## Iterative reconstruction in x-ray computed laminography from differential phase measurements |

Optics Express, Vol. 19, Issue 17, pp. 16560-16573 (2011)

http://dx.doi.org/10.1364/OE.19.016560

Acrobat PDF (955 KB)

### Abstract

Phase-contrast X-ray computed laminography is demonstrated for the volume reconstruction of extended flat objects, not suitable to the usual tomographic scan. Using a Talbot interferometer, differential phase measurements are obtained and used to reconstruct the real part of the complex refractive index. The specific geometry of laminography leads to unsampled frequencies in a double cone in the reciprocal space, which degrades the spatial resolution in the direction normal to the object plane. First, the filtered backprojection formula from differential measurements is derived. Then, reconstruction is improved by the use of prior information of compact support and limited range, included in an iterative filtered backprojection algorithm. An implementation on GPU hardware was required to handle the reconstruction of volumes within a reasonable time. A synchrotron radiation experiment on polymer meshes is reported and results of the iterative reconstruction are compared with the simpler filtered backprojection.

© 2011 OSA

## 1. Introduction

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

2. D. G. Grant, “Tomosynthesis: a three-dimensional radiographic imaging technique,” IEEE Trans. Biomed. Eng. **19**(1), 20–28 (1972). [CrossRef] [PubMed]

3. H. Matsuo, A. Iwata, I. Horiba, and N. Suzumura, “Three-dimensional image reconstruction by digital tomosynthesis using inverse filtering,” IEEE Trans. Med. Imaging **12**(2), 307–313 (1993). [CrossRef] [PubMed]

4. L. Helfen, T. Baumbach, P. Cloetens, and J. Baruchel, “Phase-contrast and holographic computed laminography,” Appl. Phys. Lett. **94**(10), 104103 (2009). [CrossRef]

5. F. Xu, L. Helfen, A. J. Moffat, G. Johnson, I. Sinclair, and T. Baumbach, “Synchrotron radiation computed laminography for polymer composite failure studies,” J. Synchrotron Radiat. **17**(2), 222–226 (2010). [CrossRef] [PubMed]

6. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. **42**(7B), 866–868 (2003). [CrossRef]

*δ*from the real part of the complex refractive index 1 –

*δ*+

*iβ*. First we will derive in section 2 the filtered backprojection formula for CL from differential phase images. The obtained filtered backprojection operator will then be used afterwards in a more effective iterative reconstruction, presented in section 3. Iterative methods generally successfully apply to ill-posed tomographic problems with various constraints like positivity, limited support or gradient sparsity (e.g. [7

7. E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. **53**, 4777 (2008). [CrossRef] [PubMed]

8. D. Lalush and B. Tsui, “Improving the convergence of iterative filtered backprojection algorithms,” Med. Phys. **21**(8), 1283–1286 (1994). [CrossRef] [PubMed]

## 2. X-ray computed laminography by Talbot interferometry

### 2.1. Scan geometry

*O*and axes (

*u⃗*,

*v⃗*,

*w⃗*), while another coordinate system local to the object is defined with same origin

*O*and axes (

*x⃗*,

*y⃗*,

*z⃗*). The object is rotated about axis

*y⃗*for 360°, as described by angle

*θ*. The object plane is tilted with a fixed acute angle

*α*from the beam path direction. Angle

*α*should be carefully chosen depending on the application. Smaller angles make the setup closer to the classical CT scan, with the limit case

*α*= 0 being CT. However, using lower angles means that X-rays are almost aligned with the object plane, which is practically infeasible because of the high absorption by the sample. Low angles may also produce phase wrapping problems in differential phase images. Conversely, higher

*α*angles produce a greater loss of resolution in the object plane normal direction.

*M*, composition of rotation by angle

_{θ}*α*around axis

*x⃗*and rotation by angle

*θ*around axis

*y⃗*: A vector

*r⃗*= [

*x,y,z*]

*expressed in object coordinates is transformed into detector coordinates*

^{T}*q⃗*= [

*u,v,w*]

*as follows :*

^{T}### 2.2. Differential phase measurements

*π*/2 phase shift to the incident rays passing through it. This produces the Talbot effect, where self images of the grating are repeatedly generated at specific distances downstream. Phase shift Φ of the rays produced by the object distorts these self images. The analysis of a self image by fringe scanning [9

9. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. **13**(11), 2693–2703 (1974). [CrossRef] [PubMed]

*φ*(

_{u}*θ*,

*u,v*) produced by the phase shift by the object, in the direction of

*u⃗*, normal to the grating lines and the beam path : where

*I*are the acquired images for the

_{k}*S*steps of the fringe scanning process,

*i*is the imaginary unit,

*d*is the pitch of the gratings and

*z*is the distance between the two gratings, given by with

_{T}*λ*the X-ray wavelength. While other distances are possible,

*z*is the distance for which the visibility of the fringe is maximum when using the

_{T}*π*/2 phase grating [10

10. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. **45**, 5254–5262 (2006). [CrossRef]

*φ*constitute the measurements of the object used for the volume reconstruction. Because the beam deflection is proportional to the derivative of the phase shift, our measurements are called differential phase measurements : Also,

_{u}*φ*can be expressed with the integral of

_{u}*δ*along the beam path, differentiated in the

*u⃗*direction (To simplify notations, the value of any multivariate function

*f*at point (

*v*

_{1},

*v*

_{2}, ⋯ ,

*v*) may be written indifferently as

_{n}*f*(

*v*

_{1},

*v*

_{2}, ⋯ ,

*v*) or

_{n}*f*(

*v⃗*) where

*v⃗*is the vector with same coordinates) :

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

11. J. Als-Nielsen and D. McMorrow, *Elements of Modern X-ray Physics*, 2nd ed. (John Wiley and Sons, 2011). [CrossRef]

*are the Fourier transform of*δ ˜

*φ*and

_{u}*δ*respectively.

### 2.3. Backprojection

*φ*(

*θ*,

*u, v*) of projections

*φ*(

_{u}*θ*,

*u, v*) along

*u⃗*is considered. For each value of

*v*, the unknown mean of the result is arbitrarily set to 0.

*φ*(

*θ*,

*u, v*) is then proportional to the phase shift produced by the object along the given ray, plus an unknown additional term depending on

*v*. In reciprocal space, the integral

*of*φ ˜

*b*(

*x,y,z*) is defined here in the object’s local coordinate system as follows : with

*u*and

*v*obtained by relation (2). Using Eq. (8), we get the expression of the backprojection directly from the Fourier transform of the beam deflection measurements

^{*}= ℝ\{0}, to take into account the special case

*ω*= 0. Using the Fourier slice theorem, Eq. (7), we obtain :

_{u}*ω*,

_{u}*ω*,

_{v}*θ*to Cartesian coordinates

*ω*

_{x}*ω*

_{y}*ω*in the object reciprocal space will give the relation between functions

_{z}*b*and

*δ*. The relation between old and new variables is : from which the following condition on

*ω*,

_{x}*ω*,

_{y}*ω*for

_{z}*ω*and

_{u}*ω*to be defined is derived : The strict nature of this inequality comes from the fact that

_{v}*ω*= 0 is excluded from the integral in Eq. (12). The frequency region not satisfying condition (14) is not acquired by the measurements and is therefore lost in the reconstruction. This is a double cone with aperture 2

_{u}*α*and axis

*ω*, as illustrated in Fig. 2.

_{y}*C*is the subset of ℝ

^{3}satisfying Eq. (14). This can then be interpreted as the filtering of the object function by H, which is the frequency response of the projection backprojection process, defined in the reciprocal space as : The obtained frequency response differs from the previous work in [3

3. H. Matsuo, A. Iwata, I. Horiba, and N. Suzumura, “Three-dimensional image reconstruction by digital tomosynthesis using inverse filtering,” IEEE Trans. Med. Imaging **12**(2), 307–313 (1993). [CrossRef] [PubMed]

13. G. Lauritsch and W. H. Härer, “A theoretical framework for filtered backprojection in tomosynthesis,” Proc. SPIE **3338**, 1127–1137 (1998). [CrossRef]

13. G. Lauritsch and W. H. Härer, “A theoretical framework for filtered backprojection in tomosynthesis,” Proc. SPIE **3338**, 1127–1137 (1998). [CrossRef]

14. 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]

*H*, expressed in the projection reciprocal space is the following function

*G*, independent of

*θ*and

*ω*: Multiplying the expression inside the integral of (11) by the inverse filter 1/

_{v}*G*gives the FBP formula from the beam deflection measurements : This means that the beam deflection data are multiplied by the following inverse filter

*F*

_{inv}in reciprocal space before backprojection :

## 3. Iterative reconstruction

4. L. Helfen, T. Baumbach, P. Cloetens, and J. Baruchel, “Phase-contrast and holographic computed laminography,” Appl. Phys. Lett. **94**(10), 104103 (2009). [CrossRef]

5. F. Xu, L. Helfen, A. J. Moffat, G. Johnson, I. Sinclair, and T. Baumbach, “Synchrotron radiation computed laminography for polymer composite failure studies,” J. Synchrotron Radiat. **17**(2), 222–226 (2010). [CrossRef] [PubMed]

13. G. Lauritsch and W. H. Härer, “A theoretical framework for filtered backprojection in tomosynthesis,” Proc. SPIE **3338**, 1127–1137 (1998). [CrossRef]

15. G. M. Stevens, R. Fahrig, and N. J. Pelc, “Filtered backprojection for modifying the impulse response of circular tomosynthesis,” Med. Phys. **28**, 372–380 (2001). [CrossRef] [PubMed]

### 3.1. Object constraints

*δ*is assumed in the vertical,

*y⃗*direction. This is interestingly in opposition to the classical limited support constraint used in CT (the limit case

*α*= 0), which is defined on axes

*x⃗*and

*z⃗*so that the object is fully contained in the field of view. In our case, It is not required that the prior support perfectly matches the actual support of the sample. An approximative rectangular box shaped prior support is empirically obtained from the FBP reconstruction.

*δ*. The expected values for

*δ*should always be positive. An upper limit is also defined depending on the object. Non negativity and support constraints have been classically used in the field of optical microscope tomography [16

16. O. Nakamura, S. Kawata, and S. Minami, “Optical microscopy tomography. II. Nonnegative constraint by a gradient-projection method,” J. Opt. Soc. Am. A **5**(4), 554–561 (1988). [CrossRef]

### 3.2. Iterative filtered backprojection

*a*×

*a*depends on the equipment. Similarly, the object space is discretized in a finite grid of voxels with equivalent size

*a*

^{3}. Let

*s*denote the estimate of the object at iteration

_{k}*k*. It is a vector with dimension

*N*, the number of voxels in the grid.

*b*is a vector representing the measured beam deflection images

*φ*. If there are

_{u}*M*different projection angles and each image has

_{p}*M*pixels, then

_{d}*b*has dimension

*M*=

*M*.

_{p}M_{d}*s*with the proposed algorithm is done in two consecutive steps as follows : where

_{k}*P*is the projection operator, modeling the imaging process leading to the measurement of a differential phase image, given an vector in object space. It is conceptually a matrix with dimension

*M*×

*N. B*is a general backprojection operator with dimension

*N*×

*M*, meaning that filtering is involved in the operator.

*𝒞*is a nonlinear function which projects the object to the prior constraint.

*λ*is a scalar factor whose value is chosen to guarantee convergence.

_{k}*P*and

*B*are of course not explicitly expressed as this is infeasible with useable data sizes because of the limited memory. Instead, projections onto pixels and backprojections onto voxels are directly calculated. In the proposed CL reconstruction from differential phase measurements, projection and backprojection operators are defined as follows :

- The operator
*P*takes the integral of the input along each ray, then differentiates the result along axis*u⃗*. - The operator
*B*performs a filtering of its input in detector space, whose action is to integrate along axis*u⃗*, compensate for the frequency response of backprojection, and damp high frequencies. The operator then backprojects values to the object space according to the geometry.

8. D. Lalush and B. Tsui, “Improving the convergence of iterative filtered backprojection algorithms,” Med. Phys. **21**(8), 1283–1286 (1994). [CrossRef] [PubMed]

*𝒞*in Eq. (20) constrains its input to the prior information of compact support and limited range of values. More specifically,

*𝒞*(

*s*) is the closest vector to

*s*satisfying the constraints. Let

*𝒮*be the set of voxel indices in [0,

*N*– 1], belonging to the prior support, and let

*δ*

_{min}and

*δ*

_{max}be the prior bounds for the values of function

*δ*.

*s*[

*i*] is the

*i*

^{th}element of vector

*s*.

### 3.3. Filtering in detector space

*F*, Eq. (21), would be by computing the fast Fourier transform (FFT) of the input, then multiplying by the filter, and finally take the inverse Fourier transform. This is a suboptimal way of proceeding because it introduces aliasing when the number of samples is low. The explicit expression of the filter kernel

*f*for

*F*in real space is a more precise approach : where 1/2

*a*is the Nyquist frequency. This can be simplified to the following function defined over the discrete values {

*na*,

*n*∈ ℕ}:

*f*, in direction

*u⃗*, produces the filtered image that is then backprojected onto the voxels in object space.

### 3.4. Steepest descent

*s*by moving along the direction

^{k}*h*with a step size

_{k}*λ*, whose value should be carefully computed to guarantee the convergence of the algorithm. Following the idea in [8

_{k}8. D. Lalush and B. Tsui, “Improving the convergence of iterative filtered backprojection algorithms,” Med. Phys. **21**(8), 1283–1286 (1994). [CrossRef] [PubMed]

*h*can be seen as the steepest descent direction of a quadratic functional of

_{k}*s*, so that the IFBP algorithms are minimizing this functional. By computing its integral with respect to

_{k}*s*,

_{k}*h*can be expressed from the gradient of a squared functional

_{k}*R*:

*λ*in Eq. (20) is so that

_{k}*R*(

*s*

_{k}_{+1}) is minimized in the direction

*h*, making the IFBP step a steepest descent step. It is computed by analyzing the partial derivative of the squared functional with respect to

_{k}*λ*:

_{k}*R*is minimum when the derivative is zero, so that the optimal step size

*λ*for steepest descent is obtained by :

_{k}*s*′

_{k}_{+1}onto the constraints with function

*𝒞*.

### 3.5. Implementation on GPU hardware

17. G. L. Zeng and G. T. Gullberg, “Unmatched projector/backprojector pairs in an iterative reconstruction algorithm,” IEEE Trans. Med. Imaging **19**, 548–555 (2000). [CrossRef] [PubMed]

18. R. Guedouar and B. Zarrad, “A comparative study between matched and mis-matched projection/back projection pairs used with ASIRT reconstruction method,” Nucl. Instrum. Methods Phys. Res. A **619**(1–3), 225–229 (2010). [CrossRef]

### 3.6. Cone beam geometry

19. 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. **90**, 224101 (2007). [CrossRef]

20. L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Am. A **1**, 612–619 (1984). [CrossRef]

## 4. Experiment

*λ*= 0.7A. Å 5.3 μm pitch gold grating was used for G1, with a nominal thickness of 1.9 μm, so that it produces a

*π*/2 phase shift at

*λ*. The absorption gold grating G2 had the same pitch, with a higher thickness of 30 μm. It was placed at a distance

*z*of 201 mm from G1. The X-ray detector was composed of a 10 μm P43 phosphor screen (Gd

_{T}_{2}O

_{2}S:Tb

^{+}powder) coupled to a cooled CCD camera (Hamamatsu Photonics C4742-98-24A), with an effective pixel size

*a*= 4.34μm. The number of pixels in acquired images was set to 1344 × 495.

*α*between the rotation axis and the beam normal plane was set to 20°. Smaller angles

*α*were found to cause strong phase wrapping in the differential phase images at some angles

*θ*. 20° was a good compromise between the quality of the measurements and the need for a small angle to reduce the region of missing frequencies. The object was rotated for 360°, with an angle step of 0.72°, so that projections along 500 angles were acquired. For each angle,

*S*= 5 steps of fringe scanning were performed, by successively acquiring a transmission image then translating the absorption grating by

*u⃗*is horizontal, from right to left. A comparison between the FBP reconstruction and the iterative method is presented in Fig. 5. The FBP algorithm, as described by Eq. (18), has the advantage of being a fast reconstruction method, requiring only one filtering step and one backprojection step. It gives an reliable first estimation of the object. However its implicit assumption that unsampled spatial frequencies are zero produces strong artifacts. Most notably, strong linear blur artifacts appears in directions with angle

*α*from the vertical axis, as can be seen in the vertical slice. Moreover, the unsampled mean value of the refractive index decrement

*δ*is also assumed to be zero, producing unrealistic negative values. The introduction of range and support constraints in the iterative algorithm helps reducing the drawbacks of FBP, at the price of a more computationally expensive method. The reduction of blur stripes can be observed, and values for

*δ*have been constrained in the range [0 10

^{−6}]. The presented result was obtained after a fixed number of 10 iterations. In Fig. 6, line profiles in horizontal and vertical directions passing through the mesh wire are presented. The improvement of the iterative reconstruction over the filtered backprojection is mainly seen in the vertical direction, where the low spatial frequencies are better estimated. From such profiles it is also possible to get a measure of the spatial resolution. Assuming that the mesh wire has a constant

*δ*, line profiles actually show the response of the imaging system to sharp edges. A simple measurement of the resolution is the 10% to 90% distance. In the case of the 215 μm wide structures of the presented mesh sample, it has been estimated to 30 μm in the horizontal direction and 40 μm in the vertical direction.

*δ*is finally presented in Fig. 8, showing details in the weave pattern.

## 5. Conclusion

*δ*through a better estimation of the unsampled frequencies. The proposed method has been demonstrated by the imaging of a polypropylene mesh sample at a synchrotron radiation facility. By an implementation of the algorithm on GPU hardware, it is now possible to obtain the reconstruction result in a few minutes. Absorption or phase laminographic imaging has already found applications in several fields, allowing for example the inspection of microsystem devices [14

14. 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]

21. K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, 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 Radiat. **15**, 55–61 (2008). [CrossRef]

5. F. Xu, L. Helfen, A. J. Moffat, G. Johnson, I. Sinclair, and T. Baumbach, “Synchrotron radiation computed laminography for polymer composite failure studies,” J. Synchrotron Radiat. **17**(2), 222–226 (2010). [CrossRef] [PubMed]

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

## Acknowledgments

## References and links

1. | F. Natterer, |

2. | D. G. Grant, “Tomosynthesis: a three-dimensional radiographic imaging technique,” IEEE Trans. Biomed. Eng. |

3. | H. Matsuo, A. Iwata, I. Horiba, and N. Suzumura, “Three-dimensional image reconstruction by digital tomosynthesis using inverse filtering,” IEEE Trans. Med. Imaging |

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

5. | F. Xu, L. Helfen, A. J. Moffat, G. Johnson, I. Sinclair, and T. Baumbach, “Synchrotron radiation computed laminography for polymer composite failure studies,” J. Synchrotron Radiat. |

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

7. | E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. |

8. | D. Lalush and B. Tsui, “Improving the convergence of iterative filtered backprojection algorithms,” Med. Phys. |

9. | J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. |

10. | A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. |

11. | J. Als-Nielsen and D. McMorrow, |

12. | G. W. Faris and R. L. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. |

13. | G. Lauritsch and W. H. Härer, “A theoretical framework for filtered backprojection in tomosynthesis,” Proc. SPIE |

14. | 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. |

15. | G. M. Stevens, R. Fahrig, and N. J. Pelc, “Filtered backprojection for modifying the impulse response of circular tomosynthesis,” Med. Phys. |

16. | O. Nakamura, S. Kawata, and S. Minami, “Optical microscopy tomography. II. Nonnegative constraint by a gradient-projection method,” J. Opt. Soc. Am. A |

17. | G. L. Zeng and G. T. Gullberg, “Unmatched projector/backprojector pairs in an iterative reconstruction algorithm,” IEEE Trans. Med. Imaging |

18. | R. Guedouar and B. Zarrad, “A comparative study between matched and mis-matched projection/back projection pairs used with ASIRT reconstruction method,” Nucl. Instrum. Methods Phys. Res. A |

19. | 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. |

20. | L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Am. A |

21. | K. Krug, L. Porra, P. Coan, A. Wallert, J. Dik, A. Coerdt, A. Bravin, M. Elyyan, P. Reischig, 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 Radiat. |

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

**OCIS Codes**

(110.6760) Imaging systems : Talbot and self-imaging effects

(110.6955) Imaging systems : Tomographic imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: June 16, 2011

Revised Manuscript: August 3, 2011

Manuscript Accepted: August 4, 2011

Published: August 12, 2011

**Virtual Issues**

Vol. 6, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Sébastien Harasse, Wataru Yashiro, and Atsushi Momose, "Iterative reconstruction in x-ray computed laminography from differential phase measurements," Opt. Express **19**, 16560-16573 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16560

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### References

- F. Natterer, The Mathematics of Computerized Tomography (Society for Industrial and Applied Mathematics, 2001). [CrossRef]
- D. G. Grant, “Tomosynthesis: a three-dimensional radiographic imaging technique,” IEEE Trans. Biomed. Eng. 19(1), 20–28 (1972). [CrossRef] [PubMed]
- H. Matsuo, A. Iwata, I. Horiba, and N. Suzumura, “Three-dimensional image reconstruction by digital tomosynthesis using inverse filtering,” IEEE Trans. Med. Imaging 12(2), 307–313 (1993). [CrossRef] [PubMed]
- L. Helfen, T. Baumbach, P. Cloetens, and J. Baruchel, “Phase-contrast and holographic computed laminography,” Appl. Phys. Lett. 94(10), 104103 (2009). [CrossRef]
- F. Xu, L. Helfen, A. J. Moffat, G. Johnson, I. Sinclair, and T. Baumbach, “Synchrotron radiation computed laminography for polymer composite failure studies,” J. Synchrotron Radiat. 17(2), 222–226 (2010). [CrossRef] [PubMed]
- A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B), 866–868 (2003). [CrossRef]
- E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777 (2008). [CrossRef] [PubMed]
- D. Lalush and B. Tsui, “Improving the convergence of iterative filtered backprojection algorithms,” Med. Phys. 21(8), 1283–1286 (1994). [CrossRef] [PubMed]
- J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974). [CrossRef] [PubMed]
- A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by X-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006). [CrossRef]
- J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics , 2nd ed. (John Wiley and Sons, 2011). [CrossRef]
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