## The relationship between wave and geometrical optics models of coded aperture type x-ray phase contrast imaging systems

Optics Express, Vol. 18, Issue 5, pp. 4103-4117 (2010)

http://dx.doi.org/10.1364/OE.18.004103

Acrobat PDF (285 KB)

### Abstract

X-ray phase contrast imaging is a very promising technique which may lead to significant advancements in medical imaging. One of the impediments to the clinical implementation of the technique is the general requirement to have an x-ray source of high coherence. The radiation physics group at UCL is currently developing an x-ray phase contrast imaging technique which works with laboratory x-ray sources. Validation of the system requires extensive modelling of relatively large samples of tissue. To aid this, we have undertaken a study of when geometrical optics may be employed to model the system in order to avoid the need to perform a computationally expensive wave optics calculation. In this paper, we derive the relationship between the geometrical and wave optics model for our system imaging an infinite cylinder. From this model we are able to draw conclusions regarding the general applicability of the geometrical optics approximation.

© 2010 Optical Society of America

## 1. Introduction

1. R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. **49**(16), 3573–3583 (2004). [CrossRef] [PubMed]

*in vivo*mammography program is in progress in Trieste, Italy, using the SYRMEP beam line [2

2. E. Castelli, F. Arfelli, D. Dreossi, R. Longo, T. Rokvic, M. Cova, E. Quaia, M. Tonutti, F. Zanconati, A. Abrami, V. Chenda, R. Menk, E. Quai, G. Tromba, P. Bregant, and F. de Guarrini, “Clinical mammography at the SYRMEP beam line,” Nucl. Instrum. Meth. A **572**(1), 237–240 (2007). [CrossRef]

*et. al*[3

3. A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. **91**(7), 074,106 (2007). [CrossRef]

5. A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. **94**(4) (2009). [CrossRef]

6. A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. **52**(22), 6555–6573 (2007). [CrossRef] [PubMed]

3. A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. **91**(7), 074,106 (2007). [CrossRef]

*in vitro*human breast tissue samples. In order to design the system and verify the experiments, it is necessary to model the entire imaging system, including the interaction of x-rays with tissue. The small refractive index contrast of tissue combined with the unpolarised x-ray source mean that a full electromagnetic calculation for the scattered x-rays can be avoided. Furthermore, the short wavelength of x-rays relative to typical cell structure dimensions means that a geometrical optics model is often sufficient. This is important as a rigorous scalar calculation of the scattered field would require prohibitively large computational resources. In this paper, we thus attempt to establish conditions under which a geometrical optics approximation can be employed to model a coded aperture XPCi system.

## 2. Wave optics model

16. A. Olivo and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of any x-ray imaging system,” Phys. Med. Biol. **51**(12), 3015–3030 (2006). [CrossRef] [PubMed]

*x*,0,−

_{s}*z*) emitting a spherical wave at wavelength

_{so}*λ*. Previous experiments have shown [16

16. A. Olivo and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of any x-ray imaging system,” Phys. Med. Biol. **51**(12), 3015–3030 (2006). [CrossRef] [PubMed]

*iωt*) sign convention, the field at position

*P*= (

*x*,

*y*,

*z*) may be given by [16

_{od}16. A. Olivo and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of any x-ray imaging system,” Phys. Med. Biol. **51**(12), 3015–3030 (2006). [CrossRef] [PubMed]

*𝓐*represents the transmitting regions of the sample apertures. In addition,

*x*,

*ξ*and

*z*are defined in Fig. 1 and (

*ξ*,

*ψ*,

*z*) and (

*x*,

*y*,

*z*) form right handed coordinate systems. The integration over

*ψ*can be performed by noting that the apertures have no dependence upon

*ψ*. We must thus evaluate:

*g*(

*x*) has a single first order stationary point,

*x*

_{0}, such that

*g*′(

*x*

_{0}) = 0,

*g*″(

*x*

_{0}) ≠ 0, can be approximated as:

*k*. Applying this approximation to Eq. (3) we find that

*y*. This result is also obtainable using Fourier theory applied to distributions [15] which reveals that Eq. (6) is in fact the solution to Eq. (3) [17]. This enables us to write Eq. (1) as:

*T*(

*ξ*) to represent the transmission function of the sample aperture. It is now easy to include the effect of a phase object with phase function

*ϕ*(

*ξ*) by following an approach similar to that of Arfelli

*et. al*[18

18. F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Val-lazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. **43**(10), 2845–2852 (1998). [CrossRef] [PubMed]

*ℬ*is the extent of the object.

## 3. Efficient evaluation of wave optics field

*T*(

*ξ*) is a periodic function with period

*L*, it can be represented as a complex Fourier series written in general as:

*et. al*[10

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

*x*′

*z*/(

_{so}*z*+

_{so}*z*) to take on values

_{od}*κL*/(2

*N*) the summation in Eq. (10) may be evaluated, for a finite number of terms, by constructing a vector of the form:

*C*may also be evaluated using the FFT, Eq. (10) may be evaluated very efficiently.

_{n}## 4. Geometrical optics model

6. A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. **52**(22), 6555–6573 (2007). [CrossRef] [PubMed]

20. A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. **53**(22), 6461–6474 (2008). [CrossRef] [PubMed]

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

*r*is the position vector of a point on the ray,

*s*the length of the ray,

*n*the refractive index of the medium and 𝓢 defines a wave front of constant phase, ie, 𝓢 (

*ξ*) = constant. It is evident from this that we assume rays are deflected in the

*ξ*direction only. Consider a phase object as depicted in Fig. 2. We define the phase function,

*ϕ*(

*ξ*), as

*n*(

*ξ*,

*z*) is the refractive index at position (

*ξ*,

*z*) and we have assumed that rays make only small angles,

*θ*, with the

_{i}*z*-axis. The angle by which the ray is deflected in then given by:

*θ*to the

_{i}*z*-axis will intercept the

*ξ*-axis at position

*ξ*=

*z*tan(

_{so}*θ*) and, if deflected by an object, will intercept the

_{i}*x*-axis at position

*z*=

*z*and

_{od}*z*= 0 is thus given by:

## 5. Modelling a finite size source

*p*th transmitting region of the detector apertures is given by [

*p*

*LM*−

*LM*/4+

*dL*,

*pLM*+

*LM*/4+

*dL*] where

*dL*is the displacement of the detector apertures relative to the projection of the sample apertures as shown in Fig. 1 and

*M*= (

*z*+

_{od}*z*)/

_{so}*z*is the system magnification. We assume that the pixels are aligned as shown in Fig. 1 such that a single pixel entirely covers a single transmitting region of the detector apertures. Before calculating the signal detected by each pixel, we introduce a source of finite size in the

_{so}*x*̄ direction. The brightness is described by

*P*(

*x*̄) which we will take to have a Gaussian profile. We can then take the signal of the

*p*th pixel to be given by:

*P*(

*x*̄) = exp(-(

*x*̄/σ)

^{2}), Eq. (19) may be expressed as:

*z*) is the error function

*K*(

*x*) may effectively be considered as a pixel sensitivity function. Figure 3 shows plots of

*K*(

*x*) for a variety of source Full Widths at Half Maximum (FWHM). This shows how a broad source leads to a broad

*K*(

*x*) thus diminishing the sensitivity of the system to fine variations in the intensity caused by phase variations in the object.

## 6. XPCi model of a dielectric fibre

### 6.1. Wave optics model

6. A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. **52**(22), 6555–6573 (2007). [CrossRef] [PubMed]

*R*, refractive index

*n*= 1−

*δ*, parallel to the

*ψ*-axis centered upon (

*ξ*,

*z*) = (

*ξ*

_{0},0). Absorbing materials can be modelled by writing

*n*= 1 −

*δ*+

*iβ*thus introducing an attenuation term in Eq. (8). We have opted to set

*β*to 0 to simplify the following analysis. Note that

*δ*is of the order of 10

^{−6}to 10

^{−7}for the range of x-ray energies and materials which we consider here. The phase function,

*ϕ*(

*ξ*), may thus be calculated as:

*ξ*= 0. Note that the cases depicted in Fig. 4 do not limit the cylinder radius, all that is important is where the cylinder boundaries lie relative to the transmitting regions of the apertures. A cylinder covering more than one sample aperture could be modelled using a combination of the cases depicted in Fig. 4. In practice our system employs a series of apertures to simultaneously image a wide field of view. For clarity, we consider here a single sample/detector aperture pair and scan the object to obtain its image. Images obtained in this way will be equivalent to those obtained in practice only when photons are not scattered between differing pre-sample/detector aperture pairs. Only a simple extension is required to model the practical system as is shown at the end of Sec. (6.2). An analysis of when this approximation is valid is given in Sec. (6.3).

*ξ*

_{0}−

*R*,

*ξ*

_{0}+

*R*] ∩ [−

*Lη*/2,

*Lη*)/2] and

*η*is the fill factor of the sample apertures. We now attempt to find asymptotic solutions, for large

*k*, to the integrals in Eq. (25) by again applying the stationary phase approximation. In this case we must also consider the end points of the integrals. It is shown by James [14, Pgs. 29–34] that when the integration in Eq. (4) is evaluated over the interval [

*a*,

*b*] and

*g*′(

*x*) is non-zero and finite at the end points, a term

*g*

_{1}(

*ξ*) and

*g*

_{2}(

*ξ*) and finding their derivatives as:

*M*,

*δ*and

*z*are limited to those values experienced in practice,

_{od}*g*′

_{1}(

*ξ*

_{1,0}) = 0 has a unique solution for every value of

*x*′. This solution must in general be calculated numerically. This may be done efficiently by evaluating

*γ*= [

*γ*] = [

_{i}*g*′

_{1}(

*ξ*)+

_{i}*x*′/

*z*] where

_{od}*ξ*= [

_{i}*ξ*] is a discretisation of the domain [

_{i}*ξ*

_{0}−

*R*+

*ε*,

*ξ*

_{0}+

*R*−

*ε*] for some small

*ε*. This corresponds to case (2) in Fig. 4 where the entire cylinder is illuminated and thus rays are refracted to all values of

*x*′. In cases (1) and (3), the bounds of integration are affected by the sample apertures which in turn affects the values of

*x*′ to which rays are refracted. The stationary point of

*g*

_{1}for a particular

*x*′ may be found by interpolation with

*γ*as the abscissa. This enables the leading term in the expansion to be calculated as

*g*′

_{2}shows that

*g*

_{2}has a single stationary point at

*ξ*

_{2,0}=

*x*′ /

*M*. In the case that

*x*′/

*M*is within the bounds of integration of

*U*

_{2}, the following term is contributed by the stationary point

*ξ*

_{2,0}:

*a*,

*b*] and that neither

*a*or

*b*are stationary points of

*g*

_{2}, the next term in the asymptotic expansion may be found as

*b*= −

*a*, becomes

### 6.2. Relationship between wave and geometrical optics models

*α*. This expression is identical to

*g*′

_{1}= 0 in Eq. (27), the stationary phase condition for integral

*U*

_{1}. It is then easy to verify that assuming identical incident field conditions, substituting Eqs. (23) and (16) into Eq. (18) results in the same magnitude as

*C*Γ(

*y*)

*I*

_{1,0}. Furthermore, substitution of the phase contributions from the phase object and the Fresnel approximation for free space propagation result in the same phase as in

*C*Γ(

*y*)

*I*

_{1,0}. This shows that the leading term in the asymptotic expansion of

*U*

_{1}gives the same field as the geometrical optics approximation to the refracted field. Examination of

*C*Γ(

*y*)

*l*

_{2,0}= 1/(

*z*+

_{so}*Z*) exp (

_{od}*ik*((

*y*

^{2}+ (

*x*−

*x*)

_{s}^{2})/(2(

*z*+

_{so}*z*)) +

_{od}*z*+

_{so}*z*)) shows that this is the geometrical optics field of the light which reaches the detector apertures without being refracted by the cylinder or blocked by the sample apertures. Closer examination of

_{od}*C*Γ(

*y*)

*I*

_{2,1}shows that this is the field due to diffraction at the edges of Ω. Note that this quantity becomes infinite at the edges of the geometrical projection of Ω onto the detector apertures. This non-physical result can be remedied by modifying the stationary phase solution [23

23. R. Buchal and J. Keller, “Boundary layer problems in diffraction theory,” Commun. Pur. Appl. Math. **13**, 85–114 (1960). [CrossRef]

### 6.3. Examples and analysis

*R*is placed with its centre at

*ξ*= 0 in the imaging system of Fig. 1, its edge will be projected onto the position

*x*′ =

*MR*in the space of the detector. We are interested in knowing how quickly the field scattered by the cylinder decays away from

*x*′ =

*MR*. Assuming that the edge of the cylinder is illuminated, photons are refracted to values of

*x*′ approaching ∞ and are described by the term

*I*

_{1,0}defined in Eq. (28). Photons reaching a position

*x*′ ≫

*MR*must be incident upon the cylinder for a value of

*ξ*very close to, but not exceeding

*R*. By writing

*ξ*=

*R*−

*ε*,

*ε*> 0, in Eqs. (27) it is easy to find a simple analytic expression giving

*I*

_{1,0}for

*x*≫

*MR*as

*ε*tends to 0. It is then simple to show that

*I*

_{1,0}

^{2}will reduce by two orders of magnitude at a position

*x*′ =

*RM*+ Δ

*x*′ where:

*x*′ for values of

*R*and

*δ*encountered in practice. Δ

*x*′ may be considered the minimum separation of adjacent sample/detector aperture pairs to ensure detector apertures principally detect photons originating from their associated sample aperture. The above analysis considers only a point source. A source of finite width may be considered by noting the definition of

*x*′ in Eqs. (2) and thus adding (

*W*/2)

*z*/

_{od}*z*to Δ

_{so}*x*′, where

*W*is the detector FWHM.

**52**(22), 6555–6573 (2007). [CrossRef] [PubMed]

*dL*, equal to half of the transmitting width of the detector apertures, thus exposing half of the pixel to directly incident radiation. We used a sample aperture periodicity of

*L*= 40

*μ*m along with

*z*= 1.6m and

_{so}*z*= .4m to match the dimensions of an experimental system currently under construction. The simulations were performed for a photon energy of 100

_{od}*k*eV.

*δ*= 10

^{−7}, a radius of 5

*μ*m and was situated with its axis at

*ξ*= − 5

*μ*m. As is expected, the wave optics intensity exhibits oscillations resulting from interference between different field components. The geometrical optics solution is physically impossible as the sharp edge occurring at

*x*= 0 would require the field to contain infinite spatial frequencies. Consideration of the angular spectrum of a propagating aperiodic field shows that such a field would require evanescent waves which, in our case, would have negligible magnitude such a distance from the sample apertures.

*equivalent*detector apertures which cause the pixels to have a spatially dependent sensitivity as described by Eq. (21). Figure 3 shows that as the source broadens, so does the width of the equivalent detector aperture sensitivity function. Because of energy conservation, one would expect the geometrical and wave optics XPCi signals to converge as the source broadens. In particular, consider the plot shown in Fig. 7. This shows the difference between the intensities, incident upon the detector apertures, predicted by wave and geometrical optics. This signal has a zero mean value as required by conservation of energy. The coded aperture XPCi signal thus depends upon the domain over which the field intensity is integrated by the detector pixel. As the sensitive part of each detector pixel increases, or equivalently, as the source broadens, the geometrical and wave optics signals thus tend to converge.

**52**(22), 6555–6573 (2007). [CrossRef] [PubMed]

*ξ*

_{0}= −

*R*. This is demonstrated in Fig. 10 where wave and geometrical optics signal traces have been plotted for a cylinder of radius 5

*μ*m and

*δ*= 10

^{−6}. The signals have been normalised by the signal for the object free case. These plots demonstrate how the signal traces converge as the FWHM of the source increases. It also shows how the peak of each trace is in the vicinity of

*ξ*

_{0}= −

*R*, as expected. Simulations run over a range of radii, values of

*δ*and source FWHM show that the peak of the signal trace does indeed occur in the region of

*ξ*

_{0}= −

*R*. This is suggests a good way of assessing the difference between the wave and geometrical optics XPCi signals as the two signals are likely to vary most at the peak. We thus calculate an error term,

*ε*(−

*R*), where

*ε*(

*ξ*

_{0}) = |

*I*

_{WO}(

*ξ*

_{0})/

*I*

_{WO}

^{N}−

*I*

_{GO}

^{N}(

*ξ*

_{0})/

*I*

_{GO}

^{N}|, and

*I*

_{WO}(

*ξ*

_{0}) and

*I*

_{GO}(

*ξ*

_{0}) are the XPCi signals for the geometrical and wave optics (full expression evaluated numerically) cases respectively, for a cylinder at position

*ξ*

_{0}.

*I*

_{GO}

^{N}and

*I*

_{WO}

^{N}are the object free XPCi signals for the geometrical and wave optics cases respectively.

*ε*it is useful to note that some approximations can provide further insight into the problem. In the case of

*ξ*

_{0}= −

*R*,

*g*

_{1}in Eq. (27) can be well approximated by

*x*′ > −

*MR*, but not too close to −

*MR*. This approximate form leads to a solution of

*ξ*

_{1,0}= −2

*δ*

^{2}

*Rz*

_{od}^{2}/

*x*′

^{2}for the stationary point of

*g*

_{1}. Substitution of

*ξ*

_{1,0}back into the approximate forms of

*g*

_{1}and

*g*″

_{1}show that both of these functions have a dependence upon

*δ*

^{2}

*R*rather than each of these independently. This suggests that it is reasonable to expect

*ε*for a particular source FWHM to be constant for constant values of

*δ*

^{2}

*R*. This is indeed the case as was verified by a large number of simulations, a small selection of which are shown in Fig. 11. This significantly simplifies the task of determining the source size for which the geometrical and wave optics signals converge. Figure 12 is a contour plot of e as a function of source FWHM and

*δ*

^{2}

*R*. The important conclusion which we can draw from this is that for our particular choice of

*z*and

_{od}*z*, as we expect a source to have a FWHM of around 50

_{so}*μ*m, the geometrical optics model will provide results consistent with those of the wave optics model. This result will make it feasible to model much larger objects.

## 7. Conclusions

## Acknowledgements

## References and links

1. | R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. |

2. | E. Castelli, F. Arfelli, D. Dreossi, R. Longo, T. Rokvic, M. Cova, E. Quaia, M. Tonutti, F. Zanconati, A. Abrami, V. Chenda, R. Menk, E. Quai, G. Tromba, P. Bregant, and F. de Guarrini, “Clinical mammography at the SYRMEP beam line,” Nucl. Instrum. Meth. A |

3. | A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. |

4. | A. Olivo and R. Speller, “Phase contrast imaging.”, International Patent WO/2008/029107, (2008). |

5. | A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, “A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously,” Appl. Phys. Lett. |

6. | A. Olivo and R. Speller, “Modelling of a novel x-ray phase contrast imaging technique based on coded apertures,” Phys. Med. Biol. |

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

8. | F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. |

9. | M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. |

10. | M. Engelhardt, C. Kottler, o. Bunk, C. David, C. Schroer, J. Baumann, m. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microsc. |

11. | A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. |

12. | A. Peterzol, A. Olivo, L. Rigon, S. Pani, and D. Dreossi, “The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging,” Med. Phys. |

13. | J. B. Keller, “Geometrical Theory of Diffraction,” J. Opt. Soc. Am. A |

14. | G. James, |

15. | J. Arsac, |

16. | A. Olivo and R. Speller, “Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of any x-ray imaging system,” Phys. Med. Biol. |

17. | A. Erdélyi, ed., |

18. | F. Arfelli, M. Assante, V. Bonvicini, A. Bravin, G. Cantatore, E. Castelli, L. Dalla Palma, M. Di Michiel, R. Longo, A. Olivo, S. Pani, D. Pontoni, P. Poropat, M. Prest, A. Rashevsky, G. Tromba, A. Vacchi, E. Val-lazza, and F. Zanconati, “Low-dose phase contrast x-ray medical imaging,” Phys. Med. Biol. |

19. | G. Arfken, |

20. | A. Olivo and R. Speller, “Image formation principles in coded-aperture based x-ray phase contrast imaging,” Phys. Med. Biol. |

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

22. | J. Murray, |

23. | R. Buchal and J. Keller, “Boundary layer problems in diffraction theory,” Commun. Pur. Appl. Math. |

**OCIS Codes**

(050.1960) Diffraction and gratings : Diffraction theory

(080.0080) Geometric optics : Geometric optics

(110.7440) Imaging systems : X-ray imaging

(340.7430) X-ray optics : X-ray coded apertures

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: July 13, 2009

Revised Manuscript: November 23, 2009

Manuscript Accepted: December 16, 2009

Published: February 17, 2010

**Virtual Issues**

Vol. 5, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Peter R. Munro, Konstantin Ignatyev, Robert D. Speller, and Alessandro Olivo, "The relationship between wave and geometrical optics models of coded aperture type x-ray phase contrast imaging systems," Opt. Express **18**, 4103-4117 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4103

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### References

- R. Lewis, "Medical phase contrast x-ray imaging: current status and future prospects," Phys. Med. Biol. 49(16), 3573-3583 (2004). [CrossRef] [PubMed]
- E. Castelli, F. Arfelli, D. Dreossi, R. Longo, T. Rokvic, M. Cova, E. Quaia, M. Tonutti, F. Zanconati, A. Abrami, V. Chenda, R. Menk, E. Quai, G. Tromba, P. Bregant, and F. de Guarrini, "Clinical mammography at the SYRMEP beam line," Nucl. Instrum. Meth. A 572(1), 237-240 (2007). [CrossRef]
- A. Olivo and R. Speller, "A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources," Appl. Phys. Lett. 91(7), 074106 (2007). [CrossRef]
- A. Olivo and R. Speller, "Phase contrast imaging," International Patent WO/2008/029107, (2008).
- A. Olivo, S. E. Bohndiek, J. A. Griffiths, A. Konstantinidis, and R. D. Speller, "A non-free-space propagation x-ray phase contrast imaging method sensitive to phase effects in two directions simultaneously," Appl. Phys. Lett. 94(4) (2009). [CrossRef]
- A. Olivo and R. Speller, "Modelling of a novel x-ray phase contrast imaging technique based on coded apertures," Phys. Med. Biol. 52(22), 6555-6573 (2007). [CrossRef] [PubMed]
- T. Gureyev and S. Wilkins, "On x-ray phase imaging with a point source," J. Opt. Soc. Am. A 15(3), 579-585 (1998). [CrossRef]
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