## Phase contrast laminography based on Talbot interferometry |

Optics Express, Vol. 20, Issue 6, pp. 6496-6508 (2012)

http://dx.doi.org/10.1364/OE.20.006496

Acrobat PDF (5214 KB)

### Abstract

Synchrotron laminography is combined with Talbot grating interferometry to address weakly absorbing specimens. Integrating both methods into one set-up provides a powerful x-ray diagnostical technique for multiple contrast screening of macroscopically large flat specimen and a subsequent non-destructive three-dimensional (3-D) inspection of regions of interest. The technique simultaneously yields the reconstruction of the 3-D absorption, phase, and the so-called dark-field contrast maps. We report on the theoretical and instrumental implementation of of this novel technique. Its broad application potential is exemplarily demonstrated for the field of cultural heritage, namely study of the historical Dead Sea parchment.

© 2012 OSA

## 1. Introduction

1. Z. des Plantes, “Eine neue Methode zur Differenzierung in der Rontgenographie,” Acta Radio. **13**, 182–192 (1932). [CrossRef]

*lateral*dimensions strongly exceed the effective field of view (like rather flat specimens such as electronic circuit boards).

2. L. Helfen, A. Myagotin, P. Pernot, M. DiMichiel, P. Mikulík, A. Berthold, and T. Baumbach, “Investigation of hybrid pixel detector arrays by synchrotron-radiation imaging,” Nucl. Inst. Meth. A **563**, 163–166 (2006). [CrossRef]

4. A. Houssaye, F. Xu, L. Helfen, C. De Buffrénil, T. Baumbach, and P. Tafforeau, “Three-dimensional pelvis and limb anatomy of the Cenomanian hind-limbed snake Eupodophis descouensi (Squamata, Ophidia) revealed by synchrotron-radiation computed laminography,” J. Vert. Paleontol. **31**, 2–6 (2011). [CrossRef]

5. S. Harasse, N. Hirayama, W. Yashiro, and A. Momose, “X-ray phase laminography with Talbot interferometer,” Proc. SPIE **7804**, 780411 (2010). [CrossRef]

7. M. Hoshino, K. Uesugi, A. Takeuchi, Y. Suzuki, and N. Yagi, “Development of x-ray laminography under an x-ray microscopic condition,” Rev. Sci. Instrum. **82**, 073706 (2011). [CrossRef] [PubMed]

8. G. Schulz, T. Weitkamp, I. Zanette, F. Pfeiffer, F. Beckmann, C. David, S. Rutishauser, E. Reznikova, and B. Müller, “High-resolution tomographic imaging of a human cerebellum: comparison of absorption and grating-based phase contrast,” J. Roy. Soc. Interf. **7**, 1665–1676 (2010). [CrossRef]

9. T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol . **68S**, 13–17 (2008). [CrossRef]

## 2. Principles of Talbot grating interferometry based laminography

10. L. Helfen, T. Baumbach, P. Mikulík, D. Kiel, P. Pernot, P. Cloetens, and J. Baruchel, “High-resolution three-dimensional imaging of flat objects by synchrotron-radiation computed laminography,” Appl. Phys. Lett. **86**, 071915 (2005). [CrossRef]

13. F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast images using polychromatic hard X-rays,” Nat. Phys. **2**, 258–261 (2006). [CrossRef]

*θ*, (2) radiography of the grating interferometry image sequences (phase-stepping technique), (3) recording the laminographic

*φ*rotation scans, if required, also in a loop together with (4) transversal (x, y) sample screening.

*for forward and backward*

_{n}*n*-dimensional Fourier transforms. The real space vectors

**r**are written in the laboratory coordinate system as

**r**=

*u*

**e**

*+*

_{u}*v*

**e**

*+*

_{v}*w*

**e**

*and in the specimen coordinate system as*

_{w}**r**=

*x*

**e**

*+*

_{x}*y*

**e**

*+*

_{y}*z*

**e**

*. For convenience we place the origin of both coordinate systems in the rotation centre of the laminography setup. A similar notation is used for reciprocal (Fourier) space vectors, writing, e.g., wave vectors in the laboratory system as*

_{z}**K**=

*K*

_{u}**ê**

*+*

_{u}*K*

_{v}**ê**

*+*

_{v}*K*

_{w}**ê**

*. Further, we define*

_{w}*w*being the optical axis and

*u*and

*v*the horizontal and vertical axes of the rectangular image plane with

**r**

_{||}or

**K**

_{||}the projections of

**r**and

**K**onto the image planes. The

*z*axis is the laminographic rotation axis, the (

*x,y*) plane is parallel to the laminographic rotation table. In laminography, the specimen’s spatial orientation is unambiguously defined by the laminographic rotation and tilt angles

*φ*and

*θ*, see Fig. 1. The laboratory system coordinates are transformed into the specimen coordinate system according to where

*R*,

_{z}*R*are conventional 3 × 3 matrices describing rotation about the

_{x}**z**and

**x**axes. The same relation holds for coordinates of all reciprocal space vectors.

### Forward problem

*n*(

**r**) = 1 –

*δ*̃(

**r**) +

*i*

*β*̃(

**r**), which considers refraction by

*δ*̃(

**r**) and absorption by

*β*̃(

**r**). Propagation of a plane wave through a grating laminography set-up involves mainly the subsequent interactions of: (a) transmission through the phase grating and diffraction during propagation, (b) transmission through the specimen in dependence on the laminographic angles

*θ*and

*φ*, by which the initially otherwise unperturbed grating diffracted wave field acquires space-dependent disturbances due to refraction, absorption, and scattering by the sample structure, (c) propagation of the perturbed wave field to the image plane (where an absorption grating is positioned), (d) transmission of the corresponding intensity field through the absorption grating. The latter generates a Moiré fringe pattern in the detector plane. Points (a) and (b) can be interchanged.

*i*

**K**

_{0}

**r**) with the wave vector

**K**

_{0}parallel to the optical axis

*w*. Passing through an object, it undergoes phase and amplitude changes, which in the eikonal or projection approximation are considered by line integrals

*p*=

*∫*{

_{L}**K**–

**K**

*n*(

**r**)}

*dl*. In the object exit plane the transmitted wave field can be written as a perturbed plane wave with inhomogeneous amplitude Here we have introduced the so-called

*x-ray transform*[15

15. F. Natterer, *The Mathematics of Computerized Tomography* (Society for Industrial and Applied Mathematics, 2001). [CrossRef]

*complex*2D disturbance function determined by the three-dimensional integration The Dirac delta functions in Eq. (3) select all allowed

**r**, contributing to the projection onto an image plane at point

**r**

_{||}=

*u*

**e**

*+*

_{u}*v*

**e**

*, where (*

_{v}*u*

**e**

*,*

_{u}*v*

**e**

*) span the rectangular 2-D projection plane.*

_{v}^{3}∈

^{3}of the three-dimensional space filled by the object corresponding to the x-ray transform (in our case the gratings or the specimen). Although at synchrotron sources the condition of parallel-beam geometry holds very well, the laminography geometry, in contrast to 3-D tomography, does

*not*allow for a split of the aquisition and reconstruction problem into a stack of 2-D problems. By use of Eq. (1) the x-ray transforms of the gratings and of the specimen may be alternatively expressed in the laboratory or specimen coordinate system, whereby only the x-ray transform of the specimen explicitly depends on the laminographic tilt and rotation angles.

16. P. Mikulík and T. Baumbach, “X-ray reflection by rough multilayer gratings: dynamical and kinematical scattering,” Phys. Rev. B **59**, 7632–7643 (1999). [CrossRef]

*T*and wave vectors

_{m}**K**

*. Coefficients*

_{m}*T*can be calculated from Eq. (2) by use of the x-ray transform Eq. (3) of the grating. The allowed wave vectors

_{m}**K**

*are selected by the intersection of the Ewald sphere with so-called grating truncation rods, which represent the grating response function*

_{m}*n*(

_{G}**r**) in the Fourier domain,

_{3}(

*n*(

_{G}**r**)), yielding (for gratings aligned in the (

*u,v*) plane) in the paraxial approximation The

**h**

*are reciprocal grating vectors, which are defined by the symmetry and period lengths of the grating (for 1-D gratings, e.g., with periods*

_{m}*D*along

**u**it is

**h**

*= 2*

_{m}*m*

*π/D*

**ê**

*).*

_{u}*p*(

_{φ,θ}**r**

_{||}) in dependence on the laminographic angles. Within a sufficiently good approximation for TGI, the specimen’s x-ray transform

*p*(

_{φ,θ}**r**

_{||})

*diffractionless*propagates into the bundle of directions

**K**

*of the plane-wave grating diffraction orders. Diffractionless propagation leads to a geometrical projection. Due to the geometric intercept theorem, a non-zero angle between the*

_{m}**K**

*and the optical axis with increasing propagation distances Δ*

_{m}*w*causes an increasing lateral shift Δ

**r**

_{||}= Δ

*w*[

**K**

_{||m}/

*K*] in the x-ray transform projected along

**K**

*onto the image plane. The total perturbed wave field in the image plane at the position of the absorption grating*

_{m}*w*consists of the superposition of field contributions of all perturbed grating diffraction orders, obtained by use of the Eq. (2)–(5): Here

_{a}*w*′ =

*w*–

_{a}*w*is the axial propagation distance between phase and absorption grating, and

_{p}*w̃*=

*w*– max(

_{a}*w*,

_{s}*w*) denotes the axial propagation distance of the specimen disturbance with the grating diffraction field. Every summand in Eq. (6) comprises the contribution of the unperturbed grating diffraction order (first two factors), which are multiplied with a disturbance factor having the specimen’s propagated x-ray transform in its exponent.

_{p}*p*(

_{φ,θ}**r**

_{||}) Eq. (6) can be rewritten in the form The sensitivity of a certain grating order with respect to the specimen is proportional to the scalar product of the reciprocal grating vector

**h**

*with the local 2-D gradient vector of the specimen’s x-ray transform ∇*

_{m}_{||}

*p*, and it is linearly magnified by the propagation distance

_{φ,θ}*w̃*. Using the self-imaging properties of unperturbed grating-diffracted fields, the image plane will favorably be positioned at such suitable fractional Talbot orders, where propagation of the initially

*phase*modulated wave field of the first grating has developed strong

*amplitude*modulation, and as a consequence essential intensity modulation. The unperturbed intensity field of the grating can be expanded in to a Fourier series,

*I*(

_{G}**r**) = ∑

*exp (*

_{n}I_{n}*i*

**h**

_{n}**r**), and after incorporating the sample, the perturbed intensity pattern can be approximated by with

**r̃**=

**r**

_{||}+ Δ

**r̃**

_{||}, and where Δ

**r̃**

_{||}=

*w̃/K*∇

_{||}

*p*′

*(*

_{φ,θ}**r**

_{||}). Here, we assumed that the effect of ∇

_{||}

*p*″

*(*

_{φ,θ}**r**

_{||}) is subdominant and can be neglected.

*A*(

**r**) = ∑

*exp (*

_{l}A_{l}*i*

**g**

_{l}**r**), creates a characteristic Moiré fringe pattern, which then can be recorded in the detector plane. This Moiré fringe pattern again becomes distorted if the specimen is introduced into the beam path. For sufficiently large pixel size, the perturbed Moiré fringe pattern (integrated over the grating periods) can be approximated as [14] and is later used for solving the inverse problem to reconstruct the object contrast functions.

### Inverse problem

*l*= ±1), well-known rules of ordinary two-beam interferometry can be applied. For 1-D gratings the phase-stepping technique is realised via displacing one of the gratings in the direction parallel to

**e**

*. The differential phase map (the gradient of the x-ray phase transform) may directly be determined from a summation over each of the*

_{u}*N*interferograms of displacement 2

*πn/N*weighted by exp(−2

*πin/N*), where

*n*= 0,1,2,...,

*N*– 1., by (see for example [14]) Analysing the series of the interferograms

*μ*(

**r**) can be retrieved together with the differential phase. One may also obtain the x-ray transforms of the effective scattering power

*f*(

**r**), which results from the superposition of coherent Fresnel diffraction and mesoscale diffuse scattering of the specimen (see for example [17

17. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E.F. Eikenberry, C. Brönniman, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. **7**, 134–137 (2008). [CrossRef] [PubMed]

*ρ*(

**r**) (which could be the linear attenuation coefficients

*μ*(

**r**), the real part of the refraction index

*δ*̃(

**r**), or the effective scattering power

*f*(

**r**)) from a series of projections reconstructed via Eq. (10) from the measured phase-stepping scans under constant

*θ*but different rotation angles

*φ*is performed by using the Fourier slice theorem. For the laminography case, the 2-D Fourier transforms 𝒫

*(*

_{φ,θ}*k*,

_{u}*k*) of the projections

_{v}*p*(

_{φ,θ}*u,v*) correspond to the cross-sections of the 3-D Fourier transform (

*k*,

_{x}*k*,

_{y}*k*) of the object functions

_{z}*ρ*(

**r**) through the reciprocal image plane (

**e**

*×*

_{u}**e**

*) ·*

_{v}**k**= 0, that is

**k**

*direction by the laminographic angle*

_{z}*θ*, the set of unbounded projections for different rotation angles does fill (and even oversamples) the region outside a double cone with the opening angle

*π*–2

*θ*, while the cone internal regions remain empty (non-sampled frequencies) [11

11. L. Helfen, A. Myagotin, P. Mikulík, P. Pernot, A. Voropaev, M. Elyyan, M. Di Michiel, J. Baruchel, and T. Baumbach, “On the implementation of computed laminography using synchrotron radiation,” Rev. Sci. Instrum. **82**, 063702 (2011). [CrossRef] [PubMed]

*ρ*(

**r**) from its x-ray transforms can be performed by generating a so-called

*compound image g*via back-projection and by the latter convolution with an

*inverse filter function ĥ*[11

11. L. Helfen, A. Myagotin, P. Mikulík, P. Pernot, A. Voropaev, M. Elyyan, M. Di Michiel, J. Baruchel, and T. Baumbach, “On the implementation of computed laminography using synchrotron radiation,” Rev. Sci. Instrum. **82**, 063702 (2011). [CrossRef] [PubMed]

*ĥ*depends on the 3-D point spread function which is determined by the experimental geometry.

*T*

^{[11. Z. des Plantes, “Eine neue Methode zur Differenzierung in der Rontgenographie,” Acta Radio. 13, 182–192 (1932). [CrossRef] ,33. J. Dik, K. Krug, L. Porra, P. Coan, G. Tauber, A. Wallert, A. Coerdt, A. Bravin, M. Elyyan, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? combining x-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Rad. 563, 163–166 (2007).]}denotes a matrix where only the 1st and 3rd row of the matrix T is retained. Since where ⊗ denotes a 3-D convolution, one obtains a

*reconstruction equation*with

*is a 2-D section of the 3-D inverse filter function ̄ in the reciprocal space through a plane defined by a projection 𝒫*

_{φ,θ}*as*

_{φ,θ}*ϕ*nor the reciprocal vector component

*k*, the projection filtering can be efficiently implemented by a series of 1-D fast Fourier transforms.

_{v}*filtered projection*. Applying Eq. (17), reconstruction equation (15) can be rewritten in the following two forms The above equation offers two fundamental ways to implement a reconstruction procedure: the reconstruction can be performed in reciprocal or directly in real space. For reconstructions from our experiments we used the second approach.

*μ*(

**r**) and

*f*(

**r**). The reconstruction of the refractive index map

*δ*

*̃*(

**r**) from the phase gradient projections requires an adaption of the filter function (Eq. (18)) to corresponding to the Hilbert transform in real space [20

20. F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. **98**, 108105 (2007). [CrossRef] [PubMed]

## 3. Instrumentation and experiments

*μ*m pixel size. It also permits detector translation along the optical axis to set the wavefield propagation distances between the object and the detector plane (up to max. 1 m downstream of the sample rotation center).

21. I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. **105**, 248102 (2010). [CrossRef]

^{2}. As a result, employing the laminographic rotation geometry, 3-D measurements of selected regions of interest (ROI) can be recorded with variable resolution, without special preparation (sample extraction). For example, the set-up allows first to measure low-resolution (preliminary) scans. After reconstruction, one is able to identify the most interesting ROIs and finally to zoom into these regions for high-resolution measurements. Also, continuous high-resolution screening of large specimen areas can be performed by repeating full-field computed laminography and systematically translating the sample through the center of the rotation table. In this way, combining the inherent full-field method with translational scanning, an enlarged effective field of view can be realized without drawbacks of additional artifacts.

*harmonic of the ID19 undulator at ESRF with the peak energy at 18 keV and energy bandwidth*

^{st}*dE/E*of about 0.1. The ID19 source size and beamline length of 145 m guarantee more than sufficient spatial (transversal) coherence for the grating interferometer. The phase and absorption gratings with pitches of 2.4

*μ*m were fabricated by the LIGA process at KIT, the first made of 4

*μ*m thick Ni (acting as a

*π*/2-shift grating), the latter made of gold with a aspect ratio of 100. For the described experiments we have operated the TGI in the 5

*Talbot order. The Moíre patterns were detected by use of the ESRF in-house developed CCD based ”Frelon 2k” camera (with a 2048 x 2048 pixel array). By use of an optical lens coupled to a thin-film crystal scintillator (LuAg 125*

^{th}*μ*m), an effective pixel size of 5.02

*μ*m was appropriately providing a field of view of about 10 mm x10 mm. All datasets were acquired via phase-stepping scans with 4 steps per projection angle, and exposure time of 0.5 s per frame. The parchment specimens were visualized at a laminographic angle of 32 degrees; 1599 projections over 360 degrees rotation angle interval were acquired.

## 4. Results

24. B.M. Murphy, M. Cotte, M. Mueller, M. Balla, and J. Gunneweg, “Degradation of parchment and ink of the Dead Sea scrolls investigated using synchrotron-based X-ray and infrared microscopy, in Holistic Qumran,” in *Holistic Qumran*, J. Gunneweg, A. Adriaens, and J. Dik, eds. (Brill Leiden, 2010), pp. 77–98. [CrossRef]

25. J. Dik, L. Helfen, P. Reischig, J. Blaas, and J. Gunneweg, “A short note on the application of synchrotron-based micro-tomography on the Dead Sea scrolls in Holistic Qumran,” in *Holistic Qumran*, J. Gunneweg, A. Adriaens, and J. Dik, eds. (Brill Leiden, 2010), pp. 21–28. [CrossRef]

24. B.M. Murphy, M. Cotte, M. Mueller, M. Balla, and J. Gunneweg, “Degradation of parchment and ink of the Dead Sea scrolls investigated using synchrotron-based X-ray and infrared microscopy, in Holistic Qumran,” in *Holistic Qumran*, J. Gunneweg, A. Adriaens, and J. Dik, eds. (Brill Leiden, 2010), pp. 77–98. [CrossRef]

## 5. Conclusions

## Acknowledgments

## References and links

1. | Z. des Plantes, “Eine neue Methode zur Differenzierung in der Rontgenographie,” Acta Radio. |

2. | L. Helfen, A. Myagotin, P. Pernot, M. DiMichiel, P. Mikulík, A. Berthold, and T. Baumbach, “Investigation of hybrid pixel detector arrays by synchrotron-radiation imaging,” Nucl. Inst. Meth. A |

3. | J. Dik, K. Krug, L. Porra, P. Coan, G. Tauber, A. Wallert, A. Coerdt, A. Bravin, M. Elyyan, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? combining x-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Rad. |

4. | A. Houssaye, F. Xu, L. Helfen, C. De Buffrénil, T. Baumbach, and P. Tafforeau, “Three-dimensional pelvis and limb anatomy of the Cenomanian hind-limbed snake Eupodophis descouensi (Squamata, Ophidia) revealed by synchrotron-radiation computed laminography,” J. Vert. Paleontol. |

5. | S. Harasse, N. Hirayama, W. Yashiro, and A. Momose, “X-ray phase laminography with Talbot interferometer,” Proc. SPIE |

6. | L. Helfen, T. Baumbach, P. Cloetens, and J. Baruchel, “Phase-contrast and holographic computed laminography,” Appl. Phys. Lett. |

7. | M. Hoshino, K. Uesugi, A. Takeuchi, Y. Suzuki, and N. Yagi, “Development of x-ray laminography under an x-ray microscopic condition,” Rev. Sci. Instrum. |

8. | G. Schulz, T. Weitkamp, I. Zanette, F. Pfeiffer, F. Beckmann, C. David, S. Rutishauser, E. Reznikova, and B. Müller, “High-resolution tomographic imaging of a human cerebellum: comparison of absorption and grating-based phase contrast,” J. Roy. Soc. Interf. |

9. | T. Weitkamp, C. David, O. Bunk, J. Bruder, P. Cloetens, and F. Pfeiffer, “X-ray phase radiography and tomography of soft tissue using grating interferometry,” Eur. J. Radiol . |

10. | L. Helfen, T. Baumbach, P. Mikulík, D. Kiel, P. Pernot, P. Cloetens, and J. Baruchel, “High-resolution three-dimensional imaging of flat objects by synchrotron-radiation computed laminography,” Appl. Phys. Lett. |

11. | L. Helfen, A. Myagotin, P. Mikulík, P. Pernot, A. Voropaev, M. Elyyan, M. Di Michiel, J. Baruchel, and T. Baumbach, “On the implementation of computed laminography using synchrotron radiation,” Rev. Sci. Instrum. |

12. | T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express |

13. | F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast images using polychromatic hard X-rays,” Nat. Phys. |

14. | A. Momose, W. Yashiroi, and Y. Takeda, “X-ray phase imaging with Talbot interferometry,” in |

15. | F. Natterer, |

16. | P. Mikulík and T. Baumbach, “X-ray reflection by rough multilayer gratings: dynamical and kinematical scattering,” Phys. Rev. B |

17. | F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E.F. Eikenberry, C. Brönniman, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. |

18. | F. Xu, L. Helfen, T. Baumbach, and H. Suhonen, “Comparison and quantification of laminography and limited-angle-tomography,” Opt. Express, submitted (2011). |

19. | A. Myagotin, A. Voropaev, L. Helfen, D. Hänschke, and T. Baumbach, “Fast volume reconstruction for parallel-beam computed laminography by filtered backprojection,” J. Parallel Distrib. Comput., submitted (2011). |

20. | F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. |

21. | I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. |

22. | D. Bradley and D. Creagh, |

23. | R. Larsen |

24. | B.M. Murphy, M. Cotte, M. Mueller, M. Balla, and J. Gunneweg, “Degradation of parchment and ink of the Dead Sea scrolls investigated using synchrotron-based X-ray and infrared microscopy, in Holistic Qumran,” in |

25. | J. Dik, L. Helfen, P. Reischig, J. Blaas, and J. Gunneweg, “A short note on the application of synchrotron-based micro-tomography on the Dead Sea scrolls in Holistic Qumran,” in |

**OCIS Codes**

(100.2960) Image processing : Image analysis

(100.5070) Image processing : Phase retrieval

(340.6720) X-ray optics : Synchrotron radiation

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

(340.7450) X-ray optics : X-ray interferometry

**ToC Category:**

Image Processing

**History**

Original Manuscript: January 31, 2012

Manuscript Accepted: February 11, 2012

Published: March 5, 2012

**Virtual Issues**

Vol. 7, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Venera Altapova, Lukas Helfen, Anton Myagotin, Daniel Hänschke, Julian Moosmann, Jan Gunneweg, and Tilo Baumbach, "Phase contrast laminography based on Talbot interferometry," Opt. Express **20**, 6496-6508 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-6496

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### References

- Z. des Plantes, “Eine neue Methode zur Differenzierung in der Rontgenographie,” Acta Radio.13, 182–192 (1932). [CrossRef]
- L. Helfen, A. Myagotin, P. Pernot, M. DiMichiel, P. Mikulík, A. Berthold, and T. Baumbach, “Investigation of hybrid pixel detector arrays by synchrotron-radiation imaging,” Nucl. Inst. Meth. A563, 163–166 (2006). [CrossRef]
- J. Dik, K. Krug, L. Porra, P. Coan, G. Tauber, A. Wallert, A. Coerdt, A. Bravin, M. Elyyan, L. Helfen, and T. Baumbach, “Relics in medieval altarpieces? combining x-ray tomographic, laminographic and phase-contrast imaging to visualize thin organic objects in paintings,” J. Synchrotron Rad.563, 163–166 (2007).
- A. Houssaye, F. Xu, L. Helfen, C. De Buffrénil, T. Baumbach, and P. Tafforeau, “Three-dimensional pelvis and limb anatomy of the Cenomanian hind-limbed snake Eupodophis descouensi (Squamata, Ophidia) revealed by synchrotron-radiation computed laminography,” J. Vert. Paleontol.31, 2–6 (2011). [CrossRef]
- S. Harasse, N. Hirayama, W. Yashiro, and A. Momose, “X-ray phase laminography with Talbot interferometer,” Proc. SPIE7804, 780411 (2010). [CrossRef]
- L. Helfen, T. Baumbach, P. Cloetens, and J. Baruchel, “Phase-contrast and holographic computed laminography,” Appl. Phys. Lett.94(10), 104103 (2009). [CrossRef]
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