## The projection approximation and edge contrast for x-ray propagation-based phase contrast imaging of a cylindrical edge

Optics Express, Vol. 18, Issue 10, pp. 9865-9878 (2010)

http://dx.doi.org/10.1364/OE.18.009865

Acrobat PDF (4019 KB)

### Abstract

We examine the projection approximation in the context of propagation-based phase contrast imaging using hard x-rays. Specifically, we consider the case of a cylinder or a rounded edge, as a simple model for the edges of many biological samples. The Argand-plane signature of a propagation-based phase contrast fringe from the edge of a cylinder is studied, and the evolution of this signature with propagation. This, along with experimental images obtained using a synchrotron source, reveals how propagation within the scattering volume is not fully described in the projection approximation's ray-based approach. This means that phase contrast fringes are underestimated by the projection approximation at a short object-to-detector propagation distance, namely a distance comparable to the free-space propagation within the volume. This failure of the projection approximation may become non-negligible in the detailed study of small anatomical features deep within a large body. Nevertheless, the projection approximation matches the exact solution for a larger propagation distance typical of those used in biomedical phase contrast imaging.

© 2010 OSA

## 1. Introduction

*e.g.*anatomical, structural) interest. Such absorptive imaging has the capacity to easily display bones and highly attenuating materials [1

1. W. C. Röntgen, “On a new kind of rays,” Nature **53**(1369), 274–276 (1896). [CrossRef]

2. P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D Appl. Phys. **29**(1), 133–146 (1996). [CrossRef]

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

2. P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D Appl. Phys. **29**(1), 133–146 (1996). [CrossRef]

*e.g.,*[5

5. M. J. Kitchen, R. A. Lewis, M. J. Morgan, M. J. Wallace, M. L. Siew, K. K. W. Siu, A. Habib, A. Fouras, N. Yagi, K. Uesugi, and S. B. Hooper, “Dynamic measures of regional lung air volume using phase contrast x-ray imaging,” Phys. Med. Biol. **53**(21), 6065–6077 (2008). [CrossRef] [PubMed]

6. D. W. Parsons, K. S. Morgan, M. Donnelley, A. Fouras, J. Crosbie, I. Williams, R. C. Boucher, K. Uesugi, N. Yagi, and K. K. W. Siu, “High-resolution visualization of airspace structures in intact mice via synchrotron phase-contrast X-ray imaging (PCXI),” J. Anat. **213**(2), 217–227 (2008). [CrossRef]

5. M. J. Kitchen, R. A. Lewis, M. J. Morgan, M. J. Wallace, M. L. Siew, K. K. W. Siu, A. Habib, A. Fouras, N. Yagi, K. Uesugi, and S. B. Hooper, “Dynamic measures of regional lung air volume using phase contrast x-ray imaging,” Phys. Med. Biol. **53**(21), 6065–6077 (2008). [CrossRef] [PubMed]

*e.g.,*[7,8

8. D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. **206**(1), 33–40 (2002). [CrossRef] [PubMed]

9. N. Yagi, Y. Suzuki, K. Umetani, Y. Kohmura, and K. Yamasaki, “Refraction-enhanced x-ray imaging of mouse lung using synchrotron radiation source,” Med. Phys. **26**(10), 2190–2193 (1999). [CrossRef] [PubMed]

*A, B, C*and

*D*of Fig. 1 . In this figure the imaging geometry is shown, with a monochromatic scalar electromagnetic plane wave propagating in the positive

*z*direction, incident on a cylinder or cylindrically modelled edge. The incident wave incurs changes in phase during its passage through the various regions of the object. As mentioned earlier, propagation based PCXI makes use of these phase changes by converting them into intensity variations through free-space propagation after the scatterer [2

2. P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D Appl. Phys. **29**(1), 133–146 (1996). [CrossRef]

10. P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Pernot-Rejmánková, M. Salomé-Pateyron, M. Schlenker, J.-Y. Buffière, E. Maire, and G. Peix, “Hard x-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D Appl. Phys. **32**(10A), 330–336 (1999). [CrossRef]

*e.g.*from each feature marked

*A-D*in Fig. 1. In other words, the diffraction between the

*z*position of edges

*A-D*and the exit plane will be sufficient for the wave that has passed on one side of the feature to interfere with the wave passing on the other side of the feature. As shown by the red traces in Fig. 1, features such as

*A*, which are further from the exit plane, will produce wider, more intense fringes than closer features such as

*B*. Application of the projection approximation to simulate this process will predict only absorption contrast at the exit plane, requiring further propagation to produce phase contrast.

## 2. X-ray image simulation using the projection approximation

11. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. A **52**(2), 116–130 (1962). [CrossRef]

*e.g.*reference 7). The inhomogeneous paraxial equation, for a single wavelength, is given in Eq. (1):where

*k*is the wavenumber,

*n*is the position dependent refractive index and

*ψ*is the envelope of the spatial wavefunction

*ψ*e

*, describing the spatial part of a coherent scalar wavefield with intensity*

^{ikz}*I = |ψ|*and phase

^{2}*φ =*arg

*ψ*. Each material is described by a complex refractive index

*n = 1-δ + iβ*, where

*β = μ/2k*(

*μ*is the linear attenuation coefficient) and

*δ*describes the refractive properties of the material.

*n*will undergo a phase shift, as denoted by the second term on the right-hand side of Eq. (2) and experience attenuation, as denoted by the third term of Eq. (2), where

*T*is the projected thickness of each material

_{j}*j*along the

*z*direction [7]:

*e.g. B*), so

*n*is defined throughout the object and the total projected thickness (

*T*) of each material may be determined by the position

*x*on the plane where the image is to be evaluated. As the projection approximation effectively projects all material to the exit surface, the diffraction and interference of waves is not described until after propagating the wavefunction from the exit plane to the image plane. This propagation was done by applying the wavefield propagator, then the modulus squared of the projected wavefield was calculated to obtain the intensity phase contrast image. The propagation was implemented using the angular spectrum representation of the Rayleigh-Sommerfeld diffraction integral of the first kind [13

13. N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A **15**(4), 857–867 (1998). [CrossRef]

*e.g*[15

15. T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. **32**(5), 563–567 (1999). [CrossRef]

*e.g*[16

16. M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and S. T. Mudie, “On the origin of speckle in x-ray phase contrast images of lung tissue,” Phys. Med. Biol. **49**(18), 4335–4348 (2004). [CrossRef] [PubMed]

17. K. K. W. Siu, K. S. Morgan, D. M. Paganin, R. Boucher, K. Uesugi, N. Yagi, and D. W. Parsons, “Phase contrast X-ray imaging for the non-invasive detection of airway surfaces and lumen characteristics in mouse models of airway disease,” Eur. J. Radiol. **68**(3Suppl), S22–S26 (2008). [CrossRef] [PubMed]

*θ*to the columns of the pixel array. Hence, to improve the signal to noise ratio, each row was shifted by aligning the fringe peaks to allow an average of the edge fringe over many rows. Since a true profile of the fringes should be taken perpendicular to the edge, correction by a factor of cos(

*θ*) was made to the horizontal axis (

*i.e.*via the pixel size).

## 3. Argand representation of the complex wavefield at the imaging plane

*i.e.*large

*x*), so has the uniform intensity and phase of the unscattered plane wave over a plane of constant

*z*. Cycling outwards around the blue spiral increases and decreases the intensity in increasingly large bands, producing the increasingly wide, intense Fresnel fringes seen when approaching (from the outside) the phase contrast fringe arising from the edge of the cylinder.

_{1}and B

_{2}, is seen as the hypocycloid; this hypocycloid results from a combination of the projection approximation and diffraction from the edge of the cylinder/opaque barrier. Note that screen A is complementary to the screen formed by B

_{1}and B

_{2}. We examine each of the diffracted components in detail below.

*C(u)*and

*S(u)*as the Fresnel integrals on the

*x*and

*y*axes respectively [18], where

*u*is a reduced variable proportional to

*z*, the distance across the imaging plane. In order to calculate the diffracted wavefield along a plane which is downstream of an opaque edge, the Cornu spiral is shifted by + 1/2 in both

*x*and

*y*directions, rotated about the origin by –π/4 and divided by √2 as in Eq. (3) [18]:

19. G. Margaritondo and G. Tromba, “Coherence-based edge diffraction sharpening of x-ray images: A simple model,” J. Appl. Phys. **85**(7), 3406–3408 (1999). [CrossRef]

19. G. Margaritondo and G. Tromba, “Coherence-based edge diffraction sharpening of x-ray images: A simple model,” J. Appl. Phys. **85**(7), 3406–3408 (1999). [CrossRef]

*cf.*Fig. 4 and Fig. 7 ).

*AB*. These can also be described by the interference of the incident plane wave and a cylindrical wave scattered from the edge of the cylinder, using the rays seen in Fig. 6. The projection approximation will incur a phase change, as given in Eq. (2), on the incident plane wave travelling through the cylinder and landing at

*C*. This distorted plane wave, as estimated via the projection approximation, will then interfere with the cylindrical edge wave

*AC*. The resulting wave, as described by Eq. (5), will then be the sum of a plane wave (

*e*) multiplied by the projection-approximation phase shift

^{ikz}*e*, and this spherical wave scattered from the edge of the cylinder;where

^{-ikδT}*S*is a coefficient describing the amplitude of the edge wave and

*R*is the distance from the edge

*A*of the cylinder to a given point on the image plane (shown in Fig. 6).

## 4. Underestimation of fringes by the PA at short object-to-detector propagation distance

*e.g*. the 0.5 Angstrom hard x-rays of Fig. 8) passes through a full 2π phase change within a micron of the cylinder edge shadow, while a longer wavelength (

*e.g*. the 1 Angstrom softer x-rays of Fig. 4) wraps more slowly (requiring 4 microns across the image plane for a 2π phase change in Fig. 4). It is for this reason that small propagation lengths on the order of millimeters (hence narrow fringes) and small pixels have been used in Fig. 8, to avoid wrapping the trace around upon itself.

*e.g.,*[2

**29**(1), 133–146 (1996). [CrossRef]

6. D. W. Parsons, K. S. Morgan, M. Donnelley, A. Fouras, J. Crosbie, I. Williams, R. C. Boucher, K. Uesugi, N. Yagi, and K. K. W. Siu, “High-resolution visualization of airspace structures in intact mice via synchrotron phase-contrast X-ray imaging (PCXI),” J. Anat. **213**(2), 217–227 (2008). [CrossRef]

## 5. Conclusion

## Appendix

*i.e*., exhibit a retrograde-like trace) will be determined by the propagation distance, refractive index and edge gradient of the cylinder. This looping will mean that the Argand-plane trace due to phase along the image plane is moving slightly backwards (due to the scattered field from the edge of the cylinder) before continuing to increase while moving further behind the increasingly thick volume of the cylinder.

*e.g.*feature α in Fig. 7) to loops (

*e.g.*feature β in Fig. 7) is seen. This is where the gradient of the cylinder is sufficiently small that the projection approximation is no longer moving the phase around the Argand plane fast enough to keep up with the fast cycles of the fringes from edge wave interference. Figure 10 shows a perfect hypocycloid, where a small circle moves along the inside of a bigger circle without any slipping. In order to avoid slipping, the distance along which it travels between peaks must be 2π

*d*, the circumference of the cylinder. If the distance is smaller (or the circle turns faster, slipping as it moves across the surface), loops will be seen.

*c.f.*Eq. (5)]:

*e.g.*dotted line in Fig. 4 or Fig. 7) is created by the first term of Eq. (A1) varying slowly in phase and the smaller circle comes from the second term varying quickly in phase. The critical case of a perfect hypocycloid with no slipping [Fig. 10(a)] will require two conditions to be fulfilled for a section of the trace (hence a section of the image plane);

- • The slowly moving phase from the first term in Eq. (A1),
*φ*, changes by the circumference of the smaller circle (2π_{object}*d*) so that a full cycloid cycle can be traced by the edge of a “non-slipping” circle. - • The fast moving phase from the second term in Eq. (A1),
*φ*, changes by a full cycle (2π) within the same distance._{edge}

*x*, in addressing the first requirement. The smaller circle will have radius

*d = S/√R*(from the second term in Eq. (A1) and will require the arc length 2π

*d*in the Argand plane on which to complete a full rotation without slipping. This arc length will come from the slow moving phase due to the slowly increasing projected thickness of the object. The change in phase across Δ

*x*is given in Eq. (A2):

*x*. This angle swept out by the phase change given in Eq. (A2) will also be the arc length traversed in the Argand plane for a circle of radius 1 (

*i.e*. intensity = 1). This means the critical case of a perfect hypocycloid is now described by Eq. (A3):

*φ*in Eq. (A1)] change by 2π (or less, to avoid loops) over the length Δ

_{edge}*x*:

*φ*and

_{object}*φ*as determined by the object and by the edge wave {Eq. (5) gives Eq. (A7)]:

_{edge}*k*is the wavenumber,

*R(x)*the distance from the edge and

*T(x)*the projected thickness of the object, as previously defined. In all edge cases,

*x*(

*i.e. x*= 0) set at the edge of the object. This will produce a requirement for a cycloid without loops, Eq. (A8):

*x*is small compared to

*z*,

*i.e*. fringes are seen close to the edge for long propagations, this simplifies to Eq. (A9) for a non-retrograde Argand-plane trace:

*z*(where the spherical edge wave has greater variations across Δ

*x*), smaller

*δ*(a weak phase object will not wrap the slow moving phase from the object around as quickly) or greater distance (

*x*) from the edge, where the reduced radius of the hypocycloid means it can complete a circuit more quickly. The increase in retrograde behaviour with position

*x*in the imaging plane when moving behind the cylinder has already been observed in Fig. 7.

*T(x)*can be described as

*a*is the radius of the cylinder and

*|x|≤ a*. Substituting into Eq. (A9) and again assuming that

*x*is close to the edge, and small compared to the radius

*a*, gives Eq. (A10);

*S*[from Eq. (5)], could be determined from a plot such as Fig. 7, simply from the

*x*coordinate at which the hypocycloid begins to loop around. With significant attenuation, this

*x*coordinate will change slightly, with the condition given in Eq. (A9) now divided by

## Acknowledgments

## References and links

1. | W. C. Röntgen, “On a new kind of rays,” Nature |

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

3. | 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,” J. Phys. D Appl. Phys. |

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

5. | M. J. Kitchen, R. A. Lewis, M. J. Morgan, M. J. Wallace, M. L. Siew, K. K. W. Siu, A. Habib, A. Fouras, N. Yagi, K. Uesugi, and S. B. Hooper, “Dynamic measures of regional lung air volume using phase contrast x-ray imaging,” Phys. Med. Biol. |

6. | D. W. Parsons, K. S. Morgan, M. Donnelley, A. Fouras, J. Crosbie, I. Williams, R. C. Boucher, K. Uesugi, N. Yagi, and K. K. W. Siu, “High-resolution visualization of airspace structures in intact mice via synchrotron phase-contrast X-ray imaging (PCXI),” J. Anat. |

7. | D. M. Paganin, |

8. | D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. |

9. | N. Yagi, Y. Suzuki, K. Umetani, Y. Kohmura, and K. Yamasaki, “Refraction-enhanced x-ray imaging of mouse lung using synchrotron radiation source,” Med. Phys. |

10. | P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Pernot-Rejmánková, M. Salomé-Pateyron, M. Schlenker, J.-Y. Buffière, E. Maire, and G. Peix, “Hard x-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D Appl. Phys. |

11. | J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. A |

12. | D. F. Lynch, M. A. O'Keefe, and A. F. Moodie, “n-beam lattice images. V. “The use of the charge-density approximation in the interpretation of lattice images,” Acta Crystallogr. |

13. | N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A |

14. | M. Nieto-Vesperinas, |

15. | T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. |

16. | M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and S. T. Mudie, “On the origin of speckle in x-ray phase contrast images of lung tissue,” Phys. Med. Biol. |

17. | K. K. W. Siu, K. S. Morgan, D. M. Paganin, R. Boucher, K. Uesugi, N. Yagi, and D. W. Parsons, “Phase contrast X-ray imaging for the non-invasive detection of airway surfaces and lumen characteristics in mouse models of airway disease,” Eur. J. Radiol. |

18. | M. Born, and E. Wolf, |

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

**OCIS Codes**

(050.5080) Diffraction and gratings : Phase shift

(110.7440) Imaging systems : X-ray imaging

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(260.0260) Physical optics : Physical optics

(340.0340) X-ray optics : X-ray optics

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

**ToC Category:**

X-ray Optics

**History**

Original Manuscript: February 19, 2010

Revised Manuscript: April 13, 2010

Manuscript Accepted: April 19, 2010

Published: April 27, 2010

**Virtual Issues**

Vol. 5, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

K. S. Morgan, K. K. W. Siu, and D. M. Paganin, "The projection approximation and edge contrast for x-ray propagation-based phase contrast imaging of a cylindrical edge," Opt. Express **18**, 9865-9878 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-10-9865

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### References

- W. C. Röntgen, “On a new kind of rays,” Nature 53(1369), 274–276 (1896). [CrossRef]
- P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D Appl. Phys. 29(1), 133–146 (1996). [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,” J. Phys. D Appl. Phys. 29, 133–146 (1995).
- S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard x-rays,” Nature 384(6607), 335–338 (1996). [CrossRef]
- M. J. Kitchen, R. A. Lewis, M. J. Morgan, M. J. Wallace, M. L. Siew, K. K. W. Siu, A. Habib, A. Fouras, N. Yagi, K. Uesugi, and S. B. Hooper, “Dynamic measures of regional lung air volume using phase contrast x-ray imaging,” Phys. Med. Biol. 53(21), 6065–6077 (2008). [CrossRef] [PubMed]
- D. W. Parsons, K. S. Morgan, M. Donnelley, A. Fouras, J. Crosbie, I. Williams, R. C. Boucher, K. Uesugi, N. Yagi, and K. K. W. Siu, “High-resolution visualization of airspace structures in intact mice via synchrotron phase-contrast X-ray imaging (PCXI),” J. Anat. 213(2), 217–227 (2008). [CrossRef]
- D. M. Paganin, Coherent X-ray Optics, Oxford University Press, New York, 2006.
- D. M. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206(1), 33–40 (2002). [CrossRef] [PubMed]
- N. Yagi, Y. Suzuki, K. Umetani, Y. Kohmura, and K. Yamasaki, “Refraction-enhanced x-ray imaging of mouse lung using synchrotron radiation source,” Med. Phys. 26(10), 2190–2193 (1999). [CrossRef] [PubMed]
- P. Cloetens, W. Ludwig, J. Baruchel, J.-P. Guigay, P. Pernot-Rejmánková, M. Salomé-Pateyron, M. Schlenker, J.-Y. Buffière, E. Maire, and G. Peix, “Hard x-ray phase imaging using simple propagation of a coherent synchrotron radiation beam,” J. Phys. D Appl. Phys. 32(10A), 330–336 (1999). [CrossRef]
- J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. A 52(2), 116–130 (1962). [CrossRef]
- D. F. Lynch, M. A. O'Keefe, and A. F. Moodie, “n-beam lattice images. V. “The use of the charge-density approximation in the interpretation of lattice images,” Acta Crystallogr. 31, 300–307 (1974).
- N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A 15(4), 857–867 (1998). [CrossRef]
- M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2nd ed. (World Scientific Publishing, New Jersey, 2006).
- T. E. Gureyev, C. Raven, A. Snigirev, I. Snigireva, and S. W. Wilkins, “Hard x-ray quantitative non-interferometric phase-contrast microscopy,” J. Phys. D Appl. Phys. 32(5), 563–567 (1999). [CrossRef]
- M. J. Kitchen, D. M. Paganin, R. A. Lewis, N. Yagi, K. Uesugi, and S. T. Mudie, “On the origin of speckle in x-ray phase contrast images of lung tissue,” Phys. Med. Biol. 49(18), 4335–4348 (2004). [CrossRef] [PubMed]
- K. K. W. Siu, K. S. Morgan, D. M. Paganin, R. Boucher, K. Uesugi, N. Yagi, and D. W. Parsons, “Phase contrast X-ray imaging for the non-invasive detection of airway surfaces and lumen characteristics in mouse models of airway disease,” Eur. J. Radiol. 68(3Suppl), S22–S26 (2008). [CrossRef] [PubMed]
- M. Born, and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999).
- G. Margaritondo and G. Tromba, “Coherence-based edge diffraction sharpening of x-ray images: A simple model,” J. Appl. Phys. 85(7), 3406–3408 (1999). [CrossRef]

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