## An approach to increasing the resolution of industrial CT images based on an aperture collimator |

Optics Express, Vol. 21, Issue 23, pp. 27946-27963 (2013)

http://dx.doi.org/10.1364/OE.21.027946

Acrobat PDF (1534 KB)

### Abstract

The spatial resolution of CT images is dominated by the focal spot size when it is large relative to the detector cells. We propose an approach to increase the spatial resolution by utilizing an aperture collimator. The aperture collimator is specially designed and placed in front of the X-ray source so that the rays penetrating the collimator form a set of narrow fan beams. Then an iterative algorithm is introduced to reconstruct CT images from the data obtained by scanning the narrow fan beams. Numerical experiments show that the proposed approach could significantly increase the resolution of the CT images. Furthermore, this approach is also robust against some challenging cases, such as the examination of low contrast object, reconstruction based on multi-energy data and perturbation of geometric errors in CT systems.

© 2013 OSA

## 1. Introduction

1. H. Hu, “Multi-slice helical CT: scan and reconstruction,” Medical Physics **26**, 5–18 (1999). [CrossRef] [PubMed]

6. J. Hsieh, M.F. Gard, and S. Gravelle, “Reconstruction technique for focal spot wobbling,”Proc. of SPIE Medical Imaging VI **1652**, 175–182 (1992). [CrossRef]

7. E. Caroli, J. B. Stephen, G. Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. **45**, 349–403 (1987). [CrossRef]

10. G. Skinner, “Imaging with coded-aperture masks,” Nucl. Instr. Meth. Phys. Res. **221**, 33–40 (1984). [CrossRef]

12. G. Knoll, W. Rogers, K. Koral, J. Stamos, and N. Clinthorne, “Application of coded apertures in tomographic head scanning,” Nucl. Instr. Meth. Phys. Res. **221**, 226–232 (1984). [CrossRef]

13. W. E. Smith, R. G. Paxman, and H. H. Barrett, “Image reconstruction from coded data: I. reconstruction algorithms and experimental results.” J. Opt. Soc. Am. A **2**, 491–500 (1985). [CrossRef] [PubMed]

20. E. E. Fenimore and T. M. Cannon, “Uniformly redundant arrays: digital reconstruction methods.” Appl. Opt. **20**, 1858–1864 (1981). [CrossRef] [PubMed]

21. A. Olivo and R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. **91**, 074106 (2007). [CrossRef]

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

24. J. Fleming and B. Goddard, “An evaluation of techniques for stationary coded aperture three-dimensional imaging in nuclear medicine,” Nucl. Instr. Meth. Phys. Res. **221**, 242–246 (1984). [CrossRef]

## 2. Imaging model with the aperture collimator

*f*(

*x*) is the linear attenuation coefficient distribution of the scanned object at point

*x*,

*β*denotes the rotation angle,

*R*(

*β*) is the relative rotation matrix,

*u*is the detector coordinate,

*s*∈ [

*s*

_{0},

*s*

_{1}] is the position on the bar-shaped spot, and

*L*(

*s*,

*u*) represents the X-rays emitted from the source position

*s*to the detector coordinate

*u*.

*I*

_{0}(

*s*) denotes the photon intensity distribution with energy

*E*

_{0}emitted from the position

*s*, and

*I*(

*β*,

*u*) denotes the photos remaining after penetrating the scanned object (the photons that were not eliminated due to attenuation while propagating from

*s*to

*u*).

*λ*is the linear attenuation coefficient of the collimator material, and

*g*(

*y*) is the characteristic function of the aperture collimator distribution. For convenience, all these symbols mentioned above are listed in Table 1.

*Î*

_{0,u,s}=

*I*

_{0,u,s}exp(−

*λg*), which represents the intensity distribution of the X-rays from each discrete focal spot after penetrating the aperture collimator, then Eq. (2) can be rewritten as

_{u,s}*Î*

_{0,u,s}of the flux after penetrating the collimator from each segmented focal spot; (2) reconstruct the CT images under the assumption that

*Î*

_{0,u,s}are known.

*Î*

_{0,u,s}is independent of the scanned object, but difficult to measure directly. The estimation of

*Î*

_{0,u,s}is somewhat similar to the coded aperture imaging in X-ray astronomy, e.g.,

*Î*

_{0,u,s}may be estimated by utilizing of specially designed masks or phantoms. For the first subproblem, we just estimate

*Î*

_{0,u,s}by forward-projection simulation in an ideal condition. Hence, the aim of this paper is to solve the second subproblem.

**Data completeness**: the combination of narrow fan beams passing through the collimator should be able to cover the whole object;**High signal-to-noise ratio (SNR)**: the apertures should be made and distributed such that there are as many X-rays as possible which penetrate the collimator, since there is a positive correlation between the noise level and the dose;**High spatial resolution**: the aperture collimator should be placed in a suitable position to modulate the fan beams such that each detector unit receives photons from only a tiny region of the bar-shaped focal spot.

## 3. The reconstruction algorithm

*f*from the measured data

_{j}*I*under the assumption that

_{n}*Î*

_{0,u,s}are known.

*F*be the image vector;

*R*, the projection matrix; and

*P*, the projection data vector, where

*R*indicates the projection matrix of

_{s}*s*-th virtual focal spot, which is the sub matrix of

*R*. Then we obtain the linear system

*p*is known for any

_{n,s}*n*,

*s*, then we can solve (4) using simultaneous algebraic reconstruction technique (SART) as follows, where

*k*-th iteration and

*r*is the

_{n,s}*n*-th row of matrix

*R*. However,

_{s}*p*cannot be measured directly, which means that we cannot calculate

_{n,s}*F*. In the following, we will propose a method to estimate

*u*for a given

*s*, such that

*Î*

_{0,u,s}is nonzero. Denoting

*k*-th iteration, then we can calculate the sum of the residual errors of the

*n*-th data

*Î*

_{0,}

*and*

_{u}*ω*is independent of the object, we have It is easy to see that Now substituting

_{u,s}*p*can be linearized by Taylor expansions as follows,

_{n}*ω*. This means that the proposed algorithm is a reasonable linear method to solve the high resolution CT imaging problem based on the collimator and virtual spots.

_{u,s}## 4. Numerical experiments

*I*

_{0,}

*= 10*

_{n}^{6}, and we set the energy of X-ray to 300 keV. The focal spot is divided into 21 virtual focal spots and the flux intensity emitted from each virtual focal spot obeys a Gaussian distribution, where

*s*denotes the distance from the focal spots to the center of the whole focus.

^{3}was selected as the material for the aperture collimator. According to data from National Institute of Standards and Technology (NIST) [25

25. J. H. Hubbell and S. M. Seltzer, “Tables of x-ray mass attenuation coefficients and mass energy-absorption coefficients from 1 kev to 20 mev for elements z = 1 to 92 and 48 additional substances of dosimetric interest,” http://www.nist.gov/pml/data/xraycoef/.

^{2}/g for 300 keV. The parameters of the aperture collimator are listed in Table 3, as mentioned in Section 2. All of the raw simulation data are polluted with Poisson noise.

### 4.1. Spatial-resolution test experiments

26. A. C. Kak and M. Slaney, *Principles of Tomographic Imaging* (IEEE Engineering in Medicine and Biology Society, 2001). [CrossRef]

### 4.2. Density-contrast test experiments

27. G. Lauritsch and H. Bruder, “Head phantom,” http://www.imp.uni-erlangen.de/phantoms/head (2012).

### 4.3. Experiments with multi-energy scanned data

### 4.4. Experiments on the perturbation of the geometrical settings

*Î*

_{0,u,s}, which may be estimated by means of masks or phantoms with special constructions, are very important in the AC-SART. We carry out some experiments to test the effect of the estimation error on

*Î*

_{0,u,s}.

*Î*

_{0,u,s}is calculated by

*g*, which indicates the intersection-length between the rays from

_{n,s}*s*to

*u*and the collimator. We add random noise to the intersection-length as follows where

*g*

_{max}denotes the maximum intersection-length, and

*t*obeys a Gaussian distribution. Figure 12 shows the simulation results for the 11-th virtual focus, to compare the intersection-length under ideal and noisy conditions (

*σ*= 0.1 and

*σ*= 0.2).

*I*

_{0,u,s}and then generate the raw data by adding noise as described above. Figure 13 shows the reconstructed images. When

*σ*= 0.1,

*t*will be distributed within [−0.3, 0.3]. The reconstructed image [Fig. 13(a)] still has high quality and high spatial resolution. When

*σ*= 0.2, the range of

*t*is from −0.6 to 0.6, and the estimation error grows rapidly which leads to large residual errors in the back-projection process. Therefore, the reconstructed CT image suffers from ring artifacts, as shown in Fig. 13(b).

## 5. Conclusion

*Î*

_{0,u,s}of the flux after penetrating the collimator from each segmented focal spot; (2) reconstruct the CT images under the assumption that

*Î*

_{0,u,s}are known. While the first subproblem is solved approximately by forward-projection simulation, an algorithm named AC-SART has been proposed to solve the second subproblem robustly. The numerical experiments show that the algorithm is able to reconstruct CT images of higher resolution than conventional algorithms. Meanwhile, it could maintain a favorable low contrast and is robust for multi-energy data as well as geometrical perturbations of CT systems.

## Acknowledgments

## References and links

1. | H. Hu, “Multi-slice helical CT: scan and reconstruction,” Medical Physics |

2. | J. Hsieh, |

3. | H. Zhang, J. Tian, M. Chen, and P. Zhang, “A novel scanning mode and image reconstruction method on super-resolution CT,” Chin. J. Stereol. Image Analysis |

4. | T.M. Peters and R.M. Lewitt, “Computed tomography with fan-beam geometry,” J. Comput. Assist. Tomogr. |

5. | A. H. Lonn, “Computed tomography system with translatable focal spot,” US Patent |

6. | J. Hsieh, M.F. Gard, and S. Gravelle, “Reconstruction technique for focal spot wobbling,”Proc. of SPIE Medical Imaging VI |

7. | E. Caroli, J. B. Stephen, G. Cocco, L. Natalucci, and A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. |

8. | T. Palmieri, “Multiplex methods and advantages in x-ray astronomy,” Astrophys. Space Sci. |

9. | P. Durouchoux, H. Hudson, J. Matteson, G. Hurford, K. Hurley, and E. Orsal, “Gamma-ray imaging with a rotating modulator,” Astron. Astrophys. |

10. | G. Skinner, “Imaging with coded-aperture masks,” Nucl. Instr. Meth. Phys. Res. |

11. | S. Webb, |

12. | G. Knoll, W. Rogers, K. Koral, J. Stamos, and N. Clinthorne, “Application of coded apertures in tomographic head scanning,” Nucl. Instr. Meth. Phys. Res. |

13. | W. E. Smith, R. G. Paxman, and H. H. Barrett, “Image reconstruction from coded data: I. reconstruction algorithms and experimental results.” J. Opt. Soc. Am. A |

14. | R. Accorsi, F. Gasparini, and R. C. Lanza, “A coded aperture for high-resolution nuclear medicine planar imaging with a conventional anger camera: experimental results,” IEEE Trans. Nucl. Sci. |

15. | Z. Mu and Y.-H. Liu, “Aperture collimation correction and maximum-likelihood image reconstruction for near-field coded aperture imaging of single photon emission computerized tomography,” IEEE Trans. Medical Imaging |

16. | R. J. Jaszczak, J. Li, H. Wang, M. R. Zalutsky, and R. E. Coleman, “Pinhole collimation for ultra-high-resolution, small-field-of-view SPECT.” Phys. Med. Biol. |

17. | K. Ogawa, T. Kawade, K. Nakamura, A. Kubo, and T. Ichihara, “Ultra high resolution pinhole spect for small animal study,” IEEE Trans. Nucl. Sci. |

18. | T. Zeniya, H. Watabe, T. Aoi, K. M. Kim, N. Teramoto, T. Hayashi, A. Sohlberg, H. Kudo, and H. Iida, “A new reconstruction strategy for image improvement in pinhole SPECT.” Eur. J. Nucl. Med. Mol. Imaging |

19. | S. D. Metzler, J. E. Bowsher, M. F. Smith, and R. J. Jaszczak, “Analytic determination of pinhole collimator sensitivity with penetration,” IEEE Trans. Medical Imaging |

20. | E. E. Fenimore and T. M. Cannon, “Uniformly redundant arrays: digital reconstruction methods.” Appl. Opt. |

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

22. | D. J. Brady, N. P. Pitsianis, and X. Sun, “Reference structure tomography,” JOSA. A. |

23. | K. Choi and D. J. Brady, “Coded aperture computed tomography,” Proc. of SPIE |

24. | J. Fleming and B. Goddard, “An evaluation of techniques for stationary coded aperture three-dimensional imaging in nuclear medicine,” Nucl. Instr. Meth. Phys. Res. |

25. | J. H. Hubbell and S. M. Seltzer, “Tables of x-ray mass attenuation coefficients and mass energy-absorption coefficients from 1 kev to 20 mev for elements z = 1 to 92 and 48 additional substances of dosimetric interest,” http://www.nist.gov/pml/data/xraycoef/. |

26. | A. C. Kak and M. Slaney, |

27. | G. Lauritsch and H. Bruder, “Head phantom,” http://www.imp.uni-erlangen.de/phantoms/head (2012). |

**OCIS Codes**

(170.7440) Medical optics and biotechnology : X-ray imaging

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

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: April 16, 2013

Revised Manuscript: September 20, 2013

Manuscript Accepted: October 23, 2013

Published: November 7, 2013

**Virtual Issues**

Vol. 9, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Yining Zhu, Defeng Chen, Yunsong Zhao, Hongwei Li, and Peng Zhang, "An approach to increasing the resolution of industrial CT images based on an aperture collimator," Opt. Express **21**, 27946-27963 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-27946

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### References

- H. Hu, “Multi-slice helical CT: scan and reconstruction,” Medical Physics 26, 5–18 (1999). [CrossRef] [PubMed]
- J. Hsieh, Computed Tomography: Principles, Design, Artifacts, and Recent Advances (SPIE Press, 2003, vol. PM188).
- H. Zhang, J. Tian, M. Chen, P. Zhang, “A novel scanning mode and image reconstruction method on super-resolution CT,” Chin. J. Stereol. Image Analysis 9, 154–157 (2005).
- T.M. Peters, R.M. Lewitt, “Computed tomography with fan-beam geometry,” J. Comput. Assist. Tomogr. 1, 429–436 (1977). [CrossRef] [PubMed]
- A. H. Lonn, “Computed tomography system with translatable focal spot,” US Patent 45, 5173852 (1990).
- J. Hsieh, M.F. Gard, S. Gravelle, “Reconstruction technique for focal spot wobbling,”Proc. of SPIE Medical Imaging VI 1652, 175–182 (1992). [CrossRef]
- E. Caroli, J. B. Stephen, G. Cocco, L. Natalucci, A. Spizzichino, “Coded aperture imaging in x- and gamma-ray astronomy,” Space Sci. Rev. 45, 349–403 (1987). [CrossRef]
- T. Palmieri, “Multiplex methods and advantages in x-ray astronomy,” Astrophys. Space Sci. 28, 277–287 (1974). [CrossRef]
- P. Durouchoux, H. Hudson, J. Matteson, G. Hurford, K. Hurley, E. Orsal, “Gamma-ray imaging with a rotating modulator,” Astron. Astrophys. 120, 150–155 (1983).
- G. Skinner, “Imaging with coded-aperture masks,” Nucl. Instr. Meth. Phys. Res. 221, 33–40 (1984). [CrossRef]
- S. Webb, The Physics of Medical Imaging (Taylor & Francis, 2010).
- G. Knoll, W. Rogers, K. Koral, J. Stamos, N. Clinthorne, “Application of coded apertures in tomographic head scanning,” Nucl. Instr. Meth. Phys. Res. 221, 226–232 (1984). [CrossRef]
- W. E. Smith, R. G. Paxman, H. H. Barrett, “Image reconstruction from coded data: I. reconstruction algorithms and experimental results.” J. Opt. Soc. Am. A 2, 491–500 (1985). [CrossRef] [PubMed]
- R. Accorsi, F. Gasparini, R. C. Lanza, “A coded aperture for high-resolution nuclear medicine planar imaging with a conventional anger camera: experimental results,” IEEE Trans. Nucl. Sci. 48, 2411–2417 (2001). [CrossRef]
- Z. Mu, Y.-H. Liu, “Aperture collimation correction and maximum-likelihood image reconstruction for near-field coded aperture imaging of single photon emission computerized tomography,” IEEE Trans. Medical Imaging 25, 701–711 (2006). [CrossRef]
- R. J. Jaszczak, J. Li, H. Wang, M. R. Zalutsky, R. E. Coleman, “Pinhole collimation for ultra-high-resolution, small-field-of-view SPECT.” Phys. Med. Biol. 39, 425–437 (1994). [CrossRef] [PubMed]
- K. Ogawa, T. Kawade, K. Nakamura, A. Kubo, T. Ichihara, “Ultra high resolution pinhole spect for small animal study,” IEEE Trans. Nucl. Sci. 45, 3122–3126 (1998). [CrossRef]
- T. Zeniya, H. Watabe, T. Aoi, K. M. Kim, N. Teramoto, T. Hayashi, A. Sohlberg, H. Kudo, H. Iida, “A new reconstruction strategy for image improvement in pinhole SPECT.” Eur. J. Nucl. Med. Mol. Imaging 31, 1166–1172 (2004). [CrossRef] [PubMed]
- S. D. Metzler, J. E. Bowsher, M. F. Smith, R. J. Jaszczak, “Analytic determination of pinhole collimator sensitivity with penetration,” IEEE Trans. Medical Imaging 20, 730–741 (2001). [CrossRef]
- E. E. Fenimore, T. M. Cannon, “Uniformly redundant arrays: digital reconstruction methods.” Appl. Opt. 20, 1858–1864 (1981). [CrossRef] [PubMed]
- A. Olivo, R. Speller, “A coded-aperture technique allowing x-ray phase contrast imaging with conventional sources,” Appl. Phys. Lett. 91, 074106 (2007). [CrossRef]
- D. J. Brady, N. P. Pitsianis, X. Sun, “Reference structure tomography,” JOSA. A. 21, 1140–1147 (2004). [CrossRef] [PubMed]
- K. Choi, D. J. Brady, “Coded aperture computed tomography,” Proc. of SPIE 74680B, 1–4 (2009).
- J. Fleming, B. Goddard, “An evaluation of techniques for stationary coded aperture three-dimensional imaging in nuclear medicine,” Nucl. Instr. Meth. Phys. Res. 221, 242–246 (1984). [CrossRef]
- J. H. Hubbell, S. M. Seltzer, “Tables of x-ray mass attenuation coefficients and mass energy-absorption coefficients from 1 kev to 20 mev for elements z = 1 to 92 and 48 additional substances of dosimetric interest,” http://www.nist.gov/pml/data/xraycoef/ .
- A. C. Kak, M. Slaney, Principles of Tomographic Imaging (IEEE Engineering in Medicine and Biology Society, 2001). [CrossRef]
- G. Lauritsch, H. Bruder, “Head phantom,” http://www.imp.uni-erlangen.de/phantoms/head (2012).

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