## Pencil beam coded aperture x-ray scatter imaging |

Optics Express, Vol. 20, Issue 15, pp. 16310-16320 (2012)

http://dx.doi.org/10.1364/OE.20.016310

Acrobat PDF (6097 KB)

### Abstract

We use coded aperture x-ray scatter imaging to interrogate scattering targets with a pencil beam. Observations from a single x-ray exposure of a flat-panel scintillation detector are used to simultaneously determine the along-beam positions and momentum transfer profiles of two crystalline powders (NaCl and Al). The system operates with a 3 cm range resolution and a momentum transfer resolution of 0.1 nm^{−1}. These results demonstrate that a single snapshot can be used to estimate scattering properties along an x-ray beam, and serve as a foundation for volumetric imaging of scattering objects.

© 2012 OSA

## 1. Background

1. D. J. Brady, *Optical Imaging and Spectroscopy* (Wiley-OSA, 2009). [CrossRef]

2. S. R. Gottesman and E. E. Fenimore, “New family of binary arrays for coded aperture imaging,” Appl. Opt. **28**, 4344–4352 (1989). [CrossRef] [PubMed]

5. D. J. Brady, N. P. Pitsianis, and X. Sun, “Reference structure tomography,” J. Opt. Soc. Am. A **21**, 1140–1147, (2004). [CrossRef]

7. P. Potuluri, M. Xu, and D. Brady, “Imaging with random 3d reference structures,” Opt. Express **11**, 2134–2141, (2003). [CrossRef] [PubMed]

8. M. Gehm, R. John, D. Brady, R. Willett, and T. Schulz, “Single-shot compressive spectral imaging with a dual-disperser architecture,” Opt. Express **15**, 14013–14027, (2007). [CrossRef] [PubMed]

9. A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. **47**, B44–B51 (2008). [CrossRef] [PubMed]

11. D. L. Batchelar and I. A. Cunningham, “Material-specific analysis using coherent-scatter imaging,” Med. Phys. **29**, 1651–1660, (2002). [CrossRef] [PubMed]

13. M. T. M. Davidson, D. L. Batchelar, S. Velupillai, J. D. Denstedt, and I. A. Cunningham, “Laboratory coherent-scatter analysis of intact urinary stones with crystalline composition: a tomographic approach,” Phys. Med. Biol. **50**, 3907 (2005). [CrossRef] [PubMed]

14. R. J. Cernik, K. H. Khor, and C. Hansson, “X-ray colour imaging,” Journal of the Royal Society Interface **5**, 477–481 (2008). [CrossRef]

15. G. Harding and B. Schreiber, “Coherent x-ray scatter imaging and its applications in biomedical science and industry,” Radiat. Phys. Chem. **56**, 229–245, (1999). [CrossRef]

18. C. Crespy, P. Duvauchelle, V. Kaftandjian, F. Soulez, and P. Ponard, “Energy dispersive x-ray diffraction to identify explosive substances: Spectra analysis procedure optimization,” Nucl. Instrum. Methods Phys. Res. A **623**, 1050 – 1060, (2010). [CrossRef]

16. G. Harding, “X-ray scatter tomography for explosives detection,” Radiat. Phys. Chem. **71**, 869–881 (2004). [CrossRef]

19. G. Harding and J. Kosanetzky, “Elastic scatter computed tomography,” Phys. Med. Biol. **30**, 183–186, (1985). [CrossRef] [PubMed]

11. D. L. Batchelar and I. A. Cunningham, “Material-specific analysis using coherent-scatter imaging,” Med. Phys. **29**, 1651–1660, (2002). [CrossRef] [PubMed]

13. M. T. M. Davidson, D. L. Batchelar, S. Velupillai, J. D. Denstedt, and I. A. Cunningham, “Laboratory coherent-scatter analysis of intact urinary stones with crystalline composition: a tomographic approach,” Phys. Med. Biol. **50**, 3907 (2005). [CrossRef] [PubMed]

20. J. Delfs and J.-P. Schlomka, “Energy-dispersive coherent scatter computed tomography,” Appl. Phys. Lett. **88**, 243506 (2006). [CrossRef]

21. G. Harding, M. Newton, and J. Kosanetzky, “Energy-dispersive x-ray diffraction tomography,” Phys. Med. Biol. **35**, 33 (1990). [CrossRef]

21. G. Harding, M. Newton, and J. Kosanetzky, “Energy-dispersive x-ray diffraction tomography,” Phys. Med. Biol. **35**, 33 (1990). [CrossRef]

22. C. Hall, P. Barnes, J. Cockcroft, S. Colston, D. Husermann, S. Jacques, A. Jupe, and M. Kunz, “Synchrotron energy-dispersive x-ray diffraction tomography,” Nucl. Instrum. Methods Phys. Res. B **140**, 253 – 257 (1998). [CrossRef]

14. R. J. Cernik, K. H. Khor, and C. Hansson, “X-ray colour imaging,” Journal of the Royal Society Interface **5**, 477–481 (2008). [CrossRef]

23. O. Lazzari, S. Jacques, T. Sochi, and P. Barnes, “Reconstructive color x-ray diffraction imaging - a novel TEDDI imaging method,” Analyst **134**, 1802–1807, (2009). [CrossRef] [PubMed]

## 2. Pencil beam system model

*z*axis. The scattering object is placed between the primary and secondary apertures so that it is penetrated by the beam. The primary beam is stopped by the secondary mask to prevent it from flooding the detector image. Scattered x-rays diverge from the main beam to strike the aperture, where they are either absorbed or transmitted to the detector plane. Each pixel in the detector array receives scattered power from multiple points along the beam, and the structure of this multiplexing is controlled by the aperture code.

**q**=

**k**′ −

**k**, where

**k**is the incident wavevector and

**k**′ is the scattered wavevector. We consider elastic scattering, for which |

**k**| = |

**k**′|. This condition holds for Rayleigh scattering (including coherent scattering) and approximates Compton scattering at low energies or at low scattering angles. Elastic scattering follows Bragg’s law

*q*= 2

*ν*sin(

*θ*/2), where

*q*= |

**q**|,

*ν*is the x-ray frequency, and

*θ*is the scattering angle as shown in Fig. 1.

*f*(

*z*,

*q*) be the ratio of scattered power to incident power at position

*z*along the beam and with momentum transfer

*q*. This model only depends on the magnitude of the momentum transfer and not its direction, so it applies to liquids, fine powders (as in this experiment), and amorphous compounds. In the absence of a coded aperture, scattering at angle

*θ*from position

*z*produces an irradiance at radius

*ρ*on the detector proportional to 1/(

*z*

^{2}+

*ρ*

^{2}). The coded aperture is modeled by the transmission function

*t*(

*ρ*,

*ϕ*) ∈ [0, 1] in the plane

*z*=

*d*, where (

*ρ*,

*ϕ*) are the polar radius and angle relative to the beam. The total irradiance at the detector point (

*ρ*,

*ϕ*) is where we assume sin(

*θ*/2) ≈

*ρ*/(2

*z*).

*P*(

*ν*) is the power spectral density of the beam, assumed independent of

*z*.

2. S. R. Gottesman and E. E. Fenimore, “New family of binary arrays for coded aperture imaging,” Appl. Opt. **28**, 4344–4352 (1989). [CrossRef] [PubMed]

*t*(

*ρ*,

*ϕ*) as a function of object position

*z*. The projected image of the aperture code is magnified by

*z*/(

*z*−

*d*). One can disambiguate images corresponding to different values of

*z*by applying aperture codes which are orthogonal under changes in scale (i. e. magnification). Harmonic codes (e.g.

*t*(

*ρ*) = cos(

*ρ*)) have this property. To also disambiguate

*q*the code must also vary as a function of

*ϕ*. As a simple easily manufactured example, we choose the square grid where

*u*is the spatial frequency and

*x*is a cartesian coordinate in the

*ρ*,

*ϕ*plane.

*z*and

*q*. For this purpose we use the function rect(

*x*) which is equal to unity for |

*x*| < 1/2 and zero everywhere else. The voxels are chosen with sampling rates Δ

*z*in

*z*and Δ

*q*in

*q*and have centers (

*z*,

_{j}*q*). The discrete model for the scattering density is where

_{j}*f*is a set of coefficients characterizing the object. The detector is partitioned into polar sections indexed by

_{j}*i*. The power measured in the section at polar coordinates (

*ρ*,

_{i}*ϕ*) and width (Δ

_{i}*ρ*, Δ

*ϕ*) in these coordinates is taking care to consider the periodicity of

*ϕ*. When Eqs. (3) and (4) are used in Eq. (1), the discrete model is expressed as the linear system where

**g**and

**f**are a vectors with components

*g*and

_{i}*f*, and

_{j}**H**is the “forward matrix”. Noting that ∫ d

*x*rect(

*x*)

**H**has components The discrete forward model (Eq. (5)) can be used with numerical methods to estimate the object vector

**f**, given measurements

**g**and functional models for

*P*(

*ν*) and

*t*(

*ρ*,

*ϕ*). We have computed the forward matrix

**H**for our experimental system and used an algorithm derived from Ref. [24

24. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. **62**, 55–59, (1972). [CrossRef]

**f**for each object configuration (the procedure is described in detail in Sec. 4). The reconstructed objects are presented in Sec. 5, but first we describe the pencil beam CAXSI experiment in the following section.

## 3. Experimental methods

25. A. Chawla and E. Samei, “Geometrical repeatability and motion blur analysis of a new multi-projection x-ray imaging system,” IEEE Nuclear Science Symposium Conference Record **5**, 3170 –3173, (2006). [CrossRef]

26. A. Chawla, S. Boyce, L. Washington, H. McAdams, and E. Samei, “Design and development of a new multi-projection x-ray system for chest imaging,” IEEE Trans. Nucl. Sci. **56**, 36–45, (2009). [CrossRef]

*K*transition doublet at 58.0–59.3 keV and from the

_{α}*K*transitions at 66.7–67.7 keV. The x-ray beam had an inherent filtration equivalent to 1.1 mm thick aluminum at 80 kVp. The normal acquisition mode was set at 116 kVp, 500 mAs [27

_{β}27. J. M. Boone and J. A. Seibert, “An accurate method for computer-generating tungsten anode x-ray spectra from 30 to 140 kv,” Med. Phys. **24**, 1661–1670, (1997). [CrossRef] [PubMed]

28. S. R. Beath and I. A. Cunningham, “Pseudomonoenergetic x-ray diffraction measurements using balanced filters for coherent-scatter computed tomography,” Med. Phys. **36**, 1839–1847, (2009). [CrossRef] [PubMed]

*K*-edge at 69.5 keV. The expected source spectrum was modeled using the semi-empirical x-ray spectrum modeling program XSPECT [29]. XSPECT produced a model for the mean spectral number density

*N*(

*ν*) of photons illuminating the object. This model is plotted with a normalized maximum value in Fig. 2, and was used to calculate the power spectral density

*P*(

*ν*).

*μ*m and a matrix size of 2048 × 1536. The source-to-image distance was 201 cm. The acquisition setup was finely calibrated to maximize the signal-to-noise ratio of the acquired images. Specifically, the detector was gain-calibrated at the expected photon flux. Furthermore, it was offset calibrated before each acquisition with 16 dark frames to correct for structured noise. Post-calibrated images were acquired using the image acquisition and processing software ViVA (Varian Medical Systems).

*u*= 9.9 cm

^{−1}so that the period was 0.64 cm.

## 4. Object reconstruction

**g**. Our images also contained a noisy background image with mean

*μ**so that the expected value at each pixel is given by*

_{b}**g**=

**Hf**+

*μ**. Treating the x-ray detection as a statistical process, our actual measurements*

_{b}**y**are approximated by the Poisson process where Poisson(

**v**) is a vector of independent Poisson observations with mean values given by the components of

**v**. Given

**y**,

**H**, and a noisy measurement of the background

**b**∼ Poisson(

*μ**), we are interested in estimating*

_{b}**f**as accurately as possible.

*μ**from*

_{b}**b**using a Poisson image denoising algorithm, and use the resulting estimate

**̂**

*μ**of*

_{b}

*μ**to reconstruct*

_{b}**f**. In particular, we estimate

*μ**using a maximum penalized likelihood estimation method discussed in [30*

_{b}30. E. Kolaczyk and R. Nowak, “Multiscale likelihood analysis and complexity penalized estimation,” The Annals of Statistics **32**, 500–527, (2004). [CrossRef]

**b**|

**g**) is given by where

*i*indexes the detector pixels. Γ is a collection of possible estimates to search from, pen(

**g**) is the penalization or the regularization function corresponding to estimate

**g**, and

*τ*is the term that balances the log-likelihood term and the penalization term. The class of estimates Γ is obtained by partitioning the image space in a recursive-dyadic (powers of two) fashion from coarser to finer cells, and fitting a constant to each partition cell. The algorithm chooses the multiscale, partition-based estimate that is the best fit to the data and also is piecewise smooth. The penalization term is proportional to the number of cells in the partition and is used to enforce our prior knowledge that

*μ**is piecewise smooth. Given*

_{b}**̂**

*μ**, we can estimate*

_{b}**f**according to a generalized maximum likelihood (GML) estimator given by The GML estimate of

**f**can be obtained using an iterative deconvolution method such as that described in Ref. [24

24. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. **62**, 55–59, (1972). [CrossRef]

*ρ*,

*ϕ*) relative to the beam position. In practice, few polar bins, relative to the size of the detector, are sufficient to reliably capture the information content in the diffraction images. In our experiments, the images were partitioned into 233 uniform radius bins between

*ρ*= 2.5 cm and

*ρ*= 11.5 cm. The polar angle was similarly segmented over its entire range into 120 bins. As a result of this strategy, we were able to reduce sampling from 2048 × 1536 to 233 × 120, which afforded significant savings in the computation of

**H**.

**f**for each test object, and we compare these reconstructions to reference data in the next section.

## 5. Results and discussion

**H**for the pencil beam system was calculated by sampling the object space with voxels of width 0.33 cm in

*z*and 0.027 nm

^{−1}in

*q*. Here and in the following we replace numerical values of

*q*with

*q*/4

*π*to agree with recent literature. From the single-frame diffraction images shown in Figs. 4(a) and 4(b), each test object’s coefficient vector

**f**representing the scattering density

*f*(

*z,q*) was estimated using the methods described in the last section. Figures 5(a) and 5(b) show the estimated spatial distribution

*f*(

*z*) = ∫ d

*q f*(

*z,q*) of the scattering density for NaCl and Al placed separately in the beam at

*z*= −60.2 cm. Although the beam penetrated only 1 cm of each sample, the spatial reconstructions both have FWHM equal to 3 cm. The peak of the spatial profile occurs at

*z*

_{0}= −59.3 cm for both samples, corresponding to a position error of 1.5%. This demonstrates the along-beam ranging capability of the coded aperture system.

*f*(

*z*

_{0},

*q*) at the spatial peak

*z*

_{0}= −59.3 cm for NaCl and Al, respectively. The dominant peaks in both profiles were accurately reconstructed, and there is evidence of some of the smaller peaks. The dominant peak for NaCl was reconstructed at

*q*= 1.767 nm

^{−1}(0.2% error) with a FWHM of 0.06 nm

^{−1}(3.6%). The dominant peak for Al was reconstructed at

*q*= 2.149 nm

^{−1}(0.4% error) and a FWHM of 0.07 nm

^{−1}(3.3%). These results show that the pencil beam coded aperture system can be used to estimate the scattering structure of target samples without

*a priori*position information.

*z*= −60.2 cm and

*z*= −52.6 cm, respectively. In this configuration the pencil beam passed first through the NaCl sample and then through the Al sample, producing the diffraction image shown in Fig. 4(c). As before, the coefficients

**f**were reconstructed and produced the spatial scattering profile

*f*(

*z*) shown in Fig. 6(a). The spatial distribution shows a peak for NaCl at

*z*= −59.3 cm (1.5% error) and another for Al at

*z*= −52.0 cm (1.1% error). Momentum transfer profiles for these two locations are shown in Figs. 6(b) and 6(c). The dominant peak for NaCl was reconstructed at

*q*= 1.79 nm

^{−1}with FWHM equal to 4.7%. The dominant peak for Al was estimated to lie at

*q*= 2.137 nm

^{−1}(0.3% error) with FWHM equal to 4.7%. The reconstructed momentum transfer profiles are consistent, whether the objects are measured separately or placed together in the beam.

*ρ*,

*ϕ*). Figure 7(b) shows the modeled diffraction pattern

**Hf**̂ based on the corresponding object estimate

**f**̂. The RMS error between these diffraction patterns is approximately 10% of the peak signal value in

**Hf**̂, indicating agreement between the two images and providing combined validity to the measurement model, the experimental process, and the reconstruction algorithm. We suspect that shot noise and dark current noise are the primary contributors to the differences between the two images.

## Acknowledgments

## References and links

1. | D. J. Brady, |

2. | S. R. Gottesman and E. E. Fenimore, “New family of binary arrays for coded aperture imaging,” Appl. Opt. |

3. | M. Harwit and N. J. A. Sloane, |

4. | A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: Mask enhanced cameras for heterodyned light fields and coded aperture refocusing,” ACM Transactions on Graphics |

5. | D. J. Brady, N. P. Pitsianis, and X. Sun, “Reference structure tomography,” J. Opt. Soc. Am. A |

6. | P. Potuluri, U. Gopinathan, J. Adleman, and D. Brady, “Lensless sensor system using a reference structure,” Opt. Express |

7. | P. Potuluri, M. Xu, and D. Brady, “Imaging with random 3d reference structures,” Opt. Express |

8. | M. Gehm, R. John, D. Brady, R. Willett, and T. Schulz, “Single-shot compressive spectral imaging with a dual-disperser architecture,” Opt. Express |

9. | A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. |

10. | K. Choi and D. J. Brady, “Coded aperture computed tomography,” in “Adaptive Coded Aperture Imaging, Non-Imaging, and Unconventional Imaging Sensor Systems ,” SPIE |

11. | D. L. Batchelar and I. A. Cunningham, “Material-specific analysis using coherent-scatter imaging,” Med. Phys. |

12. | J.-P. Schlomka, A. Harding, U. van Stevendaal, M. Grass, and G. L. Harding, “Coherent scatter computed tomography: a novel medical imaging technique,” SPIE |

13. | M. T. M. Davidson, D. L. Batchelar, S. Velupillai, J. D. Denstedt, and I. A. Cunningham, “Laboratory coherent-scatter analysis of intact urinary stones with crystalline composition: a tomographic approach,” Phys. Med. Biol. |

14. | R. J. Cernik, K. H. Khor, and C. Hansson, “X-ray colour imaging,” Journal of the Royal Society Interface |

15. | G. Harding and B. Schreiber, “Coherent x-ray scatter imaging and its applications in biomedical science and industry,” Radiat. Phys. Chem. |

16. | G. Harding, “X-ray scatter tomography for explosives detection,” Radiat. Phys. Chem. |

17. | R. W. Madden, J. Mahdavieh, R. C. Smith, and R. Subramanian, “An explosives detection system for airline security using coherent x-ray scattering technology,” SPIE |

18. | C. Crespy, P. Duvauchelle, V. Kaftandjian, F. Soulez, and P. Ponard, “Energy dispersive x-ray diffraction to identify explosive substances: Spectra analysis procedure optimization,” Nucl. Instrum. Methods Phys. Res. A |

19. | G. Harding and J. Kosanetzky, “Elastic scatter computed tomography,” Phys. Med. Biol. |

20. | J. Delfs and J.-P. Schlomka, “Energy-dispersive coherent scatter computed tomography,” Appl. Phys. Lett. |

21. | G. Harding, M. Newton, and J. Kosanetzky, “Energy-dispersive x-ray diffraction tomography,” Phys. Med. Biol. |

22. | C. Hall, P. Barnes, J. Cockcroft, S. Colston, D. Husermann, S. Jacques, A. Jupe, and M. Kunz, “Synchrotron energy-dispersive x-ray diffraction tomography,” Nucl. Instrum. Methods Phys. Res. B |

23. | O. Lazzari, S. Jacques, T. Sochi, and P. Barnes, “Reconstructive color x-ray diffraction imaging - a novel TEDDI imaging method,” Analyst |

24. | W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. |

25. | A. Chawla and E. Samei, “Geometrical repeatability and motion blur analysis of a new multi-projection x-ray imaging system,” IEEE Nuclear Science Symposium Conference Record |

26. | A. Chawla, S. Boyce, L. Washington, H. McAdams, and E. Samei, “Design and development of a new multi-projection x-ray system for chest imaging,” IEEE Trans. Nucl. Sci. |

27. | J. M. Boone and J. A. Seibert, “An accurate method for computer-generating tungsten anode x-ray spectra from 30 to 140 kv,” Med. Phys. |

28. | S. R. Beath and I. A. Cunningham, “Pseudomonoenergetic x-ray diffraction measurements using balanced filters for coherent-scatter computed tomography,” Med. Phys. |

29. | C. Dodge and M. Flynn, “Advanced integral method for the simulation of diagnostic x-ray spectra,” Med. Phys. |

30. | E. Kolaczyk and R. Nowak, “Multiscale likelihood analysis and complexity penalized estimation,” The Annals of Statistics |

**OCIS Codes**

(110.7440) Imaging systems : X-ray imaging

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

(110.1758) Imaging systems : Computational imaging

(110.3200) Imaging systems : Inverse scattering

**ToC Category:**

X-ray Optics

**History**

Original Manuscript: April 9, 2012

Revised Manuscript: June 12, 2012

Manuscript Accepted: June 13, 2012

Published: July 3, 2012

**Citation**

Kenneth MacCabe, Kalyani Krishnamurthy, Amarpreet Chawla, Daniel Marks, Ehsan Samei, and David Brady, "Pencil beam coded aperture x-ray scatter imaging," Opt. Express **20**, 16310-16320 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16310

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### References

- D. J. Brady, Optical Imaging and Spectroscopy (Wiley-OSA, 2009). [CrossRef]
- S. R. Gottesman and E. E. Fenimore, “New family of binary arrays for coded aperture imaging,” Appl. Opt.28, 4344–4352 (1989). [CrossRef] [PubMed]
- M. Harwit and N. J. A. Sloane, Hadamard Transform Optics (Academic Press, 1979).
- A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: Mask enhanced cameras for heterodyned light fields and coded aperture refocusing,” ACM Transactions on Graphics26, 69-1–69-12 (2007).
- D. J. Brady, N. P. Pitsianis, and X. Sun, “Reference structure tomography,” J. Opt. Soc. Am. A21, 1140–1147, (2004). [CrossRef]
- P. Potuluri, U. Gopinathan, J. Adleman, and D. Brady, “Lensless sensor system using a reference structure,” Opt. Express11, 965–974 (2003). [CrossRef] [PubMed]
- P. Potuluri, M. Xu, and D. Brady, “Imaging with random 3d reference structures,” Opt. Express11, 2134–2141, (2003). [CrossRef] [PubMed]
- M. Gehm, R. John, D. Brady, R. Willett, and T. Schulz, “Single-shot compressive spectral imaging with a dual-disperser architecture,” Opt. Express15, 14013–14027, (2007). [CrossRef] [PubMed]
- A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt.47, B44–B51 (2008). [CrossRef] [PubMed]
- K. Choi and D. J. Brady, “Coded aperture computed tomography,” in “Adaptive Coded Aperture Imaging, Non-Imaging, and Unconventional Imaging Sensor Systems,” SPIE 7468, 74680B-1–74680B-10, (2009).
- D. L. Batchelar and I. A. Cunningham, “Material-specific analysis using coherent-scatter imaging,” Med. Phys.29, 1651–1660, (2002). [CrossRef] [PubMed]
- J.-P. Schlomka, A. Harding, U. van Stevendaal, M. Grass, and G. L. Harding, “Coherent scatter computed tomography: a novel medical imaging technique,” SPIE5030, 256–265, (2003). [CrossRef]
- M. T. M. Davidson, D. L. Batchelar, S. Velupillai, J. D. Denstedt, and I. A. Cunningham, “Laboratory coherent-scatter analysis of intact urinary stones with crystalline composition: a tomographic approach,” Phys. Med. Biol.50, 3907 (2005). [CrossRef] [PubMed]
- R. J. Cernik, K. H. Khor, and C. Hansson, “X-ray colour imaging,” Journal of the Royal Society Interface5, 477–481 (2008). [CrossRef]
- G. Harding and B. Schreiber, “Coherent x-ray scatter imaging and its applications in biomedical science and industry,” Radiat. Phys. Chem.56, 229–245, (1999). [CrossRef]
- G. Harding, “X-ray scatter tomography for explosives detection,” Radiat. Phys. Chem.71, 869–881 (2004). [CrossRef]
- R. W. Madden, J. Mahdavieh, R. C. Smith, and R. Subramanian, “An explosives detection system for airline security using coherent x-ray scattering technology,” SPIE7079, 707915-1–707915-11, (2008).
- C. Crespy, P. Duvauchelle, V. Kaftandjian, F. Soulez, and P. Ponard, “Energy dispersive x-ray diffraction to identify explosive substances: Spectra analysis procedure optimization,” Nucl. Instrum. Methods Phys. Res. A623, 1050 – 1060, (2010). [CrossRef]
- G. Harding and J. Kosanetzky, “Elastic scatter computed tomography,” Phys. Med. Biol.30, 183–186, (1985). [CrossRef] [PubMed]
- J. Delfs and J.-P. Schlomka, “Energy-dispersive coherent scatter computed tomography,” Appl. Phys. Lett.88, 243506 (2006). [CrossRef]
- G. Harding, M. Newton, and J. Kosanetzky, “Energy-dispersive x-ray diffraction tomography,” Phys. Med. Biol.35, 33 (1990). [CrossRef]
- C. Hall, P. Barnes, J. Cockcroft, S. Colston, D. Husermann, S. Jacques, A. Jupe, and M. Kunz, “Synchrotron energy-dispersive x-ray diffraction tomography,” Nucl. Instrum. Methods Phys. Res. B140, 253 – 257 (1998). [CrossRef]
- O. Lazzari, S. Jacques, T. Sochi, and P. Barnes, “Reconstructive color x-ray diffraction imaging - a novel TEDDI imaging method,” Analyst134, 1802–1807, (2009). [CrossRef] [PubMed]
- W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am.62, 55–59, (1972). [CrossRef]
- A. Chawla and E. Samei, “Geometrical repeatability and motion blur analysis of a new multi-projection x-ray imaging system,” IEEE Nuclear Science Symposium Conference Record5, 3170 –3173, (2006). [CrossRef]
- A. Chawla, S. Boyce, L. Washington, H. McAdams, and E. Samei, “Design and development of a new multi-projection x-ray system for chest imaging,” IEEE Trans. Nucl. Sci.56, 36–45, (2009). [CrossRef]
- J. M. Boone and J. A. Seibert, “An accurate method for computer-generating tungsten anode x-ray spectra from 30 to 140 kv,” Med. Phys.24, 1661–1670, (1997). [CrossRef] [PubMed]
- S. R. Beath and I. A. Cunningham, “Pseudomonoenergetic x-ray diffraction measurements using balanced filters for coherent-scatter computed tomography,” Med. Phys.36, 1839–1847, (2009). [CrossRef] [PubMed]
- C. Dodge and M. Flynn, “Advanced integral method for the simulation of diagnostic x-ray spectra,” Med. Phys.33, 1983 (2006).
- E. Kolaczyk and R. Nowak, “Multiscale likelihood analysis and complexity penalized estimation,” The Annals of Statistics32, 500–527, (2004). [CrossRef]

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