the largest of the two values, namely the width of the PSF of the imaging system,*2

*σ*

_{sys}(

*M*)

*, and the width of the first Fresnel zone,*

*C*

_{max}(ν)/|

*φ*|

_{max}(ν) as a function of the inverse of Fresnel number calculated in accordance with Eq. (18) is presented in Fig. 5. One can see that the image contrast as a function of the inverse Fresnel number displays a behavior which is complementary to that of the spatial resolution, i.e. at large Fresnel numbers the image contrast is directly proportional to the inverse Fresnel number in agreement with Eq. (5), while at small Fresnel numbers the contrast asymptotically converges to a constant value in accordance with Eq. (19).

18. A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus X-ray source,” Rev. Sci. Instrum. **68**, 2774–2782 (1997). [CrossRef]

21. G. Margaritondo and G. Tromba, “Coherence-based edge diffraction sharpening of x-ray images: a simple model,” J. Appl. Phys. **85**, 3406–3408 (1999). [CrossRef]

10. Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. **76**, 093706 (2005). [CrossRef]

*Rule 2 (image contrast). The maximum contrast in an in-line image of a phase edge-like feature satisfying Eq. (12) can be estimated as the product of the maximum absolute phase shift generated by the feature,*|

*φ*|

_{max}(ν)

*, and the smallest of the two values, 0.2420/*

*N*and 0.4876._{F}*γ*(

*N*) varies slowly in between its two limits, 0.2717 and 0.3564, i.e. it does not change significantly over the full range of Fresnel numbers. Therefore, we can formulate the following “rule of thumb”.

_{F}*Rule 3 (signal-to-noise). The signal-to-noise in an in-line image of a phase edge-like feature satisfying Eq. (12) can be estimated as the product of approximately 0.3 times the image contrast (see Rule 2) and the square root of twice the integrated X-ray intensity incident on the feature.*

## 4. Numerical results

_{2}H

_{4}, density=1 g/cm

^{3}) sheet of thickness 100 µm, with the unsharpness parameter

*σ*

_{obj}=0.7 µm. The source was modeled as an X-ray tube with a tungsten anode operated at

*E*=50 kVp, and with 0.3 mm thick Be window. We also assumed that the lower X-ray energies were filtered out using a 1 mm thick Al filter. The normalized X-ray spectrum incident on the sample is shown in Fig. 7.

_{p}*ψ*̅|

_{max}=0.2947 Å. The source size was assumed 4 µm (FWHM), the detector resolution was 100 µm (FWHM). We modeled an X-ray system with a fixed source-to-detector distance

*R*=2 m. In the simulations we changed the value of magnification (

*M*) by changing the source-to-object distance, the effective defocus distance R′ was changing accordingly. The comparison between the results obtained by simulating the images with the help of Kirchhoff integrals and those obtained using the analytical formulae (Eqs. (6), (8) and (9)) derived in the preceding sections are presented in Tables 2 and 3. It is easy to see that the agreement between the two sets of results is quite good, with the accuracy of the analytical results improving as the necessary validity condition for the TIE,

*σ*

_{M}σ_{obj}/

*R*′≫max{

*λ*̅/(2

*π*),|

*ψ*̅|

_{max}}, becomes better satisfied.

*E*=30.78 keV (the average energy of the spectrum in Fig. 7),

*λ*=0.4 Å. Compared to the first set of calculations, we also decreased the value of the maximum phase shift to |

*φ*|

_{max}=1 rad (which corresponds to the thickness of polyethylene of 25.66 µm) in order to satisfy conditions (12) across the whole considered range of Fresnel numbers. We kept the magnification constant at

*M*=25.3, which led to the constant value of the standard deviation of the system PSF,

*σ*

_{sys}=2.341 µm. The propagation distance was varied between 2 m and 1024 m, which corresponded to the Fresnel numbers between 12.28 and 0.024. The corresponding results are presented in Table 4. In the same table we also present for comparison the results obtained with Eqs. (9) and (6) (large Fresnel numbers, or TIE approximation), and with Eqs. (17) and (19) (small Fresnel numbers). One can see that Eqs. (16) and (18) give the values which agree very well with the results obtained using Kirchhoff diffraction theory across the whole considered range of Fresnel number values. As expected, the results given by Eqs. (6) and (9) and Eqs. (16) and (19) agree well with the Kirchhoff diffraction results for large and small Fresnel numbers, respectively. One can also easily verify that the simple Rules 1 and 2 formulated above give rather good estimation of the spatial resolution and contrast, respectively, in the considered examples.

## 5. Experimental results

*R*

_{1}and object-to-detector distance

*R*

_{2}were varied, but

*R*=

*R*

_{1}+

*R*

_{2}was fixed at 2 m (magnification

*M*=

*R*/

*R*

_{1}). The detector was Fuji Imaging Plates (20 cm×25 cm; FDL-URV), scanned with a Fuji BAS-5000 scanner (using 25 µm pixel size). Under these experimental conditions, the object yielded images which possessed both absorption and phase contrast, but were dominated by the latter in the form of characteristic single black-white fringes (a typical image is presented in Fig.8). The images were subjected to flat-field correction and then analysed to provide contrast and resolution values. The observed (experimental) contrast values were obtained from the difference between the maximum and minimum intensity values divided by the sum of these quantities; the observed resolution values were obtained from the spatial separation of lines of maximum and minimum intensity, referred to the object plane. The observed data values, in both vertical and horizontal directions, are listed in Table 5.

23. D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math. **11**, 431–441 (1963). [CrossRef]

*σ*for the resolution values, i.e. monochromatic formulae. Correlation matrices and estimated standard deviations (esds) for refined-parameter values were calculated [24

_{M}24. S. Geller, “Parameter interaction in least squares structure refinement,” Acta Cryst. **14**, 1026–1035 (1961). [CrossRef]

26. W. C. Hamilton, “Significance tests on the crystallographic R factor,” Acta Cryst. **18**, 502–510 (1965). [CrossRef]

*σ*;

_{obj}*σ*;

_{src}(vert)*σ*;

_{src}(horiz)*σ*; effective X-ray energy

_{det}*E*; polyethylene-sheet thickness

*t*. Whilst the

*σ*-values associated with the object and the detector could justifiably be treated as being the same in both vertical and horizontal directions, this was not true for the X-ray source (see below) and so two parameters were required.

*σ*=4 µm and the thickness was fixed at

_{obj}*t*=100 µm. This value of

*σ*was arrived at by trial-and-error and, inasmuch as the corresponding FWHM value is of order 10% of

_{obj}*t*, is physically reasonable. The value of

*R*is consistent with a good fit to the experimental data. As the value of

_{H}*E*was refined, the value of phase shift per unit length, being energy dependent, had to be changed accordingly. These values were calculated, for polyethylene (C

_{2}H

_{4};

*ρ*=0.923 g/cm

^{3}), using X-ray data from [27

27. S. Brennan and P. L. Cowan, “A suite of programs for calculating X-ray absorption, reflection, and diffraction performance for a variety of materials at arbitrary wavelengths,” Rev. Sci. Instrum. **63**, 850–853 (1992). [CrossRef]

*R*

_{1}(with

*R*being fixed), and the resolution values increasing monotonically.

28. Y. Amemiya, K. Wakabayashi, H. Tanaka, Y. Ueno, and J. Miyahara, “Laser-stimulated luminescence used to measure X-ray diffraction of a contracting striated muscle,” Science **237**, 164–168 (1987). [CrossRef] [PubMed]

29. Y. Amemiya, “Imaging plates for use with synchrotron radiation,” J. Synch. Rad. **2**, 13–21 (1995). [CrossRef]

*E*) value given in Table 6 is entirely consistent with an X-ray tube operating at 30 kVp with minimal beam hardening from the tube window and object. In summary, the least-squares analysis has provided a good fit to experimental contrast and resolution data, and the refined-parameter values are all physically reasonable and in accord with expectations.

## 6. Conclusion

*φ*|

_{max}, and the Fresnel number,

*N*=

_{F}*kσ*

^{2}

*/*

_{M}*R*′, which is defined with respect to the total variance

*σ*

^{2}

*equal to the sum of the variance of the PSF of the imaging system,*

_{M}*σ*

^{2}

_{sys}(

*M*), and that of the unsharpness of the edge,

*σ*

^{2}

_{obj}. The spatial resolution behaves quite differently for large and small Fresnel numbers. In the case of large Fresnel numbers (short propagation distances) the spatial resolution is simply equal to the width of the PSF of the imaging system, 2

*σ*

_{sys}(

*M*), while for small Fresnel numbers (large propagation distances) the resolution is equal to the width,

*R*′, when the corresponding Fresnel number is large, the maximum image contrast is inversely proportional to the Fresnel number, while at long propagation distances (small Fresnel numbers) the maximum contrast asymptotically tends to the limit 0.4876|

*φ*|

_{max}, which does not depend on any parameters other than the maximum absolute phase shift. The signal-to-noise behaves similarly, as it is proportional to the product of maximum image contrast and the square root of the integrated X-ray intensity incident on the sample feature that is being imaged. When the relevant Fresnel number is large, the expressions for the image contrast, spatial resolution and signal-to-noise in the case of polychromatic radiation are virtually the same as in the monochromatic case with the suitable replacement of the conventional phase by the generalized eikonal of the polychromatic beam. The behaviour is much more complicated in the case of small Fresnel numbers (large propagation distances) and polychromatic radiation, where we could not obtain simple and general analytical expressions for the image characteristics.

## Acknowledgement

## References and links

1. | R. Fitzgerald, “Phase-sensitive x-ray imaging,” Physics Today |

2. | 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. |

3. | S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays:,” Nature |

4. | P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. J. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. |

5. | A. Krol, J. C. Kieffer, and E. Foerster, “Laser-driven x-ray source for diagnostic radiology. Applications of X-rays Generated from Lasers and Other Bright Sources,” Proc. SPIE |

6. | T. A. Shelkovenko, D. B. Sinars, S. A. Pikuz, K. M. Chandler, and D. A. Hammer, “Point-projection x-ray radiography using an X pinch as the radiation source,” Rev. Sci. Instrum. |

7. | C. J. Kotre, I. P. Birch, and K. J. Robson, “Anomalous image quality phantom scores in magnification mammography: evidence of phase contrast enhancement,” British J. Radiol. |

8. | J. M. Cowley, |

9. | K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X-rays,” Phys. Rev. Lett. |

10. | Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, “On the optimization of experimental parameters for x-ray in-line phase-contrast imaging,” Rev. Sci. Instrum. |

11. | M. R. Teague, “Deterministic Phase Retrieval: a Green’s Function Solution,” J. Opt. Soc. Am. |

12. | T. E. Gureyev and S. W. Wilkins, “On X-ray phase imaging with a point source,” J. Opt. Soc. Am. A |

13. | T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. |

14. | 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. |

15. | 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. |

16. | A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman “Initial experimentation with in-line holography x-ray phasecontrast imaging with ultrafast laser-based x-ray source,” Proc. SPIE |

17. | J.-P. Guigay, “Fourier transform analysis of Fresnel diffraction patterns and in-line holograms,” Optik |

18. | A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus X-ray source,” Rev. Sci. Instrum. |

19. | P. Cloetens, J.-P. Guigay, C. De Martino, and J. Baruchel, “Fractional Talbot imaging of phase gratings with hard x rays,” Opt. Lett. |

20. | V. E. Cosslett and W. C. Nixon, “The X-Ray Shadow Microscope,” J. Appl. Phys. |

21. | G. Margaritondo and G. Tromba, “Coherence-based edge diffraction sharpening of x-ray images: a simple model,” J. Appl. Phys. |

22. | K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Quart. Appl. Math. |

23. | D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math. |

24. | S. Geller, “Parameter interaction in least squares structure refinement,” Acta Cryst. |

25. | J. S. Rollett, |

26. | W. C. Hamilton, “Significance tests on the crystallographic R factor,” Acta Cryst. |

27. | S. Brennan and P. L. Cowan, “A suite of programs for calculating X-ray absorption, reflection, and diffraction performance for a variety of materials at arbitrary wavelengths,” Rev. Sci. Instrum. |

28. | Y. Amemiya, K. Wakabayashi, H. Tanaka, Y. Ueno, and J. Miyahara, “Laser-stimulated luminescence used to measure X-ray diffraction of a contracting striated muscle,” Science |

29. | Y. Amemiya, “Imaging plates for use with synchrotron radiation,” J. Synch. Rad. |

**OCIS Codes**

(110.2960) Imaging systems : Image analysis

(110.2990) Imaging systems : Image formation theory

(110.4980) Imaging systems : Partial coherence in imaging

(340.7440) X-ray optics : X-ray imaging

**ToC Category:**

X-ray Optics

**History**

Original Manuscript: November 21, 2007

Revised Manuscript: February 15, 2008

Manuscript Accepted: February 17, 2008

Published: February 22, 2008

**Citation**

Timur E. Gureyev, Yakov I. Nesterets, Andrew W. Stevenson, Peter R. Miller, Andrew Pogany, and Stephen W. Wilkins, "Some simple rules for contrast, signal-to-noise and resolution in in-line x-ray phase-contrast imaging," Opt. Express **16**, 3223-3241 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-3223

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### References

- R. Fitzgerald, "Phase-sensitive x-ray imaging," Physics Today 53, 23-26 (2000). [CrossRef]
- 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, 5486-5492 (1995). [CrossRef]
- S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany and A. W. Stevenson, "Phase-contrast imaging using polychromatic hard X-rays:," Nature 384, 335-338 (1996). [CrossRef]
- P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. J. Schlenker, "Phase objects in synchrotron radiation hard x-ray imaging," J. Phys. D: Appl. Phys. 29, 133-146 (1996). [CrossRef]
- A. Krol, J. C. Kieffer and E. Foerster, "Laser-driven x-ray source for diagnostic radiology. Applications of X-rays Generated from Lasers and Other Bright Sources," Proc. SPIE 3157, 156-163 (1997). [CrossRef]
- T. A. Shelkovenko, D. B. Sinars, S. A. Pikuz, K. M. Chandler and D. A. Hammer, "Point-projection x-ray radiography using an X pinch as the radiation source," Rev. Sci. Instrum. 72, 667-670 (2001). [CrossRef]
- C. J. Kotre, I. P. Birch and K. J. Robson, "Anomalous image quality phantom scores in magnification mammography: evidence of phase contrast enhancement," British J. Radiol. 75, 170-173 (2002).
- J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1975), Sec.3.4.2.
- K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin and Z. Barnea, "Quantitative phase imaging using hard X-rays," Phys. Rev. Lett. 77, 2961-2964 (1996). [CrossRef] [PubMed]
- Ya. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, "On the optimization of experimental parameters for x-ray in-line phase-contrast imaging," Rev. Sci. Instrum. 76, 093706 (2005). [CrossRef]
- M. R. Teague, "Deterministic Phase Retrieval: a Green's Function Solution," J. Opt. Soc. Am. 73, 1434-1441 (1983). [CrossRef]
- T. E. Gureyev and S. W. Wilkins, "On X-ray phase imaging with a point source," J. Opt. Soc. Am. A 15, 579-85 (1998). [CrossRef]
- T. E. Gureyev, A. Pogany, D. M. Paganin, and S. W. Wilkins, "Linear algorithms for phase retrieval in the Fresnel region," Opt. Commun. 231, 53-70 (2004). [CrossRef]
- 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, 068103-1 - 068103-4 (2004). [CrossRef] [PubMed]
- 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, 569-580 (2006). [CrossRef]
- A. Krol, R. Kincaid, M. Servol, J.-C. Kieffer, Y. Nesterets, T. Gureyev, A. Stevenson, S. Wilkins, H. Ye, E. Lipson, R. Toth, A. Pogany, and I. Coman, "Initial experimentation with in-line holography x-ray phase-contrast imaging with ultrafast laser-based x-ray source," Proc. SPIE 6510, 65100L (2007). [CrossRef]
- J.-P. Guigay, "Fourier transform analysis of Fresnel diffraction patterns and in-line holograms," Optik 49, 121-125 (1977).
- A. Pogany, D. Gao, and S. W. Wilkins, "Contrast and resolution in imaging with a microfocus X-ray source," Rev. Sci. Instrum. 68, 2774-2782 (1997). [CrossRef]
- P. Cloetens, J.-P. Guigay, C. De Martino, and J. Baruchel, "Fractional Talbot imaging of phase gratings with hard x rays," Opt. Lett. 22, 1059-1061 (1997). [CrossRef] [PubMed]
- V. E. Cosslett and W. C. Nixon, "The X-Ray Shadow Microscope," J. Appl. Phys. 24, 616-623 (1953). [CrossRef]
- G. Margaritondo and G. Tromba, "Coherence-based edge diffraction sharpening of x-ray images: a simple model," J. Appl. Phys. 85, 3406-3408 (1999). [CrossRef]
- K. Levenberg, "A method for the solution of certain non-linear problems in least squares," Quart. Appl. Math. 2, 164-168 (1944).
- D. Marquardt, "An algorithm for least-squares estimation of nonlinear parameters," SIAM J. Appl. Math. 11, 431-441 (1963). [CrossRef]
- S. Geller, "Parameter interaction in least squares structure refinement," Acta Cryst. 14, 1026-1035 (1961). [CrossRef]
- J. S. Rollett, Computing Methods in Crystallography (Pergamon Press, Oxford, 1965).
- W. C. Hamilton, "Significance tests on the crystallographic R factor," Acta Cryst. 18, 502-510 (1965). [CrossRef]
- S. Brennan and P. L. Cowan, "A suite of programs for calculating X-ray absorption, reflection, and diffraction performance for a variety of materials at arbitrary wavelengths," Rev. Sci. Instrum. 63, 850-853 (1992). [CrossRef]
- Y. Amemiya, K. Wakabayashi, H. Tanaka, Y. Ueno, and J. Miyahara, "Laser-stimulated luminescence used to measure X-ray diffraction of a contracting striated muscle," Science 237, 164-168 (1987). [CrossRef] [PubMed]
- Y. Amemiya, "Imaging plates for use with synchrotron radiation," J. Synch. Rad. 2, 13-21 (1995). [CrossRef]

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